hypothesis test vs significance

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StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2024 Jan-.

StatPearls [Internet].

Hypothesis testing, p values, confidence intervals, and significance.

Jacob Shreffler ; Martin R. Huecker .

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Last Update: March 13, 2023 .

Definition/Introduction

Medical providers often rely on evidence-based medicine to guide decision-making in practice. Often a research hypothesis is tested with results provided, typically with p values, confidence intervals, or both. Additionally, statistical or research significance is estimated or determined by the investigators. Unfortunately, healthcare providers may have different comfort levels in interpreting these findings, which may affect the adequate application of the data.

Issues of Concern

Without a foundational understanding of hypothesis testing, p values, confidence intervals, and the difference between statistical and clinical significance, it may affect healthcare providers' ability to make clinical decisions without relying purely on the research investigators deemed level of significance. Therefore, an overview of these concepts is provided to allow medical professionals to use their expertise to determine if results are reported sufficiently and if the study outcomes are clinically appropriate to be applied in healthcare practice.

Hypothesis Testing

Investigators conducting studies need research questions and hypotheses to guide analyses. Starting with broad research questions (RQs), investigators then identify a gap in current clinical practice or research. Any research problem or statement is grounded in a better understanding of relationships between two or more variables. For this article, we will use the following research question example:

Research Question: Is Drug 23 an effective treatment for Disease A?

Research questions do not directly imply specific guesses or predictions; we must formulate research hypotheses. A hypothesis is a predetermined declaration regarding the research question in which the investigator(s) makes a precise, educated guess about a study outcome. This is sometimes called the alternative hypothesis and ultimately allows the researcher to take a stance based on experience or insight from medical literature. An example of a hypothesis is below.

Research Hypothesis: Drug 23 will significantly reduce symptoms associated with Disease A compared to Drug 22.

The null hypothesis states that there is no statistical difference between groups based on the stated research hypothesis.

Researchers should be aware of journal recommendations when considering how to report p values, and manuscripts should remain internally consistent.

Regarding p values, as the number of individuals enrolled in a study (the sample size) increases, the likelihood of finding a statistically significant effect increases. With very large sample sizes, the p-value can be very low significant differences in the reduction of symptoms for Disease A between Drug 23 and Drug 22. The null hypothesis is deemed true until a study presents significant data to support rejecting the null hypothesis. Based on the results, the investigators will either reject the null hypothesis (if they found significant differences or associations) or fail to reject the null hypothesis (they could not provide proof that there were significant differences or associations).

To test a hypothesis, researchers obtain data on a representative sample to determine whether to reject or fail to reject a null hypothesis. In most research studies, it is not feasible to obtain data for an entire population. Using a sampling procedure allows for statistical inference, though this involves a certain possibility of error. [1] When determining whether to reject or fail to reject the null hypothesis, mistakes can be made: Type I and Type II errors. Though it is impossible to ensure that these errors have not occurred, researchers should limit the possibilities of these faults. [2]

Significance

Significance is a term to describe the substantive importance of medical research. Statistical significance is the likelihood of results due to chance. [3] Healthcare providers should always delineate statistical significance from clinical significance, a common error when reviewing biomedical research. [4] When conceptualizing findings reported as either significant or not significant, healthcare providers should not simply accept researchers' results or conclusions without considering the clinical significance. Healthcare professionals should consider the clinical importance of findings and understand both p values and confidence intervals so they do not have to rely on the researchers to determine the level of significance. [5] One criterion often used to determine statistical significance is the utilization of p values.

P values are used in research to determine whether the sample estimate is significantly different from a hypothesized value. The p-value is the probability that the observed effect within the study would have occurred by chance if, in reality, there was no true effect. Conventionally, data yielding a p<0.05 or p<0.01 is considered statistically significant. While some have debated that the 0.05 level should be lowered, it is still universally practiced. [6] Hypothesis testing allows us to determine the size of the effect.

An example of findings reported with p values are below:

Statement: Drug 23 reduced patients' symptoms compared to Drug 22. Patients who received Drug 23 (n=100) were 2.1 times less likely than patients who received Drug 22 (n = 100) to experience symptoms of Disease A, p<0.05.

Statement:Individuals who were prescribed Drug 23 experienced fewer symptoms (M = 1.3, SD = 0.7) compared to individuals who were prescribed Drug 22 (M = 5.3, SD = 1.9). This finding was statistically significant, p= 0.02.

For either statement, if the threshold had been set at 0.05, the null hypothesis (that there was no relationship) should be rejected, and we should conclude significant differences. Noticeably, as can be seen in the two statements above, some researchers will report findings with < or > and others will provide an exact p-value (0.000001) but never zero [6] . When examining research, readers should understand how p values are reported. The best practice is to report all p values for all variables within a study design, rather than only providing p values for variables with significant findings. [7] The inclusion of all p values provides evidence for study validity and limits suspicion for selective reporting/data mining.

While researchers have historically used p values, experts who find p values problematic encourage the use of confidence intervals. [8] . P-values alone do not allow us to understand the size or the extent of the differences or associations. [3] In March 2016, the American Statistical Association (ASA) released a statement on p values, noting that scientific decision-making and conclusions should not be based on a fixed p-value threshold (e.g., 0.05). They recommend focusing on the significance of results in the context of study design, quality of measurements, and validity of data. Ultimately, the ASA statement noted that in isolation, a p-value does not provide strong evidence. [9]

When conceptualizing clinical work, healthcare professionals should consider p values with a concurrent appraisal study design validity. For example, a p-value from a double-blinded randomized clinical trial (designed to minimize bias) should be weighted higher than one from a retrospective observational study [7] . The p-value debate has smoldered since the 1950s [10] , and replacement with confidence intervals has been suggested since the 1980s. [11]

Confidence Intervals

A confidence interval provides a range of values within given confidence (e.g., 95%), including the accurate value of the statistical constraint within a targeted population. [12] Most research uses a 95% CI, but investigators can set any level (e.g., 90% CI, 99% CI). [13] A CI provides a range with the lower bound and upper bound limits of a difference or association that would be plausible for a population. [14] Therefore, a CI of 95% indicates that if a study were to be carried out 100 times, the range would contain the true value in 95, [15] confidence intervals provide more evidence regarding the precision of an estimate compared to p-values. [6]

In consideration of the similar research example provided above, one could make the following statement with 95% CI:

Statement: Individuals who were prescribed Drug 23 had no symptoms after three days, which was significantly faster than those prescribed Drug 22; there was a mean difference between the two groups of days to the recovery of 4.2 days (95% CI: 1.9 – 7.8).

It is important to note that the width of the CI is affected by the standard error and the sample size; reducing a study sample number will result in less precision of the CI (increase the width). [14] A larger width indicates a smaller sample size or a larger variability. [16] A researcher would want to increase the precision of the CI. For example, a 95% CI of 1.43 – 1.47 is much more precise than the one provided in the example above. In research and clinical practice, CIs provide valuable information on whether the interval includes or excludes any clinically significant values. [14]

Null values are sometimes used for differences with CI (zero for differential comparisons and 1 for ratios). However, CIs provide more information than that. [15] Consider this example: A hospital implements a new protocol that reduced wait time for patients in the emergency department by an average of 25 minutes (95% CI: -2.5 – 41 minutes). Because the range crosses zero, implementing this protocol in different populations could result in longer wait times; however, the range is much higher on the positive side. Thus, while the p-value used to detect statistical significance for this may result in "not significant" findings, individuals should examine this range, consider the study design, and weigh whether or not it is still worth piloting in their workplace.

