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• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

• An independent variable is something the researcher changes or controls.
• A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias  will affect your results.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1. ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2. Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in  if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

• H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
• H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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The Craft of Writing a Strong Hypothesis

Table of Contents

Writing a hypothesis is one of the essential elements of a scientific research paper. It needs to be to the point, clearly communicating what your research is trying to accomplish. A blurry, drawn-out, or complexly-structured hypothesis can confuse your readers. Or worse, the editor and peer reviewers.

A captivating hypothesis is not too intricate. This blog will take you through the process so that, by the end of it, you have a better idea of how to convey your research paper's intent in just one sentence.

What is a Hypothesis?

The first step in your scientific endeavor, a hypothesis, is a strong, concise statement that forms the basis of your research. It is not the same as a thesis statement , which is a brief summary of your research paper .

The sole purpose of a hypothesis is to predict your paper's findings, data, and conclusion. It comes from a place of curiosity and intuition . When you write a hypothesis, you're essentially making an educated guess based on scientific prejudices and evidence, which is further proven or disproven through the scientific method.

The reason for undertaking research is to observe a specific phenomenon. A hypothesis, therefore, lays out what the said phenomenon is. And it does so through two variables, an independent and dependent variable.

The independent variable is the cause behind the observation, while the dependent variable is the effect of the cause. A good example of this is “mixing red and blue forms purple.” In this hypothesis, mixing red and blue is the independent variable as you're combining the two colors at your own will. The formation of purple is the dependent variable as, in this case, it is conditional to the independent variable.

Different Types of Hypotheses‌

Types of hypotheses

Some would stand by the notion that there are only two types of hypotheses: a Null hypothesis and an Alternative hypothesis. While that may have some truth to it, it would be better to fully distinguish the most common forms as these terms come up so often, which might leave you out of context.

Apart from Null and Alternative, there are Complex, Simple, Directional, Non-Directional, Statistical, and Associative and casual hypotheses. They don't necessarily have to be exclusive, as one hypothesis can tick many boxes, but knowing the distinctions between them will make it easier for you to construct your own.

1. Null hypothesis

A null hypothesis proposes no relationship between two variables. Denoted by H 0 , it is a negative statement like “Attending physiotherapy sessions does not affect athletes' on-field performance.” Here, the author claims physiotherapy sessions have no effect on on-field performances. Even if there is, it's only a coincidence.

2. Alternative hypothesis

Considered to be the opposite of a null hypothesis, an alternative hypothesis is donated as H1 or Ha. It explicitly states that the dependent variable affects the independent variable. A good  alternative hypothesis example is “Attending physiotherapy sessions improves athletes' on-field performance.” or “Water evaporates at 100 °C. ” The alternative hypothesis further branches into directional and non-directional.

• Directional hypothesis: A hypothesis that states the result would be either positive or negative is called directional hypothesis. It accompanies H1 with either the ‘<' or ‘>' sign.
• Non-directional hypothesis: A non-directional hypothesis only claims an effect on the dependent variable. It does not clarify whether the result would be positive or negative. The sign for a non-directional hypothesis is ‘≠.'

3. Simple hypothesis

A simple hypothesis is a statement made to reflect the relation between exactly two variables. One independent and one dependent. Consider the example, “Smoking is a prominent cause of lung cancer." The dependent variable, lung cancer, is dependent on the independent variable, smoking.

4. Complex hypothesis

In contrast to a simple hypothesis, a complex hypothesis implies the relationship between multiple independent and dependent variables. For instance, “Individuals who eat more fruits tend to have higher immunity, lesser cholesterol, and high metabolism.” The independent variable is eating more fruits, while the dependent variables are higher immunity, lesser cholesterol, and high metabolism.

5. Associative and casual hypothesis

Associative and casual hypotheses don't exhibit how many variables there will be. They define the relationship between the variables. In an associative hypothesis, changing any one variable, dependent or independent, affects others. In a casual hypothesis, the independent variable directly affects the dependent.

6. Empirical hypothesis

Also referred to as the working hypothesis, an empirical hypothesis claims a theory's validation via experiments and observation. This way, the statement appears justifiable and different from a wild guess.

Say, the hypothesis is “Women who take iron tablets face a lesser risk of anemia than those who take vitamin B12.” This is an example of an empirical hypothesis where the researcher  the statement after assessing a group of women who take iron tablets and charting the findings.

7. Statistical hypothesis

The point of a statistical hypothesis is to test an already existing hypothesis by studying a population sample. Hypothesis like “44% of the Indian population belong in the age group of 22-27.” leverage evidence to prove or disprove a particular statement.

Characteristics of a Good Hypothesis

Writing a hypothesis is essential as it can make or break your research for you. That includes your chances of getting published in a journal. So when you're designing one, keep an eye out for these pointers:

• A research hypothesis has to be simple yet clear to look justifiable enough.
• It has to be testable — your research would be rendered pointless if too far-fetched into reality or limited by technology.
• It has to be precise about the results —what you are trying to do and achieve through it should come out in your hypothesis.
• A research hypothesis should be self-explanatory, leaving no doubt in the reader's mind.
• If you are developing a relational hypothesis, you need to include the variables and establish an appropriate relationship among them.
• A hypothesis must keep and reflect the scope for further investigations and experiments.

Separating a Hypothesis from a Prediction

Outside of academia, hypothesis and prediction are often used interchangeably. In research writing, this is not only confusing but also incorrect. And although a hypothesis and prediction are guesses at their core, there are many differences between them.

A hypothesis is an educated guess or even a testable prediction validated through research. It aims to analyze the gathered evidence and facts to define a relationship between variables and put forth a logical explanation behind the nature of events.

Predictions are assumptions or expected outcomes made without any backing evidence. They are more fictionally inclined regardless of where they originate from.

For this reason, a hypothesis holds much more weight than a prediction. It sticks to the scientific method rather than pure guesswork. "Planets revolve around the Sun." is an example of a hypothesis as it is previous knowledge and observed trends. Additionally, we can test it through the scientific method.

Whereas "COVID-19 will be eradicated by 2030." is a prediction. Even though it results from past trends, we can't prove or disprove it. So, the only way this gets validated is to wait and watch if COVID-19 cases end by 2030.

Finally, How to Write a Hypothesis

Quick tips on writing a hypothesis

1.  Be clear about your research question

A hypothesis should instantly address the research question or the problem statement. To do so, you need to ask a question. Understand the constraints of your undertaken research topic and then formulate a simple and topic-centric problem. Only after that can you develop a hypothesis and further test for evidence.

2. Carry out a recce

Once you have your research's foundation laid out, it would be best to conduct preliminary research. Go through previous theories, academic papers, data, and experiments before you start curating your research hypothesis. It will give you an idea of your hypothesis's viability or originality.

Making use of references from relevant research papers helps draft a good research hypothesis. SciSpace Discover offers a repository of over 270 million research papers to browse through and gain a deeper understanding of related studies on a particular topic. Additionally, you can use SciSpace Copilot , your AI research assistant, for reading any lengthy research paper and getting a more summarized context of it. A hypothesis can be formed after evaluating many such summarized research papers. Copilot also offers explanations for theories and equations, explains paper in simplified version, allows you to highlight any text in the paper or clip math equations and tables and provides a deeper, clear understanding of what is being said. This can improve the hypothesis by helping you identify potential research gaps.

3. Create a 3-dimensional hypothesis

Variables are an essential part of any reasonable hypothesis. So, identify your independent and dependent variable(s) and form a correlation between them. The ideal way to do this is to write the hypothetical assumption in the ‘if-then' form. If you use this form, make sure that you state the predefined relationship between the variables.

In another way, you can choose to present your hypothesis as a comparison between two variables. Here, you must specify the difference you expect to observe in the results.

4. Write the first draft

Now that everything is in place, it's time to write your hypothesis. For starters, create the first draft. In this version, write what you expect to find from your research.

Clearly separate your independent and dependent variables and the link between them. Don't fixate on syntax at this stage. The goal is to ensure your hypothesis addresses the issue.

5. Proof your hypothesis

After preparing the first draft of your hypothesis, you need to inspect it thoroughly. It should tick all the boxes, like being concise, straightforward, relevant, and accurate. Your final hypothesis has to be well-structured as well.

Research projects are an exciting and crucial part of being a scholar. And once you have your research question, you need a great hypothesis to begin conducting research. Thus, knowing how to write a hypothesis is very important.

Now that you have a firmer grasp on what a good hypothesis constitutes, the different kinds there are, and what process to follow, you will find it much easier to write your hypothesis, which ultimately helps your research.

Now it's easier than ever to streamline your research workflow with SciSpace Discover . Its integrated, comprehensive end-to-end platform for research allows scholars to easily discover, write and publish their research and fosters collaboration.

It includes everything you need, including a repository of over 270 million research papers across disciplines, SEO-optimized summaries and public profiles to show your expertise and experience.

If you found these tips on writing a research hypothesis useful, head over to our blog on Statistical Hypothesis Testing to learn about the top researchers, papers, and institutions in this domain.

Frequently Asked Questions (FAQs)

1. what is the definition of hypothesis.

According to the Oxford dictionary, a hypothesis is defined as “An idea or explanation of something that is based on a few known facts, but that has not yet been proved to be true or correct”.

2. What is an example of hypothesis?

The hypothesis is a statement that proposes a relationship between two or more variables. An example: "If we increase the number of new users who join our platform by 25%, then we will see an increase in revenue."

3. What is an example of null hypothesis?

A null hypothesis is a statement that there is no relationship between two variables. The null hypothesis is written as H0. The null hypothesis states that there is no effect. For example, if you're studying whether or not a particular type of exercise increases strength, your null hypothesis will be "there is no difference in strength between people who exercise and people who don't."

4. What are the types of research?

• Fundamental research

• Applied research

• Qualitative research

• Quantitative research

• Mixed research

• Exploratory research

• Longitudinal research

• Cross-sectional research

• Field research

• Laboratory research

• Fixed research

• Flexible research

• Action research

• Policy research

• Classification research

• Comparative research

• Causal research

• Inductive research

• Deductive research

5. How to write a hypothesis?

• Your hypothesis should be able to predict the relationship and outcome.

• Avoid wordiness by keeping it simple and brief.

• Your hypothesis should contain observable and testable outcomes.

• Your hypothesis should be relevant to the research question.

6. What are the 2 types of hypothesis?

• Null hypotheses are used to test the claim that "there is no difference between two groups of data".

• Alternative hypotheses test the claim that "there is a difference between two data groups".

7. Difference between research question and research hypothesis?

A research question is a broad, open-ended question you will try to answer through your research. A hypothesis is a statement based on prior research or theory that you expect to be true due to your study. Example - Research question: What are the factors that influence the adoption of the new technology? Research hypothesis: There is a positive relationship between age, education and income level with the adoption of the new technology.

8. What is plural for hypothesis?

The plural of hypothesis is hypotheses. Here's an example of how it would be used in a statement, "Numerous well-considered hypotheses are presented in this part, and they are supported by tables and figures that are well-illustrated."

9. What is the red queen hypothesis?

The red queen hypothesis in evolutionary biology states that species must constantly evolve to avoid extinction because if they don't, they will be outcompeted by other species that are evolving. Leigh Van Valen first proposed it in 1973; since then, it has been tested and substantiated many times.

10. Who is known as the father of null hypothesis?

The father of the null hypothesis is Sir Ronald Fisher. He published a paper in 1925 that introduced the concept of null hypothesis testing, and he was also the first to use the term itself.

11. When to reject null hypothesis?

You need to find a significant difference between your two populations to reject the null hypothesis. You can determine that by running statistical tests such as an independent sample t-test or a dependent sample t-test. You should reject the null hypothesis if the p-value is less than 0.05.

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How to Write a Great Hypothesis

Hypothesis Format, Examples, and Tips

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Amy Morin, LCSW, is a psychotherapist and international bestselling author. Her books, including "13 Things Mentally Strong People Don't Do," have been translated into more than 40 languages. Her TEDx talk,  "The Secret of Becoming Mentally Strong," is one of the most viewed talks of all time.

Verywell / Alex Dos Diaz

• The Scientific Method

Hypothesis Format

Falsifiability of a hypothesis, operational definitions, types of hypotheses, hypotheses examples.

• Collecting Data

Frequently Asked Questions

A hypothesis is a tentative statement about the relationship between two or more  variables. It is a specific, testable prediction about what you expect to happen in a study.

One hypothesis example would be a study designed to look at the relationship between sleep deprivation and test performance might have a hypothesis that states: "This study is designed to assess the hypothesis that sleep-deprived people will perform worse on a test than individuals who are not sleep-deprived."

This article explores how a hypothesis is used in psychology research, how to write a good hypothesis, and the different types of hypotheses you might use.

The Hypothesis in the Scientific Method

In the scientific method , whether it involves research in psychology, biology, or some other area, a hypothesis represents what the researchers think will happen in an experiment. The scientific method involves the following steps:

• Forming a question
• Performing background research
• Creating a hypothesis
• Designing an experiment
• Collecting data
• Analyzing the results
• Drawing conclusions
• Communicating the results

The hypothesis is a prediction, but it involves more than a guess. Most of the time, the hypothesis begins with a question which is then explored through background research. It is only at this point that researchers begin to develop a testable hypothesis. Unless you are creating an exploratory study, your hypothesis should always explain what you  expect  to happen.

In a study exploring the effects of a particular drug, the hypothesis might be that researchers expect the drug to have some type of effect on the symptoms of a specific illness. In psychology, the hypothesis might focus on how a certain aspect of the environment might influence a particular behavior.

Remember, a hypothesis does not have to be correct. While the hypothesis predicts what the researchers expect to see, the goal of the research is to determine whether this guess is right or wrong. When conducting an experiment, researchers might explore a number of factors to determine which ones might contribute to the ultimate outcome.

In many cases, researchers may find that the results of an experiment  do not  support the original hypothesis. When writing up these results, the researchers might suggest other options that should be explored in future studies.

In many cases, researchers might draw a hypothesis from a specific theory or build on previous research. For example, prior research has shown that stress can impact the immune system. So a researcher might hypothesize: "People with high-stress levels will be more likely to contract a common cold after being exposed to the virus than people who have low-stress levels."

In other instances, researchers might look at commonly held beliefs or folk wisdom. "Birds of a feather flock together" is one example of folk wisdom that a psychologist might try to investigate. The researcher might pose a specific hypothesis that "People tend to select romantic partners who are similar to them in interests and educational level."

Elements of a Good Hypothesis

So how do you write a good hypothesis? When trying to come up with a hypothesis for your research or experiments, ask yourself the following questions:

• Is your hypothesis based on your research on a topic?
• Can your hypothesis be tested?
• Does your hypothesis include independent and dependent variables?

Before you come up with a specific hypothesis, spend some time doing background research. Once you have completed a literature review, start thinking about potential questions you still have. Pay attention to the discussion section in the  journal articles you read . Many authors will suggest questions that still need to be explored.

To form a hypothesis, you should take these steps:

• Collect as many observations about a topic or problem as you can.
• Evaluate these observations and look for possible causes of the problem.
• Create a list of possible explanations that you might want to explore.
• After you have developed some possible hypotheses, think of ways that you could confirm or disprove each hypothesis through experimentation. This is known as falsifiability.

In the scientific method ,  falsifiability is an important part of any valid hypothesis.   In order to test a claim scientifically, it must be possible that the claim could be proven false.

Students sometimes confuse the idea of falsifiability with the idea that it means that something is false, which is not the case. What falsifiability means is that  if  something was false, then it is possible to demonstrate that it is false.

One of the hallmarks of pseudoscience is that it makes claims that cannot be refuted or proven false.

A variable is a factor or element that can be changed and manipulated in ways that are observable and measurable. However, the researcher must also define how the variable will be manipulated and measured in the study.

For example, a researcher might operationally define the variable " test anxiety " as the results of a self-report measure of anxiety experienced during an exam. A "study habits" variable might be defined by the amount of studying that actually occurs as measured by time.

These precise descriptions are important because many things can be measured in a number of different ways. One of the basic principles of any type of scientific research is that the results must be replicable.   By clearly detailing the specifics of how the variables were measured and manipulated, other researchers can better understand the results and repeat the study if needed.

Some variables are more difficult than others to define. How would you operationally define a variable such as aggression ? For obvious ethical reasons, researchers cannot create a situation in which a person behaves aggressively toward others.

In order to measure this variable, the researcher must devise a measurement that assesses aggressive behavior without harming other people. In this situation, the researcher might utilize a simulated task to measure aggressiveness.

Hypothesis Checklist

• Does your hypothesis focus on something that you can actually test?
• Does your hypothesis include both an independent and dependent variable?
• Can you manipulate the variables?
• Can your hypothesis be tested without violating ethical standards?

The hypothesis you use will depend on what you are investigating and hoping to find. Some of the main types of hypotheses that you might use include:

• Simple hypothesis : This type of hypothesis suggests that there is a relationship between one independent variable and one dependent variable.
• Complex hypothesis : This type of hypothesis suggests a relationship between three or more variables, such as two independent variables and a dependent variable.
• Null hypothesis : This hypothesis suggests no relationship exists between two or more variables.
• Alternative hypothesis : This hypothesis states the opposite of the null hypothesis.
• Statistical hypothesis : This hypothesis uses statistical analysis to evaluate a representative sample of the population and then generalizes the findings to the larger group.
• Logical hypothesis : This hypothesis assumes a relationship between variables without collecting data or evidence.

A hypothesis often follows a basic format of "If {this happens} then {this will happen}." One way to structure your hypothesis is to describe what will happen to the  dependent variable  if you change the  independent variable .

The basic format might be: "If {these changes are made to a certain independent variable}, then we will observe {a change in a specific dependent variable}."

A few examples of simple hypotheses:

• "Students who eat breakfast will perform better on a math exam than students who do not eat breakfast."
• Complex hypothesis: "Students who experience test anxiety before an English exam will get lower scores than students who do not experience test anxiety."​
• "Motorists who talk on the phone while driving will be more likely to make errors on a driving course than those who do not talk on the phone."

Examples of a complex hypothesis include:

• "People with high-sugar diets and sedentary activity levels are more likely to develop depression."
• "Younger people who are regularly exposed to green, outdoor areas have better subjective well-being than older adults who have limited exposure to green spaces."

Examples of a null hypothesis include:

• "Children who receive a new reading intervention will have scores different than students who do not receive the intervention."
• "There will be no difference in scores on a memory recall task between children and adults."

Examples of an alternative hypothesis:

• "Children who receive a new reading intervention will perform better than students who did not receive the intervention."
• "Adults will perform better on a memory task than children." 

Collecting Data on Your Hypothesis

Once a researcher has formed a testable hypothesis, the next step is to select a research design and start collecting data. The research method depends largely on exactly what they are studying. There are two basic types of research methods: descriptive research and experimental research.

Descriptive Research Methods

Descriptive research such as  case studies ,  naturalistic observations , and surveys are often used when it would be impossible or difficult to  conduct an experiment . These methods are best used to describe different aspects of a behavior or psychological phenomenon.

Once a researcher has collected data using descriptive methods, a correlational study can then be used to look at how the variables are related. This type of research method might be used to investigate a hypothesis that is difficult to test experimentally.

Experimental Research Methods

Experimental methods  are used to demonstrate causal relationships between variables. In an experiment, the researcher systematically manipulates a variable of interest (known as the independent variable) and measures the effect on another variable (known as the dependent variable).

Unlike correlational studies, which can only be used to determine if there is a relationship between two variables, experimental methods can be used to determine the actual nature of the relationship—whether changes in one variable actually  cause  another to change.

A Word From Verywell

The hypothesis is a critical part of any scientific exploration. It represents what researchers expect to find in a study or experiment. In situations where the hypothesis is unsupported by the research, the research still has value. Such research helps us better understand how different aspects of the natural world relate to one another. It also helps us develop new hypotheses that can then be tested in the future.

Some examples of how to write a hypothesis include:

• "Staying up late will lead to worse test performance the next day."
• "People who consume one apple each day will visit the doctor fewer times each year."
• "Breaking study sessions up into three 20-minute sessions will lead to better test results than a single 60-minute study session."

The four parts of a hypothesis are:

• The research question
• The independent variable (IV)
• The dependent variable (DV)
• The proposed relationship between the IV and DV

Castillo M. The scientific method: a need for something better? . AJNR Am J Neuroradiol. 2013;34(9):1669-71. doi:10.3174/ajnr.A3401

Nevid J. Psychology: Concepts and Applications. Wadworth, 2013.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

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• How to Write a Strong Hypothesis | Guide & Examples

How to Write a Strong Hypothesis | Guide & Examples

Published on 6 May 2022 by Shona McCombes .

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations, and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more variables . An independent variable is something the researcher changes or controls. A dependent variable is something the researcher observes and measures.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

Prevent plagiarism, run a free check.

Step 1: ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2: Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalise more complex constructs.

Step 3: Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

Step 4: Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

Step 5: Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in if … then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

Step 6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis. The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis is not just a guess. It should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations, and statistical analysis of data).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (‘ x affects y because …’).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses. In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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What is and How to Write a Good Hypothesis in Research?

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Table of Contents

One of the most important aspects of conducting research is constructing a strong hypothesis. But what makes a hypothesis in research effective? In this article, we’ll look at the difference between a hypothesis and a research question, as well as the elements of a good hypothesis in research. We’ll also include some examples of effective hypotheses, and what pitfalls to avoid.

What is a Hypothesis in Research?

Simply put, a hypothesis is a research question that also includes the predicted or expected result of the research. Without a hypothesis, there can be no basis for a scientific or research experiment. As such, it is critical that you carefully construct your hypothesis by being deliberate and thorough, even before you set pen to paper. Unless your hypothesis is clearly and carefully constructed, any flaw can have an adverse, and even grave, effect on the quality of your experiment and its subsequent results.

Research Question vs Hypothesis

It’s easy to confuse research questions with hypotheses, and vice versa. While they’re both critical to the Scientific Method, they have very specific differences. Primarily, a research question, just like a hypothesis, is focused and concise. But a hypothesis includes a prediction based on the proposed research, and is designed to forecast the relationship of and between two (or more) variables. Research questions are open-ended, and invite debate and discussion, while hypotheses are closed, e.g. “The relationship between A and B will be C.”

A hypothesis is generally used if your research topic is fairly well established, and you are relatively certain about the relationship between the variables that will be presented in your research. Since a hypothesis is ideally suited for experimental studies, it will, by its very existence, affect the design of your experiment. The research question is typically used for new topics that have not yet been researched extensively. Here, the relationship between different variables is less known. There is no prediction made, but there may be variables explored. The research question can be casual in nature, simply trying to understand if a relationship even exists, descriptive or comparative.

How to Write Hypothesis in Research

Writing an effective hypothesis starts before you even begin to type. Like any task, preparation is key, so you start first by conducting research yourself, and reading all you can about the topic that you plan to research. From there, you’ll gain the knowledge you need to understand where your focus within the topic will lie.

Remember that a hypothesis is a prediction of the relationship that exists between two or more variables. Your job is to write a hypothesis, and design the research, to “prove” whether or not your prediction is correct. A common pitfall is to use judgments that are subjective and inappropriate for the construction of a hypothesis. It’s important to keep the focus and language of your hypothesis objective.

An effective hypothesis in research is clearly and concisely written, and any terms or definitions clarified and defined. Specific language must also be used to avoid any generalities or assumptions.

Use the following points as a checklist to evaluate the effectiveness of your research hypothesis:

• Predicts the relationship and outcome
• Simple and concise – avoid wordiness
• Clear with no ambiguity or assumptions about the readers’ knowledge
• Observable and testable results
• Relevant and specific to the research question or problem

Research Hypothesis Example

Perhaps the best way to evaluate whether or not your hypothesis is effective is to compare it to those of your colleagues in the field. There is no need to reinvent the wheel when it comes to writing a powerful research hypothesis. As you’re reading and preparing your hypothesis, you’ll also read other hypotheses. These can help guide you on what works, and what doesn’t, when it comes to writing a strong research hypothesis.

Here are a few generic examples to get you started.

Eating an apple each day, after the age of 60, will result in a reduction of frequency of physician visits.

Budget airlines are more likely to receive more customer complaints. A budget airline is defined as an airline that offers lower fares and fewer amenities than a traditional full-service airline. (Note that the term “budget airline” is included in the hypothesis.

Workplaces that offer flexible working hours report higher levels of employee job satisfaction than workplaces with fixed hours.

Each of the above examples are specific, observable and measurable, and the statement of prediction can be verified or shown to be false by utilizing standard experimental practices. It should be noted, however, that often your hypothesis will change as your research progresses.

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How to Develop a Good Research Hypothesis

The story of a research study begins by asking a question. Researchers all around the globe are asking curious questions and formulating research hypothesis. However, whether the research study provides an effective conclusion depends on how well one develops a good research hypothesis. Research hypothesis examples could help researchers get an idea as to how to write a good research hypothesis.

This blog will help you understand what is a research hypothesis, its characteristics and, how to formulate a research hypothesis

Table of Contents

What is Hypothesis?

Hypothesis is an assumption or an idea proposed for the sake of argument so that it can be tested. It is a precise, testable statement of what the researchers predict will be outcome of the study.  Hypothesis usually involves proposing a relationship between two variables: the independent variable (what the researchers change) and the dependent variable (what the research measures).

What is a Research Hypothesis?

Research hypothesis is a statement that introduces a research question and proposes an expected result. It is an integral part of the scientific method that forms the basis of scientific experiments. Therefore, you need to be careful and thorough when building your research hypothesis. A minor flaw in the construction of your hypothesis could have an adverse effect on your experiment. In research, there is a convention that the hypothesis is written in two forms, the null hypothesis, and the alternative hypothesis (called the experimental hypothesis when the method of investigation is an experiment).

