Handling Correlations Between Covariates and Random Slopes in Multilevel Models

Michael Bates, Katherine E. Castellano, Sophia Rabe-Hesketh and Anders Skrondal
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Katherine E. Castellano: Educational Testing Service
Anders Skrondal: Norwegian Institute of Public Health

Journal of Educational and Behavioral Statistics, 2014, vol. 39, issue 6, 524-549

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.

Keywords: endogeneity; hierarchical linear model; multilevel model (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (6)

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Persistent link: https://EconPapers.repec.org/RePEc:sae:jedbes:v:39:y:2014:i:6:p:524-549

DOI: 10.3102/1076998614559420

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