Evaluation of goodness-of-fit indices for structural equation models.

Abstract

Discusses how current goodness-of-fit indices fail to assess parsimony and hence disconfirmability of a model and are insensitive to misspecifications of causal relations (a) among latent variables when measurement model with many indicators is correct and (b) when causal relations corresponding to free parameters expected to be nonzero turn out to be zero or near zero. A discussion of philosophy of parsimony elucidates relations of parsimony to parameter estimation, disconfirmability, and goodness of fit. AGFI in LISREL is rejected. A method of adjusting goodness-of-fit indices by a parsimony ratio is described. Also discusses less biased estimates of goodness of fit and a relative normedfit index for testing fit of structural model exclusive of the measurement model. By a goodness-of-fit index, in structural equations modeling, we mean an index for assessing the fit of a model to data that ranges in possible value between zero and unity, with zero indicating a complete lack of fit and unity indicating perfect fit. Although chi-square statistics are often used as goodness-of-fit indices, they range between zero and infinity, with zero indicating perfect fit and a large number indicating extreme lack of fit. We prefer to call chi-square and other indices with this property lack-of-fit indices. For a recent discussion of both lack-of-fit and goodness-of-fit indices, see Wheaton (I 988). In this article we evaluate the use of goodness-of-fit indices for the assessment of the fit of structural equation models to data. Our aim is to review their rationales and to assess their strengths and weaknesses. We also consider other aspects of the problem of evaluating a structural equation model with goodness-of-fit indices. For example, are certain goodness-of-fit indices to be used only in certain stages of research (a contention of Sobel & Bohrnstedt, 1985)? Or, how biased are estimates of goodness of fit in small samples? What bearing does parsimony have on assessing the goodness of fit of the model? Can goodness-of-fit indices focus on the fit of certain aspects of a model as opposed to the fit of the overall model? For example, to what extent do current goodness-of-fit indices fail to reveal poor fits in the structural submodel among the latent variables because of good fits in the measurement model relating latent variables to manifest indicators? We describe a goodness-of-fit index now