Characteristics of a Good Research Hypothesis

As the hypothesis is specific, there is a testable prediction about what you expect to happen in a study. You may consider drawing hypothesis from previously published research based on the theory.

A good research hypothesis involves more effort than just a guess. In particular, your hypothesis may begin with a question that could be further explored through background research.

To help you formulate a promising research hypothesis, you should ask yourself the following questions:

• Is the language clear and focused?
• What is the relationship between your hypothesis and your research topic?
• Is your hypothesis testable? If yes, then how?
• What are the possible explanations that you might want to explore?
• Does your hypothesis include both an independent and dependent variable?
• Can you manipulate your variables without hampering the ethical standards?
• Does your research predict the relationship and outcome?
• Is your research simple and concise (avoids wordiness)?
• Is it clear with no ambiguity or assumptions about the readers’ knowledge
• Is your research observable and testable results?
• Is it relevant and specific to the research question or problem?

The questions listed above can be used as a checklist to make sure your hypothesis is based on a solid foundation. Furthermore, it can help you identify weaknesses in your hypothesis and revise it if necessary.

Source: Educational Hub

How to formulate a research hypothesis.

A testable hypothesis is not a simple statement. It is rather an intricate statement that needs to offer a clear introduction to a scientific experiment, its intentions, and the possible outcomes. However, there are some important things to consider when building a compelling hypothesis.

1. State the problem that you are trying to solve.

Make sure that the hypothesis clearly defines the topic and the focus of the experiment.

2. Try to write the hypothesis as an if-then statement.

Follow this template: If a specific action is taken, then a certain outcome is expected.

3. Define the variables

Independent variables are the ones that are manipulated, controlled, or changed. Independent variables are isolated from other factors of the study.

Dependent variables , as the name suggests are dependent on other factors of the study. They are influenced by the change in independent variable.

4. Scrutinize the hypothesis

Evaluate assumptions, predictions, and evidence rigorously to refine your understanding.

Types of Research Hypothesis

The types of research hypothesis are stated below:

1. Simple Hypothesis

It predicts the relationship between a single dependent variable and a single independent variable.

2. Complex Hypothesis

It predicts the relationship between two or more independent and dependent variables.

3. Directional Hypothesis

It specifies the expected direction to be followed to determine the relationship between variables and is derived from theory. Furthermore, it implies the researcher’s intellectual commitment to a particular outcome.

4. Non-directional Hypothesis

It does not predict the exact direction or nature of the relationship between the two variables. The non-directional hypothesis is used when there is no theory involved or when findings contradict previous research.

5. Associative and Causal Hypothesis

The associative hypothesis defines interdependency between variables. A change in one variable results in the change of the other variable. On the other hand, the causal hypothesis proposes an effect on the dependent due to manipulation of the independent variable.

6. Null Hypothesis

Null hypothesis states a negative statement to support the researcher’s findings that there is no relationship between two variables. There will be no changes in the dependent variable due the manipulation of the independent variable. Furthermore, it states results are due to chance and are not significant in terms of supporting the idea being investigated.

7. Alternative Hypothesis

It states that there is a relationship between the two variables of the study and that the results are significant to the research topic. An experimental hypothesis predicts what changes will take place in the dependent variable when the independent variable is manipulated. Also, it states that the results are not due to chance and that they are significant in terms of supporting the theory being investigated.

Research Hypothesis Examples of Independent and Dependent Variables

Research Hypothesis Example 1 The greater number of coal plants in a region (independent variable) increases water pollution (dependent variable). If you change the independent variable (building more coal factories), it will change the dependent variable (amount of water pollution).

Research Hypothesis Example 2 What is the effect of diet or regular soda (independent variable) on blood sugar levels (dependent variable)? If you change the independent variable (the type of soda you consume), it will change the dependent variable (blood sugar levels)

You should not ignore the importance of the above steps. The validity of your experiment and its results rely on a robust testable hypothesis. Developing a strong testable hypothesis has few advantages, it compels us to think intensely and specifically about the outcomes of a study. Consequently, it enables us to understand the implication of the question and the different variables involved in the study. Furthermore, it helps us to make precise predictions based on prior research. Hence, forming a hypothesis would be of great value to the research. Here are some good examples of testable hypotheses.

More importantly, you need to build a robust testable research hypothesis for your scientific experiments. A testable hypothesis is a hypothesis that can be proved or disproved as a result of experimentation.

Importance of a Testable Hypothesis

To devise and perform an experiment using scientific method, you need to make sure that your hypothesis is testable. To be considered testable, some essential criteria must be met:

• There must be a possibility to prove that the hypothesis is true.
• There must be a possibility to prove that the hypothesis is false.
• The results of the hypothesis must be reproducible.

Without these criteria, the hypothesis and the results will be vague. As a result, the experiment will not prove or disprove anything significant.

What are your experiences with building hypotheses for scientific experiments? What challenges did you face? How did you overcome these challenges? Please share your thoughts with us in the comments section.

Frequently Asked Questions

The steps to write a research hypothesis are: 1. Stating the problem: Ensure that the hypothesis defines the research problem 2. Writing a hypothesis as an 'if-then' statement: Include the action and the expected outcome of your study by following a ‘if-then’ structure. 3. Defining the variables: Define the variables as Dependent or Independent based on their dependency to other factors. 4. Scrutinizing the hypothesis: Identify the type of your hypothesis

Hypothesis testing is a statistical tool which is used to make inferences about a population data to draw conclusions for a particular hypothesis.

Hypothesis in statistics is a formal statement about the nature of a population within a structured framework of a statistical model. It is used to test an existing hypothesis by studying a population.

Research hypothesis is a statement that introduces a research question and proposes an expected result. It forms the basis of scientific experiments.

The different types of hypothesis in research are: • Null hypothesis: Null hypothesis is a negative statement to support the researcher’s findings that there is no relationship between two variables. • Alternate hypothesis: Alternate hypothesis predicts the relationship between the two variables of the study. • Directional hypothesis: Directional hypothesis specifies the expected direction to be followed to determine the relationship between variables. • Non-directional hypothesis: Non-directional hypothesis does not predict the exact direction or nature of the relationship between the two variables. • Simple hypothesis: Simple hypothesis predicts the relationship between a single dependent variable and a single independent variable. • Complex hypothesis: Complex hypothesis predicts the relationship between two or more independent and dependent variables. • Associative and casual hypothesis: Associative and casual hypothesis predicts the relationship between two or more independent and dependent variables. • Empirical hypothesis: Empirical hypothesis can be tested via experiments and observation. • Statistical hypothesis: A statistical hypothesis utilizes statistical models to draw conclusions about broader populations.

Wow! You really simplified your explanation that even dummies would find it easy to comprehend. Thank you so much.

Thanks a lot for your valuable guidance.

I enjoy reading the post. Hypotheses are actually an intrinsic part in a study. It bridges the research question and the methodology of the study.

Useful piece!

This is awesome.Wow.

It very interesting to read the topic, can you guide me any specific example of hypothesis process establish throw the Demand and supply of the specific product in market

Nicely explained

It is really a useful for me Kindly give some examples of hypothesis

It was a well explained content ,can you please give me an example with the null and alternative hypothesis illustrated

clear and concise. thanks.

So Good so Amazing

Good to learn

Thanks a lot for explaining to my level of understanding

Explained well and in simple terms. Quick read! Thank you

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How to Write a Research Hypothesis: Good & Bad Examples

What is a research hypothesis?

A research hypothesis is an attempt at explaining a phenomenon or the relationships between phenomena/variables in the real world. Hypotheses are sometimes called “educated guesses”, but they are in fact (or let’s say they should be) based on previous observations, existing theories, scientific evidence, and logic. A research hypothesis is also not a prediction—rather, predictions are ( should be) based on clearly formulated hypotheses. For example, “We tested the hypothesis that KLF2 knockout mice would show deficiencies in heart development” is an assumption or prediction, not a hypothesis. 

The research hypothesis at the basis of this prediction is “the product of the KLF2 gene is involved in the development of the cardiovascular system in mice”—and this hypothesis is probably (hopefully) based on a clear observation, such as that mice with low levels of Kruppel-like factor 2 (which KLF2 codes for) seem to have heart problems. From this hypothesis, you can derive the idea that a mouse in which this particular gene does not function cannot develop a normal cardiovascular system, and then make the prediction that we started with. 

What is the difference between a hypothesis and a prediction?

You might think that these are very subtle differences, and you will certainly come across many publications that do not contain an actual hypothesis or do not make these distinctions correctly. But considering that the formulation and testing of hypotheses is an integral part of the scientific method, it is good to be aware of the concepts underlying this approach. The two hallmarks of a scientific hypothesis are falsifiability (an evaluation standard that was introduced by the philosopher of science Karl Popper in 1934) and testability —if you cannot use experiments or data to decide whether an idea is true or false, then it is not a hypothesis (or at least a very bad one).

So, in a nutshell, you (1) look at existing evidence/theories, (2) come up with a hypothesis, (3) make a prediction that allows you to (4) design an experiment or data analysis to test it, and (5) come to a conclusion. Of course, not all studies have hypotheses (there is also exploratory or hypothesis-generating research), and you do not necessarily have to state your hypothesis as such in your paper. 

But for the sake of understanding the principles of the scientific method, let’s first take a closer look at the different types of hypotheses that research articles refer to and then give you a step-by-step guide for how to formulate a strong hypothesis for your own paper.

Types of Research Hypotheses

Hypotheses can be simple , which means they describe the relationship between one single independent variable (the one you observe variations in or plan to manipulate) and one single dependent variable (the one you expect to be affected by the variations/manipulation). If there are more variables on either side, you are dealing with a complex hypothesis. You can also distinguish hypotheses according to the kind of relationship between the variables you are interested in (e.g., causal or associative ). But apart from these variations, we are usually interested in what is called the “alternative hypothesis” and, in contrast to that, the “null hypothesis”. If you think these two should be listed the other way round, then you are right, logically speaking—the alternative should surely come second. However, since this is the hypothesis we (as researchers) are usually interested in, let’s start from there.

Alternative Hypothesis

If you predict a relationship between two variables in your study, then the research hypothesis that you formulate to describe that relationship is your alternative hypothesis (usually H1 in statistical terms). The goal of your hypothesis testing is thus to demonstrate that there is sufficient evidence that supports the alternative hypothesis, rather than evidence for the possibility that there is no such relationship. The alternative hypothesis is usually the research hypothesis of a study and is based on the literature, previous observations, and widely known theories. 

Null Hypothesis

The hypothesis that describes the other possible outcome, that is, that your variables are not related, is the null hypothesis ( H0 ). Based on your findings, you choose between the two hypotheses—usually that means that if your prediction was correct, you reject the null hypothesis and accept the alternative. Make sure, however, that you are not getting lost at this step of the thinking process: If your prediction is that there will be no difference or change, then you are trying to find support for the null hypothesis and reject H1. 

Directional Hypothesis

While the null hypothesis is obviously “static”, the alternative hypothesis can specify a direction for the observed relationship between variables—for example, that mice with higher expression levels of a certain protein are more active than those with lower levels. This is then called a one-tailed hypothesis. 

Another example for a directional one-tailed alternative hypothesis would be that 

H1: Attending private classes before important exams has a positive effect on performance. 

Your null hypothesis would then be that

H0: Attending private classes before important exams has no/a negative effect on performance.

Nondirectional Hypothesis

A nondirectional hypothesis does not specify the direction of the potentially observed effect, only that there is a relationship between the studied variables—this is called a two-tailed hypothesis. For instance, if you are studying a new drug that has shown some effects on pathways involved in a certain condition (e.g., anxiety) in vitro in the lab, but you can’t say for sure whether it will have the same effects in an animal model or maybe induce other/side effects that you can’t predict and potentially increase anxiety levels instead, you could state the two hypotheses like this:

H1: The only lab-tested drug (somehow) affects anxiety levels in an anxiety mouse model.

You then test this nondirectional alternative hypothesis against the null hypothesis:

H0: The only lab-tested drug has no effect on anxiety levels in an anxiety mouse model.

How to Write a Hypothesis for a Research Paper

Now that we understand the important distinctions between different kinds of research hypotheses, let’s look at a simple process of how to write a hypothesis.

Writing a Hypothesis Step:1

Ask a question, based on earlier research. Research always starts with a question, but one that takes into account what is already known about a topic or phenomenon. For example, if you are interested in whether people who have pets are happier than those who don’t, do a literature search and find out what has already been demonstrated. You will probably realize that yes, there is quite a bit of research that shows a relationship between happiness and owning a pet—and even studies that show that owning a dog is more beneficial than owning a cat ! Let’s say you are so intrigued by this finding that you wonder: 

What is it that makes dog owners even happier than cat owners? 

Let’s move on to Step 2 and find an answer to that question.

Writing a Hypothesis Step 2:

Formulate a strong hypothesis by answering your own question. Again, you don’t want to make things up, take unicorns into account, or repeat/ignore what has already been done. Looking at the dog-vs-cat papers your literature search returned, you see that most studies are based on self-report questionnaires on personality traits, mental health, and life satisfaction. What you don’t find is any data on actual (mental or physical) health measures, and no experiments. You therefore decide to make a bold claim come up with the carefully thought-through hypothesis that it’s maybe the lifestyle of the dog owners, which includes walking their dog several times per day, engaging in fun and healthy activities such as agility competitions, and taking them on trips, that gives them that extra boost in happiness. You could therefore answer your question in the following way:

Dog owners are happier than cat owners because of the dog-related activities they engage in.

Now you have to verify that your hypothesis fulfills the two requirements we introduced at the beginning of this resource article: falsifiability and testability . If it can’t be wrong and can’t be tested, it’s not a hypothesis. We are lucky, however, because yes, we can test whether owning a dog but not engaging in any of those activities leads to lower levels of happiness or well-being than owning a dog and playing and running around with them or taking them on trips.  

Writing a Hypothesis Step 3:

Make your predictions and define your variables. We have verified that we can test our hypothesis, but now we have to define all the relevant variables, design our experiment or data analysis, and make precise predictions. You could, for example, decide to study dog owners (not surprising at this point), let them fill in questionnaires about their lifestyle as well as their life satisfaction (as other studies did), and then compare two groups of active and inactive dog owners. Alternatively, if you want to go beyond the data that earlier studies produced and analyzed and directly manipulate the activity level of your dog owners to study the effect of that manipulation, you could invite them to your lab, select groups of participants with similar lifestyles, make them change their lifestyle (e.g., couch potato dog owners start agility classes, very active ones have to refrain from any fun activities for a certain period of time) and assess their happiness levels before and after the intervention. In both cases, your independent variable would be “ level of engagement in fun activities with dog” and your dependent variable would be happiness or well-being . 

Examples of a Good and Bad Hypothesis

Let’s look at a few examples of good and bad hypotheses to get you started.

Good Hypothesis Examples

Bad hypothesis examples, tips for writing a research hypothesis.

If you understood the distinction between a hypothesis and a prediction we made at the beginning of this article, then you will have no problem formulating your hypotheses and predictions correctly. To refresh your memory: We have to (1) look at existing evidence, (2) come up with a hypothesis, (3) make a prediction, and (4) design an experiment. For example, you could summarize your dog/happiness study like this:

(1) While research suggests that dog owners are happier than cat owners, there are no reports on what factors drive this difference. (2) We hypothesized that it is the fun activities that many dog owners (but very few cat owners) engage in with their pets that increases their happiness levels. (3) We thus predicted that preventing very active dog owners from engaging in such activities for some time and making very inactive dog owners take up such activities would lead to an increase and decrease in their overall self-ratings of happiness, respectively. (4) To test this, we invited dog owners into our lab, assessed their mental and emotional well-being through questionnaires, and then assigned them to an “active” and an “inactive” group, depending on… 

Note that you use “we hypothesize” only for your hypothesis, not for your experimental prediction, and “would” or “if – then” only for your prediction, not your hypothesis. A hypothesis that states that something “would” affect something else sounds as if you don’t have enough confidence to make a clear statement—in which case you can’t expect your readers to believe in your research either. Write in the present tense, don’t use modal verbs that express varying degrees of certainty (such as may, might, or could ), and remember that you are not drawing a conclusion while trying not to exaggerate but making a clear statement that you then, in a way, try to disprove . And if that happens, that is not something to fear but an important part of the scientific process.

Similarly, don’t use “we hypothesize” when you explain the implications of your research or make predictions in the conclusion section of your manuscript, since these are clearly not hypotheses in the true sense of the word. As we said earlier, you will find that many authors of academic articles do not seem to care too much about these rather subtle distinctions, but thinking very clearly about your own research will not only help you write better but also ensure that even that infamous Reviewer 2 will find fewer reasons to nitpick about your manuscript. 

Perfect Your Manuscript With Professional Editing

Now that you know how to write a strong research hypothesis for your research paper, you might be interested in our free AI proofreader , Wordvice AI, which finds and fixes errors in grammar, punctuation, and word choice in academic texts. Or if you are interested in human proofreading , check out our English editing services , including research paper editing and manuscript editing .

On the Wordvice academic resources website , you can also find many more articles and other resources that can help you with writing the other parts of your research paper , with making a research paper outline before you put everything together, or with writing an effective cover letter once you are ready to submit.

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A Practical Guide to Writing Quantitative and Qualitative Research Questions and Hypotheses in Scholarly Articles

Edward barroga.

1 Department of General Education, Graduate School of Nursing Science, St. Luke’s International University, Tokyo, Japan.

Glafera Janet Matanguihan

2 Department of Biological Sciences, Messiah University, Mechanicsburg, PA, USA.

The development of research questions and the subsequent hypotheses are prerequisites to defining the main research purpose and specific objectives of a study. Consequently, these objectives determine the study design and research outcome. The development of research questions is a process based on knowledge of current trends, cutting-edge studies, and technological advances in the research field. Excellent research questions are focused and require a comprehensive literature search and in-depth understanding of the problem being investigated. Initially, research questions may be written as descriptive questions which could be developed into inferential questions. These questions must be specific and concise to provide a clear foundation for developing hypotheses. Hypotheses are more formal predictions about the research outcomes. These specify the possible results that may or may not be expected regarding the relationship between groups. Thus, research questions and hypotheses clarify the main purpose and specific objectives of the study, which in turn dictate the design of the study, its direction, and outcome. Studies developed from good research questions and hypotheses will have trustworthy outcomes with wide-ranging social and health implications.

INTRODUCTION

Scientific research is usually initiated by posing evidenced-based research questions which are then explicitly restated as hypotheses. 1 , 2 The hypotheses provide directions to guide the study, solutions, explanations, and expected results. 3 , 4 Both research questions and hypotheses are essentially formulated based on conventional theories and real-world processes, which allow the inception of novel studies and the ethical testing of ideas. 5 , 6

It is crucial to have knowledge of both quantitative and qualitative research 2 as both types of research involve writing research questions and hypotheses. 7 However, these crucial elements of research are sometimes overlooked; if not overlooked, then framed without the forethought and meticulous attention it needs. Planning and careful consideration are needed when developing quantitative or qualitative research, particularly when conceptualizing research questions and hypotheses. 4

There is a continuing need to support researchers in the creation of innovative research questions and hypotheses, as well as for journal articles that carefully review these elements. 1 When research questions and hypotheses are not carefully thought of, unethical studies and poor outcomes usually ensue. Carefully formulated research questions and hypotheses define well-founded objectives, which in turn determine the appropriate design, course, and outcome of the study. This article then aims to discuss in detail the various aspects of crafting research questions and hypotheses, with the goal of guiding researchers as they develop their own. Examples from the authors and peer-reviewed scientific articles in the healthcare field are provided to illustrate key points.

DEFINITIONS AND RELATIONSHIP OF RESEARCH QUESTIONS AND HYPOTHESES

A research question is what a study aims to answer after data analysis and interpretation. The answer is written in length in the discussion section of the paper. Thus, the research question gives a preview of the different parts and variables of the study meant to address the problem posed in the research question. 1 An excellent research question clarifies the research writing while facilitating understanding of the research topic, objective, scope, and limitations of the study. 5

On the other hand, a research hypothesis is an educated statement of an expected outcome. This statement is based on background research and current knowledge. 8 , 9 The research hypothesis makes a specific prediction about a new phenomenon 10 or a formal statement on the expected relationship between an independent variable and a dependent variable. 3 , 11 It provides a tentative answer to the research question to be tested or explored. 4

Hypotheses employ reasoning to predict a theory-based outcome. 10 These can also be developed from theories by focusing on components of theories that have not yet been observed. 10 The validity of hypotheses is often based on the testability of the prediction made in a reproducible experiment. 8

Conversely, hypotheses can also be rephrased as research questions. Several hypotheses based on existing theories and knowledge may be needed to answer a research question. Developing ethical research questions and hypotheses creates a research design that has logical relationships among variables. These relationships serve as a solid foundation for the conduct of the study. 4 , 11 Haphazardly constructed research questions can result in poorly formulated hypotheses and improper study designs, leading to unreliable results. Thus, the formulations of relevant research questions and verifiable hypotheses are crucial when beginning research. 12

CHARACTERISTICS OF GOOD RESEARCH QUESTIONS AND HYPOTHESES

Excellent research questions are specific and focused. These integrate collective data and observations to confirm or refute the subsequent hypotheses. Well-constructed hypotheses are based on previous reports and verify the research context. These are realistic, in-depth, sufficiently complex, and reproducible. More importantly, these hypotheses can be addressed and tested. 13

There are several characteristics of well-developed hypotheses. Good hypotheses are 1) empirically testable 7 , 10 , 11 , 13 ; 2) backed by preliminary evidence 9 ; 3) testable by ethical research 7 , 9 ; 4) based on original ideas 9 ; 5) have evidenced-based logical reasoning 10 ; and 6) can be predicted. 11 Good hypotheses can infer ethical and positive implications, indicating the presence of a relationship or effect relevant to the research theme. 7 , 11 These are initially developed from a general theory and branch into specific hypotheses by deductive reasoning. In the absence of a theory to base the hypotheses, inductive reasoning based on specific observations or findings form more general hypotheses. 10

TYPES OF RESEARCH QUESTIONS AND HYPOTHESES

Research questions and hypotheses are developed according to the type of research, which can be broadly classified into quantitative and qualitative research. We provide a summary of the types of research questions and hypotheses under quantitative and qualitative research categories in Table 1 .

Research questions in quantitative research

In quantitative research, research questions inquire about the relationships among variables being investigated and are usually framed at the start of the study. These are precise and typically linked to the subject population, dependent and independent variables, and research design. 1 Research questions may also attempt to describe the behavior of a population in relation to one or more variables, or describe the characteristics of variables to be measured ( descriptive research questions ). 1 , 5 , 14 These questions may also aim to discover differences between groups within the context of an outcome variable ( comparative research questions ), 1 , 5 , 14 or elucidate trends and interactions among variables ( relationship research questions ). 1 , 5 We provide examples of descriptive, comparative, and relationship research questions in quantitative research in Table 2 .

Hypotheses in quantitative research

In quantitative research, hypotheses predict the expected relationships among variables. 15 Relationships among variables that can be predicted include 1) between a single dependent variable and a single independent variable ( simple hypothesis ) or 2) between two or more independent and dependent variables ( complex hypothesis ). 4 , 11 Hypotheses may also specify the expected direction to be followed and imply an intellectual commitment to a particular outcome ( directional hypothesis ) 4 . On the other hand, hypotheses may not predict the exact direction and are used in the absence of a theory, or when findings contradict previous studies ( non-directional hypothesis ). 4 In addition, hypotheses can 1) define interdependency between variables ( associative hypothesis ), 4 2) propose an effect on the dependent variable from manipulation of the independent variable ( causal hypothesis ), 4 3) state a negative relationship between two variables ( null hypothesis ), 4 , 11 , 15 4) replace the working hypothesis if rejected ( alternative hypothesis ), 15 explain the relationship of phenomena to possibly generate a theory ( working hypothesis ), 11 5) involve quantifiable variables that can be tested statistically ( statistical hypothesis ), 11 6) or express a relationship whose interlinks can be verified logically ( logical hypothesis ). 11 We provide examples of simple, complex, directional, non-directional, associative, causal, null, alternative, working, statistical, and logical hypotheses in quantitative research, as well as the definition of quantitative hypothesis-testing research in Table 3 .

Research questions in qualitative research

Unlike research questions in quantitative research, research questions in qualitative research are usually continuously reviewed and reformulated. The central question and associated subquestions are stated more than the hypotheses. 15 The central question broadly explores a complex set of factors surrounding the central phenomenon, aiming to present the varied perspectives of participants. 15

There are varied goals for which qualitative research questions are developed. These questions can function in several ways, such as to 1) identify and describe existing conditions ( contextual research question s); 2) describe a phenomenon ( descriptive research questions ); 3) assess the effectiveness of existing methods, protocols, theories, or procedures ( evaluation research questions ); 4) examine a phenomenon or analyze the reasons or relationships between subjects or phenomena ( explanatory research questions ); or 5) focus on unknown aspects of a particular topic ( exploratory research questions ). 5 In addition, some qualitative research questions provide new ideas for the development of theories and actions ( generative research questions ) or advance specific ideologies of a position ( ideological research questions ). 1 Other qualitative research questions may build on a body of existing literature and become working guidelines ( ethnographic research questions ). Research questions may also be broadly stated without specific reference to the existing literature or a typology of questions ( phenomenological research questions ), may be directed towards generating a theory of some process ( grounded theory questions ), or may address a description of the case and the emerging themes ( qualitative case study questions ). 15 We provide examples of contextual, descriptive, evaluation, explanatory, exploratory, generative, ideological, ethnographic, phenomenological, grounded theory, and qualitative case study research questions in qualitative research in Table 4 , and the definition of qualitative hypothesis-generating research in Table 5 .

