types of factor analysis in research methodology

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Factor Analysis: a means for theory and instrument development in support of construct validity

Mohsen tavakol.

1 School of Medicine, Medical Education Centre, the University of Nottingham, UK

Angela Wetzel

2 School of Education, Virginia Commonwealth University, USA

Introduction

Factor analysis (FA) allows us to simplify a set of complex variables or items using statistical procedures to explore the underlying dimensions that explain the relationships between the multiple variables/items. For example, to explore inter-item relationships for a 20-item instrument, a basic analysis would produce 400 correlations; it is not an easy task to keep these matrices in our heads. FA simplifies a matrix of correlations so a researcher can more easily understand the relationship between items in a scale and the underlying factors that the items may have in common. FA is a commonly applied and widely promoted procedure for developing and refining clinical assessment instruments to produce evidence for the construct validity of the measure.

In the literature, the strong association between construct validity and FA is well documented, as the method provides evidence based on test content and evidence based on internal structure, key components of construct validity. 1 From FA, evidence based on internal structure and evidence based on test content can be examined to tell us what the instrument really measures - the intended abstract concept (i.e., a factor/dimension/construct) or something else. Establishing construct validity for the interpretations from a measure is critical to high quality assessment and subsequent research using outcomes data from the measure. Therefore, FA should be a researcher’s best friend during the development and validation of a new measure or when adapting a measure to a new population. FA is also a useful companion when critiquing existing measures for application in research or assessment practice. However, despite the popularity of FA, when applied in medical education instrument development, factor analytic procedures do not always match best practice. 2 This editorial article is designed to help medical educators use FA appropriately.

The Applications of FA

The applications of FA depend on the purpose of the research. Generally speaking, there are two most important types of FA: Explorator Factor Analysis (EFA) and Confirmatory Factor Analysis (CFA).

Exploratory Factor Analysis

Exploratory Factor Analysis (EFA) is widely used in medical education research in the early phases of instrument development, specifically for measures of latent variables that cannot be assessed directly. Typically, in EFA, the researcher, through a review of the literature and engagement with content experts, selects as many instrument items as necessary to fully represent the latent construct (e.g., professionalism). Then, using EFA, the researcher explores the results of factor loadings, along with other criteria (e.g., previous theory, Minimum average partial, 3 Parallel analysis, 4 conceptual meaningfulness, etc.) to refine the measure. Suppose an instrument consisting of 30 questions yields two factors - Factor 1 and Factor 2. A good definition of a factor as a theoretical construct is to look at its factor loadings. 5 The factor loading is the correlation between the item and the factor; a factor loading of more than 0.30 usually indicates a moderate correlation between the item and the factor. Most statistical software, such as SAS, SPSS and R, provide factor loadings. Upon review of the items loading on each factor, the researcher identifies two distinct constructs, with items loading on Factor 1 all related to professionalism, and items loading on Factor 2 related, instead, to leadership. Here, EFA helps the researcher build evidence based on internal structure by retaining only those items with appropriately high loadings on Factor 1 for professionalism, the construct of interest.

It is important to note that, often, Principal Component Analysis (PCA) is applied and described, in error, as exploratory factor analysis. 2 , 6 PCA is appropriate if the study primarily aims to reduce the number of original items in the intended instrument to a smaller set. 7 However, if the instrument is being designed to measure a latent construct, EFA, using Maximum Likelihood (ML) or Principal Axis Factoring (PAF), is the appropriate method. 7   These exploratory procedures statistically analyze the interrelationships between the instrument items and domains to uncover the unknown underlying factorial structure (dimensions) of the construct of interest. PCA, by design, seeks to explain total variance (i.e., specific and error variance) in the correlation matrix. The sum of the squared loadings on a factor matrix for a particular item indicates the proportion of variance for that given item that is explained by the factors. This is called the communality. The higher the communality value, the more the extracted factors explain the variance of the item. Further, the mean score for the sum of the squared factor loadings specifies the proportion of variance explained by each factor. For example, assume four items of an instrument have produced Factor 1, factor loadings of Factor 1 are 0.86, 0.75, 0.66 and 0.58, respectively. If you square the factor loading of items, you will get the percentage of the variance of that item which is explained by Factor 1. In this example, the first principal component (PC) for item1, item2, item3 and item4 is 74%, 56%, 43% and 33%, respectively. If you sum the squared factor loadings of Factor 1, you will get the eigenvalue, which is 2.1 and dividing the eigenvalue by four (2.1/4= 0.52) we will get the proportion of variance accounted for Factor 1, which is 52 %. Since PCA does not separate specific variance and error variance, it often inflates factor loadings and limits the potential for the factor structure to be generalized and applied with other samples in subsequent study. On the other hand, Maximum likelihood and Principal Axis Factoring extraction methods separate common and unique variance (specific and error variance), which overcomes the issue attached to PCA.  Thus, the proportion of variance explained by an extracted factor more precisely reflects the extent to which the latent construct is measured by the instrument items. This focus on shared variance among items explained by the underlying factor, particularly during instrument development, helps the researcher understand the extent to which a measure captures the intended construct. It is useful to mention that in PAF, the initial communalities are not set at 1s, but they are chosen based on the squared multiple correlation coefficient. Indeed, if you run a multiple regression to predict say  item1 (dependent variable)  from other items (independent variables) and then look at the R-squared (R2), you will see R2 is equal to the communalities of item1 derived from PAF.

