Propensity Score Analysis with a Latent or Mis-Measured Covariate: Using Factor Scores from Inclusive Factor or Structural Equation Models to Reduce Bias

Introduction: Propensity score methods assume that covariates used to model treatment assignment are measured without error. This is problematic when a covariate X is measured with error or is a latent variable. For example, to estimate the effect of high school substance use on young adulthood depression, we may use propensity score methods to balance pre-exposure covariates; these, however, include depressive symptomatology, a latent X. It has been suggested that if multiple measurements Ws of X are available, rather than using Ws (or their mean/sum) in propensity score analysis, it is better to use Ws in a latent factor model, estimate a factor score (FS) and use this FS to represent X. Evaluation of this approach to date is limited.

Methods: In the context of propensity score weighting, we investigate this method (the simple FS) and two other types of FSs generated from factor or structural equation models (FA/SEM) that also include the treatment variable T (T-inclusive FSs) and other covariates Z from the propensity score model (TZ-inclusive FSs). We consider logit, probit and identity links for T. We address non-differential measurement error with respect to T, Z, and the outcome.

Results: We focus on the case where the measurement errors are independent of one another. The simple FS and the direct Ws method result in similar bias; bias is higher with the mean W method if Ws’ correlations with X are not uniform. At extreme T prevalence (near 0 or 1), the simple FS performs better than the direct Ws method in terms of variance. Relative to the simple FS, T-inclusive FSs substantially reduce bias, and when X and Z are uncorrelated, brings bias to near zero. When X is correlated with Z, this approach is also biased (but to a lesser degree), due to incompatibility between the FS model and the propensity score model with respect to inclusion/exclusion of Z. Such bias is essentially eliminated by TZ-inclusive FSs generated from models that are saturated with respect to the X-T-Z joint distribution in model fitting and in FS computation, including: (1) the linear FA model with Ws, T and Z as indicators and a residual T-Z correlation; (2) the SEM based on the true model (Ws reflecting unobserved X, and Z and  X correlated and influencing T) with logit/probit/identity link for T, fit using ML; and (3) a modified SEM with probit link fit using WLS.

When some measurement errors are correlated, all FSs’ performance is worsened if the FS model does not capture such correlations.

Conclusions: We recommend using one of these TZ-inclusive FSs to represent the mis-measured/latent X in propensity score analysis. We also recommend careful factor analysis of Ws to identify residual correlations needed in the FS model. For illustration, the method is applied to the above-mentioned example, using Add Health data.