Equivalent Models

Stelzl (1986) formalized the concept of "equivalent models." Models are equivalent when they cannot be distinguished in terms of overall fit. As an example, look at these three models for the relations between two common factors or latent constructs:

These three models imply, alternatively, that B is dependent on A, that A is dependent on B, or that A and B covary due to reasons other than a dependence relationship. While the substantive implications of the three models are very different, the fit of the three to any data set will be identical. Thus, in terms of fit, SEM cannot resolve the issue of which model should be preferred.

Stelzl (1986) introduced some basic rules for finding equivalent models, and these were generalized by Lee and Hershberger (1990). The most general rule is that if different models imply the same pattern of partial correlations among the variables in them, then those models are equivalent. One important implication of this rule is that the more saturated a model is, the more equivalent models there will be.

MacCallum et al., 1993 wrote that, based on a review of applications of SEM, researchers were not paying sufficient attention to this problem. A good fit for one model implies a good fit for all equivalent alternative models. Some equivalent models may be disregarded based on logic or theory. For example, we can be comfortable in concluding that a childhood history of disease predicts health problems in adulthood, and can disregard an equivalent model which implies the opposite--although we may not be able to disregard a model that implies that both constructs are "caused" by an outside factor.

If some alternative models are theoretically plausible, then the researcher must recognize the confound regarding the implications of their research for theory. Researchers are encouraged to look for plausible equivalent models when designing their research. If such models exist, perhaps further model development will lead to a model such that the troublesome alternative is ruled out. The problem is analogous to evaluating the power of experiment design--designing the research to yield results that will allow the researcher to distinguish between important alternative conclusions.

Lee, S., & Hershberger, S. (1990). A simple rule for generating equivalent models in structural equation modeling. Multivariate Behavioral Research, 25, 313-334.

MacCallum, R. C., Wegener, D. T., Uchino, B. N., & Fabrigar, L. R. (1993). The problem of equivalent models in applications of covariance structure analysis. Psychological Bulletin, 114, 185-199.

Stelzl, I. (1986). Changing the causal hypothesis without changing the fit: Some rules for generating equivalent path models. Multivariate Behavioral Research, 21, 309-331.

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Last updated: May 9, 1996