Confirmatory Factor Analysis

When prior EFA studies are available for your intended instrument, Confirmatory Factor Analysis extends on those findings, allowing you to confirm or disconfirm the underlying factor structures, or dimensions, extracted in prior research. CFA is a theory or model-driven approach that tests how well the data “fit” to the proposed model or theory. CFA thus departs from EFA in that researchers must first identify a factor model before analysing the data. More fundamentally, CFA is a means for statistically testing the internal structure of instruments and relies on the maximum likelihood estimation (MLE) and a different set of standards for assessing the suitability of the construct of interest. 7 , 8

Factor analysts usually use the path diagram to show the theoretical and hypothesized relationships between items and the factors to create a hypothetical model to test using the ML method. In the path diagram, circles or ovals represent factors. A rectangle represents the instrument items. Lines (→ or ↔) represent relationships between items. No line, no relationship. A single-headed arrow shows the causal relationship (the variable that the arrowhead refers to is the dependent variable), and a double-headed shows a covariance between variables or factors.

If CFA indicates the primary factors, or first-order factors, produced by the prior PAF are correlated, then the second-order factors need to be modelled and estimated to get a greater understanding of the data. It should be noted if the prior EFA applied an orthogonal rotation to the factor solution, the factors produced would be uncorrelated. Hence, the analysis of the second-order factors is not possible. Generally, in social science research, most constructs assume inter-related factors, and therefore should apply an oblique rotation. The justification for analyzing the second-order factors is that when the correlations between the primary factors exist, CFA can then statistically model a broad picture of factors not captured by the primary factors (i.e., the first-order factors). 9   The analysis of the first-order factors is like surveying mountains with a zoom lens binoculars, while the analysis of the second-order factors uses a wide-angle lens. 10 Goodness of- fit- tests need to be conducted when evaluating the hypothetical model tested by CFA. The question is: does the new data fit the hypothetical model? However, the statistical models of the goodness of- fit- tests are complex, and extend beyond the scope of this editorial paper; thus,we strongly encourage the readers consult with factors analysts to receive resources and possible advise.

Conclusions

Factor analysis methods can be incredibly useful tools for researchers attempting to establish high quality measures of those constructs not directly observed and captured by observation. Specifically, the factor solution derived from an Exploratory Factor Analysis provides a snapshot of the statistical relationships of the key behaviors, attitudes, and dispositions of the construct of interest. This snapshot provides critical evidence for the validity of the measure based on the fit of the test content to the theoretical framework that underlies the construct. Further, the relationships between factors, which can be explored with EFA and confirmed with CFA, help researchers interpret the theoretical connections between underlying dimensions of a construct and even extending to relationships across constructs in a broader theoretical model. However, studies that do not apply recommended extraction, rotation, and interpretation in FA risk drawing faulty conclusions about the validity of a measure. As measures are picked up by other researchers and applied in experimental designs, or by practitioners as assessments in practice, application of measures with subpar evidence for validity produces a ripple effect across the field. It is incumbent on researchers to ensure best practices are applied or engage with methodologists to support and consult where there are gaps in knowledge of methods. Further, it remains important to also critically evaluate measures selected for research and practice, focusing on those that demonstrate alignment with best practice for FA and instrument development. 7 , 11

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Factor analysis and how it simplifies research findings.

17 min read There are many forms of data analysis used to report on and study survey data. Factor analysis is best when used to simplify complex data sets with many variables.

What is factor analysis?

Factor analysis is the practice of condensing many variables into just a few, so that your research data is easier to work with.

For example, a retail business trying to understand customer buying behaviours might consider variables such as ‘did the product meet your expectations?’, ‘how would you rate the value for money?’ and ‘did you find the product easily?’. Factor analysis can help condense these variables into a single factor, such as ‘customer purchase satisfaction’.

The theory is that there are deeper factors driving the underlying concepts in your data, and that you can uncover and work with them instead of dealing with the lower-level variables that cascade from them. Know that these deeper concepts aren’t necessarily immediately obvious – they might represent traits or tendencies that are hard to measure, such as extraversion or IQ.

Factor analysis is also sometimes called “dimension reduction”: you can reduce the “dimensions” of your data into one or more “super-variables,” also known as unobserved variables or latent variables. This process involves creating a factor model and often yields a factor matrix that organizes the relationship between observed variables and the factors they’re associated with.

As with any kind of process that simplifies complexity, there is a trade-off between the accuracy of the data and how easy it is to work with. With factor analysis, the best solution is the one that yields a simplification that represents the true nature of your data, with minimum loss of precision. This often means finding a balance between achieving the variance explained by the model and using fewer factors to keep the model simple.

Factor analysis isn’t a single technique, but a family of statistical methods that can be used to identify the latent factors driving observable variables. Factor analysis is commonly used in market research , as well as other disciplines like technology, medicine, sociology, field biology, education, psychology and many more.

What is a factor?

In the context of factor analysis, a factor is a hidden or underlying variable that we infer from a set of directly measurable variables.

Take ‘customer purchase satisfaction’ as an example again. This isn’t a variable you can directly ask a customer to rate, but it can be determined from the responses to correlated questions like ‘did the product meet your expectations?’, ‘how would you rate the value for money?’ and ‘did you find the product easily?’.

