Factor analysis is a statistical method that is used to determine the number of underlying dimensions contained in a set of observed variables and to identify the subset of variables that corresponds to each of the underlying dimensions. The underlying dimensions are referred to as continuous latent variables or factors. The observed variables are referred to as factor indicators. There are two types of factor analysis: exploratory factor analysis (EFA) and confirmatory factor analysis (CFA).
CFA is appropriate in situations where the dimensionality of a set of variables for a given population is already known because of previous research. The task is not to determine the dimensionality of a set of variables or to find the pattern of the factor loadings. Instead, CFA may be used to investigate whether the established dimensionality and factor-loading pattern fits a new sample from the same population. This is the confirmatory aspect of the analysis. CFA may also be used to investigate whether the established dimensionality and factor-loading pattern fits a sample from a new population. In addition, the factor model can be used to study the characteristics of individuals by examining factor variances and covariances/correlations. Factor variances show the degree of heterogeneity of a factor. For example, males in their mid-twenties may be less homogeneous with respect to a factor such as alcohol abuse than females and therefore have a larger factor variance. Factor correlations show the strength of association between factors.
CFA is characterized by restrictions on factor loadings, factor variances, and factor covariances/correlations. CFA requires at least m2 restrictions where m is the number of factors. This can be compared to EFA where exactly m2 restrictions are placed. Unlike EFA, CFA can include correlated residuals that can be useful for representing the influence of minor factors on the variables. A set of background variables can be included as part of a CFA. This model is referred to as the MIMIC model. The MIMIC model can include direct effects, the effect of a background variable on a factor indicator, and indirect effects, the effect of a background variable on a factor indicator via the factor.
Mplus can estimate CFA models and CFA models with background variables for a single or multiple groups. Factor indicators for CFA models can be continuous, censored, binary, ordered categorical (ordinal), counts, or combinations of these variable types. When factor indicators are all continuous, Mplus has seven estimator choices: maximum likelihood (ML), maximum likelihood with robust standard errors and chi-square (MLR, MLF, MLM, MLMV), generalized least squares (GLS), and weighted least squares (WLS) also referred to as ADF. When at least one factor indicator is binary or ordered categorical, Mplus has seven estimator choices: weighted least squares (WLS), robust weighted least squares (WLSM, WLSMV), maximum likelihood (ML), maximum likelihood with robust standard errors and chi-square (MLR, MLF), and unweighted least squares (ULS). When at least one factor indicator is censored, unordered categorical, or a count, Mplus has six estimator choices: weighted least squares (WLS), robust weighted least squares (WLSM, WLSMV), maximum likelihood (ML), and maximum likelihood with robust standard errors and chi-square (MLR, MLF).