Similarly to p-values, 95% CIs cannot control for researchers' errors (e.g., study bias or improper data analysis). [14] In consideration of whether to report p-values or CIs, researchers should examine journal preferences. When in doubt, reporting both may be beneficial. [13] An example is below:

Reporting both: Individuals who were prescribed Drug 23 had no symptoms after three days, which was significantly faster than those prescribed Drug 22, p = 0.009. There was a mean difference between the two groups of days to the recovery of 4.2 days (95% CI: 1.9 – 7.8).

Clinical Significance

Recall that clinical significance and statistical significance are two different concepts. Healthcare providers should remember that a study with statistically significant differences and large sample size may be of no interest to clinicians, whereas a study with smaller sample size and statistically non-significant results could impact clinical practice. [14] Additionally, as previously mentioned, a non-significant finding may reflect the study design itself rather than relationships between variables.

Healthcare providers using evidence-based medicine to inform practice should use clinical judgment to determine the practical importance of studies through careful evaluation of the design, sample size, power, likelihood of type I and type II errors, data analysis, and reporting of statistical findings (p values, 95% CI or both). [4] Interestingly, some experts have called for "statistically significant" or "not significant" to be excluded from work as statistical significance never has and will never be equivalent to clinical significance. [17]

The decision on what is clinically significant can be challenging, depending on the providers' experience and especially the severity of the disease. Providers should use their knowledge and experiences to determine the meaningfulness of study results and make inferences based not only on significant or insignificant results by researchers but through their understanding of study limitations and practical implications.

Nursing, Allied Health, and Interprofessional Team Interventions

All physicians, nurses, pharmacists, and other healthcare professionals should strive to understand the concepts in this chapter. These individuals should maintain the ability to review and incorporate new literature for evidence-based and safe care.

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Disclosure: Jacob Shreffler declares no relevant financial relationships with ineligible companies.

Disclosure: Martin Huecker declares no relevant financial relationships with ineligible companies.

This book is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ), which permits others to distribute the work, provided that the article is not altered or used commercially. You are not required to obtain permission to distribute this article, provided that you credit the author and journal.

Cite this Page Shreffler J, Huecker MR. Hypothesis Testing, P Values, Confidence Intervals, and Significance. [Updated 2023 Mar 13]. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2024 Jan-.

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Understanding Hypothesis Tests: Significance Levels (Alpha) and P values in Statistics

Topics: Hypothesis Testing , Statistics

What do significance levels and P values mean in hypothesis tests? What is statistical significance anyway? In this post, I’ll continue to focus on concepts and graphs to help you gain a more intuitive understanding of how hypothesis tests work in statistics.

To bring it to life, I’ll add the significance level and P value to the graph in my previous post in order to perform a graphical version of the 1 sample t-test. It’s easier to understand when you can see what statistical significance truly means!

Here’s where we left off in my last post . We want to determine whether our sample mean (330.6) indicates that this year's average energy cost is significantly different from last year’s average energy cost of $260.

The probability distribution plot above shows the distribution of sample means we’d obtain under the assumption that the null hypothesis is true (population mean = 260) and we repeatedly drew a large number of random samples.

I left you with a question: where do we draw the line for statistical significance on the graph? Now we'll add in the significance level and the P value, which are the decision-making tools we'll need.

We'll use these tools to test the following hypotheses:

Null hypothesis: The population mean equals the hypothesized mean (260).
Alternative hypothesis: The population mean differs from the hypothesized mean (260).

What Is the Significance Level (Alpha)?

The significance level, also denoted as alpha or α, is the probability of rejecting the null hypothesis when it is true. For example, a significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference.

These types of definitions can be hard to understand because of their technical nature. A picture makes the concepts much easier to comprehend!

The significance level determines how far out from the null hypothesis value we'll draw that line on the graph. To graph a significance level of 0.05, we need to shade the 5% of the distribution that is furthest away from the null hypothesis.

Probability plot that shows the critical regions for a significance level of 0.05

In the graph above, the two shaded areas are equidistant from the null hypothesis value and each area has a probability of 0.025, for a total of 0.05. In statistics, we call these shaded areas the critical region for a two-tailed test. If the population mean is 260, we’d expect to obtain a sample mean that falls in the critical region 5% of the time. The critical region defines how far away our sample statistic must be from the null hypothesis value before we can say it is unusual enough to reject the null hypothesis.

Our sample mean (330.6) falls within the critical region, which indicates it is statistically significant at the 0.05 level.

We can also see if it is statistically significant using the other common significance level of 0.01.

Probability plot that shows the critical regions for a significance level of 0.01

The two shaded areas each have a probability of 0.005, which adds up to a total probability of 0.01. This time our sample mean does not fall within the critical region and we fail to reject the null hypothesis. This comparison shows why you need to choose your significance level before you begin your study. It protects you from choosing a significance level because it conveniently gives you significant results!

Thanks to the graph, we were able to determine that our results are statistically significant at the 0.05 level without using a P value. However, when you use the numeric output produced by statistical software , you’ll need to compare the P value to your significance level to make this determination.

Ready for a demo of Minitab Statistical Software? Just ask!

What Are P values?

P-values are the probability of obtaining an effect at least as extreme as the one in your sample data, assuming the truth of the null hypothesis.

This definition of P values, while technically correct, is a bit convoluted. It’s easier to understand with a graph!

To graph the P value for our example data set, we need to determine the distance between the sample mean and the null hypothesis value (330.6 - 260 = 70.6). Next, we can graph the probability of obtaining a sample mean that is at least as extreme in both tails of the distribution (260 +/- 70.6).

Probability plot that shows the p-value for our sample mean

In the graph above, the two shaded areas each have a probability of 0.01556, for a total probability 0.03112. This probability represents the likelihood of obtaining a sample mean that is at least as extreme as our sample mean in both tails of the distribution if the population mean is 260. That’s our P value!

When a P value is less than or equal to the significance level, you reject the null hypothesis. If we take the P value for our example and compare it to the common significance levels, it matches the previous graphical results. The P value of 0.03112 is statistically significant at an alpha level of 0.05, but not at the 0.01 level.

If we stick to a significance level of 0.05, we can conclude that the average energy cost for the population is greater than 260.

A common mistake is to interpret the P-value as the probability that the null hypothesis is true. To understand why this interpretation is incorrect, please read my blog post How to Correctly Interpret P Values .

Discussion about Statistically Significant Results

A hypothesis test evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data. A test result is statistically significant when the sample statistic is unusual enough relative to the null hypothesis that we can reject the null hypothesis for the entire population. “Unusual enough” in a hypothesis test is defined by:

The assumption that the null hypothesis is true—the graphs are centered on the null hypothesis value.
The significance level—how far out do we draw the line for the critical region?
Our sample statistic—does it fall in the critical region?

Keep in mind that there is no magic significance level that distinguishes between the studies that have a true effect and those that don’t with 100% accuracy. The common alpha values of 0.05 and 0.01 are simply based on tradition. For a significance level of 0.05, expect to obtain sample means in the critical region 5% of the time when the null hypothesis is true . In these cases, you won’t know that the null hypothesis is true but you’ll reject it because the sample mean falls in the critical region. That’s why the significance level is also referred to as an error rate!

This type of error doesn’t imply that the experimenter did anything wrong or require any other unusual explanation. The graphs show that when the null hypothesis is true, it is possible to obtain these unusual sample means for no reason other than random sampling error. It’s just luck of the draw.