While not directly observable, factors are essential for providing a clearer, more streamlined understanding of data. They enable us to capture the essence of our data’s complexity, making it simpler and more manageable to work with, and without losing lots of information.

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Key concepts in factor analysis

These concepts are the foundational pillars that guide the application and interpretation of factor analysis.

Central to factor analysis, variance measures how much numerical values differ from the average. In factor analysis, you’re essentially trying to understand how underlying factors influence this variance among your variables. Some factors will explain more variance than others, meaning they more accurately represent the variables they consist of.

The eigenvalue expresses the amount of variance a factor explains. If a factor solution (unobserved or latent variables) has an eigenvalue of 1 or above, it indicates that a factor explains more variance than a single observed variable, which can be useful in reducing the number of variables in your analysis. Factors with eigenvalues less than 1 account for less variability than a single variable and are generally not included in the analysis.

Factor score

A factor score is a numeric representation that tells us how strongly each variable from the original data is related to a specific factor. Also called the component score, it can help determine which variables are most influenced by each factor and are most important for each underlying concept.

Factor loading

Factor loading is the correlation coefficient for the variable and factor. Like the factor score, factor loadings give an indication of how much of the variance in an observed variable can be explained by the factor. High factor loadings (close to 1 or -1) mean the factor strongly influences the variable.

When to use factor analysis

Factor analysis is a powerful tool when you want to simplify complex data, find hidden patterns, and set the stage for deeper, more focused analysis.

It’s typically used when you’re dealing with a large number of interconnected variables, and you want to understand the underlying structure or patterns within this data. It’s particularly useful when you suspect that these observed variables could be influenced by some hidden factors.

For example, consider a business that has collected extensive customer feedback through surveys. The survey covers a wide range of questions about product quality, pricing, customer service and more. This huge volume of data can be overwhelming, and this is where factor analysis comes in. It can help condense these numerous variables into a few meaningful factors, such as ‘product satisfaction’, ‘customer service experience’ and ‘value for money’.

Factor analysis doesn’t operate in isolation – it’s often used as a stepping stone for further analysis. For example, once you’ve identified key factors through factor analysis, you might then proceed to a cluster analysis – a method that groups your customers based on their responses to these factors. The result is a clearer understanding of different customer segments, which can then guide targeted marketing and product development strategies.

By combining factor analysis with other methodologies, you can not only make sense of your data but also gain valuable insights to drive your business decisions.

Factor analysis assumptions

Factor analysis relies on several assumptions for accurate results. Violating these assumptions may lead to factors that are hard to interpret or misleading.

Linear relationships between variables

This ensures that changes in the values of your variables are consistent.

Sufficient variables for each factor

Because if only a few variables represent a factor, it might not be identified accurately.

Adequate sample size

The larger the ratio of cases (respondents, for instance) to variables, the more reliable the analysis.

No perfect multicollinearity and singularity

No variable is a perfect linear combination of other variables, and no variable is a duplicate of another.

Relevance of the variables

There should be some correlation between variables to make a factor analysis feasible.

Types of factor analysis

There are two main factor analysis methods: exploratory and confirmatory. Here’s how they are used to add value to your research process.

Confirmatory factor analysis

In this type of analysis, the researcher starts out with a hypothesis about their data that they are looking to prove or disprove. Factor analysis will confirm – or not – where the latent variables are and how much variance they account for.

Principal component analysis (PCA) is a popular form of confirmatory factor analysis. Using this method, the researcher will run the analysis to obtain multiple possible solutions that split their data among a number of factors. Items that load onto a single particular factor are more strongly related to one another and can be grouped together by the researcher using their conceptual knowledge or pre-existing research.

Using PCA will generate a range of solutions with different numbers of factors, from simplified 1-factor solutions to higher levels of complexity. However, the fewer number of factors employed, the less variance will be accounted for in the solution.

Exploratory factor analysis

As the name suggests, exploratory factor analysis is undertaken without a hypothesis in mind. It’s an investigatory process that helps researchers understand whether associations exist between the initial variables, and if so, where they lie and how they are grouped.

How to perform factor analysis: A step-by-step guide

Performing a factor analysis involves a series of steps, often facilitated by statistical software packages like SPSS, Stata and the R programming language . Here’s a simplified overview of the process.

Prepare your data

Start with a dataset where each row represents a case (for example, a survey respondent), and each column is a variable you’re interested in. Ensure your data meets the assumptions necessary for factor analysis.

Create an initial hypothesis

If you have a theory about the underlying factors and their relationships with your variables, make a note of this. This hypothesis can guide your analysis, but keep in mind that the beauty of factor analysis is its ability to uncover unexpected relationships.

Choose the type of factor analysis

The most common type is exploratory factor analysis, which is used when you’re not sure what to expect. If you have a specific hypothesis about the factors, you might use confirmatory factor analysis.

Form your correlation matrix

After you’ve chosen the type of factor analysis, you’ll need to create the correlation matrix of your variables. This matrix, which shows the correlation coefficients between each pair of variables, forms the basis for the extraction of factors. This is a key step in building your factor analysis model.

