Handling Correlations Between Covariates and Random Slopes in Multilevel Models. (December 2014)

• Record Type: Journal Article
• Authors: Bates, Michael David; Castellano, Katherine E.; Rabe-Hesketh, Sophia; Skrondal, Anders
• Is Part Of: Journal of educational and behavioral statistics. Volume 39:Number 6(2014)
• Page Start: 524
• Page End: 549
• Publication Date: 2014-12
• Subjects: endogeneity -- hierarchical linear model -- multilevel model; Educational statistics -- Periodicals; Social sciences -- Statistical methods -- Periodicals; 370.2
• DOI: 10.3102/1076998614559420
• Languages: English
• ISSNs: 1076-9986

Abstract

This article discusses estimation of multilevel/hierarchical linear models that include cluster-level random intercepts and random slopes. Viewing the models as structural, the random intercepts and slopes represent the effects of omitted cluster-level covariates that may be correlated with included covariates. The resulting correlations between random effects (intercepts and slopes) and included covariates, which we refer to as "cluster-level endogeneity, " lead to bias when using standard random effects (RE) estimators such as (restricted) maximum likelihood. While the problem of correlations between unit-level covariates and random intercepts is well known and can be handled by fixed-effects (FE) estimators, the problem of correlations between unit-level covariates and random slopes is rarely considered. When applied to models with random slopes, the standard FE estimator does not rely on standard cluster-level exogeneity assumptions, but requires an "uncorrelated variance assumption" that the variances of unit-level covariates are uncorrelated with their random slopes. We propose a "per-cluster regression" (PC) estimator that is straightforward to implement in standard software, and we show analytically that it is unbiased for all regression coefficients under cluster-level endogeneity and violation of the uncorrelated variance assumption. The PC, RE, and an augmented FE estimator are applied to a real data set and evaluated in a simulation study that demonstrates that our PC estimator performs well in practice.