Significance levels and P values are important tools that help you quantify and control this type of error in a hypothesis test. Using these tools to decide when to reject the null hypothesis increases your chance of making the correct decision.

If you like this post, you might want to read the other posts in this series that use the same graphical framework:

Previous: Why We Need to Use Hypothesis Tests
Next: Confidence Intervals and Confidence Levels

If you'd like to see how I made these graphs, please read: How to Create a Graphical Version of the 1-sample t-Test .

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Hypothesis Testing (cont...)

Hypothesis testing, the null and alternative hypothesis.

In order to undertake hypothesis testing you need to express your research hypothesis as a null and alternative hypothesis. The null hypothesis and alternative hypothesis are statements regarding the differences or effects that occur in the population. You will use your sample to test which statement (i.e., the null hypothesis or alternative hypothesis) is most likely (although technically, you test the evidence against the null hypothesis). So, with respect to our teaching example, the null and alternative hypothesis will reflect statements about all statistics students on graduate management courses.

The null hypothesis is essentially the "devil's advocate" position. That is, it assumes that whatever you are trying to prove did not happen ( hint: it usually states that something equals zero). For example, the two different teaching methods did not result in different exam performances (i.e., zero difference). Another example might be that there is no relationship between anxiety and athletic performance (i.e., the slope is zero). The alternative hypothesis states the opposite and is usually the hypothesis you are trying to prove (e.g., the two different teaching methods did result in different exam performances). Initially, you can state these hypotheses in more general terms (e.g., using terms like "effect", "relationship", etc.), as shown below for the teaching methods example:

Depending on how you want to "summarize" the exam performances will determine how you might want to write a more specific null and alternative hypothesis. For example, you could compare the mean exam performance of each group (i.e., the "seminar" group and the "lectures-only" group). This is what we will demonstrate here, but other options include comparing the distributions , medians , amongst other things. As such, we can state:

Now that you have identified the null and alternative hypotheses, you need to find evidence and develop a strategy for declaring your "support" for either the null or alternative hypothesis. We can do this using some statistical theory and some arbitrary cut-off points. Both these issues are dealt with next.

Significance levels

The level of statistical significance is often expressed as the so-called p -value . Depending on the statistical test you have chosen, you will calculate a probability (i.e., the p -value) of observing your sample results (or more extreme) given that the null hypothesis is true . Another way of phrasing this is to consider the probability that a difference in a mean score (or other statistic) could have arisen based on the assumption that there really is no difference. Let us consider this statement with respect to our example where we are interested in the difference in mean exam performance between two different teaching methods. If there really is no difference between the two teaching methods in the population (i.e., given that the null hypothesis is true), how likely would it be to see a difference in the mean exam performance between the two teaching methods as large as (or larger than) that which has been observed in your sample?

So, you might get a p -value such as 0.03 (i.e., p = .03). This means that there is a 3% chance of finding a difference as large as (or larger than) the one in your study given that the null hypothesis is true. However, you want to know whether this is "statistically significant". Typically, if there was a 5% or less chance (5 times in 100 or less) that the difference in the mean exam performance between the two teaching methods (or whatever statistic you are using) is as different as observed given the null hypothesis is true, you would reject the null hypothesis and accept the alternative hypothesis. Alternately, if the chance was greater than 5% (5 times in 100 or more), you would fail to reject the null hypothesis and would not accept the alternative hypothesis. As such, in this example where p = .03, we would reject the null hypothesis and accept the alternative hypothesis. We reject it because at a significance level of 0.03 (i.e., less than a 5% chance), the result we obtained could happen too frequently for us to be confident that it was the two teaching methods that had an effect on exam performance.

Whilst there is relatively little justification why a significance level of 0.05 is used rather than 0.01 or 0.10, for example, it is widely used in academic research. However, if you want to be particularly confident in your results, you can set a more stringent level of 0.01 (a 1% chance or less; 1 in 100 chance or less).

One- and two-tailed predictions

When considering whether we reject the null hypothesis and accept the alternative hypothesis, we need to consider the direction of the alternative hypothesis statement. For example, the alternative hypothesis that was stated earlier is:

The alternative hypothesis tells us two things. First, what predictions did we make about the effect of the independent variable(s) on the dependent variable(s)? Second, what was the predicted direction of this effect? Let's use our example to highlight these two points.

Sarah predicted that her teaching method (independent variable: teaching method), whereby she not only required her students to attend lectures, but also seminars, would have a positive effect (that is, increased) students' performance (dependent variable: exam marks). If an alternative hypothesis has a direction (and this is how you want to test it), the hypothesis is one-tailed. That is, it predicts direction of the effect. If the alternative hypothesis has stated that the effect was expected to be negative, this is also a one-tailed hypothesis.

Alternatively, a two-tailed prediction means that we do not make a choice over the direction that the effect of the experiment takes. Rather, it simply implies that the effect could be negative or positive. If Sarah had made a two-tailed prediction, the alternative hypothesis might have been:

In other words, we simply take out the word "positive", which implies the direction of our effect. In our example, making a two-tailed prediction may seem strange. After all, it would be logical to expect that "extra" tuition (going to seminar classes as well as lectures) would either have a positive effect on students' performance or no effect at all, but certainly not a negative effect. However, this is just our opinion (and hope) and certainly does not mean that we will get the effect we expect. Generally speaking, making a one-tail prediction (i.e., and testing for it this way) is frowned upon as it usually reflects the hope of a researcher rather than any certainty that it will happen. Notable exceptions to this rule are when there is only one possible way in which a change could occur. This can happen, for example, when biological activity/presence in measured. That is, a protein might be "dormant" and the stimulus you are using can only possibly "wake it up" (i.e., it cannot possibly reduce the activity of a "dormant" protein). In addition, for some statistical tests, one-tailed tests are not possible.

Rejecting or failing to reject the null hypothesis

Let's return finally to the question of whether we reject or fail to reject the null hypothesis.

If our statistical analysis shows that the significance level is below the cut-off value we have set (e.g., either 0.05 or 0.01), we reject the null hypothesis and accept the alternative hypothesis. Alternatively, if the significance level is above the cut-off value, we fail to reject the null hypothesis and cannot accept the alternative hypothesis. You should note that you cannot accept the null hypothesis, but only find evidence against it.

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Hypothesis Testing, P Values, Confidence Intervals, and Significance

Definition/introduction.

Issues of Concern

Hypothesis Testing

Research Question: Is Drug 23 an effective treatment for Disease A?

Research Hypothesis: Drug 23 will significantly reduce symptoms associated with Disease A compared to Drug 22.

The null hypothesis states that there is no statistical difference between groups based on the stated research hypothesis.

Researchers should be aware of journal recommendations when considering how to report p values, and manuscripts should remain internally consistent.

Significance

An example of findings reported with p values are below:

Confidence Intervals

In consideration of the similar research example provided above, one could make the following statement with 95% CI:

Clinical Significance

Nursing, Allied Health, and Interprofessional Team Interventions

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1.4: Basic Concepts of Hypothesis Testing

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John H. McDonald
University of Delaware

Learning Objectives

One of the main goals of statistical hypothesis testing is to estimate the $P$ value, which is the probability of obtaining the observed results, or something more extreme, if the null hypothesis were true. If the observed results are unlikely under the null hypothesis, reject the null hypothesis.
Alternatives to this "frequentist" approach to statistics include Bayesian statistics and estimation of effect sizes and confidence intervals.

