Nested Model Comparisons Between Common Factors and Composites

In psychological research, the common factor model is the most popular measurement model for scale items. However, there is increasing awareness that alternative measurement models, such as formative models, may make more theoretical sense for many kinds of psychological data. We demonstrate the nesting structure of three models specified in a structural equation modeling (SEM) framework: a reflective Confirmatory Factor Analysis (CFA), a formative Confirmatory Composite Analysis (CCA) and a formative Pseudo-Indicator Model (PIM). Unlike CFA, CCA and PIM allow for the specification of composites in the SEM framework. In this paper, we establish both theoretically and empirically that these three models are nested within one another, as long as the structural part of each model is saturated. As such, the three models can be compared via a chi-square difference test and other fit indices developed for nested models. We report on the results of a small simulation to evaluate whether the chi-square difference test and an RMSEA based on it RMSEA_D, can reliably discern whether data were sampled from a CFA or a formative measurement model, varying sample size, indicator weights, and the strength of the correlation with another construct. In two empirical examples, we illustrate how tools for nested model comparison can be used to distinguish among reflective and formative measurement models.