The problem of equivalent models in applications of covariance structure analysis.

Abstract

For any given covariance structure model, there will often be alternative models that are indistinguishable from the original model in terms of goodness of fit to data. The existence of such equivalent models is almost universally ignored in empirical studies. A study of 53 published applications showed that equivalent models exist routinely, often in large numbers. Detailed study of three applications showed that equivalent models may often offer substantively meaningful alternative explanations of data. The importance of the equivalent model phenomenon and recommendations for managing and confronting the problem in practice are discussed.

Investigating population heterogeneity with factor mixture models.

“…Equivalent models imply the same distribution but differ with respect to the structure they impose on the data and may differ considerably with respect to their conceptual interpretation. In the context of models for data from single homogeneous populations, the problem of equivalent models has led to a considerable body of research (see, for instance, Hershberger, 1994;Luijben, 1993;MacCallum, Wegener, Uchino, & Fabrigar, 1993;Raykov & Marcoulides, 2001). In the context of mixture models, additional problems may arise because models for different numbers of classes may imply the same mixture distribution (for examples, see McLachlan & Peel, 2000).…”

Relative contributions of parent substance use and childhood maltreatment to chronic homelessness, depression, and substance abuse problems among homeless women: mediating roles of self-esteem and abuse in adulthood

“…Although the causal hypothesis can be rejected statistically, it cannot be absolutely confirmed. For example, with cross-sectional data some alternative models might be equally plausible (MacCallum, Wegener, Uchino, & Fabrigar, 1993). However, more powerful conclusions can be drawn with this research method than in most nonexperimental research when the proposed model fits the empirical data and can be appropriately defended or justified theoretically (Bentler & Stein, 1992).…”

“…However, use of multi-indicator latent variables rather than single-item questionnaire items helped strengthen these constructs. Another limitation is that the data are cross-sectional which limits inferences about directionality of influences especially among the mediators of self-esteem and recent violence, and the outcome variables (MacCallum et al, 1993). For instance, Kilpatrick et al (1997) found reciprocal relationships between substance abuse and assaults in a longitudinal sample of women.…”

The processing-speed theory of adult age differences in cognition.

“…Path models should not be considered definitive because they can vary in the degree to which they represent or fit the data, and alternative structural models often provide equally good fits to the data (MacCallum, Wegener, Uchino, & Fabrigar, 1993). Nevertheless, path analyses can be informative about the presence or absence of particular relations, and hence they are of obvious relevance to theories postulating that a construct like processing speed mediates some of the age-related effects on measures of cognitive performance.…”