R version 4.5.1 (2025-06-13)
Platform: x86_64-pc-linux-gnu
Running under: Ubuntu 24.04.2 LTS
Matrix products: default
BLAS: /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so; LAPACK version 3.12.0
locale:
[1] LC_CTYPE=C.UTF-8 LC_NUMERIC=C LC_TIME=C.UTF-8
[4] LC_COLLATE=C.UTF-8 LC_MONETARY=C.UTF-8 LC_MESSAGES=C.UTF-8
[7] LC_PAPER=C.UTF-8 LC_NAME=C LC_ADDRESS=C
[10] LC_TELEPHONE=C LC_MEASUREMENT=C.UTF-8 LC_IDENTIFICATION=C
time zone: UTC
tzcode source: system (glibc)
attached base packages:
[1] stats graphics grDevices utils datasets methods base
loaded via a namespace (and not attached):
[1] htmlwidgets_1.6.4 compiler_4.5.1 fastmap_1.2.0 cli_3.6.5
[5] tools_4.5.1 htmltools_0.5.8.1 rmarkdown_2.29 knitr_1.50
[9] jsonlite_2.0.0 xfun_0.52 digest_0.6.37 rlang_1.1.6
[13] evaluate_1.0.4
I need your help!
I want your feedback to make the book better for you and other readers. If you find typos, errors, or places where the text may be improved, please let me know. The best ways to provide feedback are by GitHub or hypothes.is annotations.
You can leave a comment at the bottom of the page/chapter, or open an issue or submit a pull request on GitHub: https://github.com/isaactpetersen/Fantasy-Football-Analytics-Textbook
Alternatively, you can leave an annotation using hypothes.is.
To add an annotation, select some text and then click the
symbol on the pop-up menu.
To see the annotations of others, click the
symbol in the upper right-hand corner of the page.
23 Factor Analysis
23.1 Getting Started
23.1.1 Load Packages
23.2 Overview of Factor Analysis
Factor analysis involves the estimation of latent variables. Latent variables are ways of studying and operationalizing theoretical constructs that cannot be directly observed or quantified. Factor analysis is a class of latent variable models that is designated to identify the structure of a measure or set of measures, and ideally, a construct or set of constructs. It aims to identify the optimal latent structure for a group of variables. The goal of factor analysis is to identify simple, parsimonious factors that underlie the “junk” (i.e., scores filled with measurement error) that we observe.
Factor analysis encompasses two general types: confirmatory factor analysis and exploratory factor analysis. Exploratory factor analysis (EFA) is a latent variable modeling approach that is used when the researcher has no a priori hypotheses about how a set of variables is structured. EFA seeks to identify the empirically optimal-fitting model in ways that balance accuracy (i.e., variance accounted for) and parsimony (i.e., simplicity). Confirmatory factor analysis (CFA) is a latent variable modeling approach that is used when a researcher wants to evaluate how well a hypothesized model fits, and the model can be examined in comparison to alternative models. Using a CFA approach, the researcher can pit models representing two theoretical frameworks against each other to see which better accounts for the observed data.
Factor analysis involves observed (manifest) variables and unobserved (latent) factors. Factor analysis assumes that the latent factor influences the manifest variables, and the latent factor therefore reflects the common variance among the variables. A factor model potentially includes factor loadings, residuals (errors or disturbances), intercepts/means, covariances, and regression paths. When depicting a factor analysis model, rectangles represent variables we observe (i.e., manifest variables), and circles represent latent (i.e., unobserved) variables. A regression path indicates a hypothesis that one variable (or factor) influences another, and it is depicted using a single-headed arrow. The standardized regression coefficient (i.e., beta or \(\beta\)) represents the strength of association between the variables or factors. A factor loading is a regression path from a latent factor to an observed (manifest) variable. The standardized factor loading represents the strength of association between the variable and the latent factor. A residual is variance in a variable (or factor) that is unexplained by other variables or factors. A variable’s intercept is the expected value of the variable when the factor(s) (onto which it loads) is equal to zero. A covariance is the unstandardized index of the strength of association between between variables (or factors), and it is depicted with a double-headed arrow. Because a covariance is unstandardized, its scale depends on the scale of the variables. A covariance path between two variables represents omitted shared cause(s) of the variables. For instance, if you depict a covariance path between two variables, it means that there is a shared cause of the two variables that is omitted from the model (for instance, if the common cause is not known or was not assessed).
