This article is concerned with measures of fit of a model. Two types of error involved in fitting a model are considered. The first is error of approximation which involves the fit of the model, with optimally chosen but unknown parameter values, to the population covariance matrix. The second is overall error which involves the fit of the model, with parameter values estimated from the sample, to the population covariance matrix. Measures of the two types of error are proposed and point and interval estimates of the measures are suggested. These measures take the number of parameters in the model into account in order to avoid penalizing parsimonious models. Practical difficulties associated with the usual tests of exact fit or a model are discussed and a test of “close fit” of a model is suggested.

This article considers single sample approximations for the cross-validation coefficient in the analysis of covariance structures. An adjustment for predictive validity which may be employed in conjunction with any correctly specified discrepancy function is suggested. In the case of maximum likelihood estimation under normality assumptions the coefficient obtained is a simple linear function of the Akaike Information Criterion. Results of a random sampling experiment are reported.

This paper examines methods for comparing the suitability of alternative models for covariance matrices. A cross-validation procedure is suggested and its properties are examined. To motivate the discussion, a series of examples is presented using longitudinal data.

Thanks to M. W. Browne for several useful discussions that greatly clarified the ideas in this article. An anonymous referee also provided many valuable suggestions.

The total variance in any observed measure of performance can be attributed to 3 sources: (a) the correlation of the measure with the latent variable of interest'(i.e., true score variance), (b) reliable but irrelevant variance due to contamination, and (c) error. A model is proposed that specifies 3, and only 3, determinants of the relevant variance: declarative knowledge, procedural knowledge and skill, and volitional choice (motivation). The 3 determinants are defined, and their implications for performance measurement are discussed. Using data from the U.S. Army Selection and Classification Project (Project A), the authors found that the model fits a simplex pattern to the criterion data matrix. The predictor-determinant correlations are also estimated. Analyses of the data with LISREL provided strong confirmation of the model.