Decide on the extraction method

Principal component analysis is the most commonly used extraction method. If you believe your factors are correlated, you might opt for principal axis factoring, a type of factor analysis that identifies factors based on shared variance.

Determine the number of factors

Various criteria can be used here, such as Kaiser’s criterion (eigenvalues greater than 1), the scree plot method or parallel analysis. The choice depends on your data and your goals.

Interpret and validate your results

Each factor will be associated with a set of your original variables, so label each factor based on how you interpret these associations. These labels should represent the underlying concept that ties the associated variables together.

Validation can be done through a variety of methods, like splitting your data in half and checking if both halves produce the same factors.

How factor analysis can help you

As well as giving you fewer variables to navigate, factor analysis can help you understand grouping and clustering in your input variables, since they’ll be grouped according to the latent variables.

Say you ask several questions all designed to explore different, but closely related, aspects of customer satisfaction:

• How satisfied are you with our product?
• Would you recommend our product to a friend or family member?
• How likely are you to purchase our product in the future?

But you only want one variable to represent a customer satisfaction score. One option would be to average the three question responses. Another option would be to create a factor dependent variable. This can be done by running a principal component analysis (PCA) and keeping the first principal component (also known as a factor). The advantage of a PCA over an average is that it automatically weights each of the variables in the calculation.

Say you have a list of questions and you don’t know exactly which responses will move together and which will move differently; for example, purchase barriers of potential customers. The following are possible barriers to purchase:

• Price is prohibitive
• Overall implementation costs
• We can’t reach a consensus in our organization
• Product is not consistent with our business strategy
• I need to develop an ROI, but cannot or have not
• We are locked into a contract with another product
• The product benefits don’t outweigh the cost
• We have no reason to switch
• Our IT department cannot support your product
• We do not have sufficient technical resources
• Your product does not have a feature we require
• Other (please specify)

Factor analysis can uncover the trends of how these questions will move together. The following are loadings for 3 factors for each of the variables.

Notice how each of the principal components have high weights for a subset of the variables. Weight is used interchangeably with loading, and high weight indicates the variables that are most influential for each principal component. +0.30 is generally considered to be a heavy weight.

The first component displays heavy weights for variables related to cost, the second weights variables related to IT, and the third weights variables related to organizational factors. We can give our new super variables clever names.

If we were to cluster the customers based on these three components, we can see some trends. Customers tend to be high in cost barriers or organizational barriers, but not both.

The red dots represent respondents who indicated they had higher organizational barriers; the green dots represent respondents who indicated they had higher cost barriers

Considerations when using factor analysis

Factor analysis is a tool, and like any tool its effectiveness depends on how you use it. When employing factor analysis, it’s essential to keep a few key considerations in mind.

Oversimplification

While factor analysis is great for simplifying complex data sets, there’s a risk of oversimplification when grouping variables into factors. To avoid this you should ensure the reduced factors still accurately represent the complexities of your variables.

Subjectivity

Interpreting the factors can sometimes be subjective, and requires a good understanding of the variables and the context. Be mindful that multiple analysts may come up with different names for the same factor.

Supplementary techniques

Factor analysis is often just the first step. Consider how it fits into your broader research strategy and which other techniques you’ll use alongside it.

Examples of factor analysis studies

Factor analysis, including PCA, is often used in tandem with segmentation studies. It might be an intermediary step to reduce variables before using KMeans to make the segments.

Factor analysis provides simplicity after reducing variables. For long studies with large blocks of Matrix Likert scale questions, the number of variables can become unwieldy. Simplifying the data using factor analysis helps analysts focus and clarify the results, while also reducing the number of dimensions they’re clustering on.

Sample questions for factor analysis

Choosing exactly which questions to perform factor analysis on is both an art and a science. Choosing which variables to reduce takes some experimentation, patience and creativity. Factor analysis works well on Likert scale questions and Sum to 100 question types.

Factor analysis works well on matrix blocks of the following question genres:

Psychographics (Agree/Disagree):

• I value family
• I believe brand represents value

Behavioral (Agree/Disagree):

• I purchase the cheapest option
• I am a bargain shopper

Attitudinal (Agree/Disagree):

• The economy is not improving
• I am pleased with the product

Activity-Based (Agree/Disagree):

• I love sports
• I sometimes shop online during work hours

Behavioral and psychographic questions are especially suited for factor analysis.

Sample output reports

Factor analysis simply produces weights (called loadings) for each respondent. These loadings can be used like other responses in the survey.

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Institute for Digital Research and Education

A Practical Introduction to Factor Analysis

Factor analysis is a method for modeling observed variables and their covariance structure in terms of unobserved variables (i.e., factors). There are two types of factor analyses, exploratory and confirmatory. Exploratory factor analysis (EFA) is method to explore the underlying structure of a set of observed variables, and is a crucial step in the scale development process. The first step in EFA is factor extraction. During this seminar, we will discuss how principal components analysis and common factor analysis differ in their approach to variance partitioning. Common factor analysis models can be estimated using various estimation methods such as principal axis factoring and maximum likelihood, and we will compare the practical differences between these two methods. After extracting the best factor structure, we can obtain a more interpretable factor solution through factor rotation. Here is where we will discuss the difference between orthogonal and oblique rotations, and finally how to the final solution to generate a factor score. For the latter portion of the seminar we will introduce confirmatory factor analysis (CFA), which is a method to verify a factor structure that has already been defined. Topics to discuss include identification, model fit, and degrees of freedom demonstrated through a three-item, two-item and eight-item one factor CFA and a two-factor CFA. SPSS will be used for the EFA portion of the seminar and R (lavaan) will be used for the CFA portion. You can click on the main link to access each portion of the seminar.