Qualitative studies usually pose at least one central research question and several subquestions starting with How or What . These research questions use exploratory verbs such as explore or describe . These also focus on one central phenomenon of interest, and may mention the participants and research site. 15

Hypotheses in qualitative research

Hypotheses in qualitative research are stated in the form of a clear statement concerning the problem to be investigated. Unlike in quantitative research where hypotheses are usually developed to be tested, qualitative research can lead to both hypothesis-testing and hypothesis-generating outcomes. 2 When studies require both quantitative and qualitative research questions, this suggests an integrative process between both research methods wherein a single mixed-methods research question can be developed. 1

FRAMEWORKS FOR DEVELOPING RESEARCH QUESTIONS AND HYPOTHESES

Research questions followed by hypotheses should be developed before the start of the study. 1 , 12 , 14 It is crucial to develop feasible research questions on a topic that is interesting to both the researcher and the scientific community. This can be achieved by a meticulous review of previous and current studies to establish a novel topic. Specific areas are subsequently focused on to generate ethical research questions. The relevance of the research questions is evaluated in terms of clarity of the resulting data, specificity of the methodology, objectivity of the outcome, depth of the research, and impact of the study. 1 , 5 These aspects constitute the FINER criteria (i.e., Feasible, Interesting, Novel, Ethical, and Relevant). 1 Clarity and effectiveness are achieved if research questions meet the FINER criteria. In addition to the FINER criteria, Ratan et al. described focus, complexity, novelty, feasibility, and measurability for evaluating the effectiveness of research questions. 14

The PICOT and PEO frameworks are also used when developing research questions. 1 The following elements are addressed in these frameworks, PICOT: P-population/patients/problem, I-intervention or indicator being studied, C-comparison group, O-outcome of interest, and T-timeframe of the study; PEO: P-population being studied, E-exposure to preexisting conditions, and O-outcome of interest. 1 Research questions are also considered good if these meet the “FINERMAPS” framework: Feasible, Interesting, Novel, Ethical, Relevant, Manageable, Appropriate, Potential value/publishable, and Systematic. 14

As we indicated earlier, research questions and hypotheses that are not carefully formulated result in unethical studies or poor outcomes. To illustrate this, we provide some examples of ambiguous research question and hypotheses that result in unclear and weak research objectives in quantitative research ( Table 6 ) 16 and qualitative research ( Table 7 ) 17 , and how to transform these ambiguous research question(s) and hypothesis(es) into clear and good statements.

a These statements were composed for comparison and illustrative purposes only.

b These statements are direct quotes from Higashihara and Horiuchi. 16

a This statement is a direct quote from Shimoda et al. 17

The other statements were composed for comparison and illustrative purposes only.

CONSTRUCTING RESEARCH QUESTIONS AND HYPOTHESES

To construct effective research questions and hypotheses, it is very important to 1) clarify the background and 2) identify the research problem at the outset of the research, within a specific timeframe. 9 Then, 3) review or conduct preliminary research to collect all available knowledge about the possible research questions by studying theories and previous studies. 18 Afterwards, 4) construct research questions to investigate the research problem. Identify variables to be accessed from the research questions 4 and make operational definitions of constructs from the research problem and questions. Thereafter, 5) construct specific deductive or inductive predictions in the form of hypotheses. 4 Finally, 6) state the study aims . This general flow for constructing effective research questions and hypotheses prior to conducting research is shown in Fig. 1 .

Research questions are used more frequently in qualitative research than objectives or hypotheses. 3 These questions seek to discover, understand, explore or describe experiences by asking “What” or “How.” The questions are open-ended to elicit a description rather than to relate variables or compare groups. The questions are continually reviewed, reformulated, and changed during the qualitative study. 3 Research questions are also used more frequently in survey projects than hypotheses in experiments in quantitative research to compare variables and their relationships.

Hypotheses are constructed based on the variables identified and as an if-then statement, following the template, ‘If a specific action is taken, then a certain outcome is expected.’ At this stage, some ideas regarding expectations from the research to be conducted must be drawn. 18 Then, the variables to be manipulated (independent) and influenced (dependent) are defined. 4 Thereafter, the hypothesis is stated and refined, and reproducible data tailored to the hypothesis are identified, collected, and analyzed. 4 The hypotheses must be testable and specific, 18 and should describe the variables and their relationships, the specific group being studied, and the predicted research outcome. 18 Hypotheses construction involves a testable proposition to be deduced from theory, and independent and dependent variables to be separated and measured separately. 3 Therefore, good hypotheses must be based on good research questions constructed at the start of a study or trial. 12

In summary, research questions are constructed after establishing the background of the study. Hypotheses are then developed based on the research questions. Thus, it is crucial to have excellent research questions to generate superior hypotheses. In turn, these would determine the research objectives and the design of the study, and ultimately, the outcome of the research. 12 Algorithms for building research questions and hypotheses are shown in Fig. 2 for quantitative research and in Fig. 3 for qualitative research.

EXAMPLES OF RESEARCH QUESTIONS FROM PUBLISHED ARTICLES

• EXAMPLE 1. Descriptive research question (quantitative research)
• - Presents research variables to be assessed (distinct phenotypes and subphenotypes)
• “BACKGROUND: Since COVID-19 was identified, its clinical and biological heterogeneity has been recognized. Identifying COVID-19 phenotypes might help guide basic, clinical, and translational research efforts.
• RESEARCH QUESTION: Does the clinical spectrum of patients with COVID-19 contain distinct phenotypes and subphenotypes? ” 19
• EXAMPLE 2. Relationship research question (quantitative research)
• - Shows interactions between dependent variable (static postural control) and independent variable (peripheral visual field loss)
• “Background: Integration of visual, vestibular, and proprioceptive sensations contributes to postural control. People with peripheral visual field loss have serious postural instability. However, the directional specificity of postural stability and sensory reweighting caused by gradual peripheral visual field loss remain unclear.
• Research question: What are the effects of peripheral visual field loss on static postural control ?” 20
• EXAMPLE 3. Comparative research question (quantitative research)
• - Clarifies the difference among groups with an outcome variable (patients enrolled in COMPERA with moderate PH or severe PH in COPD) and another group without the outcome variable (patients with idiopathic pulmonary arterial hypertension (IPAH))
• “BACKGROUND: Pulmonary hypertension (PH) in COPD is a poorly investigated clinical condition.
• RESEARCH QUESTION: Which factors determine the outcome of PH in COPD?
• STUDY DESIGN AND METHODS: We analyzed the characteristics and outcome of patients enrolled in the Comparative, Prospective Registry of Newly Initiated Therapies for Pulmonary Hypertension (COMPERA) with moderate or severe PH in COPD as defined during the 6th PH World Symposium who received medical therapy for PH and compared them with patients with idiopathic pulmonary arterial hypertension (IPAH) .” 21
• EXAMPLE 4. Exploratory research question (qualitative research)
• - Explores areas that have not been fully investigated (perspectives of families and children who receive care in clinic-based child obesity treatment) to have a deeper understanding of the research problem
• “Problem: Interventions for children with obesity lead to only modest improvements in BMI and long-term outcomes, and data are limited on the perspectives of families of children with obesity in clinic-based treatment. This scoping review seeks to answer the question: What is known about the perspectives of families and children who receive care in clinic-based child obesity treatment? This review aims to explore the scope of perspectives reported by families of children with obesity who have received individualized outpatient clinic-based obesity treatment.” 22
• EXAMPLE 5. Relationship research question (quantitative research)
• - Defines interactions between dependent variable (use of ankle strategies) and independent variable (changes in muscle tone)
• “Background: To maintain an upright standing posture against external disturbances, the human body mainly employs two types of postural control strategies: “ankle strategy” and “hip strategy.” While it has been reported that the magnitude of the disturbance alters the use of postural control strategies, it has not been elucidated how the level of muscle tone, one of the crucial parameters of bodily function, determines the use of each strategy. We have previously confirmed using forward dynamics simulations of human musculoskeletal models that an increased muscle tone promotes the use of ankle strategies. The objective of the present study was to experimentally evaluate a hypothesis: an increased muscle tone promotes the use of ankle strategies. Research question: Do changes in the muscle tone affect the use of ankle strategies ?” 23

EXAMPLES OF HYPOTHESES IN PUBLISHED ARTICLES

• EXAMPLE 1. Working hypothesis (quantitative research)
• - A hypothesis that is initially accepted for further research to produce a feasible theory
• “As fever may have benefit in shortening the duration of viral illness, it is plausible to hypothesize that the antipyretic efficacy of ibuprofen may be hindering the benefits of a fever response when taken during the early stages of COVID-19 illness .” 24
• “In conclusion, it is plausible to hypothesize that the antipyretic efficacy of ibuprofen may be hindering the benefits of a fever response . The difference in perceived safety of these agents in COVID-19 illness could be related to the more potent efficacy to reduce fever with ibuprofen compared to acetaminophen. Compelling data on the benefit of fever warrant further research and review to determine when to treat or withhold ibuprofen for early stage fever for COVID-19 and other related viral illnesses .” 24
• EXAMPLE 2. Exploratory hypothesis (qualitative research)
• - Explores particular areas deeper to clarify subjective experience and develop a formal hypothesis potentially testable in a future quantitative approach
• “We hypothesized that when thinking about a past experience of help-seeking, a self distancing prompt would cause increased help-seeking intentions and more favorable help-seeking outcome expectations .” 25
• “Conclusion
• Although a priori hypotheses were not supported, further research is warranted as results indicate the potential for using self-distancing approaches to increasing help-seeking among some people with depressive symptomatology.” 25
• EXAMPLE 3. Hypothesis-generating research to establish a framework for hypothesis testing (qualitative research)
• “We hypothesize that compassionate care is beneficial for patients (better outcomes), healthcare systems and payers (lower costs), and healthcare providers (lower burnout). ” 26
• Compassionomics is the branch of knowledge and scientific study of the effects of compassionate healthcare. Our main hypotheses are that compassionate healthcare is beneficial for (1) patients, by improving clinical outcomes, (2) healthcare systems and payers, by supporting financial sustainability, and (3) HCPs, by lowering burnout and promoting resilience and well-being. The purpose of this paper is to establish a scientific framework for testing the hypotheses above . If these hypotheses are confirmed through rigorous research, compassionomics will belong in the science of evidence-based medicine, with major implications for all healthcare domains.” 26
• EXAMPLE 4. Statistical hypothesis (quantitative research)
• - An assumption is made about the relationship among several population characteristics ( gender differences in sociodemographic and clinical characteristics of adults with ADHD ). Validity is tested by statistical experiment or analysis ( chi-square test, Students t-test, and logistic regression analysis)
• “Our research investigated gender differences in sociodemographic and clinical characteristics of adults with ADHD in a Japanese clinical sample. Due to unique Japanese cultural ideals and expectations of women's behavior that are in opposition to ADHD symptoms, we hypothesized that women with ADHD experience more difficulties and present more dysfunctions than men . We tested the following hypotheses: first, women with ADHD have more comorbidities than men with ADHD; second, women with ADHD experience more social hardships than men, such as having less full-time employment and being more likely to be divorced.” 27
• “Statistical Analysis
• ( text omitted ) Between-gender comparisons were made using the chi-squared test for categorical variables and Students t-test for continuous variables…( text omitted ). A logistic regression analysis was performed for employment status, marital status, and comorbidity to evaluate the independent effects of gender on these dependent variables.” 27

EXAMPLES OF HYPOTHESIS AS WRITTEN IN PUBLISHED ARTICLES IN RELATION TO OTHER PARTS

• EXAMPLE 1. Background, hypotheses, and aims are provided
• “Pregnant women need skilled care during pregnancy and childbirth, but that skilled care is often delayed in some countries …( text omitted ). The focused antenatal care (FANC) model of WHO recommends that nurses provide information or counseling to all pregnant women …( text omitted ). Job aids are visual support materials that provide the right kind of information using graphics and words in a simple and yet effective manner. When nurses are not highly trained or have many work details to attend to, these job aids can serve as a content reminder for the nurses and can be used for educating their patients (Jennings, Yebadokpo, Affo, & Agbogbe, 2010) ( text omitted ). Importantly, additional evidence is needed to confirm how job aids can further improve the quality of ANC counseling by health workers in maternal care …( text omitted )” 28
• “ This has led us to hypothesize that the quality of ANC counseling would be better if supported by job aids. Consequently, a better quality of ANC counseling is expected to produce higher levels of awareness concerning the danger signs of pregnancy and a more favorable impression of the caring behavior of nurses .” 28
• “This study aimed to examine the differences in the responses of pregnant women to a job aid-supported intervention during ANC visit in terms of 1) their understanding of the danger signs of pregnancy and 2) their impression of the caring behaviors of nurses to pregnant women in rural Tanzania.” 28
• EXAMPLE 2. Background, hypotheses, and aims are provided
• “We conducted a two-arm randomized controlled trial (RCT) to evaluate and compare changes in salivary cortisol and oxytocin levels of first-time pregnant women between experimental and control groups. The women in the experimental group touched and held an infant for 30 min (experimental intervention protocol), whereas those in the control group watched a DVD movie of an infant (control intervention protocol). The primary outcome was salivary cortisol level and the secondary outcome was salivary oxytocin level.” 29
• “ We hypothesize that at 30 min after touching and holding an infant, the salivary cortisol level will significantly decrease and the salivary oxytocin level will increase in the experimental group compared with the control group .” 29
• EXAMPLE 3. Background, aim, and hypothesis are provided
• “In countries where the maternal mortality ratio remains high, antenatal education to increase Birth Preparedness and Complication Readiness (BPCR) is considered one of the top priorities [1]. BPCR includes birth plans during the antenatal period, such as the birthplace, birth attendant, transportation, health facility for complications, expenses, and birth materials, as well as family coordination to achieve such birth plans. In Tanzania, although increasing, only about half of all pregnant women attend an antenatal clinic more than four times [4]. Moreover, the information provided during antenatal care (ANC) is insufficient. In the resource-poor settings, antenatal group education is a potential approach because of the limited time for individual counseling at antenatal clinics.” 30
• “This study aimed to evaluate an antenatal group education program among pregnant women and their families with respect to birth-preparedness and maternal and infant outcomes in rural villages of Tanzania.” 30
• “ The study hypothesis was if Tanzanian pregnant women and their families received a family-oriented antenatal group education, they would (1) have a higher level of BPCR, (2) attend antenatal clinic four or more times, (3) give birth in a health facility, (4) have less complications of women at birth, and (5) have less complications and deaths of infants than those who did not receive the education .” 30

Research questions and hypotheses are crucial components to any type of research, whether quantitative or qualitative. These questions should be developed at the very beginning of the study. Excellent research questions lead to superior hypotheses, which, like a compass, set the direction of research, and can often determine the successful conduct of the study. Many research studies have floundered because the development of research questions and subsequent hypotheses was not given the thought and meticulous attention needed. The development of research questions and hypotheses is an iterative process based on extensive knowledge of the literature and insightful grasp of the knowledge gap. Focused, concise, and specific research questions provide a strong foundation for constructing hypotheses which serve as formal predictions about the research outcomes. Research questions and hypotheses are crucial elements of research that should not be overlooked. They should be carefully thought of and constructed when planning research. This avoids unethical studies and poor outcomes by defining well-founded objectives that determine the design, course, and outcome of the study.

Disclosure: The authors have no potential conflicts of interest to disclose.

Author Contributions:

• Conceptualization: Barroga E, Matanguihan GJ.
• Methodology: Barroga E, Matanguihan GJ.
• Writing - original draft: Barroga E, Matanguihan GJ.
• Writing - review & editing: Barroga E, Matanguihan GJ.

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Home » What is a Hypothesis – Types, Examples and Writing Guide

What is a Hypothesis – Types, Examples and Writing Guide

Table of Contents

Definition:

Hypothesis is an educated guess or proposed explanation for a phenomenon, based on some initial observations or data. It is a tentative statement that can be tested and potentially proven or disproven through further investigation and experimentation.

Hypothesis is often used in scientific research to guide the design of experiments and the collection and analysis of data. It is an essential element of the scientific method, as it allows researchers to make predictions about the outcome of their experiments and to test those predictions to determine their accuracy.

Types of Hypothesis

Types of Hypothesis are as follows:

Research Hypothesis

A research hypothesis is a statement that predicts a relationship between variables. It is usually formulated as a specific statement that can be tested through research, and it is often used in scientific research to guide the design of experiments.

Null Hypothesis

The null hypothesis is a statement that assumes there is no significant difference or relationship between variables. It is often used as a starting point for testing the research hypothesis, and if the results of the study reject the null hypothesis, it suggests that there is a significant difference or relationship between variables.

Alternative Hypothesis

An alternative hypothesis is a statement that assumes there is a significant difference or relationship between variables. It is often used as an alternative to the null hypothesis and is tested against the null hypothesis to determine which statement is more accurate.

Directional Hypothesis

A directional hypothesis is a statement that predicts the direction of the relationship between variables. For example, a researcher might predict that increasing the amount of exercise will result in a decrease in body weight.

Non-directional Hypothesis

A non-directional hypothesis is a statement that predicts the relationship between variables but does not specify the direction. For example, a researcher might predict that there is a relationship between the amount of exercise and body weight, but they do not specify whether increasing or decreasing exercise will affect body weight.

Statistical Hypothesis

A statistical hypothesis is a statement that assumes a particular statistical model or distribution for the data. It is often used in statistical analysis to test the significance of a particular result.

Composite Hypothesis

A composite hypothesis is a statement that assumes more than one condition or outcome. It can be divided into several sub-hypotheses, each of which represents a different possible outcome.

Empirical Hypothesis

An empirical hypothesis is a statement that is based on observed phenomena or data. It is often used in scientific research to develop theories or models that explain the observed phenomena.

Simple Hypothesis

A simple hypothesis is a statement that assumes only one outcome or condition. It is often used in scientific research to test a single variable or factor.

Complex Hypothesis

A complex hypothesis is a statement that assumes multiple outcomes or conditions. It is often used in scientific research to test the effects of multiple variables or factors on a particular outcome.

Applications of Hypothesis

Hypotheses are used in various fields to guide research and make predictions about the outcomes of experiments or observations. Here are some examples of how hypotheses are applied in different fields:

• Science : In scientific research, hypotheses are used to test the validity of theories and models that explain natural phenomena. For example, a hypothesis might be formulated to test the effects of a particular variable on a natural system, such as the effects of climate change on an ecosystem.
• Medicine : In medical research, hypotheses are used to test the effectiveness of treatments and therapies for specific conditions. For example, a hypothesis might be formulated to test the effects of a new drug on a particular disease.
• Psychology : In psychology, hypotheses are used to test theories and models of human behavior and cognition. For example, a hypothesis might be formulated to test the effects of a particular stimulus on the brain or behavior.
• Sociology : In sociology, hypotheses are used to test theories and models of social phenomena, such as the effects of social structures or institutions on human behavior. For example, a hypothesis might be formulated to test the effects of income inequality on crime rates.
• Business : In business research, hypotheses are used to test the validity of theories and models that explain business phenomena, such as consumer behavior or market trends. For example, a hypothesis might be formulated to test the effects of a new marketing campaign on consumer buying behavior.
• Engineering : In engineering, hypotheses are used to test the effectiveness of new technologies or designs. For example, a hypothesis might be formulated to test the efficiency of a new solar panel design.

How to write a Hypothesis

Here are the steps to follow when writing a hypothesis:

Identify the Research Question

The first step is to identify the research question that you want to answer through your study. This question should be clear, specific, and focused. It should be something that can be investigated empirically and that has some relevance or significance in the field.

Conduct a Literature Review

Before writing your hypothesis, it’s essential to conduct a thorough literature review to understand what is already known about the topic. This will help you to identify the research gap and formulate a hypothesis that builds on existing knowledge.

Determine the Variables

The next step is to identify the variables involved in the research question. A variable is any characteristic or factor that can vary or change. There are two types of variables: independent and dependent. The independent variable is the one that is manipulated or changed by the researcher, while the dependent variable is the one that is measured or observed as a result of the independent variable.

Formulate the Hypothesis

Based on the research question and the variables involved, you can now formulate your hypothesis. A hypothesis should be a clear and concise statement that predicts the relationship between the variables. It should be testable through empirical research and based on existing theory or evidence.

Write the Null Hypothesis

The null hypothesis is the opposite of the alternative hypothesis, which is the hypothesis that you are testing. The null hypothesis states that there is no significant difference or relationship between the variables. It is important to write the null hypothesis because it allows you to compare your results with what would be expected by chance.

Refine the Hypothesis

After formulating the hypothesis, it’s important to refine it and make it more precise. This may involve clarifying the variables, specifying the direction of the relationship, or making the hypothesis more testable.

Examples of Hypothesis

Here are a few examples of hypotheses in different fields:

• Psychology : “Increased exposure to violent video games leads to increased aggressive behavior in adolescents.”
• Biology : “Higher levels of carbon dioxide in the atmosphere will lead to increased plant growth.”
• Sociology : “Individuals who grow up in households with higher socioeconomic status will have higher levels of education and income as adults.”
• Education : “Implementing a new teaching method will result in higher student achievement scores.”
• Marketing : “Customers who receive a personalized email will be more likely to make a purchase than those who receive a generic email.”
• Physics : “An increase in temperature will cause an increase in the volume of a gas, assuming all other variables remain constant.”
• Medicine : “Consuming a diet high in saturated fats will increase the risk of developing heart disease.”

Purpose of Hypothesis

The purpose of a hypothesis is to provide a testable explanation for an observed phenomenon or a prediction of a future outcome based on existing knowledge or theories. A hypothesis is an essential part of the scientific method and helps to guide the research process by providing a clear focus for investigation. It enables scientists to design experiments or studies to gather evidence and data that can support or refute the proposed explanation or prediction.

The formulation of a hypothesis is based on existing knowledge, observations, and theories, and it should be specific, testable, and falsifiable. A specific hypothesis helps to define the research question, which is important in the research process as it guides the selection of an appropriate research design and methodology. Testability of the hypothesis means that it can be proven or disproven through empirical data collection and analysis. Falsifiability means that the hypothesis should be formulated in such a way that it can be proven wrong if it is incorrect.

In addition to guiding the research process, the testing of hypotheses can lead to new discoveries and advancements in scientific knowledge. When a hypothesis is supported by the data, it can be used to develop new theories or models to explain the observed phenomenon. When a hypothesis is not supported by the data, it can help to refine existing theories or prompt the development of new hypotheses to explain the phenomenon.

When to use Hypothesis

Here are some common situations in which hypotheses are used:

• In scientific research , hypotheses are used to guide the design of experiments and to help researchers make predictions about the outcomes of those experiments.
• In social science research , hypotheses are used to test theories about human behavior, social relationships, and other phenomena.
• I n business , hypotheses can be used to guide decisions about marketing, product development, and other areas. For example, a hypothesis might be that a new product will sell well in a particular market, and this hypothesis can be tested through market research.

Characteristics of Hypothesis

Here are some common characteristics of a hypothesis:

• Testable : A hypothesis must be able to be tested through observation or experimentation. This means that it must be possible to collect data that will either support or refute the hypothesis.
• Falsifiable : A hypothesis must be able to be proven false if it is not supported by the data. If a hypothesis cannot be falsified, then it is not a scientific hypothesis.
• Clear and concise : A hypothesis should be stated in a clear and concise manner so that it can be easily understood and tested.
• Based on existing knowledge : A hypothesis should be based on existing knowledge and research in the field. It should not be based on personal beliefs or opinions.
• Specific : A hypothesis should be specific in terms of the variables being tested and the predicted outcome. This will help to ensure that the research is focused and well-designed.
• Tentative: A hypothesis is a tentative statement or assumption that requires further testing and evidence to be confirmed or refuted. It is not a final conclusion or assertion.
• Relevant : A hypothesis should be relevant to the research question or problem being studied. It should address a gap in knowledge or provide a new perspective on the issue.

Advantages of Hypothesis

Hypotheses have several advantages in scientific research and experimentation:

• Guides research: A hypothesis provides a clear and specific direction for research. It helps to focus the research question, select appropriate methods and variables, and interpret the results.
• Predictive powe r: A hypothesis makes predictions about the outcome of research, which can be tested through experimentation. This allows researchers to evaluate the validity of the hypothesis and make new discoveries.
• Facilitates communication: A hypothesis provides a common language and framework for scientists to communicate with one another about their research. This helps to facilitate the exchange of ideas and promotes collaboration.
• Efficient use of resources: A hypothesis helps researchers to use their time, resources, and funding efficiently by directing them towards specific research questions and methods that are most likely to yield results.
• Provides a basis for further research: A hypothesis that is supported by data provides a basis for further research and exploration. It can lead to new hypotheses, theories, and discoveries.
• Increases objectivity: A hypothesis can help to increase objectivity in research by providing a clear and specific framework for testing and interpreting results. This can reduce bias and increase the reliability of research findings.

