In multivariate statistics, exploratory factor analysis (EFA) is a statistical method used to uncover the underlying structure of a relatively large set of variables. EFA is a technique within factor analysis whose overarching goal is to identify the underlying relationships between measured variables. It is commonly used by researchers when developing a scale (a scale is a collection of questions used to measure a particular research topic) and serves to identify a set of latent constructs underlying a battery of measured variables. It should be used when the researcher has no a priori hypothesis about factors or patterns of measured variables. Measured variables are any one of several attributes of people that may be observed and measured. Examples of measured variables could be the physical height, weight, and pulse rate of a human being. Usually, researchers would have a large number of measured variables, which are assumed to be related to a smaller number of "unobserved" factors. Researchers must carefully consider the number of measured variables to include in the analysis. EFA procedures are more accurate when each factor is represented by multiple measured variables in the analysis.
EFA is based on the common factor model. In this model, manifest variables are expressed as a function of common factors, unique factors, and errors of measurement. Each unique factor influences only one manifest variable, and does not explain correlations between manifest variables. Common factors influence more than one manifest variable and "factor loadings" are measures of the influence of a common factor on a manifest variable. For the EFA procedure, we are more interested in identifying the common factors and the related manifest variables.
EFA assumes that any indicator/measured variable may be associated with any factor. When developing a scale, researchers should use EFA first before moving on to confirmatory factor analysis. EFA is essential to determine underlying factors/constructs for a set of measured variables; while confirmatory factor analysis allows the researcher to test the hypothesis that a relationship between the observed variables and their underlying latent exists.Suhr, D. D. (2006). Exploratory or confirmatory factor analysis? (pp. 1-17). Cary: SAS Institute. EFA requires the researcher to make a number of important decisions about how to conduct the analysis because there is no one set method.
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”.
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. 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).
Overfactoring occurs when too many factors are included in a model and may lead researchers to put forward constructs with little theoretical value.
Underfactoring occurs when too few factors are included in a model. 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. Broadly speaking, most of the existing procedures approach the determination of the appropriate number of factors (1) by inspecting patterns of eigenvalues of the covariance matrix, or (2) treating it as a model selection problem. Existing approaches include: Kaiser's (1960) eigenvalue-greater-than-one rule (or K1 rule), Cattell's (1966) scree plot,Cattell, R. B. (1966). The scree test for the number of factors. Multivariate Behavioral Research, I, 245-276. Revelle and Rocklin's (1979) very simple structure criterion, model comparison techniques, Raiche, Roipel, and Blais's (2006) acceleration factor and optimal coordinates,Raiche, G., Roipel, M., & Blais, J. G.|Non graphical solutions for the Cattell’s scree test. Paper presented at The International Annual Meeting of the Psychometric Society, Montreal|date=2006|Retrieved December 10, 2012 from Velicer's (1976) minimum average partial, Horn's (1965) parallel analysis, and Ruscio and Roche's (2012) comparison data. Recent simulation studies assessing the robustness of such techniques suggest that the latter five can better assist practitioners to 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)Courtney, M. G. R. (2013). Determining the number of factors to retain in EFA: Using the SPSS R-Menu v2.0 to make more judicious estimations. Practical Assessment, Research and Evaluation, 18(8). Available online: 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 explaining the variance of 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) Kaiser's criterion, named after the American psychologist Henry Felix Kaiser, determines the number of factors by compute the eigenvalues for the correlation matrix and counting 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, and is widely regarded as suboptimal in comparison to more modern methods. 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:
Likelihood ratio statistic:Lawley, D. N. (1940). The estimation of factor loadings by the method of maximumlikelihood. Proceedings of the Royal Society ofedinborough, 60A, 64-82. Used to test the null hypothesis that a model has perfect model fit. It should be applied to models with an increasing number of factors until the result is nonsignificant, indicating that the model is not rejected as good model fit of the population. This statistic should be used with a large sample size and normally distributed data. There are some drawbacks to the likelihood ratio test. First, when there is a large sample size, even small discrepancies between the model and the data result in model rejection. When there is a small sample size, even large discrepancies between the model and data may not be significant, which leads to underfactoring. Another disadvantage of the likelihood ratio test is that the null hypothesis of perfect fit is an unrealistic standard.