I. Exploratory Factor Analysis (EFA)

• Motivating example: The SAQ
• Pearson correlation formula
• Partitioning the variance in factor analysis
• principal components analysis
• principal axis factoring
• maximum likelihood
• Simple Structure
• Orthogonal rotation (Varimax)
• Oblique (Direct Oblimin)
• Generating factor scores

II. Confirmatory Factor Analysis (CFA)

• Motivating example SPSS Anxiety Questionairre
• The factor analysis model
• The model-implied covariance matrix
• The path diagram
• Known values, parameters, and degrees of freedom
• Three-item (one) factor analysis
• Identification of a three-item one factor CFA
• Running a one-factor CFA in lavaan
• (Optional) How to manually obtain the standardized solution
• (Optional) Degrees of freedom with means
• One factor CFA with two items
• One factor CFA with more than three items (SAQ-8)
• Model chi-square
• A note on sample size
• (Optional) Model test of the baseline or null model
• Incremental versus absolute fit index
• CFI (Confirmatory Factor Index)
• TLI (Tucker Lewis Index)
• Uncorrelated factors
• Correlated factors
• Second-Order CFA
• (Optional) Warning message with second-order CFA
• (Optional) Obtaining the parameter table

DiStefano, C., Zhu, M., & Mindrila, D. (2009). Understanding and using factor scores: Considerations for the applied researcher. Practical Assessment, Research & Evaluation , 14 (20), 2.

Field, A. (2009). Discovering statistics using SPSS . Sage publications.

Geiser, C. (2012). Data analysis with Mplus . Guilford Press.

Lawley, D. N., Maxwell, A. E. (1971). Factor Analysis as a Statistical Method.  Second Edition. American Elsevier Publishing Company, Inc. New York.

Pett, M. A., Lackey, N. R., & Sullivan, J. J. (2003). Making sense of factor analysis: The use of factor analysis for instrument development in health care research . Sage.

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Understanding Factor Analysis in Psychology

John Loeppky is a freelance journalist based in Regina, Saskatchewan, Canada, who has written about disability and health for outlets of all kinds.

Steven Gans, MD is board-certified in psychiatry and is an active supervisor, teacher, and mentor at Massachusetts General Hospital.

Skynesher / Getty Images

What Is Factor Analysis and What Does It Do?

Types of factor analysis, advantages and disadvantages of factor analysis, how is factor analysis used in psychology.

Like many methods encountered by those studying psychology , factor analysis has a long history.

The primary goal of factor analysis is to distill a large data set into a working set of connections or factors.

It was originally discussed by British psychologist Charles Spearman in the early 20th century and has gone on to be used in not only psychology but in other fields that often rely on statistical analyses,

But what is it, what are some real-world examples, and what are the different types? In this article, we'll answer all of those questions.

The primary goal of factor analysis is to distill a large data set into a working set of connections or factors. Dr. Jessie Borelli, PhD , who works at the University of California-Irvine, uses factor analysis in her work on attachment.

She is doing research that looks into how people perceive relationships and how they connect to one another. She gives the example of providing a hypothetical questionnaire with 100 items on it and using factor analysis to drill deeper into the data. "So, rather than looking at each individual item on its own I'd rather say, 'Is there is there any way in which these items kind of cluster together or go together so that I can... create units of analysis that are bigger than the individual items."

Factor analysis is looking to identify patterns where it is assumed that there are already connections between areas of the data.

An Example Where Factor Analysis Is Useful

One common example of a factor analysis is when you are taking something not easily quantifiable, like socio-economic status , and using it to group together highly correlated variables like income level and types of jobs.

Factor analysis isn't just used in psychology but also deployed in fields like sociology, business, and technology sector fields like machine learning.

There are two types of factor analysis that are most commonly referred to: exploratory factor analysis and confirmatory factor analysis.

Here are the two types of factor analysis:

• Exploratory analysis : The goal of this analysis is to find general patterns in a set of data points.
• Confirmatory factor analysis : The goal of this analysis is to test various hypothesized relationships among certain variables.

Exploratory Analysis

In an exploratory analysis, you are being a little bit more open-minded as a researcher because you are using this type of analysis to provide some clarity in your data set that you haven't yet found. It's an approach that Borelli uses in her own research.

Confirmatory Factor Analysis

On the other hand, if you're using a confirmatory factor analysis you are using the assumptions or theoretical findings you have already identified to drive your statistical model.

Unlike in an exploratory factor analysis, where the relationships between factors and variables are more open, a confirmatory factor analysis requires you to select which variables you are testing for. In Borelli's words:

"When you do a confirmatory factor analysis, you kind of tell your analytic program what you think the data should look like, in terms of, 'I think it should have these two factors and this is the way I think it should look.'"

Let's take a look at the advantages and disadvantages of factor analysis.

A main advantage of a factor analysis is that it allows researchers to reduce a number of variables by combining them into a single factor.