Limitations of Hypothesis

Some Limitations of the Hypothesis are as follows:

• Limited to observable phenomena: Hypotheses are limited to observable phenomena and cannot account for unobservable or intangible factors. This means that some research questions may not be amenable to hypothesis testing.
• May be inaccurate or incomplete: Hypotheses are based on existing knowledge and research, which may be incomplete or inaccurate. This can lead to flawed hypotheses and erroneous conclusions.
• May be biased: Hypotheses may be biased by the researcher’s own beliefs, values, or assumptions. This can lead to selective interpretation of data and a lack of objectivity in research.
• Cannot prove causation: A hypothesis can only show a correlation between variables, but it cannot prove causation. This requires further experimentation and analysis.
• Limited to specific contexts: Hypotheses are limited to specific contexts and may not be generalizable to other situations or populations. This means that results may not be applicable in other contexts or may require further testing.
• May be affected by chance : Hypotheses may be affected by chance or random variation, which can obscure or distort the true relationship between variables.

About the author

Muhammad Hassan

Researcher, Academic Writer, Web developer

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What Is A Research (Scientific) Hypothesis? A plain-language explainer + examples

By:  Derek Jansen (MBA)  | Reviewed By: Dr Eunice Rautenbach | June 2020

If you’re new to the world of research, or it’s your first time writing a dissertation or thesis, you’re probably noticing that the words “research hypothesis” and “scientific hypothesis” are used quite a bit, and you’re wondering what they mean in a research context .

“Hypothesis” is one of those words that people use loosely, thinking they understand what it means. However, it has a very specific meaning within academic research. So, it’s important to understand the exact meaning before you start hypothesizing. 

Research Hypothesis 101

• What is a hypothesis ?
• What is a research hypothesis (scientific hypothesis)?
• Requirements for a research hypothesis
• Definition of a research hypothesis
• The null hypothesis

What is a hypothesis?

Let’s start with the general definition of a hypothesis (not a research hypothesis or scientific hypothesis), according to the Cambridge Dictionary:

Hypothesis: an idea or explanation for something that is based on known facts but has not yet been proved.

In other words, it’s a statement that provides an explanation for why or how something works, based on facts (or some reasonable assumptions), but that has not yet been specifically tested . For example, a hypothesis might look something like this:

Hypothesis: sleep impacts academic performance.

This statement predicts that academic performance will be influenced by the amount and/or quality of sleep a student engages in – sounds reasonable, right? It’s based on reasonable assumptions , underpinned by what we currently know about sleep and health (from the existing literature). So, loosely speaking, we could call it a hypothesis, at least by the dictionary definition.

But that’s not good enough…

Unfortunately, that’s not quite sophisticated enough to describe a research hypothesis (also sometimes called a scientific hypothesis), and it wouldn’t be acceptable in a dissertation, thesis or research paper . In the world of academic research, a statement needs a few more criteria to constitute a true research hypothesis .

What is a research hypothesis?

A research hypothesis (also called a scientific hypothesis) is a statement about the expected outcome of a study (for example, a dissertation or thesis). To constitute a quality hypothesis, the statement needs to have three attributes – specificity , clarity and testability .

Let’s take a look at these more closely.

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Hypothesis Essential #1: Specificity & Clarity

A good research hypothesis needs to be extremely clear and articulate about both what’ s being assessed (who or what variables are involved ) and the expected outcome (for example, a difference between groups, a relationship between variables, etc.).

Let’s stick with our sleepy students example and look at how this statement could be more specific and clear.

Hypothesis: Students who sleep at least 8 hours per night will, on average, achieve higher grades in standardised tests than students who sleep less than 8 hours a night.

As you can see, the statement is very specific as it identifies the variables involved (sleep hours and test grades), the parties involved (two groups of students), as well as the predicted relationship type (a positive relationship). There’s no ambiguity or uncertainty about who or what is involved in the statement, and the expected outcome is clear.

Contrast that to the original hypothesis we looked at – “Sleep impacts academic performance” – and you can see the difference. “Sleep” and “academic performance” are both comparatively vague , and there’s no indication of what the expected relationship direction is (more sleep or less sleep). As you can see, specificity and clarity are key.

Hypothesis Essential #2: Testability (Provability)

A statement must be testable to qualify as a research hypothesis. In other words, there needs to be a way to prove (or disprove) the statement. If it’s not testable, it’s not a hypothesis – simple as that.

For example, consider the hypothesis we mentioned earlier:

Hypothesis: Students who sleep at least 8 hours per night will, on average, achieve higher grades in standardised tests than students who sleep less than 8 hours a night.  

We could test this statement by undertaking a quantitative study involving two groups of students, one that gets 8 or more hours of sleep per night for a fixed period, and one that gets less. We could then compare the standardised test results for both groups to see if there’s a statistically significant difference. 

Again, if you compare this to the original hypothesis we looked at – “Sleep impacts academic performance” – you can see that it would be quite difficult to test that statement, primarily because it isn’t specific enough. How much sleep? By who? What type of academic performance?

So, remember the mantra – if you can’t test it, it’s not a hypothesis 🙂

Defining A Research Hypothesis

You’re still with us? Great! Let’s recap and pin down a clear definition of a hypothesis.

A research hypothesis (or scientific hypothesis) is a statement about an expected relationship between variables, or explanation of an occurrence, that is clear, specific and testable.

So, when you write up hypotheses for your dissertation or thesis, make sure that they meet all these criteria. If you do, you’ll not only have rock-solid hypotheses but you’ll also ensure a clear focus for your entire research project.

What about the null hypothesis?

You may have also heard the terms null hypothesis , alternative hypothesis, or H-zero thrown around. At a simple level, the null hypothesis is the counter-proposal to the original hypothesis.

For example, if the hypothesis predicts that there is a relationship between two variables (for example, sleep and academic performance), the null hypothesis would predict that there is no relationship between those variables.

At a more technical level, the null hypothesis proposes that no statistical significance exists in a set of given observations and that any differences are due to chance alone.

And there you have it – hypotheses in a nutshell. 

If you have any questions, be sure to leave a comment below and we’ll do our best to help you. If you need hands-on help developing and testing your hypotheses, consider our private coaching service , where we hold your hand through the research journey.

Psst… there’s more (for free)

This post is part of our dissertation mini-course, which covers everything you need to get started with your dissertation, thesis or research project. 

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15 Comments

Very useful information. I benefit more from getting more information in this regard.

Very great insight,educative and informative. Please give meet deep critics on many research data of public international Law like human rights, environment, natural resources, law of the sea etc

In a book I read a distinction is made between null, research, and alternative hypothesis. As far as I understand, alternative and research hypotheses are the same. Can you please elaborate? Best Afshin

This is a self explanatory, easy going site. I will recommend this to my friends and colleagues.

Very good definition. How can I cite your definition in my thesis? Thank you. Is nul hypothesis compulsory in a research?

It’s a counter-proposal to be proven as a rejection

Please what is the difference between alternate hypothesis and research hypothesis?

It is a very good explanation. However, it limits hypotheses to statistically tasteable ideas. What about for qualitative researches or other researches that involve quantitative data that don’t need statistical tests?

In qualitative research, one typically uses propositions, not hypotheses.

could you please elaborate it more

I’ve benefited greatly from these notes, thank you.

This is very helpful

well articulated ideas are presented here, thank you for being reliable sources of information

Excellent. Thanks for being clear and sound about the research methodology and hypothesis (quantitative research)

I have only a simple question regarding the null hypothesis. – Is the null hypothesis (Ho) known as the reversible hypothesis of the alternative hypothesis (H1? – How to test it in academic research?

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Research Hypothesis In Psychology: Types, & Examples

Saul Mcleod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

Learn about our Editorial Process

Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

On This Page:

A research hypothesis, in its plural form “hypotheses,” is a specific, testable prediction about the anticipated results of a study, established at its outset. It is a key component of the scientific method .

Hypotheses connect theory to data and guide the research process towards expanding scientific understanding

Some key points about hypotheses:

• A hypothesis expresses an expected pattern or relationship. It connects the variables under investigation.
• It is stated in clear, precise terms before any data collection or analysis occurs. This makes the hypothesis testable.
• A hypothesis must be falsifiable. It should be possible, even if unlikely in practice, to collect data that disconfirms rather than supports the hypothesis.
• Hypotheses guide research. Scientists design studies to explicitly evaluate hypotheses about how nature works.
• For a hypothesis to be valid, it must be testable against empirical evidence. The evidence can then confirm or disprove the testable predictions.
• Hypotheses are informed by background knowledge and observation, but go beyond what is already known to propose an explanation of how or why something occurs.

Predictions typically arise from a thorough knowledge of the research literature, curiosity about real-world problems or implications, and integrating this to advance theory. They build on existing literature while providing new insight.

Types of Research Hypotheses

Alternative hypothesis.

The research hypothesis is often called the alternative or experimental hypothesis in experimental research.

It typically suggests a potential relationship between two key variables: the independent variable, which the researcher manipulates, and the dependent variable, which is measured based on those changes.

The alternative hypothesis states a relationship exists between the two variables being studied (one variable affects the other).

A hypothesis is a testable statement or prediction about the relationship between two or more variables. It is a key component of the scientific method. Some key points about hypotheses:

• Important hypotheses lead to predictions that can be tested empirically. The evidence can then confirm or disprove the testable predictions.

In summary, a hypothesis is a precise, testable statement of what researchers expect to happen in a study and why. Hypotheses connect theory to data and guide the research process towards expanding scientific understanding.

An experimental hypothesis predicts what change(s) will occur in the dependent variable when the independent variable is manipulated.

It states that the results are not due to chance and are significant in supporting the theory being investigated.

The alternative hypothesis can be directional, indicating a specific direction of the effect, or non-directional, suggesting a difference without specifying its nature. It’s what researchers aim to support or demonstrate through their study.

Null Hypothesis

The null hypothesis states no relationship exists between the two variables being studied (one variable does not affect the other). There will be no changes in the dependent variable due to manipulating the independent variable.

It states results are due to chance and are not significant in supporting the idea being investigated.

The null hypothesis, positing no effect or relationship, is a foundational contrast to the research hypothesis in scientific inquiry. It establishes a baseline for statistical testing, promoting objectivity by initiating research from a neutral stance.

Many statistical methods are tailored to test the null hypothesis, determining the likelihood of observed results if no true effect exists.

This dual-hypothesis approach provides clarity, ensuring that research intentions are explicit, and fosters consistency across scientific studies, enhancing the standardization and interpretability of research outcomes.

Nondirectional Hypothesis

A non-directional hypothesis, also known as a two-tailed hypothesis, predicts that there is a difference or relationship between two variables but does not specify the direction of this relationship.

It merely indicates that a change or effect will occur without predicting which group will have higher or lower values.

For example, “There is a difference in performance between Group A and Group B” is a non-directional hypothesis.

Directional Hypothesis

A directional (one-tailed) hypothesis predicts the nature of the effect of the independent variable on the dependent variable. It predicts in which direction the change will take place. (i.e., greater, smaller, less, more)

It specifies whether one variable is greater, lesser, or different from another, rather than just indicating that there’s a difference without specifying its nature.

For example, “Exercise increases weight loss” is a directional hypothesis.

Falsifiability

The Falsification Principle, proposed by Karl Popper , is a way of demarcating science from non-science. It suggests that for a theory or hypothesis to be considered scientific, it must be testable and irrefutable.

Falsifiability emphasizes that scientific claims shouldn’t just be confirmable but should also have the potential to be proven wrong.

It means that there should exist some potential evidence or experiment that could prove the proposition false.

However many confirming instances exist for a theory, it only takes one counter observation to falsify it. For example, the hypothesis that “all swans are white,” can be falsified by observing a black swan.

For Popper, science should attempt to disprove a theory rather than attempt to continually provide evidence to support a research hypothesis.

Can a Hypothesis be Proven?

Hypotheses make probabilistic predictions. They state the expected outcome if a particular relationship exists. However, a study result supporting a hypothesis does not definitively prove it is true.

All studies have limitations. There may be unknown confounding factors or issues that limit the certainty of conclusions. Additional studies may yield different results.

In science, hypotheses can realistically only be supported with some degree of confidence, not proven. The process of science is to incrementally accumulate evidence for and against hypothesized relationships in an ongoing pursuit of better models and explanations that best fit the empirical data. But hypotheses remain open to revision and rejection if that is where the evidence leads.

• Disproving a hypothesis is definitive. Solid disconfirmatory evidence will falsify a hypothesis and require altering or discarding it based on the evidence.
• However, confirming evidence is always open to revision. Other explanations may account for the same results, and additional or contradictory evidence may emerge over time.

We can never 100% prove the alternative hypothesis. Instead, we see if we can disprove, or reject the null hypothesis.

If we reject the null hypothesis, this doesn’t mean that our alternative hypothesis is correct but does support the alternative/experimental hypothesis.

Upon analysis of the results, an alternative hypothesis can be rejected or supported, but it can never be proven to be correct. We must avoid any reference to results proving a theory as this implies 100% certainty, and there is always a chance that evidence may exist which could refute a theory.

How to Write a Hypothesis

• Identify variables . The researcher manipulates the independent variable and the dependent variable is the measured outcome.
• Operationalized the variables being investigated . Operationalization of a hypothesis refers to the process of making the variables physically measurable or testable, e.g. if you are about to study aggression, you might count the number of punches given by participants.
• Decide on a direction for your prediction . If there is evidence in the literature to support a specific effect of the independent variable on the dependent variable, write a directional (one-tailed) hypothesis. If there are limited or ambiguous findings in the literature regarding the effect of the independent variable on the dependent variable, write a non-directional (two-tailed) hypothesis.
• Make it Testable : Ensure your hypothesis can be tested through experimentation or observation. It should be possible to prove it false (principle of falsifiability).
• Clear & concise language . A strong hypothesis is concise (typically one to two sentences long), and formulated using clear and straightforward language, ensuring it’s easily understood and testable.

Consider a hypothesis many teachers might subscribe to: students work better on Monday morning than on Friday afternoon (IV=Day, DV= Standard of work).

Now, if we decide to study this by giving the same group of students a lesson on a Monday morning and a Friday afternoon and then measuring their immediate recall of the material covered in each session, we would end up with the following:

• The alternative hypothesis states that students will recall significantly more information on a Monday morning than on a Friday afternoon.
• The null hypothesis states that there will be no significant difference in the amount recalled on a Monday morning compared to a Friday afternoon. Any difference will be due to chance or confounding factors.

More Examples

• Memory : Participants exposed to classical music during study sessions will recall more items from a list than those who studied in silence.
• Social Psychology : Individuals who frequently engage in social media use will report higher levels of perceived social isolation compared to those who use it infrequently.
• Developmental Psychology : Children who engage in regular imaginative play have better problem-solving skills than those who don’t.
• Clinical Psychology : Cognitive-behavioral therapy will be more effective in reducing symptoms of anxiety over a 6-month period compared to traditional talk therapy.
• Cognitive Psychology : Individuals who multitask between various electronic devices will have shorter attention spans on focused tasks than those who single-task.
• Health Psychology : Patients who practice mindfulness meditation will experience lower levels of chronic pain compared to those who don’t meditate.
• Organizational Psychology : Employees in open-plan offices will report higher levels of stress than those in private offices.
• Behavioral Psychology : Rats rewarded with food after pressing a lever will press it more frequently than rats who receive no reward.

15 Hypothesis Examples

A hypothesis is defined as a testable prediction , and is used primarily in scientific experiments as a potential or predicted outcome that scientists attempt to prove or disprove (Atkinson et al., 2021; Tan, 2022).

In my types of hypothesis article, I outlined 13 different hypotheses, including the directional hypothesis (which makes a prediction about an effect of a treatment will be positive or negative) and the associative hypothesis (which makes a prediction about the association between two variables).

This article will dive into some interesting examples of hypotheses and examine potential ways you might test each one.

Hypothesis Examples

1. “inadequate sleep decreases memory retention”.

Field: Psychology

Type: Causal Hypothesis A causal hypothesis explores the effect of one variable on another. This example posits that a lack of adequate sleep causes decreased memory retention. In other words, if you are not getting enough sleep, your ability to remember and recall information may suffer.

How to Test:

To test this hypothesis, you might devise an experiment whereby your participants are divided into two groups: one receives an average of 8 hours of sleep per night for a week, while the other gets less than the recommended sleep amount.

During this time, all participants would daily study and recall new, specific information. You’d then measure memory retention of this information for both groups using standard memory tests and compare the results.

Should the group with less sleep have statistically significant poorer memory scores, the hypothesis would be supported.

Ensuring the integrity of the experiment requires taking into account factors such as individual health differences, stress levels, and daily nutrition.

Relevant Study: Sleep loss, learning capacity and academic performance (Curcio, Ferrara & De Gennaro, 2006)

2. “Increase in Temperature Leads to Increase in Kinetic Energy”

Field: Physics

Type: Deductive Hypothesis The deductive hypothesis applies the logic of deductive reasoning – it moves from a general premise to a more specific conclusion. This specific hypothesis assumes that as temperature increases, the kinetic energy of particles also increases – that is, when you heat something up, its particles move around more rapidly.

This hypothesis could be examined by heating a gas in a controlled environment and capturing the movement of its particles as a function of temperature.

You’d gradually increase the temperature and measure the kinetic energy of the gas particles with each increment. If the kinetic energy consistently rises with the temperature, your hypothesis gets supporting evidence.

Variables such as pressure and volume of the gas would need to be held constant to ensure validity of results.

3. “Children Raised in Bilingual Homes Develop Better Cognitive Skills”

Field: Psychology/Linguistics

Type: Comparative Hypothesis The comparative hypothesis posits a difference between two or more groups based on certain variables. In this context, you might propose that children raised in bilingual homes have superior cognitive skills compared to those raised in monolingual homes.

Testing this hypothesis could involve identifying two groups of children: those raised in bilingual homes, and those raised in monolingual homes.

Cognitive skills in both groups would be evaluated using a standard cognitive ability test at different stages of development. The examination would be repeated over a significant time period for consistency.

If the group raised in bilingual homes persistently scores higher than the other, the hypothesis would thereby be supported.

The challenge for the researcher would be controlling for other variables that could impact cognitive development, such as socio-economic status, education level of parents, and parenting styles.

Relevant Study: The cognitive benefits of being bilingual (Marian & Shook, 2012)

4. “High-Fiber Diet Leads to Lower Incidences of Cardiovascular Diseases”

Field: Medicine/Nutrition

Type: Alternative Hypothesis The alternative hypothesis suggests an alternative to a null hypothesis. In this context, the implied null hypothesis could be that diet has no effect on cardiovascular health, which the alternative hypothesis contradicts by suggesting that a high-fiber diet leads to fewer instances of cardiovascular diseases.

To test this hypothesis, a longitudinal study could be conducted on two groups of participants; one adheres to a high-fiber diet, while the other follows a diet low in fiber.

After a fixed period, the cardiovascular health of participants in both groups could be analyzed and compared. If the group following a high-fiber diet has a lower number of recorded cases of cardiovascular diseases, it would provide evidence supporting the hypothesis.

Control measures should be implemented to exclude the influence of other lifestyle and genetic factors that contribute to cardiovascular health.

Relevant Study: Dietary fiber, inflammation, and cardiovascular disease (King, 2005)

5. “Gravity Influences the Directional Growth of Plants”

Field: Agronomy / Botany

Type: Explanatory Hypothesis An explanatory hypothesis attempts to explain a phenomenon. In this case, the hypothesis proposes that gravity affects how plants direct their growth – both above-ground (toward sunlight) and below-ground (towards water and other resources).

The testing could be conducted by growing plants in a rotating cylinder to create artificial gravity.

Observations on the direction of growth, over a specified period, can provide insights into the influencing factors. If plants consistently direct their growth in a manner that indicates the influence of gravitational pull, the hypothesis is substantiated.

It is crucial to ensure that other growth-influencing factors, such as light and water, are uniformly distributed so that only gravity influences the directional growth.

6. “The Implementation of Gamified Learning Improves Students’ Motivation”

Field: Education

Type: Relational Hypothesis The relational hypothesis describes the relation between two variables. Here, the hypothesis is that the implementation of gamified learning has a positive effect on the motivation of students.

To validate this proposition, two sets of classes could be compared: one that implements a learning approach with game-based elements, and another that follows a traditional learning approach.

The students’ motivation levels could be gauged by monitoring their engagement, performance, and feedback over a considerable timeframe.

If the students engaged in the gamified learning context present higher levels of motivation and achievement, the hypothesis would be supported.

Control measures ought to be put into place to account for individual differences, including prior knowledge and attitudes towards learning.

Relevant Study: Does educational gamification improve students’ motivation? (Chapman & Rich, 2018)

7. “Mathematics Anxiety Negatively Affects Performance”

Field: Educational Psychology

Type: Research Hypothesis The research hypothesis involves making a prediction that will be tested. In this case, the hypothesis proposes that a student’s anxiety about math can negatively influence their performance in math-related tasks.

To assess this hypothesis, researchers must first measure the mathematics anxiety levels of a sample of students using a validated instrument, such as the Mathematics Anxiety Rating Scale.

Then, the students’ performance in mathematics would be evaluated through standard testing. If there’s a negative correlation between the levels of math anxiety and math performance (meaning as anxiety increases, performance decreases), the hypothesis would be supported.

It would be crucial to control for relevant factors such as overall academic performance and previous mathematical achievement.

8. “Disruption of Natural Sleep Cycle Impairs Worker Productivity”

Field: Organizational Psychology

Type: Operational Hypothesis The operational hypothesis involves defining the variables in measurable terms. In this example, the hypothesis posits that disrupting the natural sleep cycle, for instance through shift work or irregular working hours, can lessen productivity among workers.

To test this hypothesis, you could collect data from workers who maintain regular working hours and those with irregular schedules.

Measuring productivity could involve examining the worker’s ability to complete tasks, the quality of their work, and their efficiency.

If workers with interrupted sleep cycles demonstrate lower productivity compared to those with regular sleep patterns, it would lend support to the hypothesis.

Consideration should be given to potential confounding variables such as job type, worker age, and overall health.

9. “Regular Physical Activity Reduces the Risk of Depression”

Field: Health Psychology

Type: Predictive Hypothesis A predictive hypothesis involves making a prediction about the outcome of a study based on the observed relationship between variables. In this case, it is hypothesized that individuals who engage in regular physical activity are less likely to suffer from depression.

Longitudinal studies would suit to test this hypothesis, tracking participants’ levels of physical activity and their mental health status over time.

The level of physical activity could be self-reported or monitored, while mental health status could be assessed using standard diagnostic tools or surveys.

If data analysis shows that participants maintaining regular physical activity have a lower incidence of depression, this would endorse the hypothesis.

However, care should be taken to control other lifestyle and behavioral factors that could intervene with the results.

Relevant Study: Regular physical exercise and its association with depression (Kim, 2022)

10. “Regular Meditation Enhances Emotional Stability”

Type: Empirical Hypothesis In the empirical hypothesis, predictions are based on amassed empirical evidence . This particular hypothesis theorizes that frequent meditation leads to improved emotional stability, resonating with numerous studies linking meditation to a variety of psychological benefits.

Earlier studies reported some correlations, but to test this hypothesis directly, you’d organize an experiment where one group meditates regularly over a set period while a control group doesn’t.

Both groups’ emotional stability levels would be measured at the start and end of the experiment using a validated emotional stability assessment.

If regular meditators display noticeable improvements in emotional stability compared to the control group, the hypothesis gains credit.

You’d have to ensure a similar emotional baseline for all participants at the start to avoid skewed results.

11. “Children Exposed to Reading at an Early Age Show Superior Academic Progress”

Type: Directional Hypothesis The directional hypothesis predicts the direction of an expected relationship between variables. Here, the hypothesis anticipates that early exposure to reading positively affects a child’s academic advancement.

A longitudinal study tracking children’s reading habits from an early age and their consequent academic performance could validate this hypothesis.

Parents could report their children’s exposure to reading at home, while standardized school exam results would provide a measure of academic achievement.

If the children exposed to early reading consistently perform better acadically, it gives weight to the hypothesis.

However, it would be important to control for variables that might impact academic performance, such as socioeconomic background, parental education level, and school quality.

12. “Adopting Energy-efficient Technologies Reduces Carbon Footprint of Industries”

Field: Environmental Science

Type: Descriptive Hypothesis A descriptive hypothesis predicts the existence of an association or pattern related to variables. In this scenario, the hypothesis suggests that industries adopting energy-efficient technologies will resultantly show a reduced carbon footprint.

Global industries making use of energy-efficient technologies could track their carbon emissions over time. At the same time, others not implementing such technologies continue their regular tracking.

After a defined time, the carbon emission data of both groups could be compared. If industries that adopted energy-efficient technologies demonstrate a notable reduction in their carbon footprints, the hypothesis would hold strong.

In the experiment, you would exclude variations brought by factors such as industry type, size, and location.

13. “Reduced Screen Time Improves Sleep Quality”

Type: Simple Hypothesis The simple hypothesis is a prediction about the relationship between two variables, excluding any other variables from consideration. This example posits that by reducing time spent on devices like smartphones and computers, an individual should experience improved sleep quality.

A sample group would need to reduce their daily screen time for a pre-determined period. Sleep quality before and after the reduction could be measured using self-report sleep diaries and objective measures like actigraphy, monitoring movement and wakefulness during sleep.

If the data shows that sleep quality improved post the screen time reduction, the hypothesis would be validated.

Other aspects affecting sleep quality, like caffeine intake, should be controlled during the experiment.

Relevant Study: Screen time use impacts low‐income preschool children’s sleep quality, tiredness, and ability to fall asleep (Waller et al., 2021)

14. Engaging in Brain-Training Games Improves Cognitive Functioning in Elderly

Field: Gerontology

Type: Inductive Hypothesis Inductive hypotheses are based on observations leading to broader generalizations and theories. In this context, the hypothesis deduces from observed instances that engaging in brain-training games can help improve cognitive functioning in the elderly.