Introduction

There are different ways of doing statistics. The technique used by the vast majority of biologists, and the technique that most of this handbook describes, is sometimes called "frequentist" or "classical" statistics. It involves testing a null hypothesis by comparing the data you observe in your experiment with the predictions of a null hypothesis. You estimate what the probability would be of obtaining the observed results, or something more extreme, if the null hypothesis were true. If this estimated probability (the $P$ value) is small enough (below the significance value), then you conclude that it is unlikely that the null hypothesis is true; you reject the null hypothesis and accept an alternative hypothesis.

Many statisticians harshly criticize frequentist statistics, but their criticisms haven't had much effect on the way most biologists do statistics. Here I will outline some of the key concepts used in frequentist statistics, then briefly describe some of the alternatives.

Null Hypothesis

The null hypothesis is a statement that you want to test. In general, the null hypothesis is that things are the same as each other, or the same as a theoretical expectation. For example, if you measure the size of the feet of male and female chickens, the null hypothesis could be that the average foot size in male chickens is the same as the average foot size in female chickens. If you count the number of male and female chickens born to a set of hens, the null hypothesis could be that the ratio of males to females is equal to a theoretical expectation of a $1:1$ ratio.

The alternative hypothesis is that things are different from each other, or different from a theoretical expectation.

For example, one alternative hypothesis would be that male chickens have a different average foot size than female chickens; another would be that the sex ratio is different from $1:1$.

Usually, the null hypothesis is boring and the alternative hypothesis is interesting. For example, let's say you feed chocolate to a bunch of chickens, then look at the sex ratio in their offspring. If you get more females than males, it would be a tremendously exciting discovery: it would be a fundamental discovery about the mechanism of sex determination, female chickens are more valuable than male chickens in egg-laying breeds, and you'd be able to publish your result in Science or Nature . Lots of people have spent a lot of time and money trying to change the sex ratio in chickens, and if you're successful, you'll be rich and famous. But if the chocolate doesn't change the sex ratio, it would be an extremely boring result, and you'd have a hard time getting it published in the Eastern Delaware Journal of Chickenology . It's therefore tempting to look for patterns in your data that support the exciting alternative hypothesis. For example, you might look at $48$ offspring of chocolate-fed chickens and see $31$ females and only $17$ males. This looks promising, but before you get all happy and start buying formal wear for the Nobel Prize ceremony, you need to ask "What's the probability of getting a deviation from the null expectation that large, just by chance, if the boring null hypothesis is really true?" Only when that probability is low can you reject the null hypothesis. The goal of statistical hypothesis testing is to estimate the probability of getting your observed results under the null hypothesis.

Biological vs. Statistical Null Hypotheses

It is important to distinguish between biological null and alternative hypotheses and statistical null and alternative hypotheses. "Sexual selection by females has caused male chickens to evolve bigger feet than females" is a biological alternative hypothesis; it says something about biological processes, in this case sexual selection. "Male chickens have a different average foot size than females" is a statistical alternative hypothesis; it says something about the numbers, but nothing about what caused those numbers to be different. The biological null and alternative hypotheses are the first that you should think of, as they describe something interesting about biology; they are two possible answers to the biological question you are interested in ("What affects foot size in chickens?"). The statistical null and alternative hypotheses are statements about the data that should follow from the biological hypotheses: if sexual selection favors bigger feet in male chickens (a biological hypothesis), then the average foot size in male chickens should be larger than the average in females (a statistical hypothesis). If you reject the statistical null hypothesis, you then have to decide whether that's enough evidence that you can reject your biological null hypothesis. For example, if you don't find a significant difference in foot size between male and female chickens, you could conclude "There is no significant evidence that sexual selection has caused male chickens to have bigger feet." If you do find a statistically significant difference in foot size, that might not be enough for you to conclude that sexual selection caused the bigger feet; it might be that males eat more, or that the bigger feet are a developmental byproduct of the roosters' combs, or that males run around more and the exercise makes their feet bigger. When there are multiple biological interpretations of a statistical result, you need to think of additional experiments to test the different possibilities.

Testing the Null Hypothesis

The primary goal of a statistical test is to determine whether an observed data set is so different from what you would expect under the null hypothesis that you should reject the null hypothesis. For example, let's say you are studying sex determination in chickens. For breeds of chickens that are bred to lay lots of eggs, female chicks are more valuable than male chicks, so if you could figure out a way to manipulate the sex ratio, you could make a lot of chicken farmers very happy. You've fed chocolate to a bunch of female chickens (in birds, unlike mammals, the female parent determines the sex of the offspring), and you get $25$ female chicks and $23$ male chicks. Anyone would look at those numbers and see that they could easily result from chance; there would be no reason to reject the null hypothesis of a $1:1$ ratio of females to males. If you got $47$ females and $1$ male, most people would look at those numbers and see that they would be extremely unlikely to happen due to luck, if the null hypothesis were true; you would reject the null hypothesis and conclude that chocolate really changed the sex ratio. However, what if you had $31$ females and $17$ males? That's definitely more females than males, but is it really so unlikely to occur due to chance that you can reject the null hypothesis? To answer that, you need more than common sense, you need to calculate the probability of getting a deviation that large due to chance.

In the figure above, I used the BINOMDIST function of Excel to calculate the probability of getting each possible number of males, from $0$ to $48$, under the null hypothesis that $0.5$ are male. As you can see, the probability of getting $17$ males out of $48$ total chickens is about $0.015$. That seems like a pretty small probability, doesn't it? However, that's the probability of getting exactly $17$ males. What you want to know is the probability of getting $17$ or fewer males. If you were going to accept $17$ males as evidence that the sex ratio was biased, you would also have accepted $16$, or $15$, or $14$,… males as evidence for a biased sex ratio. You therefore need to add together the probabilities of all these outcomes. The probability of getting $17$ or fewer males out of $48$, under the null hypothesis, is $0.030$. That means that if you had an infinite number of chickens, half males and half females, and you took a bunch of random samples of $48$ chickens, $3.0\%$ of the samples would have $17$ or fewer males.

This number, $0.030$, is the $P$ value. It is defined as the probability of getting the observed result, or a more extreme result, if the null hypothesis is true. So "$P=0.030$" is a shorthand way of saying "The probability of getting $17$ or fewer male chickens out of $48$ total chickens, IF the null hypothesis is true that $50\%$ of chickens are male, is $0.030$."

False Positives vs. False Negatives

After you do a statistical test, you are either going to reject or accept the null hypothesis. Rejecting the null hypothesis means that you conclude that the null hypothesis is not true; in our chicken sex example, you would conclude that the true proportion of male chicks, if you gave chocolate to an infinite number of chicken mothers, would be less than $50\%$.

When you reject a null hypothesis, there's a chance that you're making a mistake. The null hypothesis might really be true, and it may be that your experimental results deviate from the null hypothesis purely as a result of chance. In a sample of $48$ chickens, it's possible to get $17$ male chickens purely by chance; it's even possible (although extremely unlikely) to get $0$ male and $48$ female chickens purely by chance, even though the true proportion is $50\%$ males. This is why we never say we "prove" something in science; there's always a chance, however miniscule, that our data are fooling us and deviate from the null hypothesis purely due to chance. When your data fool you into rejecting the null hypothesis even though it's true, it's called a "false positive," or a "Type I error." So another way of defining the $P$ value is the probability of getting a false positive like the one you've observed, if the null hypothesis is true.