You Can Analyze Fewer Data Points

When answering your research questions, it's a lot easier to be working with three variables than thirty, for example.

Disadvantages

Disadvantages include that the factor analysis relies on the quality of the data, and also may allow for different interpretations of the data. For example, during one study, Borelli found that after deploying a factor analysis, she was still left with results that didn't connect well with what had been found in hundreds of other studies .

Due to the nature of the sample being new and being more culturally diverse than others being explored, she used an exploratory factor analysis that left her with more questions than answers.

The goal of factor analysis in psychology is often to make connections that allow researchers to develop models with common factors in ways that might be hard or impossible to observe otherwise.

So, for example, intelligence is a difficult concept to directly observe. However, it can be inferred from factors that we can directly measure on specific tests.

Factor analysis has often been used in the field of psychology to help us better understand the structure of personality.

This is due to the multitude of factors researchers have to consider when it comes to understanding the concept of personality. This area of personality research is certainly not new, with easily findable research dating as far back as 1942 recognizing its power in personality research.

Britannica. Charles E. Spearman .

United State Environmental Protection Agency. Exploratory Data Analysis .

Flora DB, Curran PJ. An empirical evaluation of alternative methods of estimation for confirmatory factor analysis with ordinal data .  Psychol Methods . 2004;9(4):466-491. doi:10.1037/1082-989X.9.4.466

Wolfle D. Factor analysis in the study of personality .  The Journal of Abnormal and Social Psychology. 1942;37(3):393–397.

By John Loeppky John Loeppky is a freelance journalist based in Regina, Saskatchewan, Canada, who has written about disability and health for outlets of all kinds.

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Intoduction to Factor Analysis

In this tutorial, I’ll introduce to you the multivariate statistical method , factor analysis (FA), used for uncovering the underlying relationships between a set of observed variables. It is a widely utilized technique in social and behavioral science for identifying factors such as intelligence, socio-economic status, or personality traits.

The table of contents is structured as follows:

Let’s dive into it!

What is Factor Analysis?

Factor analysis (FA) is a technique used to identify the underlying structure of the data in terms of a smaller set of *unobserved factors ( latents ). These factors are linear combinations of the *observed variables and help to explain the correlations among them.

This linear relation can be formalized for the observed variable x as x = λx * ξ + δx , where λx refers to factor loadings associated with x , ξ refers to underlying factors, and δx refers to error associated with x . This error is the portion of the variable that can’t be explained by the underlying factors.

The factor analysis model can be visualized in a diagram, where the direction of the causal effect is indicated by arrows. See the figure below.

As with all statistical techniques, it relies on some statistical assumptions . Let’s take a look at them!

Assumptions of FA

It is important that your data meets the following assumptions to obtain reliable results in FA.

• Sufficient Sample Size : Generally, a larger sample size yields more reliable results.
• Linearity: Assumes that the relationship between observed variables and factors is linear.
• Adequacy of Correlations: There should be some correlations between the observed variables, or FA will be ineffective.

Let’s see next what the main FA types and their uses are!

Main Types of FA

In the realm of FA, two prominent approaches stand out: Exploratory Factor Analysis (EFA) and Confirmatory Factor Analysis (CFA).

• Exploratory Factor Analysis (EFA) : EFA is used when the researcher does not have a specific idea of the underlying structure of data. It is employed to explore the possible underlying factor structure without imposing any preconceived structure on the outcome.
• Confirmatory Factor Analysis (CFA) : CFA, on the other hand, is used when the researcher has a specific idea of the underlying structure based on theory or previous studies. In CFA, the researcher tests a hypothesized model to see how well it fits the data.

The choice between these two methods should be guided by the specific needs of your study or the research question, allowing for either an exploration of underlying patterns or a focused test of predefined hypotheses.

So far, we have had an overview of the FA concept. Next, we will explore the steps of conducting an FA.

Steps of FA

• Gather and Prepare Your Data: Collect your dataset and ensure it is cleaned and preprocessed.
• Choose the Number of Factors: Use criteria to decide how many factors to retain, e.g., scree plot .
• Estimate the Factor Model: Use software (like R , SPSS , or Python ‘s scikit-learn) to perform FA. The software will estimate the factor loadings (associations between variables and factors), factor variances (variable variance explained by the factors), and unique variances (variable variance not explained by the factors).
• Interpret the Factors: In EFA, examine the factor loadings to interpret what each factor represents; in CFA, check if the variables are associated with the factors as hypothesized. Generally, high loadings (positive or negative) of a variable on a factor indicate strong associations.
• Rotate the Factors (if necessary, in EFA): Rotate results to make the interpretation of factors clearer by simplifying the structure of loadings.
• Confirm the Solution (if necessary, in CFA): Confirm whether the data fits the model well using various fit indices .
• Use the Factors: The factors can be used in further analyses as summary variables or to test hypotheses about the relationships between the factors and other variables.
• Validate Your Model: It’s essential to validate your factor structure on a different sample to ensure that it is generalizable.

Examples of FA

To learn some more specific information about EFA and CFA, you are welcome to visit our tutorials, Exploratory Factor Analysis and Confirmatory Factor Analysis . However, FA is not limited to these two. There are various specialized or extended types of FA, e.g., Principal Component Analysis , Canonical Factor Analysis .