In factor analysis, the relation between an indicator (\(\text{X}\)) and its underlying latent factor(s) (\(\text{F}\)) can be represented with a regression formula as in Equation 23.1:
\[ X = \lambda \cdot \text{F} + \text{Item Intercept} + \text{Error Term} \tag{23.1}\]
where:
- \(\text{X}\) is the observed value of the indicator
- \(\lambda\) is the factor loading, indicating the strength of the association between the indicator and the latent factor(s)
- \(\text{F}\) is the person’s value on the latent factor(s)
- \(\text{Item Intercept}\) represents the constant term that accounts for the expected value of the indicator when the latent factor(s) are zero
- \(\text{Error Term}\) is the residual, indicating the extent of variance in the indicator that is not explained by the latent factor(s)
When the latent factors are uncorrelated, the (standardized) error term for an indicator is calculated as 1 minus the sum of squared standardized factor loadings for a given item (including cross-loadings). A cross-loadings is when a variable loads onto more than one latent factor.
Factor analysis is a powerful technique to help identify the factor structure that underlies a measure or construct. However, given the extensive method variance that influences scores on measure, factor analysis (and principal component analysis) tends to extract method factors. Method factors are factors that are related to the methods being assessed rather than the construct of interest. To better estimate construct factors, it is sometimes necessary to estimate both construct and method factors.
23.3 Exploratory Factor Analysis
Exploratory factor analysis (EFA) is used if you have no a priori hypotheses about the factor structure of the model, but you would like to understand the latent variables represented by your items.
EFA is partly induced from the data. You feed in the data and let the program build the factor model. You can set some parameters going in, including how to extract or rotate the factors. The factors are extracted from the data without specifying the number and pattern of loadings between the items and the latent factors (Bollen, 2002). All cross-loadings are freely estimated.
23.3.1 Factor Rotation
When using EFA or principal component analysis, an important step is, possibly, to rotate the factors to make them more interpretable and simple, which is the whole goal. To interpret the results of a factor analysis, we examine the factor matrix. The columns refer to the different factors; the rows refer to the different observed variables. The cells in the table are the factor loadings—they are basically the correlation between the variable and the factor. Our goal is to achieve a model with simple structure because it is easily interpretable. Simple structure means that every variable loads perfectly on one and only one factor, as operationalized by a matrix of factor loadings with values of one and zero and nothing else. An example of a factor matrix that follows simple structure is depicted in Figure 23.1.
An example of a factor analysis model that follows simple structure is depicted in Figure 23.2. Each variable loads onto one and only one factor, which makes it easy to interpret the meaning of each factor, because a given factor represents the common variance among the items that load onto it.
However, pure simple structure only occurs in simulations, not in real-life data. In reality, our unrotated factor analysis model might look like the model in Figure 23.3. In this example, the factor analysis model does not show simple structure because the items have cross-loadings—that is, the items load onto more than one factor. The cross-loadings make it difficult to interpret the factors, because all of the items load onto all of the factors, so the factors are not very distinct from each other, which makes it difficult to interpret what the factors mean.
As a result of the challenges of interpretability caused by cross-loadings, factor rotations are often performed. An example of an unrotated factor matrix is in Figure 23.4.
In the example factor matrix in Figure 23.5, the factor analysis is not very helpful—it tells us very little because it did not distinguish between the two factors. The variables have similar loadings on Factor 1 and Factor 2. An example of a unrotated factor solution is in Figure 23.5. In the figure, all of the variables are in the midst of the quadrants—they are not on the factors’ axes. Thus, the factors are not very informative.
As a result, to improve the interpretability of the factor analysis, we can do what is called rotation. Rotation leverages the idea that there are infinite solutions to the factor analysis model that fit equally well. Rotation involves changing the orientation of the factors by changing the axes so that variables end up with very high (close to one or negative one) or very low (close to zero) loadings, so that it is clear which factors include which variables. That is, rotation rescales the factors and tries to identify the ideal solution (factor) for each variable. It searches for simple structure and keeps searching until it finds a minimum. After rotation, if the rotation was successful for imposing simple structure, each factor will have loadings close to one (or negative one) for some variables and close to zero for other variables. The goal of factor rotation is to achieve simple structure, to help make it easier to interpret the meaning of the factors.
To perform factor rotation, orthogonal rotations are often used. Orthogonal rotations make the rotated factors uncorrelated. An example of a commonly used orthogonal rotation is varimax rotation. Varimax rotation maximizes the sum of the variance of the squared loadings (i.e., so that items have either a very high or very low loading on a factor) and yields axes with a 90-degree angle.
