Sampling weights in latent variable modeling

This article reviews several basic statistical tools needed for modeling data with sam-pling weights that are implemented in Mplus Version 3. These tools are illustrated in simulation studies for several latent variable models including factor analysis with continuous and categorical indicators, latent class analysis, and growth models. The pseudomaximum likelihood estimation method is reviewed and illustrated with strat-ified cluster sampling. Additionally, the weighted least squares method for estimat-ing structural equation models with categorical and continuous outcomes imple-mented in Mplus extended to incorporate sampling weights is also illustrated. The performance of several chi-square tests under unequal probability sampling is evalu-ated. Simulation studies compare the methods used in several statistical packages such as Mplus, HLM, SAS Proc Mixed, MLwiN, and the weighted sample statistics method used in other software packages. Unequal probability of selection is an inevitable feature of complex sampling sur-veys. This can be the result of stratified sampling, cluster sampling, subpopulation oversampling, designed unequal probability sampling, and so on. If the unequal probability of selection is not incorporated in the analysis, a substantial bias in the parameter estimates may arise. This bias is commonly known as selection bias. If the probability of selection is known and incorporated in the analysis, the selection bias can be eliminated. An unbiased estimator for the mean under unequal proba-bility sampling was first developed by Horvitz and Thompson (1952). Skinner (1989) developed the pseudomaximum likelihood (PML) method, which under unequal probability sampling can be used to estimate any statistical model includ-ing the latent variable models discussed here. This article describes several basic statistical tools needed to deal with unequal probability of selection that are implemented in Mplus version 3 (Muthén &amp