A longitudinal study could be conducted where an experimental group of elderly people partakes in regular brain-training games.

Their cognitive functioning could be assessed at the start of the study and at regular intervals using standard neuropsychological tests.

If the group engaging in brain-training games shows better cognitive functioning scores over time compared to a control group not playing these games, the hypothesis would be supported.

15. Farming Practices Influence Soil Erosion Rates

Type: Null Hypothesis A null hypothesis is a negative statement assuming no relationship or difference between variables. The hypothesis in this context asserts there’s no effect of different farming practices on the rates of soil erosion.

Comparing soil erosion rates in areas with different farming practices over a considerable timeframe could help test this hypothesis.

If, statistically, the farming practices do not lead to differences in soil erosion rates, the null hypothesis is accepted.

However, if marked variation appears, the null hypothesis is rejected, meaning farming practices do influence soil erosion rates. It would be crucial to control for external factors like weather, soil type, and natural vegetation.

The variety of hypotheses mentioned above underscores the diversity of research constructs inherent in different fields, each with its unique purpose and way of testing.

While researchers may develop hypotheses primarily as tools to define and narrow the focus of the study, these hypotheses also serve as valuable guiding forces for the data collection and analysis procedures, making the research process more efficient and direction-focused.

Hypotheses serve as a compass for any form of academic research. The diverse examples provided, from Psychology to Educational Studies, Environmental Science to Gerontology, clearly demonstrate how certain hypotheses suit specific fields more aptly than others.

It is important to underline that although these varied hypotheses differ in their structure and methods of testing, each endorses the fundamental value of empiricism in research. Evidence-based decision making remains at the heart of scholarly inquiry, regardless of the research field, thus aligning all hypotheses to the core purpose of scientific investigation.

Testing hypotheses is an essential part of the scientific method . By doing so, researchers can either confirm their predictions, giving further validity to an existing theory, or they might uncover new insights that could potentially shift the field’s understanding of a particular phenomenon. In either case, hypotheses serve as the stepping stones for scientific exploration and discovery.

Atkinson, P., Delamont, S., Cernat, A., Sakshaug, J. W., & Williams, R. A. (2021).  SAGE research methods foundations . SAGE Publications Ltd.

Curcio, G., Ferrara, M., & De Gennaro, L. (2006). Sleep loss, learning capacity and academic performance.  Sleep medicine reviews ,  10 (5), 323-337.

Kim, J. H. (2022). Regular physical exercise and its association with depression: A population-based study short title: Exercise and depression.  Psychiatry Research ,  309 , 114406.

King, D. E. (2005). Dietary fiber, inflammation, and cardiovascular disease.  Molecular nutrition & food research ,  49 (6), 594-600.

Marian, V., & Shook, A. (2012, September). The cognitive benefits of being bilingual. In Cerebrum: the Dana forum on brain science (Vol. 2012). Dana Foundation.

Tan, W. C. K. (2022). Research Methods: A Practical Guide For Students And Researchers (Second Edition) . World Scientific Publishing Company.

Waller, N. A., Zhang, N., Cocci, A. H., D’Agostino, C., Wesolek‐Greenson, S., Wheelock, K., … & Resnicow, K. (2021). Screen time use impacts low‐income preschool children’s sleep quality, tiredness, and ability to fall asleep. Child: care, health and development, 47 (5), 618-626.

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Dr. Chris Drew is the founder of the Helpful Professor. He holds a PhD in education and has published over 20 articles in scholarly journals. He is the former editor of the Journal of Learning Development in Higher Education. [Image Descriptor: Photo of Chris]

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How to Write a Research Paper Introduction (with Examples)

The research paper introduction section, along with the Title and Abstract, can be considered the face of any research paper. The following article is intended to guide you in organizing and writing the research paper introduction for a quality academic article or dissertation.

The research paper introduction aims to present the topic to the reader. A study will only be accepted for publishing if you can ascertain that the available literature cannot answer your research question. So it is important to ensure that you have read important studies on that particular topic, especially those within the last five to ten years, and that they are properly referenced in this section. 1 What should be included in the research paper introduction is decided by what you want to tell readers about the reason behind the research and how you plan to fill the knowledge gap. The best research paper introduction provides a systemic review of existing work and demonstrates additional work that needs to be done. It needs to be brief, captivating, and well-referenced; a well-drafted research paper introduction will help the researcher win half the battle.

The introduction for a research paper is where you set up your topic and approach for the reader. It has several key goals:

• Present your research topic
• Capture reader interest
• Summarize existing research
• Position your own approach
• Define your specific research problem and problem statement
• Highlight the novelty and contributions of the study
• Give an overview of the paper’s structure

The research paper introduction can vary in size and structure depending on whether your paper presents the results of original empirical research or is a review paper. Some research paper introduction examples are only half a page while others are a few pages long. In many cases, the introduction will be shorter than all of the other sections of your paper; its length depends on the size of your paper as a whole.

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Table of Contents

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The introduction in a research paper is placed at the beginning to guide the reader from a broad subject area to the specific topic that your research addresses. They present the following information to the reader

• Scope: The topic covered in the research paper
• Context: Background of your topic
• Importance: Why your research matters in that particular area of research and the industry problem that can be targeted

The research paper introduction conveys a lot of information and can be considered an essential roadmap for the rest of your paper. A good introduction for a research paper is important for the following reasons:

• It stimulates your reader’s interest: A good introduction section can make your readers want to read your paper by capturing their interest. It informs the reader what they are going to learn and helps determine if the topic is of interest to them.
• It helps the reader understand the research background: Without a clear introduction, your readers may feel confused and even struggle when reading your paper. A good research paper introduction will prepare them for the in-depth research to come. It provides you the opportunity to engage with the readers and demonstrate your knowledge and authority on the specific topic.
• It explains why your research paper is worth reading: Your introduction can convey a lot of information to your readers. It introduces the topic, why the topic is important, and how you plan to proceed with your research.
• It helps guide the reader through the rest of the paper: The research paper introduction gives the reader a sense of the nature of the information that will support your arguments and the general organization of the paragraphs that will follow. It offers an overview of what to expect when reading the main body of your paper.

What are the parts of introduction in the research?

A good research paper introduction section should comprise three main elements: 2

• What is known: This sets the stage for your research. It informs the readers of what is known on the subject.
• What is lacking: This is aimed at justifying the reason for carrying out your research. This could involve investigating a new concept or method or building upon previous research.
• What you aim to do: This part briefly states the objectives of your research and its major contributions. Your detailed hypothesis will also form a part of this section.

How to write a research paper introduction?

The first step in writing the research paper introduction is to inform the reader what your topic is and why it’s interesting or important. This is generally accomplished with a strong opening statement. The second step involves establishing the kinds of research that have been done and ending with limitations or gaps in the research that you intend to address. Finally, the research paper introduction clarifies how your own research fits in and what problem it addresses. If your research involved testing hypotheses, these should be stated along with your research question. The hypothesis should be presented in the past tense since it will have been tested by the time you are writing the research paper introduction.

The following key points, with examples, can guide you when writing the research paper introduction section:

• Highlight the importance of the research field or topic
• Describe the background of the topic
• Present an overview of current research on the topic

Example: The inclusion of experiential and competency-based learning has benefitted electronics engineering education. Industry partnerships provide an excellent alternative for students wanting to engage in solving real-world challenges. Industry-academia participation has grown in recent years due to the need for skilled engineers with practical training and specialized expertise. However, from the educational perspective, many activities are needed to incorporate sustainable development goals into the university curricula and consolidate learning innovation in universities.

• Reveal a gap in existing research or oppose an existing assumption
• Formulate the research question

Example: There have been plausible efforts to integrate educational activities in higher education electronics engineering programs. However, very few studies have considered using educational research methods for performance evaluation of competency-based higher engineering education, with a focus on technical and or transversal skills. To remedy the current need for evaluating competencies in STEM fields and providing sustainable development goals in engineering education, in this study, a comparison was drawn between study groups without and with industry partners.

• State the purpose of your study
• Highlight the key characteristics of your study
• Describe important results
• Highlight the novelty of the study.
• Offer a brief overview of the structure of the paper.

Example: The study evaluates the main competency needed in the applied electronics course, which is a fundamental core subject for many electronics engineering undergraduate programs. We compared two groups, without and with an industrial partner, that offered real-world projects to solve during the semester. This comparison can help determine significant differences in both groups in terms of developing subject competency and achieving sustainable development goals.

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You can use the same process to develop each section of your article, and finally your research paper in half the time and without any of the stress.

The purpose of the research paper introduction is to introduce the reader to the problem definition, justify the need for the study, and describe the main theme of the study. The aim is to gain the reader’s attention by providing them with necessary background information and establishing the main purpose and direction of the research.

The length of the research paper introduction can vary across journals and disciplines. While there are no strict word limits for writing the research paper introduction, an ideal length would be one page, with a maximum of 400 words over 1-4 paragraphs. Generally, it is one of the shorter sections of the paper as the reader is assumed to have at least a reasonable knowledge about the topic. 2 For example, for a study evaluating the role of building design in ensuring fire safety, there is no need to discuss definitions and nature of fire in the introduction; you could start by commenting upon the existing practices for fire safety and how your study will add to the existing knowledge and practice.

When deciding what to include in the research paper introduction, the rest of the paper should also be considered. The aim is to introduce the reader smoothly to the topic and facilitate an easy read without much dependency on external sources. 3 Below is a list of elements you can include to prepare a research paper introduction outline and follow it when you are writing the research paper introduction. Topic introduction: This can include key definitions and a brief history of the topic. Research context and background: Offer the readers some general information and then narrow it down to specific aspects. Details of the research you conducted: A brief literature review can be included to support your arguments or line of thought. Rationale for the study: This establishes the relevance of your study and establishes its importance. Importance of your research: The main contributions are highlighted to help establish the novelty of your study Research hypothesis: Introduce your research question and propose an expected outcome. Organization of the paper: Include a short paragraph of 3-4 sentences that highlights your plan for the entire paper

Cite only works that are most relevant to your topic; as a general rule, you can include one to three. Note that readers want to see evidence of original thinking. So it is better to avoid using too many references as it does not leave much room for your personal standpoint to shine through. Citations in your research paper introduction support the key points, and the number of citations depend on the subject matter and the point discussed. If the research paper introduction is too long or overflowing with citations, it is better to cite a few review articles rather than the individual articles summarized in the review. A good point to remember when citing research papers in the introduction section is to include at least one-third of the references in the introduction.

The literature review plays a significant role in the research paper introduction section. A good literature review accomplishes the following: Introduces the topic – Establishes the study’s significance – Provides an overview of the relevant literature – Provides context for the study using literature – Identifies knowledge gaps However, remember to avoid making the following mistakes when writing a research paper introduction: Do not use studies from the literature review to aggressively support your research Avoid direct quoting Do not allow literature review to be the focus of this section. Instead, the literature review should only aid in setting a foundation for the manuscript.

Remember the following key points for writing a good research paper introduction: 4

• Avoid stuffing too much general information: Avoid including what an average reader would know and include only that information related to the problem being addressed in the research paper introduction. For example, when describing a comparative study of non-traditional methods for mechanical design optimization, information related to the traditional methods and differences between traditional and non-traditional methods would not be relevant. In this case, the introduction for the research paper should begin with the state-of-the-art non-traditional methods and methods to evaluate the efficiency of newly developed algorithms.
• Avoid packing too many references: Cite only the required works in your research paper introduction. The other works can be included in the discussion section to strengthen your findings.
• Avoid extensive criticism of previous studies: Avoid being overly critical of earlier studies while setting the rationale for your study. A better place for this would be the Discussion section, where you can highlight the advantages of your method.
• Avoid describing conclusions of the study: When writing a research paper introduction remember not to include the findings of your study. The aim is to let the readers know what question is being answered. The actual answer should only be given in the Results and Discussion section.

To summarize, the research paper introduction section should be brief yet informative. It should convince the reader the need to conduct the study and motivate him to read further. If you’re feeling stuck or unsure, choose trusted AI academic writing assistants like Paperpal to effortlessly craft your research paper introduction and other sections of your research article.

1. Jawaid, S. A., & Jawaid, M. (2019). How to write introduction and discussion. Saudi Journal of Anaesthesia, 13(Suppl 1), S18.

2. Dewan, P., & Gupta, P. (2016). Writing the title, abstract and introduction: Looks matter!. Indian pediatrics, 53, 235-241.

3. Cetin, S., & Hackam, D. J. (2005). An approach to the writing of a scientific Manuscript1. Journal of Surgical Research, 128(2), 165-167.

4. Bavdekar, S. B. (2015). Writing introduction: Laying the foundations of a research paper. Journal of the Association of Physicians of India, 63(7), 44-6.

Paperpal is a comprehensive AI writing toolkit that helps students and researchers achieve 2x the writing in half the time. It leverages 21+ years of STM experience and insights from millions of research articles to provide in-depth academic writing, language editing, and submission readiness support to help you write better, faster.  

Get accurate academic translations, rewriting support, grammar checks, vocabulary suggestions, and generative AI assistance that delivers human precision at machine speed. Try for free or upgrade to Paperpal Prime starting at US$19 a month to access premium features, including consistency, plagiarism, and 30+ submission readiness checks to help you succeed.  

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Confidence distributions and hypothesis testing

• Regular Article
• Open access
• Published: 29 March 2024

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• Eugenio Melilli   ORCID: orcid.org/0000-0003-2542-5286 1 &
• Piero Veronese   ORCID: orcid.org/0000-0002-4416-2269 1  

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The traditional frequentist approach to hypothesis testing has recently come under extensive debate, raising several critical concerns. Additionally, practical applications often blend the decision-theoretical framework pioneered by Neyman and Pearson with the inductive inferential process relied on the p -value, as advocated by Fisher. The combination of the two methods has led to interpreting the p -value as both an observed error rate and a measure of empirical evidence for the hypothesis. Unfortunately, both interpretations pose difficulties. In this context, we propose that resorting to confidence distributions can offer a valuable solution to address many of these critical issues. Rather than suggesting an automatic procedure, we present a natural approach to tackle the problem within a broader inferential context. Through the use of confidence distributions, we show the possibility of defining two statistical measures of evidence that align with different types of hypotheses under examination. These measures, unlike the p -value, exhibit coherence, simplicity of interpretation, and ease of computation, as exemplified by various illustrative examples spanning diverse fields. Furthermore, we provide theoretical results that establish connections between our proposal, other measures of evidence given in the literature, and standard testing concepts such as size, optimality, and the p -value.

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1 Introduction

In applied research, the standard frequentist approach to hypothesis testing is commonly regarded as a straightforward, coherent, and automatic method for assessing the validity of a conjecture represented by one of two hypotheses, denoted as \({{{\mathcal {H}}}_{0}}\) and \({{{\mathcal {H}}}_{1}}\) . The probabilities \(\alpha \) and \(\beta \) of committing type I and type II errors (reject \({{{\mathcal {H}}}_{0}}\) , when it is true and accept \({{{\mathcal {H}}}_{0}}\) when it is false, respectively) are controlled through a carefully designed experiment. After having fixed \(\alpha \) (usually at 0.05), the p -value is used to quantify the measure of evidence against the null hypothesis. If the p -value is less than \(\alpha \) , the conclusion is deemed significant , suggesting that it is unlikely that the null hypothesis holds. Regrettably, this methodology is not as secure as it may seem, as evidenced by a large literature, see the ASA’s Statement on p -values (Wasserstein and Lazar 2016 ) and The American Statistician (2019, vol. 73, sup1) for a discussion of various principles, misconceptions, and recommendations regarding the utilization of p -values. The standard frequentist approach is, in fact, a blend of two different views on hypothesis testing presented by Neyman-Pearson and Fisher. The first authors approach hypothesis testing within a decision-theoretic framework, viewing it as a behavioral theory. In contrast, Fisher’s perspective considers testing as a component of an inductive inferential process that does not necessarily require an alternative hypothesis or concepts from decision theory such as loss, risk or admissibility, see Hubbard and Bayarri ( 2003 ). As emphasized by Goodman ( 1993 ) “the combination of the two methods has led to a reinterpretation of the p -value simultaneously as an ‘observed error rate’ and as a ‘measure of evidence’. Both of these interpretations are problematic...”.

It is out of our scope to review the extensive debate on hypothesis testing. Here, we briefly touch upon a few general points, without delving into the Bayesian approach.

i) The long-standing caution expressed by Berger and Sellke ( 1987 ) and Berger and Delampady ( 1987 ) that a p -value of 0.05 provides only weak evidence against the null hypothesis has been further substantiated by recent investigations into experiment reproducibility, see e.g., Open Science Collaboration OSC ( 2015 ) and Johnson et al. ( 2017 ). In light of this, 72 statisticians have stated “For fields where the threshold for defining statistical significance for new discoveries is \(p<0.05\) , we propose a change to \(p<0.005\) ”, see Benjamin et al. ( 2018 ).

ii) The ongoing debate regarding the selection of a one-sided or two-sided test leaves the standard practice of doubling the p-value , when moving from the first to the second type of test, without consistent support, see e.g., Freedman ( 2008 ).

iii) There has been a longstanding argument in favor of integrating hypothesis testing with estimation, see e.g. Yates ( 1951 , pp. 32–33) or more recently, Greenland et al. ( 2016 ) who emphasize that “... statistical tests should never constitute the sole input to inferences or decisions about associations or effects ... in most scientific settings, the arbitrary classification of results into significant and non-significant is unnecessary for and often damaging to valid interpretation of data”.

iv) Finally, the p -value is incoherent when it is regarded as a statistical measure of the evidence provided by the data in support of a hypothesis \({{{\mathcal {H}}}_{0}}\) . As shown by Schervish ( 1996 ), it is possible that the p -value for testing the hypothesis \({{{\mathcal {H}}}_{0}}\) is greater than that for testing \({{{\mathcal {H}}}_{0}}^{\prime } \supset {{{\mathcal {H}}}_{0}}\) for the same observed data.

While theoretical insights into hypothesis testing are valuable for elucidating various aspects, we believe they cannot be compelled to serve as a unique, definitive practical guide for real-world applications. For example, uniformly most powerful (UMP) tests for discrete models not only rarely exist, but nobody uses them because they are randomized. On the other hand, how can a test of size 0.05 be considered really different from one of size 0.047 or 0.053? Moreover, for one-sided hypotheses, why should the first type error always be much more severe than the second type one? Alternatively, why should the test for \({{{\mathcal {H}}}_{0}}: \theta \le \theta _0\) versus \({{{\mathcal {H}}}_{1}}: \theta >\theta _0\) always be considered equivalent to the test for \({{{\mathcal {H}}}_{0}}: \theta = \theta _0\) versus \({{{\mathcal {H}}}_{1}}: \theta >\theta _0\) ? Furthermore, the decision to test \({{{\mathcal {H}}}_{0}}: \theta =\theta _0\) rather than \({{{\mathcal {H}}}_{0}}: \theta \in [\theta _0-\epsilon , \theta _0+\epsilon ]\) , for a suitable positive \(\epsilon \) , should be driven by the specific requirements of the application and not solely by the existence of a good or simple test. In summary, we concur with Fisher ( 1973 ) that “no scientific worker has a fixed level of significance at which from year to year, and in all circumstances, he rejects hypotheses; he rather gives his mind to each particular case in the light of his evidence and his ideas”.

Considering all these crucial aspects, we believe it is essential to seek an applied hypothesis testing approach that encourages researchers to engage more deeply with the specific problem, avoids relying on standardized procedures, and is consistently integrated into a broader framework of inference. One potential solution can be found resorting to the “confidence distribution” (CD) approach. The modern CD theory was introduced by Schweder and Hjort ( 2002 ) and Singh et al. ( 2005 ) and relies on the idea of constructing a data-depending distribution for the parameter of interest to be used for inferential purposes. A CD should not be confused with a Bayesian posterior distribution. It is not derived through the Bayes theorem, and it does not require any prior distributions. Similar to the conventional practice in point or interval estimation, where one seeks a point or interval estimator, the objective of this theory is to discover a distribution estimator . Thanks to a clarification of this concept and a formalized definition of the CD within a purely frequentist setting, a wide literature on the topic has been developed encompassing both theoretical developments and practical applications, see e.g. for a general overview Schweder and Hjort ( 2016 ), Singh et al. ( 2007 ), and Xie and Singh ( 2013 ). We also remark that when inference is required for a real parameter, it is possible to establish a relationship between CDs and fiducial distributions, originally introduced by Fisher ( 1930 ). For a modern and general presentation of the fiducial inference see Hannig ( 2009 ) and Hannig et al. ( 2016 ), while for a connection with the CDs see Schweder and Hjort ( 2016 ) and Veronese and Melilli ( 2015 , 2018a ). Some results about the connection between CDs and hypothesis testing are presented in Singh et al. ( 2007 , Sec. 3.3) and Xie & Singh ( 2013 , Sec. 4.3), but the focus is only on the formal relationships between the support that a CD can provide for a hypothesis and the p -value.

In this paper we discuss in details the application of CDs in hypothesis testing. We show how CDs can offer valuable solutions to address the aforementioned difficulties and how a test can naturally be viewed as a part of a more extensive inferential process. Once a CD has been specified, everything can be developed straightforwardly, without any particular technical difficulties. The core of our approach centers on the notion of support provided by the data to a hypothesis through a CD. We introduce two distinct but related types of support, the choice of which depends on the hypothesis under consideration. They are always coherent, easy to interpret and to compute, even in case of interval hypotheses, contrary to what happens for the p -value. The flexibility, simplicity, and effectiveness of our proposal are illustrated by several examples from various fields and a simulation study. We have postponed the presentation of theoretical results, comparisons with other proposals found in the literature, as well as the connections with standard hypothesis testing concepts such as size, significance level, optimality, and p -values to the end of the paper to enhance its readability.

The paper is structured as follows: In Sect. 2 , we provide a review of the CD’s definition and the primary methods for its construction, with a particular focus on distinctive aspects that arise when dealing with discrete models (Sect. 2.1 ). Section 3 explores the application of the CD in hypothesis testing and introduces the two notions of support. In Sect. 4 , we discuss several examples to illustrate the benefits of utilizing the CD in various scenarios, offering comparisons with traditional p -values. Theoretical results about tests based on the CD and comparisons with other measures of support or plausibility for hypotheses are presented in Sect. 5 . Finally, in Sect. 6 , we summarize the paper’s findings and provide concluding remarks. For convenience, a table of CDs for some common statistical models can be found in Appendix A, while all the proofs of the propositions are presented in Appendix B.

2 Confidence distributions

The modern definition of confidence distribution for a real parameter \(\theta \) of interest, see Schweder & Hjort ( 2002 ; 2016 , sec. 3.2) and Singh et al. ( 2005 ; 2007 ) can be formulated as follows:

Definition 1

Let \(\{P_{\theta ,\varvec{\lambda }},\theta \in \Theta \subseteq \mathbb {R}, \varvec{\lambda }\in \varvec{\Lambda }\}\) be a parametric model for data \(\textbf{X}\in {\mathcal {X}}\) ; here \(\theta \) is the parameter of interest and \(\varvec{\lambda }\) is a nuisance parameter. A function H of \(\textbf{X}\) and \(\theta \) is called a confidence distribution for \(\theta \) if: i) for each value \(\textbf{x}\) of \(\textbf{X}\) , \(H(\textbf{x},\cdot )=H_{\textbf{x}}(\cdot )\) is a continuous distribution function on \(\Theta \) ; ii) \(H(\textbf{X},\theta )\) , seen as a function of the random element \(\textbf{X}\) , has the uniform distribution on (0, 1), whatever the true parameter value \((\theta , \varvec{\lambda })\) . The function H is an asymptotic confidence distribution if the continuity requirement in i) is removed and ii) is replaced by: ii) \(^{\prime }\) \(H(\textbf{X},\theta )\) converges in law to the uniform distribution on (0, 1) for the sample size going to infinity, whatever the true parameter value \((\theta , \varvec{\lambda })\) .

The CD theory is placed in a purely frequentist context and the uniformity of the distribution ensures the correct coverage of the confidence intervals. The CD should be regarded as a distribution estimator of a parameter \(\theta \) and its mean, median or mode can serve as point estimates of \(\theta \) , see Xie and Singh ( 2013 ) for a detailed discussion. In essence, the CD can be employed in a manner similar to a Bayesian posterior distribution, but its interpretation differs and does not necessitate any prior distribution. Closely related to the CD is the confidence curve (CC) which, given an observation \(\textbf{x}\) , is defined as \( CC_{\textbf{x}}(\theta )=|1-2H_{\textbf{x}}(\theta )|\) ; see Schweder and Hjort ( 2002 ). This function provides the boundary points of equal-tailed confidence intervals for any level \(1-\alpha \) , with \(0<\alpha <1\) , and offers an immediate visualization of their length.

Various procedures can be adopted to obtain exact or asymptotic CDs starting, for example, from pivotal functions, likelihood functions and bootstrap distributions, as detailed in Singh et al. ( 2007 ), Xie and Singh ( 2013 ), Schweder and Hjort ( 2016 ). A CD (or an asymptotic CD) can also be derived directly from a real statistic T , provided that its exact or asymptotic distribution function \(F_{\theta }(t)\) is a continuously monotonic function in \(\theta \) and its limits are 0 and 1 as \(\theta \) approaches its boundaries. For example, if \(F_{\theta }(t)\) is nonincreasing, we can define

Furthermore, if \(H_t(\theta )\) is differentiable in \(\theta \) , we can obtain the CD-density \(h_t(\theta )=-({\partial }/{\partial \theta }) F_{\theta }(t)\) , which coincides with the fiducial density suggested by Fisher. In particular, when the statistical model belongs to the real regular natural exponential family (NEF) with natural parameter \(\theta \) and sufficient statistic T , there always exists an “optimal” CD for \(\theta \) which is given by ( 1 ), see Veronese and Melilli ( 2015 ).

