Estimating the variance-covariance matrix of two-step estimates of latent variable models: a general simulation-based approach

We propose a general procedure for estimating the variance–covariance matrix of two-step estimates of structural parameters in latent variable models. The method is partially simulation-based, in that it includes drawing simulated values of the measurement parameters of the model from their sampling distribution obtained from the first step of two-step estimation, and using them to quantify part of the variability in the parameter estimates from the second step. This is asymptotically equal to the standard closed-form estimate of the variance–covariance matrix of two-step estimates, but it avoids the need to evaluate a cross-derivative matrix which is the most inconvenient element of the standard estimate. The method can be applied to any type of latent variable models. We present it in more detail in the context of two common models where the measurement items are categorical: latent-class models with categorical latent variables and latent trait models with continuous latent variables. The good performance of the proposed procedure is demonstrated with simulation studies and illustrated with two applied examples.