An example of a factor matrix following an orthogonal rotation is depicted in Figure 23.6. An example of a factor solution following an orthogonal rotation is depicted in Figure 23.7.
An example of a factor matrix from SPSS following an orthogonal rotation is depicted in Figure 23.8.
An example of a factor structure from an orthogonal rotation is in Figure 23.9.
Sometimes, however, the two factors and their constituent variables may be correlated. Examples of two correlated factors may be depression and anxiety. When the two factors are correlated in reality, if we make them uncorrelated, this would result in an inaccurate model. Oblique rotation allows for factors to be correlated and yields axes with less an angle of less than 90 degrees. However, if the factors have low correlation (e.g., .2 or less), you can likely continue with orthogonal rotation. Nevertheless, just because an oblique rotation allows for correlated factors does not mean that the factors will be correlated, so oblique rotation provides greater flexibility than orthogonal rotation. An example of a factor structure from an oblique rotation is in Figure 23.10. Results from an oblique rotation are more complicated than orthogonal rotation—they provide lots of output and are more complicated to interpret. In addition, oblique rotation might not yield a smooth answer if you have a relatively small sample size.
As an example of rotation based on interpretability, consider the Five-Factor Model of Personality (the Big Five), which goes by the acronym, OCEAN: Openness, Conscientiousness, Extraversion, Agreeableness, and Neuroticism. Although the five factors of personality are somewhat correlated, we can use rotation to ensure they are maximally independent. Upon rotation, extraversion and neuroticism are essentially uncorrelated, as depicted in Figure 23.11. The other pole of extraversion is intraversion and the other pole of neuroticism might be emotional stability or calmness.
Simple structure is achieved when each variable loads highly onto as few factors as possible (i.e., each item has only one significant or primary loading). Oftentimes this is not the case, so we choose our rotation method in order to decide if the factors can be correlated (an oblique rotation) or if the factors will be uncorrelated (an orthogonal rotation). If the factors are not correlated with each other, use an orthogonal rotation. The correlation between an item and a factor is a factor loading, which is simply a way to ask how much a variable is correlated with the underlying factor. However, its interpretation is more complicated if there are correlated factors!
An orthogonal rotation (e.g., varimax) can help with simplicity of interpretation because it seeks to yield simple structure without cross-loadings. Cross-loadings are instances where a variable loads onto multiple factors. My recommendation would always be to use an orthogonal rotation if you have reason to believe that finding simple structure in your data is possible; otherwise, the factors are extremely difficult to interpret—what exactly does a cross-loading even mean? However, you should always try an oblique rotation, too, to see how strongly the factors are correlated. Examples of oblique rotations include oblimin and promax.
23.4 Confirmatory Factor Analysis
Confirmatory factor analysis (CFA) is used to (dis)confirm a priori hypotheses about the factor structure of the model. CFA is a test of the hypothesis. In CFA, you specify the model and ask how well this model represents the data. The researcher specifies the number, meaning, associations, and pattern of free parameters in the factor loading matrix (Bollen, 2002). A key advantage of CFA is the ability to directly compare alternative models (i.e., factor structures), which is valuable for theory testing (Strauss & Smith, 2009). For instance, you could use CFA to test whether the variance in several measures’ scores is best explained with one factor or two factors. In CFA, cross-loadings are not estimated unless the researcher specifies them.
23.5 Determining the Number of Factors to Retain
A goal of factor analysis and principal component analysis is simplification or parsimony, while still explaining as much variance as possible. The hope is that you can have fewer factors that explain the associations between the variables than the number of observed variables. But how do you decide on the number of factors?
There are a number of criteria that one can use to help determine how many factors/components to keep:
- Kaiser-Guttman criterion: factors with eigenvalues greater than zero
- or, for principal component analysis, components with eigenvalues greater than 1
- Cattell’s scree test: the “elbow” in a scree plot minus one; sometimes operationalized with optimal coordinates (OC) or the acceleration factor (AF)
- Parallel analysis: factors that explain more variance than randomly simulated data
- Very simple structure (VSS) criterion: larger is better
- Velicer’s minimum average partial (MAP) test: smaller is better
- Akaike information criterion (AIC): smaller is better
- Bayesian information criterion (BIC): smaller is better
- Sample size-adjusted BIC (SABIC): smaller is better
- Root mean square error of approximation (RMSEA): smaller is better
- Chi-square difference test: smaller is better; a significant test indicates that the more complex model is significantly better fitting than the less complex model
- Standardized root mean square residual (SRMR): smaller is better
- Comparative Fit Index (CFI): larger is better
- Tucker Lewis Index (TLI): larger is better
There is not necessarily a “correct” criterion to use in determining how many factors to keep, so it is generally recommended that researchers use multiple criteria in combination with theory and interpretability.