The CDs based on a real statistic play an important role in hypothesis testing. In this setting remarkable results are obtained when the model has monotone likelihood ratio (MLR). We recall that if \(\textbf{X}\) is a random vector distributed according to the family \(\{p_\theta , \theta \in \Theta \subseteq \mathbb {R}\}\) , this family is said to have MLR in the real statistic \(T(\textbf{X})\) if, for any \(\theta _1 <\theta _2\) , the ratio \(p_{\theta _2}(\textbf{x})/p_{\theta _1}(\textbf{x})\) is a nondecreasing function of \(T(\textbf{x})\) for values of \(\textbf{x}\) that induce at least one of \(p_{\theta _1}\) and \(p_{\theta _2}\) to be positive. Furthermore, for such families, it holds that \(F_{\theta _2}(t) \le F_{\theta _1}(t)\) for each t , see Shao ( 2003 , Sec. 6.1.2). Families with MLR not only allow the construction of Uniformly Most Powerful (UMP) tests in various scenarios but also identify the statistic T , which can be employed in constructing the CD for \(\theta \) . Indeed, because \(F_\theta (t)\) is nonincreasing in \(\theta \) for each t , \(H_t(\theta )\) can be defined as in ( 1 ) provided the conditions of continuity and limits of \(F_{\theta }(t)\) are met. Of course, if the MLR is nonincreasing in T a similar result holds and the CD for \(\theta \) is \(H_t(\theta )=F_\theta (t)\) .

An interesting characteristic of the CD that validates its suitability for use in a testing problem is its consistency , meaning that it increasingly concentrates around the “true” value of \(\theta \) as the sample size grows, leading to the correct decision.

Definition 2

The sequence of CDs \(H(\textbf{X}_n, \cdot )\) is consistent at some \(\theta _0 \in \Theta \) if, for every neighborhood U of \(\theta _0\) , \(\int _U dH(\textbf{X}_n, \theta ) \rightarrow 1\) , as \(n\rightarrow \infty \) , in probability under \(\theta _0\) .

The following proposition provides some useful asymptotic properties of a CD for independent identically distributed (i.i.d.) random variables.

Proposition 1

Let \(X_1,X_2,\ldots \) be a sequence of i.i.d. random variables from a distribution function \(F_{\theta }\) , parameterized by a real parameter \(\theta \) , and let \(H_{\textbf{x}_n}\) be the CD for \(\theta \) based on \(\textbf{x}_n=(x_1, \ldots , x_n)\) . If \(\theta _0\) denotes the true value of \(\theta \) , then \(H(\textbf{X}_n, \cdot )\) is consistent at \(\theta _0\) if one of the following conditions holds:

\(F_{\theta }\) belongs to a NEF;

\(F_{\theta }\) is a continuous distribution function and standard regularity assumptions hold;

its expected value and variance converge for \(n\rightarrow \infty \) to \(\theta _0\) , and 0, respectively, in probability under \(\theta _0\) .

Finally, if i) or ii) holds the CD is asymptotically normal.

Table 8 in Appendix A provides a list of CDs for various standard models. Here, we present two basic examples, while numerous others will be covered in Sect. 4 within an inferential and testing framework.

( Normal model ) Let \(\textbf{X}=(X_1,\ldots ,X_n)\) be an i.i.d. sample from a normal distribution N \((\mu ,\sigma ^2)\) , with \(\sigma ^2\) known. A standard pivotal function is \(Q({\bar{X}}, \mu )=\sqrt{n}({\bar{X}}-\mu )/ \sigma \) , where \(\bar{X}=\sum X_i/n\) . Since \(Q({\bar{X}}, \mu )\) is decreasing in \(\mu \) and has the standard normal distribution \(\Phi \) , the CD for \(\mu \) is \(H_{\bar{x}}(\mu )=1-\Phi (\sqrt{n}({\bar{x}}-\mu )/ \sigma )=\Phi (\sqrt{n}(\mu -{\bar{x}})/ \sigma )\) , that is a N \(({\bar{x}},\sigma /\sqrt{n})\) . When the variance is unknown we can use the pivotal function \(Q({\bar{X}}, \mu )=\sqrt{n}({\bar{X}}-\mu )/S\) , where \(S^2=\sum (X_i-\bar{X})^2/(n-1)\) , and the CD for \(\mu \) is \(H_{{\bar{x}},s}(\mu )=1-F^{T_{n-1}}(\sqrt{n}({\bar{x}}-\mu )/ \sigma )=F^{T_{n-1}}(\sqrt{n}(\mu -{\bar{x}})/ \sigma )\) , where \(F^{T_{n-1}}\) is the t-distribution function with \(n-1\) degrees of freedom.

( Uniform model ) Let \(\textbf{X}=(X_1,\ldots ,X_n)\) be an i.i.d. sample from the uniform distribution on \((0,\theta )\) , \(\theta >0\) . Consider the (sufficient) statistic \(T=\max (X_1, \ldots ,X_n)\) whose distribution function is \(F_\theta (t)=(t/\theta )^n\) , for \(0<t<\theta \) . Because \(F_\theta (t)\) is decreasing in \(\theta \) and the limit conditions are satisfied for \(\theta >t\) , the CD for \(\theta \) is \(H_t(\theta )=1-(t/\theta )^n\) , i.e. a Pareto distribution \(\text {Pa}(n, t)\) with parameters n (shape) and t (scale). Since the uniform distribution is not regular, the consistency of the CD follows from condition iii) of Proposition 1 . This is because \(E^{H_{t}}(\theta )=nt/(n-1)\) and \(Var^{H_{t}}(\theta )=nt^2/((n-2)(n-1)^2)\) , so that, for \(n\rightarrow \infty \) , \(E^{H_{t}}(\theta ) \rightarrow \theta _0\) (from the strong consistency of the estimator T of \(\theta \) , see e.g. Shao 2003 , p.134) and \(Var^{H_{t}}(\theta )\rightarrow 0\) trivially.

2.1 Peculiarities of confidence distributions for discrete models

When the model is discrete, clearly we can only derive asymptotic CDs. However, a crucial question arises regarding uniqueness. Since \(F_{\theta }(t)=\text{ Pr}_\theta \{T \le t\}\) does not coincide with Pr \(_\theta \{T<t\}\) for any value t within the support \({\mathcal {T}}\) of T , it is possible to define two distinct “extreme” CDs. If \(F_\theta (t)\) is non increasing in \(\theta \) , we refer to the right CD as \(H_{t}^r(\theta )=1-\text{ Pr}_\theta \{T\le t\}\) and to the left CD as \(H_{t}^\ell (\theta )=1-\text{ Pr}_\theta \{T<t\}\) . Note that \(H_{t}^r(\theta ) < H_{t}^\ell (\theta )\) , for every \(t \in {{\mathcal {T}}}\) and \(\theta \in \Theta \) , so that the center (i.e. the mean or the median) of \(H_{t}^r(\theta )\) is greater than that of \(H_{t}^\ell (\theta )\) . If \(F_\theta (t)\) is increasing in \(\theta \) , we define \( H_{t}^\ell (\theta )=F_\theta (t)\) and \(H^r_t(\theta )=\text{ Pr}_\theta \{T<t\}\) and one again \(H_{t}^r(\theta ) < H_{t}^\ell (\theta )\) . Veronese & Melilli ( 2018b , sec. 3.2) suggest overcoming this nonuniqueness by averaging the CD-densities \(h_t^r\) and \(h_t^\ell \) using the geometric mean \(h_t^g(\theta )\propto \sqrt{h_t^r(\theta )h_t^\ell (\theta )}\) . This typically results in a simpler CD compared to the one obtained through the arithmetic mean, with smaller confidence intervals. Note that the (asymptotic) CD defined in ( 1 ) for discrete models corresponds to the right CD, and it is more appropriately referred to as \(H_t^r(\theta )\) hereafter. Clearly, \(H_{t}^\ell (\theta )\) can be obtained from \(H_{t}^r(\theta )\) by replacing t with its preceding value in the support \({\mathcal {T}}\) . For discrete models, the table in Appendix A reports \(H_{t}^r(\theta )\) , \(H_{t}^\ell (\theta )\) and \(H_t^g(\theta )\) . Compared to \(H^{\ell }_t\) and \(H^r_t\) , \(H^g_t\) offers the advantage of closely approximating a uniform distribution when viewed as a function of the random variable T .

Proposition 2

Given a discrete statistic T with distribution indexed by a real parameter \(\theta \in \Theta \) and support \({{\mathcal {T}}}\) independent of \(\theta \) , assume that, for each \(\theta \in \Theta \) and \(t\in {\mathcal {T}}\) , \(H^r_t(\theta )< H^g_t(\theta ) < H^{\ell }_t(\theta )\) . Then, denoting by \(G^j\) the distribution function of \(H^j_T\) , with \(j=\ell ,g,r\) , we have \(G^\ell (u) \le u \le G^r(u)\) . Furthermore,

Notice that the assumption in Proposition 2 is always satisfied when the model belongs to a NEF, see Veronese and Melilli ( 2018a ).

The possibility of constructing different CDs using the same discrete statistic T plays an important role in connection with standard p -values, as we will see in Sect. 5 .

(Binomial model) Let \(\textbf{X}=(X_1,\ldots , X_n)\) be an i.i.d. sample from a binomial distributions Bi(1,  p ) with success probability p . Then \(T=\sum _{i=1}^n X_i\) is distributed as a Bi( n ,  p ) and by ( 1 ), recalling the well-known relationship between the binomial and beta distributions, it follows that the right CD for p is a Be( \(t+1,n-t\) ) for \(t=0,1,\ldots , n-1\) . Furthermore, the left CD is a Be( \(t,n-t+1\) ) and it easily follows that \(H_t^g(p)\) is a Be( \(t+1/2,n-t+1/2\) ). Figure 1 shows the corresponding three CD-densities along with their respective CCs, emphasizing the central position of \(h_t^g(p)\) and its confidence intervals in comparison to \(h_t^\ell (p)\) and \(h^r_t(p)\) .

(Binomial model) CD-densities (left plot) and CCs (right plot) corresponding to \(H_t^g(p)\) (solid lines), \(H_t^{\ell }(p)\) (dashed lines) and \(H_t^r(p)\) (dotted lines) for the parameter p with n = 15 and \(t=5\) . In the CC plot, the horizontal dotted line is at level 0.95

3 Confidence distributions in testing problems

As mentioned in Sect. 1 , we believe that introducing a CD can serve as a valuable and unifying approach, compelling individuals to think more deeply about the specific problem they aim to address rather than resorting to automatic rules. In fact, the availability of a whole distribution for the parameter of interest equips statisticians and practitioners with a versatile tool for handling a wide range of inference tasks, such as point and interval estimation, hypothesis testing, and more, without the need for ad hoc procedures. Here, we will address the issue in the simplest manner, referring to Sect. 5 for connections with related ideas in the literature and additional technical details.

Given a set \(A \subseteq \Theta \subseteq \mathbb {R}\) , it seems natural to measure the “support” that the data \(\textbf{x}\) provide to A through the CD \(H_{\textbf{x}}\) , as \(CD(A)=H_{\textbf{x}}(A)= \int _{A} dH_{\textbf{x}}(\theta )\) . Notice that, with a slight abuse of notation widely used in literature (see e.g., Singh et al. 2007 , who call \(H_{\textbf{x}}(A)\) strong-support ), we use \(H_{\textbf{x}}(\theta )\) to indicate the distribution function on \(\Theta \subseteq \mathbb {R}\) evaluated at \(\theta \) and \(H_{\textbf{x}}(A)\) to denote the mass that \(H_{\textbf{x}}\) induces on a (measurable) subset \(A\subseteq \Theta \) . It immediately follows that to compare the plausibility of k different hypotheses \({{\mathcal {H}}}_{i}: \theta \in \Theta _i\) , \(i=1,\ldots ,k\) , with \(\Theta _i \subseteq \Theta \) not being a singleton, it is enough to compute each \(H_{\textbf{x}}(\Theta _i)\) . We will call \(H_{\textbf{x}}(\Theta _i)\) the CD-support provided by \(H_{\textbf{x}}\) to the set \(\Theta _i\) . In particular, consider the usual case in which we have two hypotheses \({{{\mathcal {H}}}_{0}}: \theta \in \Theta _0\) and \({{{\mathcal {H}}}_{1}}: \theta \in \Theta _1\) , with \(\Theta _0 \cap \Theta _1= \emptyset \) , \(\Theta _0 \cup \Theta _1 = \Theta \) and assume that \({{{\mathcal {H}}}_{0}}\) is not a precise hypothesis (i.e. is not of type \(\theta =\theta _0\) ). As in the Bayesian approach one can compute the posterior odds, here we can evaluate the confidence odds \(CO_{0,1}\) of \({{{\mathcal {H}}}_{0}}\) against \({{{\mathcal {H}}}_{1}}\)

If \(CO_{0,1}\) is greater than one, the data support \({{{\mathcal {H}}}_{0}}\) more than \({{{\mathcal {H}}}_{1}}\) and this support clearly increases with \(CO_{0,1}\) . Sometimes this type of information can be sufficient to have an idea of the reasonableness of the hypotheses, but if we need to take a decision, we can include the confidence odds in a full decision setting. Thus, writing the decision space as \({{\mathcal {D}}}=\{0,1\}\) , where i indicates accepting \({{{\mathcal {H}}}}_i\) , for \(i=0,1\) , a penalization for the two possible errors must be specified. A simple loss function is

where \(\delta \) denotes the decision taken and \(a_i >0\) , \(i=0,1\) . The optimal decision is the one that minimizes the (expected) confidence loss

Therefore, we will choose \({{{\mathcal {H}}}_{0}}\) if \(a_0 H_{\textbf{x}}(\Theta _0) > a_1 H_{\textbf{x}}(\Theta _1)\) , that is if \(CO_{0,1}>a_1/a_0\) or equivalently if \(H_{\textbf{x}}(\Theta _0)>a_1/(a_0+a_1)=\gamma \) . Clearly, if there is no reason to penalize differently the two errors by setting an appropriate value for the ratio \(a_1/a_0\) , we assume \(a_0=a_1\) so that \(\gamma =0.5\) . This implies that the chosen hypothesis will be the one receiving the highest level of the CD-support. Therefore, we state the following

Definition 3

Given the two (non precise) hypotheses \({{\mathcal {H}}}_i: \theta \in \Theta _i\) , \(i=0,1\) , the CD-support of \({{\mathcal {H}}}_i\) is defined as \(H_{\textbf{x}}(\Theta _i)\) . The hypothesis \({{{\mathcal {H}}}_{0}}\) is rejected according to the CD-test if the CD-support is less than a fixed threshold \(\gamma \) depending on the loss function ( 3 ) or, equivalently, if the confidence odds \(CO_{0,1}\) are less than \(a_1/a_0=\gamma /(1-\gamma )\) .

Unfortunately, the previous notion of CD-support fails for a precise hypothesis \({{{\mathcal {H}}}_{0}}:\theta =\theta _0\) , since in this case \(H_{\textbf{x}}(\{\theta _0\})\) trivially equals zero. Notice that the problem cannot be solved by transforming \({{{\mathcal {H}}}_{0}}:\theta =\theta _0\) into the seemingly more reasonable \({{{\mathcal {H}}}_{0}}^{\prime }:\theta \in [\theta _0-\epsilon , \theta _0+\epsilon ]\) because, apart from the arbitrariness of \(\epsilon \) , the CD-support for very narrow range intervals would typically remain negligible. We thus introduce an alternative way to assess the plausibility of a precise hypothesis or, more generally, of a “small” interval hypothesis.

Consider first \({{{\mathcal {H}}}_{0}}:\theta =\theta _0\) and assume, as usual, that \(H_{\textbf{x}}(\theta )\) is a CD for \(\theta \) , based on the data \(\textbf{x}\) . Looking at the confidence curve \(CC_{\textbf{x}}(\theta )=|1-2H_{\textbf{x}}(\theta )|\) in Fig. 2 , it is reasonable to assume that the closer \(\theta _0\) is to the median \(\theta _m\) of the CD, the greater the consistency of the value of \(\theta _0\) with respect to \(\textbf{x}\) . Conversely, the complement to 1 of the CC represents the unconsidered confidence relating to both tails of the distribution. We can thus define a measure of plausibility for \({{{\mathcal {H}}}_{0}}:\theta =\theta _0\) as \((1-CC_{\textbf{x}}(\theta ))/2\) and this measure will be referred to as the CD*-support given by \(\textbf{x}\) to the hypothesis. It is immediate to see that

In other words, if \(\theta _0 < \theta _m\) \([\theta _0 > \theta _m]\) the CD*-support is \(H_{\textbf{x}}(\theta _0)\) \([1-H_{\textbf{x}}(\theta _0)]\) and corresponds to the CD-support of all \(\theta \) ’s that are less plausible than \(\theta _0\) among those located on the left [right] side of the CC . Clearly, if \(\theta _0 = \theta _m\) the CD*-support equals 1/2, its maximum value. Notice that in this case no alternative hypothesis is considered and that the CD*-support provides a measure of plausibility for \(\theta _0\) by examining “the direction of the observed departure from the null hypothesis”. This quotation is derived from Gibbons and Pratt ( 1975 ) and was originally stated to support their preference for reporting a one-tailed p -value over a two-tailed one. Here we are in a similar context and we refer to their paper for a detailed discussion of this recommendation.

The CD*-supports of the points \(\theta _0\) , \(\theta _1\) , \(\theta _m\) and \(\theta _2\) correspond to half of the solid vertical lines and are given by \(H_{\textbf{x}}(\theta _0)\) , \(H_{\textbf{x}}(\theta _1)\) , \(H_{\textbf{x}}(\theta _m)=1/2\) e \(1-H_{\textbf{x}}(\theta _2)\) , respectively

An alternative way to intuitively justify formula ( 4 ) is as follows. Since \(H_{\textbf{x}}(\{\theta _0\})=0\) , we can look at the set K of values of \(\theta \) which are in some sense “more consistent” with the observed data \(\textbf{x}\) than \(\theta _0\) , and define the plausibility of \({{{\mathcal {H}}}_{0}}\) as \(1-H_{\textbf{x}}(K)\) . This procedure was followed in a Bayesian framework by Pereira et al. ( 1999 ) and Pereira et al. ( 2008 ) who, in order to identify K , relay on the posterior distribution of \(\theta \) and focus on its mode. We refer to these papers for a more detailed discussion of this idea. Here we emphasize only that the evidence \(1-H_{\textbf{x}}(K)\) supporting \({{{\mathcal {H}}}_{0}}\) cannot be considered as evidence against a possible alternative hypothesis. In our context, the set K can be identified as the set \(\{\theta \in \Theta : \theta < \theta _0\}\) if \(H_{\textbf{x}}(\theta _0)>1-H_{\textbf{x}}(\theta _0)\) or as \(\{\theta \in \Theta : \theta >\theta _0\}\) if \(H_{\textbf{x}}(\theta _0)\le 1-H_{\textbf{x}}(\theta _0)\) . It follows immediately that \(1-H_{\textbf{x}}(K)=\min \{H_{\textbf{x}}(\theta _0), 1-H_{\textbf{x}}(\theta _0)\}\) , which coincides with the CD*-support given in ( 4 ).

We can readily extend the previous definition of CD*-support to interval hypotheses \({{{\mathcal {H}}}_{0}}:\theta \in [\theta _1, \theta _2]\) . This extension becomes particularly pertinent when dealing with small intervals, where the CD-support may prove ineffective. In such cases, the set K of \(\theta \) values that are “more consistent” with the data \(\textbf{x}\) than those falling within the interval \([\theta _1, \theta _2]\) should clearly exclude this interval. Instead, it should include one of the two tails, namely, either \({\theta \in \Theta : \theta < \theta _1}\) or \({\theta \in \Theta : \theta > \theta _2}\) , depending on which one receives a greater mass from the CD. Then

so that the CD*-support of the interval \([\theta _1,\theta _2]\) is \(\text{ CD* }([\theta _1,\theta _2])=1-H_{\textbf{x}}(K)=\min \{H_{\textbf{x}}(\theta _2), 1-H_{\textbf{x}}(\theta _1)\}\) , which reduces to ( 4 ) in the case of a degenerate interval (i.e., when \(\theta _1=\theta _2=\theta _0\) ). Therefore, we can establish the following

Definition 4

Given the hypothesis \({{{\mathcal {H}}}_{0}}: \theta \in [\theta _1,\theta _2]\) , with \(\theta _1 \le \theta _2 \) , the CD*-support of \({{{\mathcal {H}}}_{0}}\) is defined as \(\min \{H_{\textbf{x}}(\theta _2), 1-H_{\textbf{x}}(\theta _1)\}\) . If \(H_{\textbf{x}}(\theta _2) <1-H_{\textbf{x}}(\theta _1)\) it is more reasonable to consider values of \(\theta \) greater than those specified by \({{{\mathcal {H}}}_{0}}\) , and conversely, the opposite holds true in the reverse situation. Furthermore, the hypothesis \({{{\mathcal {H}}}_{0}}\) is rejected according to the CD*-test if its CD*-support is less than a fixed threshold \(\gamma ^*\) .

The definition of CD*-support has been established for bounded interval (or precise) hypothesis. However, it can be readily extended to one-sided intervals such as \((-\infty , \theta _0]\) or \([\theta _0, +\infty )\) , but in these cases, it is evident that the CD*- and the CD-support are equivalent. For a general interval hypothesis we observe that \(H_{\textbf{x}}([\theta _1, \theta _2])\le \min \{H_{\textbf{x}}(\theta _2), 1-H_{\textbf{x}}(\theta _1)\}\) . Consequently, the CD-support can never exceed the CD*-support, even though they exhibit significant similarity when \(\theta _1\) or \(\theta _2\) resides in the extreme region of one tail of the CD or when the CD is highly concentrated (see examples 4 , 6 and 7 ).

It is crucial to emphasize that both CD-support and CD*-support are coherent measures of the evidence provided by the data for a hypothesis. This coherence arises from the fact that if \({{{\mathcal {H}}}_{0}}\subset {{{\mathcal {H}}}_{0}}^{\prime }\) , both the supports for \({{{\mathcal {H}}}_{0}}^{\prime }\) cannot be less than those for \({{{\mathcal {H}}}_{0}}\) . This is in stark contrast to the behavior of p -values, as demonstrated in Schervish ( 1996 ), Peskun ( 2020 ), and illustrated in Examples 4 and 7 .

Finally, as seen in Sect. 2.1 , various options for CDs are available for discrete models. Unless a specific problem suggests otherwise (see Sect. 5.1 ), we recommend using the geometric mean \(H_t^g\) as it offers a more impartial treatment of \({{{\mathcal {H}}}_{0}}\) and e \({{{\mathcal {H}}}_{1}}\) , as shown in Proposition 2 .

In this section, we illustrate the behavior, effectiveness, and simplicity of CD- and CD*-supports in an inferential context through several examples. We examine various contexts to assess the flexibility and consistency of our approach and compare it with the standard one. It is worth noting that the computation of the p -value for interval hypotheses is challenging and does not have a closed form.

( Normal model ) As seen in Example 1 , the CD for the mean \(\mu \) of a normal model is N \(({\bar{x}},\sigma /\sqrt{n})\) , for \(\sigma \) known. For simplicity, we assume this case; otherwise, the CD would be a t-distribution. Figure 3 shows the CD-density and the corresponding CC for \({\bar{x}}=2.7\) with three different values of \(\sigma /\sqrt{n}\) : \(1/\sqrt{50}=0.141\) , \(1/\sqrt{25}=0.2\) and \(1/\sqrt{10}=0.316\) .

The observed \({\bar{x}}\) specifies the center of both the CD and the CC, and values of \(\mu \) that are far from it receive less support the smaller the dispersion \(\sigma /\sqrt{n}\) of the CD. Alternatively, values of \(\mu \) within the CC, i.e., within the confidence interval of a specific level, are more reasonable than values outside it. These values become more plausible as the level of the interval decreases. Table 1 clarifies these points by providing the CD-support, confidence odds, CD*-support, and the p -value of the UMPU test for different interval hypotheses and different values of \(\sigma /\sqrt{n}\) .

(Normal model) CD-densities (left plot) and CCs (right plot) for \(\mu \) with \({\bar{x}}=2.7\) and three values of \(\sigma /\sqrt{n}\) : \(1/\sqrt{50}\) (solid line), \(1/\sqrt{25}\) (dashed line) and \(1/\sqrt{10}\) (dotted line). In the CC plot the dotted horizontal line is at level 0.95

It can be observed that when the interval is sufficiently large, e.g., [2.0, 2.5], the CD- and the CD*-supports are similar. However, for smaller intervals, as in the other three cases, the difference between the CD- and the CD*-support increases with the variance of the CD, \(\sigma /\sqrt{n}\) , regardless of whether the interval contains the observation \({\bar{x}}\) or not. These aspects are general depending on the form of the CD. Therefore, a comparison between these two measures can be useful to clarify whether an interval is smaller or not, according to the problem under analysis. Regarding the p -value of the UMPU test (see Schervish 1996 , equation 2), it is similar to the CD*-support when the interval is large (first case). However, the difference increases with the growth of the variance in the other cases. Furthermore, enlarging the interval from [2.4, 2.6] to [2.3, 2.6], not reported in Table 1 , while the CD*-supports remain unchanged, results in p -values reducing to 0.241, 0.331, and 0.479 for the three considered variances. This once again highlights the incoherence of the p -value as a measure of the plausibility of a hypothesis.