In this tutorial, we didn’t get into mathematical details, I suggest you check Brown’s book Confirmatory Factor Analysis for Applied Research (2006) for a deeper understanding.

*Observed variables can also be referred to as manifest variables, indicators, and endogenous variables, whereas latent variables can be referred to as factors, constructs, unobserved/underlying variables, and exogenous variables in the context of FA.

Video, Further Resources & Summary

This tutorial is introductory and simplifies many complex aspects of FA. For deeper study, it is recommended to consult more analysis-specific tutorials.

Do you need more explanations on what the factor analysis is? Then you might check out the following video of the Statistics Globe YouTube channel.

In the video tutorial, we introduce factor analysis as a general concept.

The YouTube video will be added soon.

Furthermore, you could have a look at some of the other tutorials on Statistics Globe:

• What is a Principal Component Analysis (PCA)?
• Scree Plot for PCA Explained
• Exploratory Factor Analysis
• Confirmatory Factor Analysis

This article has explained the factor analysis concept. If you have further questions, you may leave a comment below.

This page was created in collaboration with Cansu Kebabci. You might have a look at Cansu’s author page to get more information about academic background and the other articles she has written for Statistics Globe.

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I’m Joachim Schork. On this website, I provide statistics tutorials as well as code in Python and R programming.

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3 Types of Factor Analysis

Factor analysis is a statistical technique used to identify the underlying structure of a set of variables.

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Market research and analysis of large volumes of data are necessary when it comes to analyzing and determining the right market segment, potential demand, and potential areas of competition, product development requirements and all other facets of the business marketing portfolio. One of the most common tools used to deal with the vast amounts of data is Factor Analysis.

What is Factor Analysis?

Factor analysi s is a statistical technique used to identify the underlying structure of a set of variables. In layman’s terms, it is used to analyze the relationship between two observable variables and how it is affected by another smaller set of unobservable variables. For example, factor analysis can be used in market segmentation to identify the underlying variables according to which customers can be grouped.

Uses of factor analysis in market research and analysis

Factor analysis has proved to be very beneficial in market research and analysis of variables that determine consumer behavior:

• It helps to make sense of large data with interlinked relationships
• It may point out relationships that may not have been obvious
• It can point out to the underlying relationships with respect to consumer tastes, preferences, etc.
• It is easier to condense and correlate data through factor analysis and also to draw conclusions from the data gathered in market research and analysis.
• It can be used to form empirical clusters of variables and underlying factors that affect them

Types of factor analysis

A factor analysis is mainly used for interpretation of data and in analyzing the underlying relationships between variable and other underlying factors that may determine consumer behavior. Instead of grouping responses and response types, factor analysis segregates the variable and groups these according to their co relevance.There are mainly three types of factor analysis that are used for different kinds of market research and analysis.

• Exploratory factor analysis
• Confirmatory factor analysis
• Structural equation modeling

Exploratory factor analysis is used to measure the underlying factors that affect the variables in a data structure without setting any predefined structure to the outcome. Confirmatory factor analysis on the other hand is used as tool in market research and analysis to reconfirm the effects and correlation of an existing set of predetermined factors and variables that affect these factors. Structural equation modeling hypothesizes a relationship between a set of variables and factors and tests these casual relationships on the linear equation model.  Structural equation modeling can be used for exploratory and confirmatory modeling alike, and hence it can be used for confirming results as well as testing hypotheses.

Factor analysis will only yield accurate and useful results if done by a researcher who has adequate knowledge to select data and assign attributes. Selecting factors and variables so as to avoid too much similarity of characteristics is also important. Factor analysis, if done correctly, can allow for market research and analysis that helps in various areas of decision making like product features, product development, pricing, market segmentation, penetration and even with targeting.

Applications of Factor Analysis

Factor analysis has several applications in different fields, including:

• Market Research: it is widely used in market research to identify the underlying factors that influence customer preferences and behavior. For example, a market research study may use factor analysis to identify the key factors that influence consumers' purchasing decisions.
• Psychology: it is used in psychology to identify the underlying dimensions of personality traits, intelligence, and cognitive abilities. For example, a factor analysis of a personality test may identify factors such as extraversion, neuroticism, and openness to experience.
• Education: it is used in education to identify the underlying factors that contribute to academic achievement. For example, a factor analysis of academic test scores may identify factors such as verbal reasoning, mathematical ability, and spatial ability.

Factor analysis is a powerful statistical technique that can help in identifying the underlying factors that contribute to the patterns of the data. By reducing large amounts of data into a smaller number of factors, it can simplify the data and help in making meaningful conclusions. Whether you're conducting market research, psychological studies, or educational research, factor analysis can be a valuable tool for understanding the relationships between variables and identifying the underlying factors that explain the patterns in the data.

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• Factor Analysis

Factor analysis is a technique in mathematics that we use to reduce a larger number into a smaller number. Moreover, in this topic, we will talk about it and its various aspects.

What is Factor Analysis?

It refers to a method that reduces a large variable into a smaller variable factor. Furthermore, this technique takes out maximum ordinary variance from all the variables and put them in common score.

Moreover, it is a part of General Linear Model (GLM) and it believes several theories that contain no multicollinearity, linear relationship, true correlation , and relevant variables into the analysis among factors and variables.