Another way your data can fool you is when you don't reject the null hypothesis, even though it's not true. If the true proportion of female chicks is $51\%$, the null hypothesis of a $50\%$ proportion is not true, but you're unlikely to get a significant difference from the null hypothesis unless you have a huge sample size. Failing to reject the null hypothesis, even though it's not true, is a "false negative" or "Type II error." This is why we never say that our data shows the null hypothesis to be true; all we can say is that we haven't rejected the null hypothesis.

Significance Levels

Does a probability of $0.030$ mean that you should reject the null hypothesis, and conclude that chocolate really caused a change in the sex ratio? The convention in most biological research is to use a significance level of $0.05$. This means that if the $P$ value is less than $0.05$, you reject the null hypothesis; if $P$ is greater than or equal to $0.05$, you don't reject the null hypothesis. There is nothing mathematically magic about $0.05$, it was chosen rather arbitrarily during the early days of statistics; people could have agreed upon $0.04$, or $0.025$, or $0.071$ as the conventional significance level.

The significance level (also known as the "critical value" or "alpha") you should use depends on the costs of different kinds of errors. With a significance level of $0.05$, you have a $5\%$ chance of rejecting the null hypothesis, even if it is true. If you try $100$ different treatments on your chickens, and none of them really change the sex ratio, $5\%$ of your experiments will give you data that are significantly different from a $1:1$ sex ratio, just by chance. In other words, $5\%$ of your experiments will give you a false positive. If you use a higher significance level than the conventional $0.05$, such as $0.10$, you will increase your chance of a false positive to $0.10$ (therefore increasing your chance of an embarrassingly wrong conclusion), but you will also decrease your chance of a false negative (increasing your chance of detecting a subtle effect). If you use a lower significance level than the conventional $0.05$, such as $0.01$, you decrease your chance of an embarrassing false positive, but you also make it less likely that you'll detect a real deviation from the null hypothesis if there is one.

The relative costs of false positives and false negatives, and thus the best $P$ value to use, will be different for different experiments. If you are screening a bunch of potential sex-ratio-changing treatments and get a false positive, it wouldn't be a big deal; you'd just run a few more tests on that treatment until you were convinced the initial result was a false positive. The cost of a false negative, however, would be that you would miss out on a tremendously valuable discovery. You might therefore set your significance value to $0.10$ or more for your initial tests. On the other hand, once your sex-ratio-changing treatment is undergoing final trials before being sold to farmers, a false positive could be very expensive; you'd want to be very confident that it really worked. Otherwise, if you sell the chicken farmers a sex-ratio treatment that turns out to not really work (it was a false positive), they'll sue the pants off of you. Therefore, you might want to set your significance level to $0.01$, or even lower, for your final tests.

The significance level you choose should also depend on how likely you think it is that your alternative hypothesis will be true, a prediction that you make before you do the experiment. This is the foundation of Bayesian statistics, as explained below.

You must choose your significance level before you collect the data, of course. If you choose to use a different significance level than the conventional $0.05$, people will be skeptical; you must be able to justify your choice. Throughout this handbook, I will always use $P< 0.05$ as the significance level. If you are doing an experiment where the cost of a false positive is a lot greater or smaller than the cost of a false negative, or an experiment where you think it is unlikely that the alternative hypothesis will be true, you should consider using a different significance level.

One-tailed vs. Two-tailed Probabilities

The probability that was calculated above, $0.030$, is the probability of getting $17$ or fewer males out of $48$. It would be significant, using the conventional $P< 0.05$criterion. However, what about the probability of getting $17$ or fewer females? If your null hypothesis is "The proportion of males is $17$ or more" and your alternative hypothesis is "The proportion of males is less than $0.5$," then you would use the $P=0.03$ value found by adding the probabilities of getting $17$ or fewer males. This is called a one-tailed probability, because you are adding the probabilities in only one tail of the distribution shown in the figure. However, if your null hypothesis is "The proportion of males is $0.5$", then your alternative hypothesis is "The proportion of males is different from $0.5$." In that case, you should add the probability of getting $17$ or fewer females to the probability of getting $17$ or fewer males. This is called a two-tailed probability. If you do that with the chicken result, you get $P=0.06$, which is not quite significant.

You should decide whether to use the one-tailed or two-tailed probability before you collect your data, of course. A one-tailed probability is more powerful, in the sense of having a lower chance of false negatives, but you should only use a one-tailed probability if you really, truly have a firm prediction about which direction of deviation you would consider interesting. In the chicken example, you might be tempted to use a one-tailed probability, because you're only looking for treatments that decrease the proportion of worthless male chickens. But if you accidentally found a treatment that produced $87\%$ male chickens, would you really publish the result as "The treatment did not cause a significant decrease in the proportion of male chickens"? I hope not. You'd realize that this unexpected result, even though it wasn't what you and your farmer friends wanted, would be very interesting to other people; by leading to discoveries about the fundamental biology of sex-determination in chickens, in might even help you produce more female chickens someday. Any time a deviation in either direction would be interesting, you should use the two-tailed probability. In addition, people are skeptical of one-tailed probabilities, especially if a one-tailed probability is significant and a two-tailed probability would not be significant (as in our chocolate-eating chicken example). Unless you provide a very convincing explanation, people may think you decided to use the one-tailed probability after you saw that the two-tailed probability wasn't quite significant, which would be cheating. It may be easier to always use two-tailed probabilities. For this handbook, I will always use two-tailed probabilities, unless I make it very clear that only one direction of deviation from the null hypothesis would be interesting.

Reporting your results

In the olden days, when people looked up $P$ values in printed tables, they would report the results of a statistical test as "$P< 0.05$", "$P< 0.01$", "$P>0.10$", etc. Nowadays, almost all computer statistics programs give the exact $P$ value resulting from a statistical test, such as $P=0.029$, and that's what you should report in your publications. You will conclude that the results are either significant or they're not significant; they either reject the null hypothesis (if $P$ is below your pre-determined significance level) or don't reject the null hypothesis (if $P$ is above your significance level). But other people will want to know if your results are "strongly" significant ($P$ much less than $0.05$), which will give them more confidence in your results than if they were "barely" significant ($P=0.043$, for example). In addition, other researchers will need the exact $P$ value if they want to combine your results with others into a meta-analysis.

Computer statistics programs can give somewhat inaccurate $P$ values when they are very small. Once your $P$ values get very small, you can just say "$P< 0.00001$" or some other impressively small number. You should also give either your raw data, or the test statistic and degrees of freedom, in case anyone wants to calculate your exact $P$ value.

Effect Sizes and Confidence Intervals

A fairly common criticism of the hypothesis-testing approach to statistics is that the null hypothesis will always be false, if you have a big enough sample size. In the chicken-feet example, critics would argue that if you had an infinite sample size, it is impossible that male chickens would have exactly the same average foot size as female chickens. Therefore, since you know before doing the experiment that the null hypothesis is false, there's no point in testing it.

This criticism only applies to two-tailed tests, where the null hypothesis is "Things are exactly the same" and the alternative is "Things are different." Presumably these critics think it would be okay to do a one-tailed test with a null hypothesis like "Foot length of male chickens is the same as, or less than, that of females," because the null hypothesis that male chickens have smaller feet than females could be true. So if you're worried about this issue, you could think of a two-tailed test, where the null hypothesis is that things are the same, as shorthand for doing two one-tailed tests. A significant rejection of the null hypothesis in a two-tailed test would then be the equivalent of rejecting one of the two one-tailed null hypotheses.