Now, consider a precise hypothesis, for instance, \({{{\mathcal {H}}}_{0}}:\mu =2.35\) . For the three values used for \(\sigma /\sqrt{n}\) , the CD*-supports are 0.007, 0.040, and 0.134, respectively. From Fig. 3 , it is evident that the point \(\mu =2.35\) lies to the left of the median of the CD. Consequently, the data suggest values of \(\mu \) larger than 2.35. Furthermore, looking at the CC, it becomes apparent that 2.35 is not encompassed within the confidence interval of level 0.95 when \(\sigma /\sqrt{n}=1/\sqrt{50}\) , contrary to what occurs in the other two cases. Due to the symmetry of the normal model, the UMPU test coincides with the equal tailed test, so that the p -value is equal to 2 times the CD*-support (see Remark 4 in Sect. 5.2 ). Furthermore, the size of the CD*-test is \(2\gamma ^*\) , where \(\gamma ^*\) is the threshold fixed to decide whether to reject the hypothesis or not (see Proposition 5 . Thus, if a test of level 0.05 is desired, it is sufficient to fix \(\gamma ^*=0.025\) , and both the CD*-support and the p -value lead to the same decision, namely, rejecting \({{{\mathcal {H}}}_{0}}\) only for the case \(\sigma /\sqrt{n}=0.141\) .

To assess the effectiveness of the CD*-support, we conduct a brief simulation study. For different values of \(\mu \) , we generate 100000 values of \({\bar{x}}\) from a normal distribution with mean \(\mu \) and various standard deviation \(\sigma /\sqrt{n}\) . We obtain the corresponding CDs with the CD*-supports and compute also the p -values. In Table 2 , we consider \({{{\mathcal {H}}}_{0}}: \mu \in [2.0, 2.5]\) and the performance of the CD*-support can be evaluated looking for example at the proportions of values in the intervals [0, 0.4), [0.4, 0.6) and [0.6, 1]. Values of the CD*-support in the first interval suggest a low plausibility of \({{{\mathcal {H}}}_{0}}\) in the light of the data, while values in the third one suggest a high plausibility. We highlight the proportions of incorrect evaluations in boldface. The last column of the table reports the proportion of errors resulting from the use of the standard procedure based on the p -value for a threshold of 0.05. Note how the proportion of errors related to the CD*-support is generally quite low with a maximum value of 0.301, contrary to what happens for the automatic procedure based on the p -value, which reaches a proportion of error of 0.845. Notice that the maximum error due to the CD*-support is obtained when \({{{\mathcal {H}}}_{0}}\) is true, while that due to the p -value is obtained in the opposite, as expected.

We consider now the two hypotheses \({{{\mathcal {H}}}_{0}}:\mu =2.35\) and \({{{\mathcal {H}}}_{0}}: \mu \in [2.75,2.85]\) . Notice that the interval in the second hypothesis should be regarded as small, because it can be checked that the CD- and CD*-supports consistently differ, as can be seen for example in Table 1 for the case \({\bar{x}}=2.7\) . Thus, this hypothesis can be considered not too different from a precise one. Because for a precise hypothesis the CD*-support cannot be larger than 0.5, to evaluate the performance of the CD*-support we can consider the three intervals [0, 0.2), [0.2, 0.3) and [0.3, 0.5].

Table 3 reports the results of the simulation including again the proportion of errors resulting from the use of the p -value with threshold 0.05. For the precise hypothesis \({{{\mathcal {H}}}_{0}}: \mu =2.35\) , the proportion of values of the CD*-support less than 0.2 when \(\mu =2.35\) is, whatever the standard deviation, approximately equal to 0.4. This depends on the fact that for a precise hypothesis, the CD*-support has a uniform distribution on the interval [0, 0.5], see Proposition 5 . This aspect must be taken into careful consideration when setting a threshold for a CD*-test. On the other hand, the proportion of values of the CD*-support in the interval [0.3, 0.5], which wrongly support \({{{\mathcal {H}}}_{0}}\) when it is false, goes from 0.159 to 0.333 for \(\mu =2.55\) and from 0.010 to 0.193 for \(\mu =2.75\) , which are surely better than those obtained from the standard procedure based on the p -value. Take now the hypothesis \({{{\mathcal {H}}}_{0}}: \mu \in [2.75,2.85]\) . Since it can be considered not too different from a precise hypothesis, we consider the proportion of values of the CD*-support in the intervals [0, 0.2), [0.2, 0.3) and [0.3, 1]. Notice that, for simplicity, we assume 1 as the upper bound of the third interval, even though for small intervals, the values of the CD*-support can not be much larger than 0.5. In our simulation it does not exceed 0.635. For the different values of \(\mu \) considered the behavior of the CD*-support and p -value is not too different from the previous case of a precise hypothesis even if the proportion of errors when \({{{\mathcal {H}}}_{0}}\) is true decreases for both while it increases when \({{{\mathcal {H}}}_{0}}\) is false.

Binomial model Suppose we are interested in assessing the chances of candidate A winning the next ballot for a certain administrative position. The latest election poll based on a sample of size \(n=20\) , yielded \(t=9\) votes in favor of A . What can we infer? Clearly, we have a binomial model where the parameter p denotes the probability of having a vote in favor of A . The standard estimate of p is \(\hat{p}=9/20=0.45\) , which might suggest that A will lose the ballot. However, the usual (Wald) confidence interval of level 0.95 based on the normal approximation, i.e. \(\hat{p} \pm 1.96 \sqrt{\hat{p}(1-\hat{p})/n}\) , is (0.232, 0.668). Given its considerable width, this interval suggests that the previous estimate is unreliable. We could perform a statistical test with a significance level \(\alpha \) , but what is \({{{\mathcal {H}}}_{0}}\) , and what value of \(\alpha \) should we consider? If \({{{\mathcal {H}}}_{0}}: p \ge 0.5\) , implying \({{{\mathcal {H}}}_{1}}: p <0.5\) , the p -value is 0.327. This suggests not rejecting \({{{\mathcal {H}}}_{0}}\) for any usual value \(\alpha \) . However, if we choose \({{{\mathcal {H}}}_{0}}^\prime : p \le 0.5\) the p -value is 0.673, and in this case, we would not reject \({{{\mathcal {H}}}_{0}}^\prime \) . These results provide conflicting indications. As seen in Example 3 , the CD for p , \(H_t^g(p)\) , is Be(9.5,11.5) and Fig. 4 shows its CD-density along with the corresponding CC, represented by solid lines. The dotted horizontal line at 0.95 in the CC plot highlights the (non asymptotic) equal-tailed confidence interval (0.251, 0.662), which is shorter than the Wald interval. Note that our interval can be easily obtained by computing the quantiles of order 0.025 and 0.975 of the beta distribution.

(Binomial model) CD-densities (left plot) and CCs (right plot) corresponding to \(H_t^g(p)\) , for the parameter p , with \(\hat{p}=t/n=0.45\) : \(n=20\) , \(t=9\) (solid lines) and \(n=60\) , \(t=27\) (dashed lines). In the CC plot the horizontal dotted line is at level 0.95

The CD-support provided by the data for the two hypotheses \({{{\mathcal {H}}}_{0}}:p \ge 0.5\) and \({{{\mathcal {H}}}_{1}}:p < 0.5\) (the choice of what is called \(H_0\) being irrelevant), is \(1-H_t^g(0.5)=0.328\) and \(H_t^g(0.5)=0.672\) respectively. Therefore, the confidence odds are \(CO_{0,1}=0.328/0.672=0.488\) , suggesting that the empirical evidence in favor of the victory of A is half of that of its defeat. Now, consider a sample of size \(n=60\) with \(t=27\) , so that again \(\hat{p}=0.45\) . While a standard analysis leads to the same conclusions (the p -values for \({{{\mathcal {H}}}_{0}}\) and \({{{\mathcal {H}}}_{0}}^{\prime }\) are 0.219 and 0.781, respectively), the use of the CD clarifies the differences between the two cases. The corresponding CD-density and CC are also reported in Fig. 4 (dashed lines) and, as expected, they are more concentrated around \(\hat{p}\) . Thus, the accuracy of the estimates of p is greater for the larger n and the length of the confidence intervals is smaller. Furthermore, for \(n=60\) , \(CO_{0,1}=0.281\) reducing the chance that A wins to about 1 to 4.

As a second application on the binomial model, we follow Johnson and Rossell ( 2010 ) and consider a stylized phase II trial of a new drug designed to improve the overall response rate from 20% to 40% for a specific population of patients with a common disease. The hypotheses are \({{{\mathcal {H}}}_{0}}:p \le 0.2\) versus \({{{\mathcal {H}}}_{1}}: p>0.2\) . It is assumed that patients are accrued and the trial continues until one of the two events occurs: (a) data clearly support one of the two hypotheses (indicated by a CD-support greater than 0.9) or (b) 50 patients have entered the trial. Trials that are not stopped before the 51st patient accrues are assumed to be inconclusive.

Based on a simulation of 1000 trials, Table 4 reports the proportions of trials that conclude in favor of each hypothesis, along with the average number of patients observed before each trial is stopped, for \(\theta =0.1\) (the central value of \({{{\mathcal {H}}}_{0}}\) ) and for \(\theta =0.4\) . A comparison with the results reported by Johnson and Rossell ( 2010 ) reveals that our approach is clearly superior with respect to Bayesian inferences performed with standard priors and comparable to that obtained under their non-local prior carefully specified. Although there is a slight reduction in the proportion of trials stopped for \({{\mathcal {H}}}_0\) (0.814 compared to 0.91), the average number of involved patients is lower (12.7 compared to 17.7), and the power is higher (0.941 against 0.812).

(Exponential distribution) Suppose an investigator aims to compare the performance of a new item, measured in terms of average lifetime, with that of the one currently in use, which is 0.375. To model the item lifetime, it is common to use the exponential distribution with rate parameter \(\lambda \) , so that the mean is \(1/\lambda \) . The typical testing problem is defined by \({{\mathcal {H}}}_0: \lambda =1/0.375=2.667\) versus \({{\mathcal {H}}}_1: \lambda \ne 2.667\) . In many cases, it would be more realistic and interesting to consider hypotheses of the form \({{\mathcal {H}}}_0: \lambda \in [\lambda _1,\lambda _2]\) versus \({{\mathcal {H}}}_1: \lambda \notin [\lambda _1,\lambda _2]\) , and if \({{{\mathcal {H}}}_{0}}\) is rejected, it becomes valuable to know whether the new item is better or worse than the old one. Note that, although an UMPU test exists for this problem, calculating its p -value is not simple and cannot be expressed in a closed form. Here we consider two different null hypotheses: \({{\mathcal {H}}}_0: \lambda \in [2, 4]\) and \({{\mathcal {H}}}_0: \lambda \in [2.63, 2.70]\) , corresponding to a tolerance in the difference between the mean lifetimes of the new and old items equal to 0.125 and 0.005, respectively. Given a sample of n new items with mean \({\bar{x}}\) , it follows from Table 8 in Appendix A that the CD for \(\lambda \) is Ga( n ,  t ), where \(t=n\bar{x}\) . Assuming \(n=10\) , we consider two values of t , namely, 1.5 and 4.5. The corresponding CD-densities are illustrated in Fig. 5 showing how the observed value t significantly influences the shape of the distribution, altering both its center and its dispersion, in contrast to the normal model. Specifically, for \(t=1.5\) , the potential estimates of \(\lambda \) , represented by the mean and median of the CD, are 6.67 and 6.45, respectively. For \(t=4.5\) , these values change to 2.22 and 2.15.

Table 5 provides the CD- and the CD*-supports corresponding to the two null hypotheses considered, along with the p -values of the UMPU test. Figure 5 and Table 5 together make it evident that, for \(t=1.5\) , the supports of both interval null hypotheses are very low and leading to their rejection, unless the problem requires a loss function that strongly penalizes a wrong rejection. Furthermore, it is immediately apparent that the data suggest higher values of \(\lambda \) , indicating a lower average lifetime of the new item. Note that the standard criterion “ p -value \(< 0.05\) ” would imply not rejecting \({{{\mathcal {H}}}_{0}}: \lambda \in [2,4]\) . For \(t=4.5\) , when \({{{\mathcal {H}}}_{0}}: \lambda \in [2,4]\) , the median 2.15 of the CD falls within the interval [2, 4]. Consequently, both the CD- and the CD*-supports are greater than 0.5, leading to the acceptance of \({{{\mathcal {H}}}_{0}}\) , as also suggested by the p -value. When \({{{\mathcal {H}}}_{0}}: \lambda \in [2.63, 2.70]\) , the CD-support becomes meaningless, whereas the CD*-support is not negligible (0.256) and should be carefully evaluated in accordance with the problem under analysis. This contrasts with the indication provided by the p -value (0.555).

For the point null hypothesis \(\lambda =2.67\) , the analysis is similar to that for the interval [2.63, 2.70]. Note that, in this case, in addition to the UMPU test, it is also possible to consider the simpler and most frequently used equal-tailed test. The corresponding p -value is 0.016 for \(t=1.5\) and 0.484 for \(t=4.5\) ; these values are exactly two times the CD*-support, see Remark 4 .

(Exponential model) CD-densities for the rate parameter \(\lambda \) , with \(n=10\) and \(t=1.5\) (dashed line) and \(t=4.5\) (solid line)

( Uniform model ) As seen in Example 2 , the CD for the parameter \(\theta \) of the uniform distribution \(\text {U}(0, \theta )\) is a Pareto distribution \(\text {Pa}(n, t)\) , where t is the sample maximum. Figure 6 shows the CD-density for \(n=10\) and \(t=2.1\) .

Consider now \({{{\mathcal {H}}}_{0}}: \theta \in [\theta _1, \theta _2]\) versus \({{{\mathcal {H}}}_{1}}: \theta \notin [\theta _1, \theta _2]\) . As usual, we can identify the interval \([\theta _1, \theta _2]\) on the plot of the CD-density and immediately recognize when the CD-test trivially rejects \({{{\mathcal {H}}}_{0}}\) (the interval lies on the left of t , i.e. \(\theta _2<t\) ), when the value of \(\theta _1\) is irrelevant and only the CD-support of \([t,\theta _2]\) determines the decision ( \(\theta _1<t<\theta _2\) ), or when the whole CD-support of \([\theta _1,\theta _2]\) must be considered ( \(t<\theta _1<\theta _2\) ). These facts are not as intuitive when the p -value is used. Indeed, for this problem, there exists the UMP test of level \(\alpha \) (see Eftekharian and Taheri 2015 ) and it is possible to write the p -value as

(we are not aware of previous mention of it). Table 6 reports the p -value of the UMP test, as well as the CD and CD*-supports, for the two hypotheses \({{{\mathcal {H}}}_{0}}: \theta \in [1.5, 2.2]\) and \({{{\mathcal {H}}}_{0}}^\prime : \theta \in [2.0, 2.2]\) for a sample of size \(n=10\) and various values of t .

It can be observed that, when t belongs to the interval \([\theta _1, \theta _2]\) , the CD- and CD*-supports do not depend on \(\theta _1\) , as previously remarked, while the p -value does. This reinforces the incoherence of the p -value shown by Schervish ( 1996 ). For instance, when \(t=2.19\) , the p -value for \({{{\mathcal {H}}}_{0}}\) is 0.046, while that for \({{{\mathcal {H}}}_{0}}^{\prime }\) (included in \({{{\mathcal {H}}}_{0}}\) ) is larger, namely 0.072. Thus, assuming \(\alpha =0.05\) , the UMP test leads to the rejection of \({{{\mathcal {H}}}_{0}}\) but it results in the acceptance of the smaller hypothesis \({{{\mathcal {H}}}_{0}}^{\prime }\) .

(Uniform model) CD-density for \(\theta \) with \(n=10\) and \(t=2.1\)

( Sharpe ratio ) The Sharpe ratio is one of the most widely used measures of performance of stocks and funds. It is defined as the average excess return relative to the volatility, i.e. \(SR=\theta =(\mu _R-R_f)/\sigma _R\) , where \(\mu _R\) and \(\sigma _R\) are the mean and standard deviation of a return R and \(R_f\) is a risk-free rate. Under the typical assumption of constant risk-free rate, the excess returns \(X_1, X_2, \ldots , X_n\) of the fund over a period of length n are considered, leading to \(\theta =\mu /\sigma \) , where \(\mu \) and \(\sigma \) are the mean and standard deviation of each \(X_i\) . If the sample is not too small, the distribution and the dependence of the \(X_i\) ’s are not so crucial, and the inference on \(\theta \) is similar to that obtained under the basic assumption of i.i.d. normal random variables, as discussed in Opdyke ( 2007 ). Following this article, we consider the weekly returns of the mutual fund Fidelity Blue Chip Growth from 12/24/03 to 12/20/06 (these data are available for example on Yahoo! Finance, https://finance.yahoo.com/quote/FBGRX ) and assume that the excess returns are i.i.d. normal with a risk-free rate equal to 0.00052. Two different samples are analyzed: the first one includes all \(n_1=159\) observations from the entire period, while the second one is limited to the \(n_2=26\) weeks corresponding to the fourth quarter of 2005 and the first quarter of 2006. The sample mean, the standard deviation, and the corresponding sample Sharpe ratio for the first sample are \(\bar{x}_1=0.00011\) , \(s_1=0.01354\) , \(t_1=\bar{x}_1/s_1=0.00842\) . For the second sample, the values are \(\bar{x}_2=0.00280\) , \(s_2=0.01048\) , \(t_2=\bar{x}_2/s_2=0.26744\) .

We can derive the CD for \(\theta \) starting from the sampling distribution of the statistic \(W=\sqrt{n}T=\sqrt{n}\bar{X}/S\) , which has a noncentral t-distribution with \(n-1\) degrees of freedom and noncentrality parameter \(\tau =\sqrt{n}\mu /\sigma =\sqrt{n}\theta \) . This family has MLR (see Lehmann and Romano 2005 , p. 224) and the distribution function \(F^W_\tau \) of W is continuous in \(\tau \) with \(\lim _{\tau \rightarrow +\infty } F^W_\tau (w)=0\) and \(\lim _{\tau \rightarrow -\infty } F^W_\tau (w)=1\) , for each w in \(\mathbb {R}\) . Thus, from ( 1 ), the CD for \(\tau \) is \(H^\tau _w(\tau )=1-F^W_\tau (w)\) . Recalling that \(\theta =\tau /\sqrt{n}\) , the CD for \(\theta \) can be obtained using a trivial transformation which leads to \(H^\theta _w(\theta )=H^\tau _{w}(\sqrt{n}\theta )=1-F_{\sqrt{n}\theta }^W(w)\) , where \(w=\sqrt{n}t\) . In Figure 7 , the CD-densities for \(\theta \) relative to the two samples are plotted: they are symmetric and centered on the estimate t of \(\theta \) , and the dispersion is smaller for the one with the larger n .

Now, let us consider the typical hypotheses for the Sharpe ratio \({{\mathcal {H}}}_0: \theta \le 0\) versus \({{\mathcal {H}}}_1: \theta >0\) . From Table 7 , which reports the CD-supports and the corresponding odds for the two samples, and from Fig. 7 , it appears that the first sample clearly favors neither hypothesis, while \({{{\mathcal {H}}}_{1}}\) is strongly supported by the second one. Here, the p -value coincides with the CD-support (see Proposition 3 ), but choosing the the usual values 0.05 or 0.01 to decide whether to reject \({{{\mathcal {H}}}_{0}}\) or not may lead to markedly different conclusions.

When the assumption of i.i.d. normal returns does not hold, it is possible to show (Opdyke 2007 ) that the asymptotic distribution of T is normal with mean and variance \(\theta \) and \(\sigma ^2_T=(1+\theta ^2(\gamma _4-1)/4-\theta \gamma _3)/n\) , where \(\gamma _3\) and \(\gamma _4\) are the skewness and kurtosis of the \(X_i\) ’s. Thus, the CD for \(\theta \) can be derived from the asymptotic distribution of T and is N( \(t,\hat{\sigma }^2_T)\) , where \(\hat{\sigma }^2_T\) is obtained by estimating the population moments using the sample counterparts. The last column of Table 7 shows that the asymptotic CD-supports for \({{{\mathcal {H}}}_{0}}\) are not too different from the previous ones.

(Sharpe ratio) CD-densities for \(\theta =\mu /\sigma \) with \(n_1=159, t_1=0.008\) (solid line) and \(n_2\) =26, \(t_2=0.267\) (dashed line)

( Ratio of Poisson rates ) The comparison of Poisson rates \(\mu _1\) and \(\mu _2\) is important in various contexts, as illustrated for example by Lehmann & Romano ( 2005 , sec. 4.5), who also derive the UMPU test for the ratio \(\phi =\mu _1/\mu _2\) . Given two i.i.d. samples of sizes \(n_1\) and \(n_2\) from independent Poisson distributions, we can summarize the data with the two sufficient sample sums \(S_1\) and \(S_2\) , where \(S_i \sim \) Po( \(n_i\mu _i\) ), \(i=1,2\) . Reparameterizing the joint density of \((S_1, S_2)\) with \(\phi =\mu _1/\mu _2\) and \(\lambda =n_1\mu _1+n_2\mu _2\) , it is simple to verify that the conditional distribution of \(S_1\) given \(S_1+S_2=s_1+s_2\) is Bi( \(s_1+s_2, w\phi /(1+w\phi )\) ), with \(w=n_1/n_2\) , while the marginal distribution of \(S_1+S_2\) depends only on \(\lambda \) . Thus, for making inference on \(\phi \) , it is reasonable to use the CD for \(\phi \) obtained from the previous conditional distribution. Referring to the table in Appendix A, the CD \(H^g_{s_1,s_2}\) for \(w\phi /(1+w\phi )\) is Be \((s_1+1/2, s_2+1/2)\) , enabling us to determine the CD-density for \(\phi \) through the change of variable rule:

We compare our results with those derived by the standard conditional test implemented through the function poisson.test in R. We use the “eba1977” data set available in the package ISwR, ( https://CRAN.R-project.org/package=ISwR ), which contains counts of incident lung cancer cases and population size in four neighboring Danish cities by age group. Specifically, we compare the \(s_1=11\) lung cancer cases in a population of \(n_1=800\) people aged 55–59 living in Fredericia with the \(s_2=21\) cases observed in the other cities, which have a total of \(n_2=3011\) residents. For the hypothesis \({{{\mathcal {H}}}_{0}}: \phi =1\) versus \({{{\mathcal {H}}}_{1}}: \phi \ne 1\) , the R-output provides a p -value of 0.080 and a 0.95 confidence interval of (0.858, 4.277). If a significance level \(\alpha =0.05\) is chosen, \({{{\mathcal {H}}}_{0}}\) is not rejected, leading to the conclusion that there should be no reason for the inhabitants of Fredericia to worry.

Looking at the three CD-densities for \(\phi \) in Fig. 8 , it is evident that values of \(\phi \) greater than 1 are more supported than values less than 1. Thus, one should test the hypothesis \({{{\mathcal {H}}}_{0}}: \phi \le 1\) versus \({{{\mathcal {H}}}_{1}}: \phi >1\) . Using ( 5 ), it follows that the CD-support of \({{{\mathcal {H}}}_{0}}\) is \(H^g_{s_1,s_2}(1)=0.037\) , and the confidence odds are \(CO_{0,1}=0.037/(1-0.037)=0.038\) . To avoid rejecting \({{{\mathcal {H}}}_{0}}\) , a very asymmetric loss function should be deemed suitable. Finally, we observe that the confidence interval computed in R, is the Clopper-Pearson one, which has exact coverage but, as generally recognized, is too wide. In our context, this corresponds to taking the lower bound of the interval using the CC generated by \(H^\ell _{s_1, s_2}\) and the upper bound using that generated by \(H^r_{s_1, s_2}\) (see Veronese and Melilli 2015 ). It includes the interval generated by \(H_{s_1, s_2}^g\) , namely (0.931, 4.026), as shown in the right plot of Fig. 8 .

(Poisson-rates) CD-densities (left plot) and CCs (right plot) corresponding to \(H^g_{s_1,s_2}(\phi )\) (solid lines), \(H^\ell _{s_1,s_2}(\phi )\) (dashed lines) and \(H^r_{s_1,s_2}(\phi )\) (dotted lines) for the parameter \(\phi \) . In the CC plot the vertical lines identify the Clopper-Pearson confidence interval (dashed and dotted lines) and that based on \(H^g_{s_1,s_2}(\phi )\) (solid lines). The dotted horizontal line is at level 0.95

5 Properties of CD-support and CD*-support

5.1 one-sided hypotheses.

The CD-support of a set is the mass assigned to it by the CD, making it a fundamental component in all inferential problems based on CDs. Nevertheless, its direct utilization in hypothesis testing is rare, with the exception of Xie and Singh ( 2013 ). It can also be viewed as a specific instance of evidential support , a notion introduced by Bickel ( 2022 ) within a broader category of models known as evidential models , which encompass both posterior distributions and confidence distributions as specific cases.

