Summary Classical diagnostics for structural equation models are based on aggregate forms of the data and are ill suited for checking distributional or linearity assumptions. We extend recently developed goodness-of-fit tests for correlated data based on subject-specific residuals to structural equation models with latent variables. The proposed tests lend themselves to graphical displays and are designed to detect misspecified distributional or linearity assumptions. To complement graphical displays, test statistics are defined; the null distributions of the test statistics are approximated using computationally efficient simulation techniques. The properties of the proposed tests are examined via simulation studies. We illustrate the methods using data from a study of in utero lead exposure.