Exploratory factor analysis

Fitting procedures

Fitting procedures are used to estimate the factor loadings and unique variances of the model (Factor loadings are the regression coefficients between items and factors and measure the influence of a common factor on a measured variable). There are several factor analysis fitting methods to choose from, however there is little information on all of their strengths and weaknesses and many don’t even have an exact name that is used consistently. Principal axis factoring (PAF) and maximum likelihood (ML) are two extraction methods that are generally recommended. In general, ML or PAF give the best results, depending on whether data are normally-distributed or if the assumption of normality has been violated.

Maximum likelihood (ML)

The maximum likelihood method has many advantages in that it allows researchers to compute of a wide range of indexes of the goodness of fit of the model, it allows researchers to test the statistical significance of factor loadings, calculate correlations among factors and compute confidence intervals for these parameters. ML is the best choice when data are normally distributed because “it allows for the computation of a wide range of indexes of the goodness of fit of the model [and] permits statistical significance testing of factor loadings and correlations among factors and the computation of confidence intervals”. ML should not be used if the data are not normally distributed.

Principal axis factoring (PAF)

Called “principal” axis factoring because the first factor accounts for as much common variance as possible, then the second factor next most variance, and so on. PAF is a descriptive procedure so it is best to use when the focus is just on your sample and you do not plan to generalize the results beyond your sample. An advantage of PAF is that it can be used when the assumption of normality has been violated. Another advantage of PAF is that it is less likely than ML to produce improper solutions. A downside of PAF is that it provides a limited range of goodness-of-fit indexes compared to ML and does not allow for the computation of confidence intervals and significance tests.

Selecting the appropriate number of factors

When selecting how many factors to include in a model, researchers must try to balance parsimony (a model with relatively few factors) and plausibility (that there are enough factors to adequately account for correlations among measured variables). It is better to include too many factors (overfactoring) than too few factors (underfactoring).

Overfactoring occurs when too many factors are included in a model. It is not as bad as underfactoring because major factors will usually be accurately represented and extra factors will have no measured variables load onto them. Still, it should be avoided because overfactoring may lead researchers to put forward constructs with little theoretical value.

Underfactoring occurs when too few factors are included in a model. This is considered to be a greater error than overfactoring. If not enough factors are included in a model, there is likely to be substantial error. Measured variables that load onto a factor not included in the model can falsely load on factors that are included, altering true factor loadings . This can result in rotated solutions in which two factors are combined into a single factor, obscuring the true factor structure.

There are a number of procedures designed to determine the optimal number of factors to retain in EFA. These include Kaiser's (1960) eigenvalue-greater-than-one rule (or K1 rule), Cattell's (1966) scree plot, Revelle and Rocklin's (1979) very simple structure criterion, model comparison techniques, Raiche, Roipel, and Blais's (2006) acceleration factor and optimal coordinates, Velicer's (1976) minimum average partial, Horn's (1965) parallel analysis, and Ruscio and Roche's (2012) comparison data. However, more recent simulation studies concerning the robustness of such techniques suggests that the latter five can assist practitioners to more judiciously model data. These five modern techniques are now easily accessible through integrated use of IBM SPSS Statistics software (SPSS) and R (R Development Core Team, 2011). See Courtney (2013) for guidance on how to carry out these procedures for continuous, ordinal, and heterogenous (continuous and ordinal) data.

With the exception of Revelle and Rocklin's (1979) very simple structure criterion, model comparison techniques, and Velicer's (1976) minimum average partial, all other procedures rely on the analysis of eigenvalues. The eigenvalue of a factor represents the amount of variance of the variables accounted for by that factor. The lower the eigenvalue, the less that factor contributes to the explanation of variances in the variables.

A short description of each of the nine procedures mentioned above is provided below.

Kaiser's (1960) eigenvalue-greater-than-one rule (K1 or Kaiser criterion)

Compute the eigenvalues for the correlation matrix and determine how many of these eigenvalues are greater than 1. This number is the number of factors to include in the model. A disadvantage of this procedure is that it is quite arbitrary (e.g., an eigenvalue of 1.01 is included whereas an eigenvalue of .99 is not). This procedure often leads to overfactoring and sometimes underfactoring. Therefore, this procedure should not be used. A variation of the K1 criterion has been created to lessen the severity of the criterion's problems where a researcher calculates confidence intervals for each eigenvalue and retains only factors which have the entire confidence interval greater than 1.0.

