Stepwise Variable Selection in Factor Analysis

It is very important to choose appropriate variables to be analyzed in multivariate analysis when there are many observed variables such as those in a questionnaire. What is actually done in scale construction with factor analysis is nothing but variable selection. In this paper, we take several goodness-of-fit statistics as measures of variable selection and develop backward elimination and forward selection procedures in exploratory factor analysis. Once factor analysis is done for a certain number p of observed variables (the p -variable model is labeled the current model), simple formulas for predicted fit measures such as chi-square, GFI, CFI, IFI and RMSEA, developed in the field of the structural equation modeling, are provided for all models obtained by adding an external variable (so that the number of variables is p + 1) and for those by deleting an internal variable (so that the number is p − 1), provided that the number of factors is held constant. A program SEFA (Stepwise variable selection in Exploratory Factor Analysis) is developed to actually obtain a list of the fit measures for all such models. The list is very useful in determining which variable should be dropped from the current model to improve the fit of the current model. It is also useful in finding a suitable variable that may be added to the current model. A model with more appropriate variables makes more stable inference in general. The criteria traditionally often used for variable selection is magnitude of communalities. This criteria gives a different choice of variables and does not improve fit of the model in most cases.