Let us now consider a classical testing problem. Let \(\textbf{X}\) be an i.i.d. sample with a distribution depending on a real parameter \(\theta \) and let \({{{\mathcal {H}}}_{0}}: \theta \le \theta _0\) versus \({{{\mathcal {H}}}_{1}}: \theta >\theta _0\) , where \(\theta _0\) is a fixed value (the case \({{{\mathcal {H}}}_{0}}^\prime : \theta \ge \theta _0\) versus \({{{\mathcal {H}}}_{1}}^\prime : \theta <\theta _0\) is perfectly specular and will not be analyzed). In order to compare our test with the standard one, we assume that the model has MLR in \(T=T(\textbf{X})\) . Suppose first that the distribution function \(F_\theta (t)\) of T is continuous and that the CD for \(\theta \) is \(H_t(\theta )=1- F_{\theta }(t)\) . From Sect. 3 , the CD-support for \({{{\mathcal {H}}}_{0}}\) (which coincides with the CD*-support) is \(H_t(\theta _0)\) . In this case, the UMP test exists, as established by the Karlin-Rubin theorem, and rejects \({{{\mathcal {H}}}_{0}}\) if \(t > t_\alpha \) , where \(t_\alpha \) depends on the chosen significance level \(\alpha \) , or alternatively, if the p -value \(\text{ Pr}_{\theta _0}(T\ge t)\) is less than \(\alpha \) . Since \(\text{ Pr}_{\theta _0}(T\ge t)=1-F_{\theta _0}(t)=H_t(\theta _0)\) , the p -value coincides with the CD-support. Thus, to define a CD-test with size \(\alpha \) , it is enough to fix its rejection region as \(\{t: H_t(\theta _0)<\alpha \}\) , and both tests lead to the same conclusion.

When the statistic T is discrete, we have seen that various choices of CDs are possible. Assuming that \(H^r_t(\theta )< H^g_t(\theta ) < H^{\ell }_t(\theta )\) , as occurs for models belonging to a real NEF, it follows immediately that \(H^{r}_t\) provides stronger support for \({{\mathcal {H}}}_0: \theta \le \theta _0\) than \(H^g_t\) does, while \(H^{\ell }_t\) provides stronger support for \({{\mathcal {H}}}_0^\prime : \theta \ge \theta _0\) than \(H^g_t\) does. In other words, \(H_t^{\ell }\) is more conservative than \(H^g_t\) for testing \({{{\mathcal {H}}}_{0}}\) and the same happens to \(H^r_t\) for \({{{\mathcal {H}}}_{0}}^{\prime }\) . Therefore, selecting the appropriate CD can lead to the standard testing result. For example, in the case of \({{{\mathcal {H}}}_{0}}:\theta \le \theta _0\) versus \({{{\mathcal {H}}}_{1}}: \theta > \theta _0\) , the p -value is \(\text{ Pr}_{\theta _0}(T\ge t)=1-\text{ Pr}_{\theta _0}(T<t)=H^{\ell }_t(\theta _0)\) , and the rejection region of the standard test and that of the CD-test based on \(H_t^{\ell }\) coincide if the threshold is the same. However, as both tests are non-randomized, their size is typically strictly less than the fixed threshold.

The following proposition summarizes the previous considerations.

Proposition 3

Consider a model indexed by a real parameter \(\theta \) with MLR in the statistic T and the one-sided hypotheses \({{{\mathcal {H}}}_{0}}: \theta \le \theta _0\) versus \({{{\mathcal {H}}}_{1}}: \theta >\theta _0\) , or \({{{\mathcal {H}}}_{0}}^\prime : \theta \ge \theta _0\) versus \({{{\mathcal {H}}}_{1}}^\prime : \theta <\theta _0\) . If T is continuous, then the CD-support and the p -value associated with the UMP test are equal. Thus, if a common threshold \(\alpha \) is set for both rejection regions, the two tests have size \(\alpha \) . If T is discrete, the CD-support coincides with the usual p -value if \(H^\ell _t [H^r_t]\) is chosen when \({{{\mathcal {H}}}_{0}}: \theta \le \theta _0\) \([{{{\mathcal {H}}}_{0}}^\prime : \theta \ge \theta _0]\) . For a fixed threshold \(\alpha \) , the two tests have a size not greater than \(\alpha \) .

The CD-tests with threshold \(\alpha \) mentioned in the previous proposition have significance level \(\alpha \) and are, therefore, valid , that is \(\sup _{\theta \in \Theta _0} Pr_\theta (H(T)\le \alpha ) \le \alpha \) (see Martin and Liu 2013 ). This is no longer true if, for a discrete T , we choose \(H^g_t\) . However, Proposition 2 implies that its average size is closer to \(\alpha \) compared to those of the tests obtained using \(H^\ell _t\) \([H^r_t]\) , making \(H^g_t\) more appropriate when the problem does not strongly suggest that the null hypothesis should be considered true “until proven otherwise”.

5.2 Precise and interval hypotheses

The notion of CD*-support surely demands more attention than that of CD-support. Recalling that the CD*-support only accounts for one direction of deviation from the precise or interval hypothesis, we will first briefly explore its connections with similar notions.

While the CD-support is an additive measure, meaning that for any set \(A \subseteq \Theta \) and its complement \(A^c\) , we always have \(\text{ CD }(A) +\text{ CD }(A^c)=1\) , the CD*-support is only a sub-additive measure, that is \(\text{ CD* }(A) +\text{ CD* }(A^c)\le 1\) , as can be easily checked. This suggests that the CD*-support can be related to a belief function. In essence, a belief function \(\text{ bel}_\textbf{x}(A)\) measures the evidence in \(\textbf{x}\) that supports A . However, due to its sub-additivity, it alone cannot provide sufficient information; it must be coupled with the plausibility function, defined as \(\text {pl}_\textbf{x}(A) = 1 - \text {bel}_\textbf{x}(A^c)\) . We refer to Martin and Liu ( 2013 ) for a detailed treatment of these notions within the general framework of Inferential Models , which admits a CD as a very specific case. We only mention here that they show that when \(A=\{\theta _0\}\) (i.e. a singleton), \(\text{ bel}_\textbf{x}(\{\theta _0\})=0\) , but \(\text{ bel}_\textbf{x}(\{\theta _0\}^c)\) can be different from 1. In particular, for the normal model N \((\theta ,1)\) , they found that, under some assumptions, \(\text{ bel}_\textbf{x}(\{\theta _0\}^c) =|2\Phi (x-\theta _0)-1|\) . Recalling the definition of the CC and the CD provided in Example 1 , it follows that the plausibility of \(\theta _0\) is \(\text {pl}_\textbf{x}(\{\theta _0\})=1-\text{ bel}_\textbf{x}(\{\theta _0\}^c)=1-|2\Phi (x-\theta _0)-1|= 1-CC_\textbf{x}(\theta _0)\) , and using ( 4 ), we can conclude that the CD*-support of \(\theta _0\) corresponds to half their plausibility.

The CD*-support for a precise hypothesis \({{{\mathcal {H}}}_{0}}: \theta =\theta _0\) is related to the notion of evidence, as defined in a Bayesian context by Pereira et al. ( 2008 ). Evidence is the posterior probability of the set \(\{\theta \in \Theta : p(\theta |\textbf{x})<p(\theta _0|\textbf{x})\}\) , where \(p(\theta |\textbf{x})\) is the posterior density of \(\theta \) . In particular, when a unimodal and symmetric CD is used as a posterior distribution, it is easy to check that the CD*-support coincides with half of the evidence.

The CD*-support is also related to the notion of weak-support defined by Singh et al. ( 2007 ) as \(\sup _{\theta \in [\theta _1,\theta _2]} 2 \min \{H_{\textbf{x}}(\theta ), 1-H_{\textbf{x}}(\theta )\}\) , but important differences exist. If data give little support to \({{{\mathcal {H}}}_{0}}\) , our definition highlights better whether values of \(\theta \) on the right or on the left of \({{{\mathcal {H}}}_{0}}\) are more reasonable. Moreover, if \({{{\mathcal {H}}}_{0}}\) is highly supported, that is \(\theta _m \in [\theta _1,\theta _2]\) , the weak-support is always equal to one, while the CD*-support assumes values in the interval [0.5, 1], allowing to better discriminate between different cases. Only if \({{{\mathcal {H}}}_{0}}\) is a precise hypothesis the two definitions agree, leaving out the multiplicative constant of two.

There exists a strong connection between the CD*-support and the e-value introduced by Peskun ( 2020 ). Under certain regularity assumptions, the e -value can be expressed in terms of a CD and coincides with the CD*-support, so that the properties and results originally established by Peskun for the e -value also apply to the CD*-support. More precisely, let us first consider the case of an observation x generated by the normal model \(\text {N}(\mu ,1)\) . Peskun shows that for the hypothesis \({{{\mathcal {H}}}_{0}}: \mu \in [\mu _1,\mu _2]\) , the e -value is equal to \(\min \{\Phi (x-\mu _1), \Phi (\mu _2-x)\}\) . Since, as shown in Example 1 , \(H_x(\mu )=1-\Phi (x-\mu )=\Phi (\mu -x)\) , it immediately follows that \(\min \{H_x(\mu _2),1-H_x(\mu _1)\}= \min \{\Phi (\mu _2-x), \Phi (x-\mu _1)\}\) , so that the e -value and the CD*-support coincide. For a more general case, we present the following result.

Proposition 4

Let \(\textbf{X}\) be a random vector distributed according to the family of densities \(\{p_\theta , \theta \in \Theta \subseteq \mathbb {R}\}\) with a MLR in the real continuous statistic \(T=T(\textbf{X})\) , with distribution function \(F_\theta (t)\) . If \(F_\theta (t)\) is continuous in \(\theta \) with limits 0 and 1 for \(\theta \) tending to \(\sup (\Theta )\) and \(\inf (\Theta )\) , respectively, then the CD*-support and the e -value for the hypothesis \({{{\mathcal {H}}}_{0}}: \theta \in [\theta _1,\theta _2]\) , \(\theta _1 \le \theta _2\) , are equivalent.

We emphasize, however, that the advantage of the CD*-support over the e -value relies on the fact that knowledge of the entire CD allows us to naturally encompass the testing problem into a more comprehensive and coherent inferential framework, in which the e -value is only one of the aspects to be taken into consideration.

Suppose now that a test of significance for \({{\mathcal {H}}}_0: \theta \in [\theta _1,\theta _2]\) , with \(\theta _1 \le \theta _2\) , is desired and that the CD for \(\theta \) is \(H_t(\theta )\) . Recall that the CD-support for \({{{\mathcal {H}}}_{0}}\) is \(H_t([\theta _1,\theta _2]) = \int _{\theta _1}^{\theta _2} dH_{t}(\theta ) = H_t(\theta _2)-H_t(\theta _1)\) , and that when \(\theta _1=\theta _2=\theta _0\) , or the interval \([\theta _1,\theta _2]\) is “small”, it becomes ineffective, and the CD*-support must be employed. The following proposition establishes some results about the CD- and the CD*-tests.

Proposition 5

Given a statistical model parameterized by the real parameter \(\theta \) with MLR in the continuous statistic T , consider the hypothesis \({{{\mathcal {H}}}_{0}}: \theta \in [\theta _1,\theta _2]\) with \( \theta _1 \le \theta _2\) . Then,

both the CD- and the CD*-tests reject \({{{\mathcal {H}}}_{0}}\) for all values of T that are smaller or larger than suitable values;

if a threshold \(\gamma \) is fixed for the CD-test, its size is not less than \(\gamma \) ;

for a precise hypothesis, i.e., \(\theta _1=\theta _2\) , the CD*-support, seen as function of the random variable T , has the uniform distribution on (0, 0.5);

if a threshold \(\gamma ^*\) is fixed for the CD*-test, its size falls within the interval \([\gamma ^*, \min (2\gamma ^*,1)]\) and equals \(\min (2\gamma ^*,1)\) when \(\theta _1=\theta _2\) , (i.e. when \({{{\mathcal {H}}}_{0}}\) is a precise hypothesis);

the CD-support is never greater than the CD*-support, and if a common threshold is fixed for both tests, the size of the CD-test is not smaller than that of the CD*-test.

Point i) highlights that the rejection regions generated by the CD- and CD*-tests are two-sided, resembling standard tests for hypotheses of this kind. However, even when \(\gamma = \gamma ^*\) , the rejection regions differ, with the CD-test being more conservative for \({{{\mathcal {H}}}_{0}}\) . This becomes crucial for small intervals, where the CD-test tends to reject the null hypothesis almost invariably.

Under the assumption of Proposition 5 , the p -value corresponding to the commonly used equal tailed test for a precise hypothesis \({{{\mathcal {H}}}_{0}}:\theta =\theta _0\) is \(2\min \{F_{\theta _0}(t), 1-F_{\theta _0}(t)\}\) , so that it coincides with 2 times the CD*-support.

For interval hypotheses, a UMPU test essentially exists only for models within a NEF, and an interesting relationship can be established with the CD-test.

Proposition 6

Given the CD based on the sufficient statistic of a continuous real NEF with natural parameter \(\theta \) , consider the hypothesis \({{\mathcal {H}}}_0: \theta \in [\theta _1,\theta _2]\) versus \({{\mathcal {H}}}_1: \theta \notin [\theta _1,\theta _2]\) , with \(\theta _1 < \theta _2\) . If the CD-test has size \(\alpha _{CD}\) , it is the UMPU test among all \(\alpha _{CD}\) -level tests.

For interval hypotheses, unlike one-sided hypotheses, when the statistic T is discrete, there is no clear reason to prefer either \(H_t^{\ell }\) or \(H_t^r\) . Neither test is more conservative, as their respective rejection regions are shifted by just one point in the support of T . Thus, \(H^g_t\) can be considered again a reasonable compromise, due to its greater proximity to the uniform distribution. Moreover, while the results stated for continuous statistics may not hold exactly for discrete statistics, they remain approximately valid for not too small sample sizes, thanks to the asymptotic normality of CDs, as stated in Proposition 1 .

6 Conclusions

In this article, we propose the use of confidence distributions to address a hypothesis testing problem concerning a real parameter of interest. Specifically, we introduce the CD- and CD*-supports, which are suitable for evaluating one-sided or large interval null hypotheses and precise or small interval null hypotheses, respectively. This approach does not necessarily require identifying the first and second type errors or fixing a significance level a priori. We do not propose an automatic procedure; instead, we suggest a careful and more general inferential analysis of the problem based on CDs. CD- and CD*-supports are two simple coherent measures of evidence for a hypothesis with a clear meaning and interpretation. None of these features are owned by the p -value, which is more complex and generally does not exist in closed form for interval hypothesis.

It is well known that the significance level \(\alpha \) of a test, which is crucial to take a decision, should be adjusted according to the sample size, but this is almost never done in practice. In our approach, the support provided by the CD to a hypothesis trivially depends on the sample size through the dispersion of the CD. For example, if \({{{\mathcal {H}}}_{0}}: \theta \in [\theta _1,\theta _2]\) , you can easily observe the effect of sample size on the CD-support of \({{{\mathcal {H}}}_{0}}\) by examining the interval \([\theta _1, \theta _2]\) on the CD-density plot. The CD-support can be non-negligible also when the length \(\Delta =\theta _2-\theta _1\) is small for a CD that is sufficiently concentrated on the interval. The relationship between \(\Delta \) and the dispersion of the CD highlights again the importance of a thoughtful choice of the threshold used for decision-making and the unreasonableness of using standard values. Note that the CD- and CD*-tests are similar in many standard situations, as shown in the examples presented.

Finally, we have investigated some theoretical aspects of the CD- and CD*-tests which are crucial in standard approach. While for one-sided hypotheses, an agreement with standard tests can be established, there are some distinctions to be made for two-sided hypotheses. If a threshold \(\gamma \) is fixed for a CD- or CD*-test, then its size exceeds \(\gamma \) reaching \(2\gamma \) for a CD*-test relative to a precise hypothesis. This is because the CD*-support only considers the appropriate tail suggested by the data and it does not adhere to the typical procedure of doubling the one-sided p -value, a procedure that can be criticized, as seen in Sect. 1 . Of course, if one is convinced of the need to double the p -value, in our context, it is sufficient to double the CD*-support. In the case of a precise hypothesis \({{{\mathcal {H}}}_{0}}: \theta = \theta _0\) , this leads to a valid test because \(Pr_{\theta _0}\left( 2\min \{H_{\textbf{x}}(\theta _0),1-H_{\textbf{x}}(\theta _0)\}\le \alpha \right) \le \alpha \) , as can be deduced by considering the relationship of the CD*-support with the e -value and the results in Peskun ( 2020 , Sec. 2).

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Acknowledgements

Partial financial support was received from Bocconi University. The authors would like to thank the referees for their valuable comments, suggestions and references, which led to a significantly improved version of the manuscript

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Appendix A. Table of confidence distributions

Appendix b. proof of propositions, proof of proposition 1.

The asymptotic normality and the consistency of the CD in i) and ii) follow from Veronese & Melilli ( 2015 , Thm. 3) for models belonging to a NEF and from Veronese & Melilli ( 2018b , Thm. 1) for continuous arbitrary models. Part iii) of the proposition follows directly using the Chebyshev’s inequality. \(\diamond \)

Proof of Proposition 2

Denote by \(F_{\theta }(t)\) the distribution function of T , assume that its support \({{\mathcal {T}}}=\{t_1,t_2,\ldots ,t_k\}\) is finite for simplicity and let \(p_j=p_j(\theta )=\text{ Pr}_\theta (T=t_j)\) , \(j=1,2,\ldots ,k\) for a fixed \(\theta \) . Consider the case \(H_t^r(\theta )=1-F_{\theta }(t)\) (if \(H_t^r(\theta )=F_{\theta }(t)\) the proof is similar) so that, for each \(j=2,\ldots ,k\) , \(H_{t_j}^\ell (\theta )=H_{t_{j-1}}^r(\theta )\) and \(H_{t_1}^\ell (\theta )=1\) . The supports of the random variables \(H^r_T(\theta )\) , \(H^\ell _T(\theta )\) and \(H^g_T(\theta )\) are, respectively,

where ( 6 ) holds because \(H^r_{t_j}(\theta )< H^g_{t_j}(\theta ) < H^{\ell }_{t_j}(\theta )\) . The probabilities corresponding to the points included in the three supports are of course the same, that is \(p_k,p_{k-1},\ldots ,p_1\) , in this order, so that \(G^\ell (u) \le u \le G^r(u)\) .

Let \(d(Q,R)=\int |Q(x)-R(x)|dx\) be the distance between the two arbitrary distribution functions Q and R . Denoting \(G^u\) as the uniform distribution function on (0, 1), we have

where the last inequality follows from ( 6 ). Thus, the distance from uniformity of \(H_T^g(\theta )\) is less than that of \(H_T^\ell (\theta )\) and of \(H_T^r(\theta )\) and ( 2 ) is proven. \(\diamond \)

Proof of Proposition 4

Given the statistic T and the hypothesis \({{{\mathcal {H}}}_{0}}: \theta \in [\theta _1,\theta _2]\) , the e -value, see Peskun 2020 , equation 12), is \(\min \bigg \{\max _{\theta \in [\theta _1,\theta _2]} F_\theta (t), \max _{\theta \in [\theta _1,\theta _2]} (1-F_\theta (t))\bigg \}\) . Under the assumptions of the proposition, it follows that \(F_t(\theta )\) is monotonically nonincreasing in \(\theta \) for each t (see Section 2 ). As a result, the e -value simplifies to:

where the last expression coincides with the CD*-support of \({{{\mathcal {H}}}_{0}}\) . Note that the same result holds if the MLR is nondecreasing in T ensuring that \(F_t(\theta )\) is monotonically nondecreasing. \(\diamond \)

Proof of Proposition 5

Point i). Consider first the CD-test and let \(g(t)=H_t([\theta _1,\theta _2])=H_t(\theta _2)-H_t(\theta _1)=F_{\theta _1}(t)-F_{\theta _2}(t)\) , which is a nonnegative, continuous function with \(\lim _{t\rightarrow \pm \infty }g(t)=0\) and with derivative \(g^\prime (t)=f_{\theta _1}(t)- f_{\theta _2}(t)\) . Let \(t_0 \in \mathbb {R}\) be a point such that g is nondecreasing for \(t<t_0\) and strictly decreasing for \(t \in (t_0,t_1)\) , for a suitable \(t_1>t_0\) ; the existence of \(t_0\) is guaranteed by the properties of g . It follows that \(g^\prime (t) \ge 0\) for \(t<t_0\) and \(g^\prime (t)<0\) in \((t_0,t_1)\) . We show that \(t_0\) is the unique point at which the function \(g^\prime \) changes sign. Indeed, if \(t_2\) were a point greater than \(t_1\) such that \(g^\prime (t)>0\) for t in a suitable interval \((t_2,t_3)\) , with \(t_3> t_2\) , we would have, in this interval, \(f_{\theta _1}(t)>f_{\theta _2}(t)\) . Since \(f_{\theta _1}(t)<f_{\theta _2}(t)\) for \(t \in (t_0,t_1)\) , this implies \(f_{\theta _2}(t)/f_{\theta _1}(t)>1\) for \(t \in (t_0,t_1)\) and \(f_{\theta _2}(t)/f_{\theta _1}(t)<1\) for \(t \in (t_2,t_3)\) , which contradicts the assumption of the (nondecreasing) MLR in T . Thus, g ( t ) is nondecreasing for \(t<t_0\) and nonincreasing for \(t>t_0\) , and the set \(\{t: H_t([\theta _1,\theta _2])< \gamma \}\) coincides with \( \{t: t<t^\prime \) or \(t>t^{\prime \prime }\}\) for suitable \(t^\prime \) and \(t^{\prime \prime }\) .

Consider now the CD*-test. The corresponding support is \(\min \{H_t(\theta _2), 1-H_t(\theta _1)\}= \min \{1-F_{\theta _2}(t), F_{\theta _1}(t)\}\) , which is a continuous function of t and approaches zero as \(t \rightarrow \pm \infty \) . Moreover, it equals \(F_{\theta _1}(t)\) for \(t\le t^*=\inf \{t: F_{\theta _1}(t)=1-F_{\theta _2}(t)\}\) and \(1-F_{\theta _2}(t)\) for \(t\ge t^*\) . Thus, the function is nondecreasing for \(t \le t^*\) and nonincreasing for \(t \ge t^*\) , and the result is proven.

Point ii). Suppose having observed \(t^\prime = F_{\theta _1}^{-1}(\gamma )\) , then the CD-support for \({{{\mathcal {H}}}_{0}}\) is

so that \(t^\prime \) belongs to the rejection region defined by the threshold \(\gamma \) . Due to the structure of this region specified in point i), all \(t\le t^{\prime }\) belong to it. Now,

because \(F_{\theta }(t) \le F_{\theta _1}(t)\) for each t and \(\theta \in [\theta _1,\theta _2]\) . It follows that the size of the CD-test with threshold \(\gamma \) is not smaller than \(\gamma \) .

Point iii). The result follows from the equality of the CD*-support with the e -value, as stated in Proposition 4 , and the uniformity of the e -value as proven in Peskun ( 2020 , Sec. 2).

Point iv). The size of the CD*-test with threshold \(\gamma ^*\) is the supremum on \([\theta _1,\theta _2]\) of the following probability

under the assumption that \(F_{\theta _1}^{-1}(\gamma ^*) <F_{\theta _2}^{-1}(1-\gamma ^*)\) , otherwise the probability is one. Because \(F_{\theta _2}(t) \le F_{\theta }(t) \le F_{\theta _1}(t)\) for each t and \(\theta \in [\theta _1,\theta _2]\) , it follows that \(F_{\theta }(F_{\theta _1}^{-1}(\gamma ^*)) \le F_{\theta _1}(F_{\theta _1}^{-1}(\gamma ^*))=\gamma ^*\) , and \(F_{\theta }(F_{\theta _2}^{-1}(1-\gamma ^*)) \ge F_{\theta _2}(F_{\theta _2}^{-1}(1-\gamma ^*)) = 1-\gamma ^*\) so that the size is

Finally, if \(\theta =\theta _2\) , from ( 7 ) we have

and thus the size of the CD*-test must be included in the interval \([\gamma ^*,2\gamma ^*]\) , provided that \(2\gamma ^*\) is less than 1. For the case \(\theta _1=\theta _2\) , it follows from ( 7 ) that the size of the CD*-test is \(2\gamma ^*\) .

Point v). Because \(H_t([\theta _1,\theta _2]=H_t(\theta _2)-H_t(\theta _1)\le H_t(\theta _2)\) and also \(H_t(\theta _2)-H_t(\theta _1) \le 1-H_t(\theta _1)\) , recalling Definition 4 , it immediately follows that the CD-support is not greater than the CD*-support. Thus if the same threshold is fixed for the two tests, the rejection region of the CD-test includes that of the CD*-test, and the size of the first test is not smaller than that of the second one. \(\diamond \)

Proof of Proposition 6

Recall from point i) of Proposition 5 , that the CD-test with threshold \(\gamma \) rejects \({{{\mathcal {H}}}_{0}}: \theta \in [\theta _1,\theta _2]\) for values of T less than \(t^\prime \) or greater than \(t^{\prime \prime }\) , with \(t^\prime \) and \(t^{\prime \prime }\) solutions of the equation \(F_{\theta _1}(t)-F_{\theta _2}(t)=\gamma \) . Denoting with \(\pi _{CD}\) its power function, we have

Thus the power function of the CD-test is equal in \(\theta _1\) and \(\theta _2\) and this condition characterizes the UMPU test for the exponential families, see Lehmann & Romano ( 2005 , p. 135). \(\diamond \)

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Melilli, E., Veronese, P. Confidence distributions and hypothesis testing. Stat Papers (2024). https://doi.org/10.1007/s00362-024-01542-4

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DOI : https://doi.org/10.1007/s00362-024-01542-4

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