Cattell's (1966) scree plot

Compute the eigenvalues for the correlation matrix and plot the values from largest to smallest. Examine the graph to determine the last substantial drop in the magnitude of eigenvalues. The number of plotted points before the last drop is the number of factors to include in the model. This method has been criticized because of its subjective nature (i.e., there is no clear objective definition of what constitutes a substantial drop). As this procedure is subjective, Courtney (2013) does not recommend it.

Revelle and Rocklin (1979) very simple structure

Revelle and Rocklin’s (1979) VSS criterion operationalizes this tendency by assessing the extent to which the original correlation matrix is reproduced by a simplified pattern matrix, in which only the highest loading for each item is retained, all other loadings being set to zero. The VSS criterion for assessing the extent of replication can take values between 0 and 1, and is a measure of the goodness-of-fit of the factor solution. The VSS criterion is gathered from factor solutions that involve one factor (k = 1) to a user-specified theoretical maximum number of factors. Thereafter, the factor solution that provides the highest VSS criterion determines the optimal number of interpretable factors in the matrix. In an attempt to accommodate datasets where items covary with more than one factor (i.e., more factorially complex data), the criterion can also be carried out with simplified pattern matrices in which the highest two loadings are retained, with the rest set to zero (Max VSS complexity 2). Courtney also does not recommend VSS because of lack of robust simulation research concerning the performance of the VSS criterion.

Model comparison techniques

Choose the best model from a series of models that differ in complexity. Researchers use goodness-of-fit measures to fit models beginning with a model with zero factors and gradually increase the number of factors. The goal is to ultimately choose a model that explains the data significantly better than simpler models (with fewer factors) and explains the data as well as more complex models (with more factors).

There are different methods that can be used to assess model fit:

Optimal Coordinate and Acceleration Factor

In an attempt to overcome the subjective weakness of Cattell’s (1966) scree test, presented two families of non-graphical solutions. The first method, coined the optimal coordinate (OC), attempts to determine the location of the scree by measuring the gradients associated with eigenvalues and their preceding coordinates. The second method, coined the acceleration factor (AF), pertains to a numerical solution for determining the coordinate where the slope of the curve changes most abruptly. Both of these methods have out-performed the K1 method in simulation. In the Ruscio and Roche study (2012), > the OC method was correct 74.03% of the time rivaling the PA technique (76.42%). The AF method was correct 45.91% of the time with a tendency toward under-estimation. Both the OC and AF methods, generated with the use of Pearson correlation coefficients, were reviewed in Ruscio and Roche’s (2012) simulation study. Results suggested that both techniques performed quite well under ordinal response categories of two to seven (C = 2-7)and quasi-continuous (C = 10 or 20) data situations. Given the accuracy of these procedures under simulation, they are highly recommended for determining the number of factors to retain in EFA. It is one of Courtney's 5 recommended modern procedures.

Velicer's Minimum Average Partial test (MAP)

Velicer’s (1976) MAP test “involves a complete principal components analysis followed by the examination of a series of matrices of partial correlations” (p. 397). The squared correlation for Step “0” (see Figure 4) is the average squared off-diagonal correlation for the unpartialed correlation matrix. On Step 1, the first principal component and its associated items are partialed out. Thereafter, the average squared off-diagonal correlation for the subsequent correlation matrix is then computed for Step 1. On Step 2, the first two principal components are partialed out and the resultant average squared off-diagonal correlation is again computed. The computations are carried out for k minus one step (k representing the total number of variables in the matrix). Thereafter, all of the average squared correlations for each step are lined up and the step number in the analyses that resulted in the lowest average squared partial correlation determines the number of components or factors to retain (Velicer, 1976). By this method, components are maintained as long as the variance in the correlation matrix represents systematic variance, as opposed to residual or error variance. Although methodologically akin to principal components analysis, the MAP technique has been shown to perform quite well in determining the number of factors to retain in multiple simulation studies. However, in a very small minority of cases MAP may grossly overestimate the number of factors in a dataset for unknown reasons. This procedure is made available through SPSS's user interface. See Courtney (2013) for guidance. This is one of his five recommended modern procedures.

