Cross-validation by downweighting inuential cases in structural equation modelling Ke-Hai Yuan 1 *, Linda L. Marshall 2 and Rebecca Weston 2 1 University of Notre Dame, USA 2 University of North Texas, USA In the social and behavioural sciences, structural equation modelling has been widely used to test a substantive theory or causal relationship among latent constructs. Cross- validation (CV) is a valuable tool for selecting the best model among competing structural models. Inuential cases or outliers are often present in practical data. Therefore, even the correct model for the majority of the data may not cross-validate well. This paper discusses various drawbacks of CV based on sample covariance matrices, and develops a procedure for using robust covariance matrices in the model calibration and validation stages. Examples illustrate that the CV index based on sample covariance matrices is very sensitive to inuential cases, and even a single outlier can cause the CV index to support a wrong model. The CV index based on robust covariance matrices is much less sensitive to inuential cases and thus leads to a more valid conclusion about the practical value of a model structure. 1. Introduction Structural equation modelling (SEM) is one of the most popular methods in multivariate analysis, with extensive applications in the social and behavioural sciences (Bentler & Dudgeon, 1996). The advantage of SEM is that manifest variables, latent variables and measurement errors can be modelled and tested simultaneously. However, due to the complexity of the structural relationship and measurement errors, an SEM model is, at best, only an approximation of the real world. When the sample size is large, most models will be rejected because of the high power of the commonly used goodness-of-t chi-square test statistics. On the other hand, when the same data are used for both model estimation and testing, a non-signicant chi-square statistic does not necessarily imply 125 British Journal of Mathematical and Statistical Psychology (2002), 55, 125–143 © 2002 The British Psychological Society www.bps.org.uk * Requests for reprints should be addressed to Ke-Hai Yuan, Dept of Psychology, University of Notre Dame, Notre Dame, IN 46556, USA (e-mail: kyuan@nd.edu).