A related criticism is that a significant rejection of a null hypothesis might not be biologically meaningful, if the difference is too small to matter. For example, in the chicken-sex experiment, having a treatment that produced $49.9\%$ male chicks might be significantly different from $50\%$, but it wouldn't be enough to make farmers want to buy your treatment. These critics say you should estimate the effect size and put a confidence interval on it, not estimate a $P$ value. So the goal of your chicken-sex experiment should not be to say "Chocolate gives a proportion of males that is significantly less than $50\%$ (($P=0.015$)" but to say "Chocolate produced $36.1\%$ males with a $95\%$ confidence interval of $25.9\%$ to $47.4\%$." For the chicken-feet experiment, you would say something like "The difference between males and females in mean foot size is $2.45mm$, with a confidence interval on the difference of $\pm 1.98mm$."

Estimating effect sizes and confidence intervals is a useful way to summarize your results, and it should usually be part of your data analysis; you'll often want to include confidence intervals in a graph. However, there are a lot of experiments where the goal is to decide a yes/no question, not estimate a number. In the initial tests of chocolate on chicken sex ratio, the goal would be to decide between "It changed the sex ratio" and "It didn't seem to change the sex ratio." Any change in sex ratio that is large enough that you could detect it would be interesting and worth follow-up experiments. While it's true that the difference between $49.9\%$ and $50\%$ might not be worth pursuing, you wouldn't do an experiment on enough chickens to detect a difference that small.

Often, the people who claim to avoid hypothesis testing will say something like "the $95\%$ confidence interval of $25.9\%$ to $47.4\%$ does not include $50\%$, so we conclude that the plant extract significantly changed the sex ratio." This is a clumsy and roundabout form of hypothesis testing, and they might as well admit it and report the $P$ value.

Bayesian statistics

Another alternative to frequentist statistics is Bayesian statistics. A key difference is that Bayesian statistics requires specifying your best guess of the probability of each possible value of the parameter to be estimated, before the experiment is done. This is known as the "prior probability." So for your chicken-sex experiment, you're trying to estimate the "true" proportion of male chickens that would be born, if you had an infinite number of chickens. You would have to specify how likely you thought it was that the true proportion of male chickens was $50\%$, or $51\%$, or $52\%$, or $47.3\%$, etc. You would then look at the results of your experiment and use the information to calculate new probabilities that the true proportion of male chickens was $50\%$, or $51\%$, or $52\%$, or $47.3\%$, etc. (the posterior distribution).

I'll confess that I don't really understand Bayesian statistics, and I apologize for not explaining it well. In particular, I don't understand how people are supposed to come up with a prior distribution for the kinds of experiments that most biologists do. With the exception of systematics, where Bayesian estimation of phylogenies is quite popular and seems to make sense, I haven't seen many research biologists using Bayesian statistics for routine data analysis of simple laboratory experiments. This means that even if the cult-like adherents of Bayesian statistics convinced you that they were right, you would have a difficult time explaining your results to your biologist peers. Statistics is a method of conveying information, and if you're speaking a different language than the people you're talking to, you won't convey much information. So I'll stick with traditional frequentist statistics for this handbook.

Having said that, there's one key concept from Bayesian statistics that is important for all users of statistics to understand. To illustrate it, imagine that you are testing extracts from $1000$ different tropical plants, trying to find something that will kill beetle larvae. The reality (which you don't know) is that $500$ of the extracts kill beetle larvae, and $500$ don't. You do the $1000$ experiments and do the $1000$ frequentist statistical tests, and you use the traditional significance level of $P< 0.05$. The $500$ plant extracts that really work all give you $P< 0.05$; these are the true positives. Of the $500$ extracts that don't work, $5\%$ of them give you $P< 0.05$ by chance (this is the meaning of the $P$ value, after all), so you have $25$ false positives. So you end up with $525$ plant extracts that gave you a $P$ value less than $0.05$. You'll have to do further experiments to figure out which are the $25$ false positives and which are the $500$ true positives, but that's not so bad, since you know that most of them will turn out to be true positives.

Now imagine that you are testing those extracts from $1000$ different tropical plants to try to find one that will make hair grow. The reality (which you don't know) is that one of the extracts makes hair grow, and the other $999$ don't. You do the $1000$ experiments and do the $1000$ frequentist statistical tests, and you use the traditional significance level of $P< 0.05$. The one plant extract that really works gives you P <0.05; this is the true positive. But of the $999$ extracts that don't work, $5\%$ of them give you $P< 0.05$ by chance, so you have about $50$ false positives. You end up with $51$ $P$ values less than $0.05$, but almost all of them are false positives.

Now instead of testing $1000$ plant extracts, imagine that you are testing just one. If you are testing it to see if it kills beetle larvae, you know (based on everything you know about plant and beetle biology) there's a pretty good chance it will work, so you can be pretty sure that a $P$ value less than $0.05$ is a true positive. But if you are testing that one plant extract to see if it grows hair, which you know is very unlikely (based on everything you know about plants and hair), a $P$ value less than $0.05$ is almost certainly a false positive. In other words, if you expect that the null hypothesis is probably true, a statistically significant result is probably a false positive. This is sad; the most exciting, amazing, unexpected results in your experiments are probably just your data trying to make you jump to ridiculous conclusions. You should require a much lower $P$ value to reject a null hypothesis that you think is probably true.

A Bayesian would insist that you put in numbers just how likely you think the null hypothesis and various values of the alternative hypothesis are, before you do the experiment, and I'm not sure how that is supposed to work in practice for most experimental biology. But the general concept is a valuable one: as Carl Sagan summarized it, "Extraordinary claims require extraordinary evidence."

Recommendations

Here are three experiments to illustrate when the different approaches to statistics are appropriate. In the first experiment, you are testing a plant extract on rabbits to see if it will lower their blood pressure. You already know that the plant extract is a diuretic (makes the rabbits pee more) and you already know that diuretics tend to lower blood pressure, so you think there's a good chance it will work. If it does work, you'll do more low-cost animal tests on it before you do expensive, potentially risky human trials. Your prior expectation is that the null hypothesis (that the plant extract has no effect) has a good chance of being false, and the cost of a false positive is fairly low. So you should do frequentist hypothesis testing, with a significance level of $0.05$.

In the second experiment, you are going to put human volunteers with high blood pressure on a strict low-salt diet and see how much their blood pressure goes down. Everyone will be confined to a hospital for a month and fed either a normal diet, or the same foods with half as much salt. For this experiment, you wouldn't be very interested in the $P$ value, as based on prior research in animals and humans, you are already quite certain that reducing salt intake will lower blood pressure; you're pretty sure that the null hypothesis that "Salt intake has no effect on blood pressure" is false. Instead, you are very interested to know how much the blood pressure goes down. Reducing salt intake in half is a big deal, and if it only reduces blood pressure by $1mm$ Hg, the tiny gain in life expectancy wouldn't be worth a lifetime of bland food and obsessive label-reading. If it reduces blood pressure by $20mm$ with a confidence interval of $\pm 5mm$, it might be worth it. So you should estimate the effect size (the difference in blood pressure between the diets) and the confidence interval on the difference.