Types of Factor Analysis

There are different methods that we use in factor analysis from the data set:

1. Principal component analysis

It is the most common method which the researchers use. Also, it extracts the maximum variance and put them into the first factor. Subsequently, it removes the variance explained by the first factor and extracts the second factor. Moreover, it goes on until the last factor.

2. Common Factor Analysis

It’s the second most favoured technique by researchers. Also, it extracts common variance and put them into factors . Furthermore, this technique doesn’t include the variance of all variables and is used in SEM.

3. Image Factoring

It is on the basis of the correlation matrix and makes use of OLS regression technique in order to predict the factor in image factoring.

4. Maximum likelihood method

It also works on the correlation matrix but uses a maximum likelihood method to factor.

5. Other methods of factor analysis

Alfa factoring outweighs least squares. Weight square is another regression-based method that we use for factoring.

Factor loading- Basically it the correlation coefficient for the factors and variables. Also, it explains the variable on a particular factor shown by variance.

Eigenvalues- Characteristics roots are its other name. Moreover, it explains the variance shown by that particular factor out of the total variance. Furthermore, commonality column helps to know how much variance the first factor explained out of total variance.

Factor Score- It’s another name is the component score. Besides, it’s the score of all rows and columns that we can use as an index for all variables and for further analysis. Moreover, we can standardize it by multiplying it with a common term.

Rotation method- This method makes it more reliable to understand the output. Also, it affects the eigenvalues method but the eigenvalues method doesn’t affect it. Besides, there are 5 rotation methods: (1) No Rotation Method, (2) Varimax Rotation Method, (3) Quartimax Rotation Method, (4) Direct Oblimin Rotation Method, and (5) Promax Rotation Method.

Assumptions of Factor Analysis

Factor analysis has several assumptions. These include:

• There are no outliers in the data.
• The sample size is supposed to be greater than the factor.
• It is an interdependency method so there should be no perfect multicollinearity between the variables.
• Factor analysis is a linear function thus it doesn’t require homoscedasticity between variables.
• It is also based on the linearity assumption. So, we can also use non-linear variables. However, after a transfer, they change into a linear variable.
• Moreover, it assumes interval data.

Key Concepts of Factor Analysis

It includes the following key concept:

Exploratory factor analysis- It assumes that any variable or indicator can be associated with any factor. Moreover, it is the most common method used by researchers. Furthermore, it isn’t based on any prior theory.

Confirmatory Factor Analysis- It is used to determine the factors loading and factors of measured variables, and to confirm what it expects on the basis of pre-established assumption. Besides, it uses two approaches:

• The Traditional Method
• The SEM Approach

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Question. How many types of Factor analysis are there?

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Best practices for your confirmatory factor analysis: A JASP and lavaan tutorial

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Confirmatory factor analysis (CFA) is a fundamental method for evaluating the internal structural validity of measurement instruments. In most CFA applications, the measurement model serves as a means to an end rather than an end in itself. To select the appropriate model, prior validity evidence is crucial, and items are typically assessed on an ordinal scale, which has been used in the applied social sciences. However, textbooks on structural equation modeling (SEM) often overlook this specific case, focusing on applications estimable using maximum likelihood (ML) instead. Unfortunately, several popular commercial SEM software packages lack suitable solutions for handling this ‘typical CFA’, leading to confusion and suboptimal decision-making when conducting CFA in this context. This article conceptually contributes to this ongoing discussion by presenting a set of guidelines for conducting a typical CFA, drawing from recent empirical research. We provide a practical contribution by introducing and developing a tutorial example within the JASP and lavaan  software platforms. Supplementary materials such as videos, files, and scripts are freely available.

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Introduction to Factor Analysis

• By: Jae-On Kim & Charles W. Mueller
• Publisher: SAGE Publications, Inc.
• Series: Quantitative Applications in the Social Sciences
• Publication year: 1978
• Online pub date: January 01, 2011
• Discipline: Anthropology
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Describes the mathematical and logical foundations at a level which does not presume advanced mathematical or statistical skills, illustrating how to do factor analysis with several of the more popular packaged computer programmes.

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• Obtaining Factor Analysis Solutions

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Computer Science > Computer Vision and Pattern Recognition

Title: mm1: methods, analysis & insights from multimodal llm pre-training.

Abstract: In this work, we discuss building performant Multimodal Large Language Models (MLLMs). In particular, we study the importance of various architecture components and data choices. Through careful and comprehensive ablations of the image encoder, the vision language connector, and various pre-training data choices, we identified several crucial design lessons. For example, we demonstrate that for large-scale multimodal pre-training using a careful mix of image-caption, interleaved image-text, and text-only data is crucial for achieving state-of-the-art (SOTA) few-shot results across multiple benchmarks, compared to other published pre-training results. Further, we show that the image encoder together with image resolution and the image token count has substantial impact, while the vision-language connector design is of comparatively negligible importance. By scaling up the presented recipe, we build MM1, a family of multimodal models up to 30B parameters, including both dense models and mixture-of-experts (MoE) variants, that are SOTA in pre-training metrics and achieve competitive performance after supervised fine-tuning on a range of established multimodal benchmarks. Thanks to large-scale pre-training, MM1 enjoys appealing properties such as enhanced in-context learning, and multi-image reasoning, enabling few-shot chain-of-thought prompting.

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