Parallel analysis

To carry out the PA test, users compute the eigenvalues for the correlation matrix and plot the values from largest to smallest and then plot a set of random eigenvalues. The number of eigenvalues before the intersection points indicates how many factors to include in your model. This procedure can be somewhat artbitrary (i.e. a factor just meeting the cutoff will be included and one just below will not). Moreover, the method is very sensitive to sample size, with PA suggesting more factors in datasets with larger sample sizes. Despite its shortcomings, this procedure performs very well in simulation studies and is one of Courtney's recommended procedures. See Courtney (2013) concerning how to perform this procedure from within the SPSS interface.

Ruscio and Roche’s Comparison Data

In 2012 Ruscio and Roche introduced the comparative data (CD) procedure in an attempt improve upon the PA method. The authors state that “rather than generating random datasets, which only take into account sampling error, multiple datasets with known factorial structures are analyzed to determine which best reproduces the profile of eigenvalues for the actual data” (p. 258). The strength of the procedure is its ability to not only incorporate sampling error, but also the factorial structure and multivariate distribution of the items. Ruscio and Roche’s (2012) simulation study determined that the CD procedure outperformed many other methods aimed at determining the correct number of factors to retain. In that study, the CD technique, making use of Pearson correlations accurately predicted the correct number of factors 87.14% of the time. Although, it should be noted that the simulated study did not involve more than five factors. Therefore, the applicability of the CD procedure to estimate factorial structures beyond five factors is yet to be tested. Courtney includes this procedure in his recommended list and gives guidelines showing how it can be easily carried out from within SPSS's user interface.

Convergence of multiple tests

A review of 60 journal articles by Henson and Roberts (2006) found that none used multiple modern techniques in an attempt to find convergence, such as PA and Velicer’s (1976) minimum average partial (MAP) procedures. Ruscio and Roche (2012) simulation study demonstrated the empirical advantage of seeking convergence. When the CD and PA procedures agreed, the accuracy of the estimated number of factors was correct 92.2% of the time. Ruscio and Roche (2012) demonstrated that when further tests were in agreement, the accuracy of the estimation could be increased even further.

Tailoring Courtney's recommended procedures for ordinal and continuous data

Recent simulation studies in the field of psychometrics suggest that the parallel analysis, minimum average partial, and comparative data techniques can be improved for different data situations. For example, in simulation studies, the performance of the minimum average partial test, when ordinal data is concerned, can be improved by utilizing polychoric correlations, as opposed to Pearson correlations. Courtney (2013) details how each of these three procedures can be optimized and carried out simultaneously from within the SPSS interface.

Factor rotation

Factor rotation is the process for interpreting factor matrixes. For any solution with two or more factors there are an infinite number of orientations of the factors that will explain the data equally well. Because there is no unique solution, a researcher must select a single solution from the infinite possibilities. The goal of factor rotation is to rotate factors in multidimensional space to arrive at a solution with best simple structure. There are two types of factor rotation: orthogonal and oblique rotation.

Orthogonal rotation

Orthogonal rotations constrain factors to be uncorrelated. Varimax is considered the best orthogonal rotation and consequently is used the most often in psychology research. An advantage of orthogonal rotation is its simplicity and conceptual clarity, although there are several disadvantages. In the social sciences, there is often a theoretical basis for expecting constructs to be correlated, therefore orthogonal rotations may not be very realistic because it ignores this possibility. Also, because orthogonal rotations require factors to be uncorrelated, they are less likely to produce solutions with simple structure.

Oblique rotation

Oblique rotations permit correlations among factors, though the factors thus identified may not correlate. If factors do not correlate (correlation estimates approximate zero), these rotations may produce solutions similar to orthogonal rotation. Several oblique rotation procedures are commonly used, such as direct oblimin rotation, direct quartimin rotation, promax rotation, and Harris-Kaiser orthoblique rotation. An advantage of oblique rotation is that it produces solutions with better simple structure because it allows factors to correlate, and produces estimates of correlations among factors.

Factor interpretation

Factor loadings are numerical values that indicate the strength and direction of a factor on a measured variable. Factor loadings indicate how strongly the factor influences the measured variable. In order to label the factors in the model, researchers should examine the factor pattern to see which items load highly on which factors and then determine what those items have in common. Whatever the items have in common will indicate the meaning of the factor.