In the third experiment, you are going to put magnetic hats on guinea pigs and see if their blood pressure goes down (relative to guinea pigs wearing the kind of non-magnetic hats that guinea pigs usually wear). This is a really goofy experiment, and you know that it is very unlikely that the magnets will have any effect (it's not impossible—magnets affect the sense of direction of homing pigeons, and maybe guinea pigs have something similar in their brains and maybe it will somehow affect their blood pressure—it just seems really unlikely). You might analyze your results using Bayesian statistics, which will require specifying in numerical terms just how unlikely you think it is that the magnetic hats will work. Or you might use frequentist statistics, but require a $P$ value much, much lower than $0.05$ to convince yourself that the effect is real.

Picture of giant concrete chicken from Sue and Tony's Photo Site.
Picture of guinea pigs wearing hats from all over the internet; if you know the original photographer, please let me know.

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Hypothesis testing.

Key Topics:

Basic approach
Null and alternative hypothesis
Decision making and the p -value
Z-test & Nonparametric alternative

Basic approach to hypothesis testing

State a model describing the relationship between the explanatory variables and the outcome variable(s) in the population and the nature of the variability. State all of your assumptions .
Specify the null and alternative hypotheses in terms of the parameters of the model.
Invent a test statistic that will tend to be different under the null and alternative hypotheses.
Using the assumptions of step 1, find the theoretical sampling distribution of the statistic under the null hypothesis of step 2. Ideally the form of the sampling distribution should be one of the “standard distributions”(e.g. normal, t , binomial..)
Calculate a p -value , as the area under the sampling distribution more extreme than your statistic. Depends on the form of the alternative hypothesis.
Choose your acceptable type 1 error rate (alpha) and apply the decision rule : reject the null hypothesis if the p-value is less than alpha, otherwise do not reject.
$\frac{\bar{X}-\mu_0}{\sigma / \sqrt{n}}$
general form is: (estimate - value we are testing)/(st.dev of the estimate)
z-statistic follows N(0,1) distribution
2 × the area above |z|, area above z,or area below z, or
compare the statistic to a critical value, |z| ≥ z α/2 , z ≥ z α , or z ≤ - z α
Choose the acceptable level of Alpha = 0.05, we conclude …. ?

Making the Decision

It is either likely or unlikely that we would collect the evidence we did given the initial assumption. (Note: “likely” or “unlikely” is measured by calculating a probability!)

If it is likely , then we “ do not reject ” our initial assumption. There is not enough evidence to do otherwise.

If it is unlikely , then:

either our initial assumption is correct and we experienced an unusual event or,
our initial assumption is incorrect

In statistics, if it is unlikely, we decide to “ reject ” our initial assumption.

Example: Criminal Trial Analogy

First, state 2 hypotheses, the null hypothesis (“H 0 ”) and the alternative hypothesis (“H A ”)

H 0 : Defendant is not guilty.
H A : Defendant is guilty.

Usually the H 0 is a statement of “no effect”, or “no change”, or “chance only” about a population parameter.

While the H A , depending on the situation, is that there is a difference, trend, effect, or a relationship with respect to a population parameter.

It can one-sided and two-sided.
In two-sided we only care there is a difference, but not the direction of it. In one-sided we care about a particular direction of the relationship. We want to know if the value is strictly larger or smaller.

Then, collect evidence, such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, handwriting samples, etc. (In statistics, the data are the evidence.)

Next, you make your initial assumption.

Defendant is innocent until proven guilty.

In statistics, we always assume the null hypothesis is true .

Then, make a decision based on the available evidence.

If there is sufficient evidence (“beyond a reasonable doubt”), reject the null hypothesis . (Behave as if defendant is guilty.)
If there is not enough evidence, do not reject the null hypothesis . (Behave as if defendant is not guilty.)

If the observed outcome, e.g., a sample statistic, is surprising under the assumption that the null hypothesis is true, but more probable if the alternative is true, then this outcome is evidence against H 0 and in favor of H A .

An observed effect so large that it would rarely occur by chance is called statistically significant (i.e., not likely to happen by chance).

Using the p -value to make the decision

The p -value represents how likely we would be to observe such an extreme sample if the null hypothesis were true. The p -value is a probability computed assuming the null hypothesis is true, that the test statistic would take a value as extreme or more extreme than that actually observed. Since it's a probability, it is a number between 0 and 1. The closer the number is to 0 means the event is “unlikely.” So if p -value is “small,” (typically, less than 0.05), we can then reject the null hypothesis.

Significance level and p -value

Significance level, α, is a decisive value for p -value. In this context, significant does not mean “important”, but it means “not likely to happened just by chance”.

α is the maximum probability of rejecting the null hypothesis when the null hypothesis is true. If α = 1 we always reject the null, if α = 0 we never reject the null hypothesis. In articles, journals, etc… you may read: “The results were significant ( p <0.05).” So if p =0.03, it's significant at the level of α = 0.05 but not at the level of α = 0.01. If we reject the H 0 at the level of α = 0.05 (which corresponds to 95% CI), we are saying that if H 0 is true, the observed phenomenon would happen no more than 5% of the time (that is 1 in 20). If we choose to compare the p -value to α = 0.01, we are insisting on a stronger evidence!

So, what kind of error could we make? No matter what decision we make, there is always a chance we made an error.

Errors in Criminal Trial:

Errors in Hypothesis Testing

Type I error (False positive): The null hypothesis is rejected when it is true.

α is the maximum probability of making a Type I error.

Type II error (False negative): The null hypothesis is not rejected when it is false.

β is the probability of making a Type II error

There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!

The power of a statistical test is its probability of rejecting the null hypothesis if the null hypothesis is false. That is, power is the ability to correctly reject H 0 and detect a significant effect. In other words, power is one minus the type II error risk.

$\text{Power }=1-\beta = P\left(\text{reject} H_0 | H_0 \text{is false } \right)$

Which error is worse?

Type I = you are innocent, yet accused of cheating on the test. Type II = you cheated on the test, but you are found innocent.

This depends on the context of the problem too. But in most cases scientists are trying to be “conservative”; it's worse to make a spurious discovery than to fail to make a good one. Our goal it to increase the power of the test that is to minimize the length of the CI.

We need to keep in mind:

the effect of the sample size,
the correctness of the underlying assumptions about the population,
statistical vs. practical significance, etc…

(see the handout). To study the tradeoffs between the sample size, α, and Type II error we can use power and operating characteristic curves.

What type of error might we have made?

Type I error is claiming that average student height is not 65 inches, when it really is. Type II error is failing to claim that the average student height is not 65in when it is.

We rejected the null hypothesis, i.e., claimed that the height is not 65, thus making potentially a Type I error. But sometimes the p -value is too low because of the large sample size, and we may have statistical significance but not really practical significance! That's why most statisticians are much more comfortable with using CI than tests.

There is a need for a further generalization. What if we can't assume that σ is known? In this case we would use s (the sample standard deviation) to estimate σ.

If the sample is very large, we can treat σ as known by assuming that σ = s . According to the law of large numbers, this is not too bad a thing to do. But if the sample is small, the fact that we have to estimate both the standard deviation and the mean adds extra uncertainty to our inference. In practice this means that we need a larger multiplier for the standard error.

We need one-sample t -test.

One sample t -test

Assume data are independently sampled from a normal distribution with unknown mean μ and variance σ 2 . Make an initial assumption, μ 0 .
t-statistic: $\frac{\bar{X}-\mu_0}{s / \sqrt{n}}$ where s is a sample st.dev.
t-statistic follows t -distribution with df = n - 1
Alpha = 0.05, we conclude ….

