Propensity score analysis (PSA) is often used by researchers to control for selection bias due to multiple covariates in quasi-experimental studies. However, covariates with low reliability have been shown to lead to biased treatment effects estimates in PSA. Latent variable analysis is a promising strategy to reduce the negative effects of observed variables’ measurement error. This Monte Carlo simulation study compared the performance of five methods to adjust propensity scores for unreliability. The results indicate that the latent variable model with inclusive factor score (PSIF) generated the lowest relative bias of treatment effect estimates, followed by propensity score estimation with structural equation model (PS-SEM). However, only PSIF provided unbiased treatment effect estimates across conditions with high, medium and low reliability. The results also show that evaluation of covariate balance can be misleading when there are unreliable covariates, because treatment effect estimates can be biased when covariate balanced is deemed adequate.

The likelihood ratio test (LRT) is widely used for comparing the relative fit of nested latent variable models. Following Wilks' theorem, the LRT is conducted by comparing the LRT statistic with its asymptotic distribution under the restricted model, a χ 2 -distribution with degrees of freedom equal to the difference in the number of free parameters between the two nested models under comparison. For models with latent variables such as factor analysis, structural equation models and random effects models, however, it is often found that the χ 2 approximation does not hold. In this note, we show how the regularity conditions of the Wilks' theorem may be violated using three examples of models with latent variables.In addition, a more general theory for LRT is given that provides the correct asymptotic theory for these LRTs. This general theory was first established in Chernoff (1954) and discussed in both van der Vaart (2000) and Drton ( 2009), but it does not seem to have received enough attention. We illustrate this general theory with the three examples.