We provide a basic review of the data screening and assumption testing issues relevant to exploratory and confirmatory factor analysis along with practical advice for conducting analyses that are sensitive to these concerns. Historically, factor analysis was developed for explaining the relationships among many continuous test scores, which led to the expression of the common factor model as a multivariate linear regression model with observed, continuous variables serving as dependent variables, and unobserved factors as the independent, explanatory variables. Thus, we begin our paper with a review of the assumptions for the common factor model and data screening issues as they pertain to the factor analysis of continuous observed variables. In particular, we describe how principles from regression diagnostics also apply to factor analysis. Next, because modern applications of factor analysis frequently involve the analysis of the individual items from a single test or questionnaire, an important focus of this paper is the factor analysis of items. Although the traditional linear factor model is well-suited to the analysis of continuously distributed variables, commonly used item types, including Likert-type items, almost always produce dichotomous or ordered categorical variables. We describe how relationships among such items are often not well described by product-moment correlations, which has clear ramifications for the traditional linear factor analysis. An alternative, non-linear factor analysis using polychoric correlations has become more readily available to applied researchers and thus more popular. Consequently, we also review the assumptions and data-screening issues involved in this method. Throughout the paper, we demonstrate these procedures using an historic data set of nine cognitive ability variables. Keywords: exploratory factor analysis, confirmatory factor analysis, item factor analysis, structural equation modeling, regression diagnostics, data screening, assumption testing Like any statistical modeling procedure, factor analysis carries a set of assumptions and the accuracy of results is vulnerable not only to violation of these assumptions but also to disproportionate influence from unusual observations. Nonetheless, the importance of data screening and assumption testing is often ignored or misconstrued in empirical research articles utilizing factor analysis. Perhaps some researchers have an overly indiscriminate impression that, as a large sample procedure, factor analysis is "robust" to assumption violation and the influence of unusual observations. Or, researchers may simply be unaware of these issues. Thus, with the applied psychology researcher in mind, our primary goal for this paper is to provide a general review of the data screening and assumption testing issues relevant to factor analysis along with practical advice for conducting analyses that are sensitive to these concerns. Although presentation of some matrix-based formulas is necessary, we aim to keep the paper relatively non-technical and didactic. To make the statistical concepts concrete, we provide data analytic demonstrations using different factor analyses based on an historic data set of nine cognitive ability variables. First, focusing on factor analyses of continuous observed variables, we review the common factor model and its assumptions and show how principles from regression diagnostics can be applied to determine the presence of influential observations. Next, we move to the analysis of categorical observed variables, because treating ordered, categorical variables as continuous variables is an extremely common form of assumption violation involving factor analysis in the substantive research literature. Thus, a key aspect of the paper focuses on how the linear common factor model is not well-suited to the analysis of categorical, ordinally scaled item-level variables, such as Likert-type items. We then describe an alternative approach to item factor analysis based on polychoric correlations along with its assumptions and limitations. We begin with a review the linear common factor model which forms the basis for both exploratory factor analysis (EFA) and confirmatory factor analysis (CFA). The major goal of both EFA and CFA is to model the relationships among a potentially large number of observed variables using a smaller number of unobserved, or latent, variables. The latent variables are the factors. In EFA, a researcher does not have a strong prior theory about the number of factors or how each observed variable relates to the factors. In CFA, the number of factors is hypothesized a priori along with hypothesized relationships between factors and observed variables. With both EFA and CFA, the factors influence the observed variables to account for www.frontiersin.org
The linearity assumption of FA and PCA is, for example, necessarily violated when the common factor model is fitted to Pearson's product-moment correlations amongst categorical, ordinal scaled items, including Likert-style scales REF .
Old and New Ideas for Data Screening and Assumption Testing for Exploratory and Confirmatory Factor Analysis
Front. Psychology
Psychology Medicine