Testing for the population proportion

Let's go back to our CNN poll. Assume we have a SRS of 1,017 adults.

We are interested in testing the following hypothesis: H 0 : p = 0.50 vs. p > 0.50

What is the test statistic?

If alpha = 0.05, what do we conclude?

We will see more details in the next lesson on proportions, then distributions, and possible tests.

IMAGES

Significance Level and Power of a Hypothesis Test Tutorial
An easy-to-understand summary of significance level
Introduction to Hypothesis Testing in R
Hypothesis Testing: Upper, Lower, and Two Tailed Tests
Hypothesis testing tutorial using p value method
Significance level and Statistically significant

VIDEO

Probability and Statistics
Chapter 09: Hypothesis testing: non-directional worked example
Hypothesis Testing Vs Parameter Estimation
Hypothesis Testing
TEST OF SIGNIFICANCE
Statistical Significance and Hypothesis Testing

COMMENTS

Hypothesis Testing, P Values, Confidence Intervals, and Significance
Medical providers often rely on evidence-based medicine to guide decision-making in practice. Often a research hypothesis is tested with results provided, typically with p values, confidence intervals, or both. Additionally, statistical or research significance is estimated or determined by the investigators. Unfortunately, healthcare providers may have different comfort levels in interpreting ...
How Hypothesis Tests Work: Significance Levels (Alpha) and P values
For example, to be statistically significant at the 0.01 significance level requires more substantial evidence than the 0.05 significance level. However, there is a tradeoff in hypothesis tests. Lower significance levels also reduce the power of a hypothesis test to detect a difference that does exist.
Understanding Hypothesis Tests: Significance Levels (Alpha) and P
The P value of 0.03112 is statistically significant at an alpha level of 0.05, but not at the 0.01 level. If we stick to a significance level of 0.05, we can conclude that the average energy cost for the population is greater than 260. A common mistake is to interpret the P-value as the probability that the null hypothesis is true.
Significance tests (hypothesis testing)
Unit test. Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values to make conclusions about hypotheses.
An Easy Introduction to Statistical Significance (With Examples)
The p value determines statistical significance. An extremely low p value indicates high statistical significance, while a high p value means low or no statistical significance. Example: Hypothesis testing. To test your hypothesis, you first collect data from two groups. The experimental group actively smiles, while the control group does not.
Hypothesis Testing
Let's return finally to the question of whether we reject or fail to reject the null hypothesis. If our statistical analysis shows that the significance level is below the cut-off value we have set (e.g., either 0.05 or 0.01), we reject the null hypothesis and accept the alternative hypothesis. Alternatively, if the significance level is above ...
Understanding Significance Levels in Statistics
While this post looks at significance levels from a conceptual standpoint, learn about the significance level and p-values using a graphical representation of how hypothesis tests work. Additionally, my post about the types of errors in hypothesis testing takes a deeper look at both Type 1 and Type II errors, and the tradeoffs between them.
Hypothesis Testing
Table of contents. Step 1: State your null and alternate hypothesis. Step 2: Collect data. Step 3: Perform a statistical test. Step 4: Decide whether to reject or fail to reject your null hypothesis. Step 5: Present your findings. Other interesting articles. Frequently asked questions about hypothesis testing.
Statistical Significance: Definition & Meaning
The definition of statistically significant is that the sample effect is unlikely to be caused by chance (i.e., sampling error). In other words, what we see in the sample likely reflects an effect or relationship that exists in the population. Using a more statistically correct technical definition, statistical significance relates to the ...
P-values and significance tests (video)
About. Transcript. We compare a P-value to a significance level to make a conclusion in a significance test. Given the null hypothesis is true, a p-value is the probability of getting a result as or more extreme than the sample result by random chance alone. If a p-value is lower than our significance level, we reject the null hypothesis.
7.5: Critical values, p-values, and significance level
When we use z z -scores in this way, the obtained value of z z (sometimes called z z -obtained) is something known as a test statistic, which is simply an inferential statistic used to test a null hypothesis. The formula for our z z -statistic has not changed: z = X¯¯¯¯ − μ σ¯/ n−−√ (7.5.1) (7.5.1) z = X ¯ − μ σ ¯ / n.
11.8: Significance Testing and Confidence Intervals
Since zero is lower than $2.00$, it is rejected as a plausible value and a test of the null hypothesis that there is no difference between means is significant. It turns out that the $p$ value is $0.0057$. There is a similar relationship between the $99\%$ confidence interval and significance at the $0.01$ level.
Hypothesis Testing
Hypothesis Testing Step 1: State the Hypotheses. In all three examples, our aim is to decide between two opposing points of view, Claim 1 and Claim 2. In hypothesis testing, Claim 1 is called the null hypothesis (denoted " Ho "), and Claim 2 plays the role of the alternative hypothesis (denoted " Ha ").
Choosing the Right Statistical Test
Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test. Significance is usually denoted by a p-value, or probability value. Statistical significance is arbitrary - it depends on the threshold, or alpha value, chosen by the ...
Hypothesis testing and p-values (video)
In this video there was no critical value set for this experiment. In the last seconds of the video, Sal briefly mentions a p-value of 5% (0.05), which would have a critical of value of z = (+/-) 1.96. Since the experiment produced a z-score of 3, which is more extreme than 1.96, we reject the null hypothesis.
S.3.1 Hypothesis Testing (Critical Value Approach)
The critical value for conducting the left-tailed test H0 : μ = 3 versus HA : μ < 3 is the t -value, denoted -t( α, n - 1), such that the probability to the left of it is α. It can be shown using either statistical software or a t -table that the critical value -t0.05,14 is -1.7613. That is, we would reject the null hypothesis H0 : μ = 3 ...
What is the difference between "testing of hypothesis" and "test of
In contrast, testing of significance is a purely statistical concept. In essence, one has two hypotheses - the null hypothesis, which states that there is no difference between your two (or more) collections of data. The alternative hypothesis is that there is a difference between your two samples that did not occur by chance.
11.2: Significance Testing
There are two approaches (at least) to conducting significance tests. In one (favored by R. Fisher), a significance test is conducted and the probability value reflects the strength of the evidence against the null hypothesis. If the probability is below 0.01 0.01, the data provide strong evidence that the null hypothesis is false.
Hypothesis Testing and Confidence Intervals
The relationship between the confidence level and the significance level for a hypothesis test is as follows: Confidence level = 1 - Significance level (alpha) For example, if your significance level is 0.05, the equivalent confidence level is 95%. Both of the following conditions represent statistically significant results: The P-value in a ...
Hypothesis Testing, P Values, Confidence Intervals, and Significance
Issues of Concern. Without a foundational understanding of hypothesis testing, p values, confidence intervals, and the difference between statistical and clinical significance, it may affect healthcare providers' ability to make clinical decisions without relying purely on the research investigators deemed level of significance.
1.4: Basic Concepts of Hypothesis Testing
So you should do frequentist hypothesis testing, with a significance level of $0.05$. In the second experiment, you are going to put human volunteers with high blood pressure on a strict low-salt diet and see how much their blood pressure goes down. Everyone will be confined to a hospital for a month and fed either a normal diet, or the same ...
Hypothesis Testing
The p-value is a probability computed assuming the null hypothesis is true, that the test statistic would take a value as extreme or more extreme than that actually observed. Since it's a probability, it is a number between 0 and 1. ... less than 0.05), we can then reject the null hypothesis. Significance level and p-value. Significance level ...