A scree plot provides lots of information. A scree plot has the factor number on the x-axis and the eigenvalue on the y-axis. The eigenvalue is the variance accounted for by a factor; when using a varimax (orthogonal) rotation, an eigenvalue (or factor variance) is calculated as the sum of squared standardized factor (or component) loadings on that factor. An example of a scree plot is in Figure 23.12.
The total variance is equal to the number of variables you have, so one eigenvalue is approximately one variable’s worth of variance. The first factor accounts for the most variance, the second factor accounts for the second-most variance, and so on. The more factors you add, the less variance is explained by the additional factor.
One criterion for how many factors to keep is the Kaiser-Guttman criterion. According to the Kaiser-Guttman criterion, you should keep any factors whose eigenvalue is greater than 1. That is, for the sake of simplicity, parsimony, and data reduction, you should take any factors that explain more than a single variable would explain. According to the Kaiser-Guttman criterion, we would keep three factors from Figure 23.12 that have eigenvalues greater than 1. The default in SPSS is to retain factors with eigenvalues greater than 1. However, keeping factors whose eigenvalue is greater than 1 is not the most correct rule. If you let SPSS do this, you may get many factors with eigenvalues around 1 (e.g., factors with an eigenvalue ~ 1.0001) that are not adding so much that it is worth the added complexity. The Kaiser-Guttman criterion usually results in keeping too many factors. Factors with small eigenvalues around 1 could reflect error shared across variables. For instance, factors with small eigenvalues could reflect method variance (i.e., method factor), such as a self-report factor that turns up as a factor in factor analysis, but that may be useless to you as a conceptual factor of a construct of interest.
Another criterion is Cattell’s scree test, which involves selecting the number of factors from looking at the scree plot. “Scree” refers to the rubble of stones at the bottom of a mountain. According to Cattell’s scree test, you should keep the factors before the last steep drop in eigenvalues—i.e., the factors before the rubble, where the slope approaches zero. The beginning of the scree (or rubble), where the slope approaches zero, is called the “elbow” of a scree plot. Using Cattell’s scree test, you retain the number of factors that explain the most variance prior to the explained variance drop-off, because, ultimately, you want to include only as many factors in which you gain substantially more by the inclusion of these factors. That is, you would keep the number of factors at the elbow of the scree plot minus one. If the last steep drop occurs from Factor 4 to Factor 5 and the elbow is at Factor 5, we would keep four factors. In Figure 23.12, the last steep drop in eigenvalues occurs from Factor 3 to Factor 4; the elbow of the scree plot occurs at Factor 4. We would keep the number of factors at the elbow minus one. Thus, using Cattell’s scree test, we would keep three factors based on Figure 23.12.
There are more sophisticated ways of using a scree plot, but they usually end up at a similar decision. Examples of more sophisticated tests include parallel analysis and very simple structure (VSS) plots. In a parallel analysis, you examine where the eigenvalues from observed data and random data converge, so you do not retain a factor that explains less variance than would be expected by random chance.
In general, my recommendation is to use Cattell’s scree test, and then test the factor solutions with plus or minus one factor. You should never accept factors with eigenvalues less than zero (or components from principal component analysis with eigenvalues less than one), because they are likely to be largely composed of error. If you are using maximum likelihood factor analysis, you can compare the fit of various models with model fit criteria to see which model fits best for its parsimony. A model will always fit better when you add additional parameters or factors, so you examine if there is significant improvement in model fit when adding the additional factor—that is, we keep adding complexity until additional complexity does not buy us much. Always try a factor solution that is one less and one more than suggested by Cattell’s scree test to buffer your final solution because the purpose of factor analysis is to explain things and to have interpretability. Even if all rules or indicators suggest to keep X number of factors, maybe \(\pm\) one factor helps clarify things. Even though factor analysis is empirical, theory and interpretatability should also inform decisions.
23.6 Interpreting and Using Latent Factors
The next step is interpreting the model and latent factors. One data matrix can lead to many different (correct) models—you must choose one based on the factor structure and theory. Use theory to interpret the model and label the factors.