A Latent Cluster-Mean Approach to the Contextual Effects Model With Missing Data

In organizational studies involving multiple levels, the association between a covariate and an outcome often differs at different levels of aggregation, giving rise to widespread interest in "contextual effects models." Such models partition the regression into within- and between-cluster components. The conventional approach uses each cluster's sample average of the covariate as a regressor to identify the between-cluster component of the regression. This procedure, however, yields biased estimates of contextual effects unless the cluster sizes are large. Moreover, bias in estimation of such contextual coefficients in turn introduces bias in estimated coefficients of other correlated cluster-level covariates. Missing data further complicate valid inferences. This article proposes an alternative approach that conditions on the latent "true" cluster means of covariates having contextual effects while taking into account ignorable missing data with a general missing pattern at each level. The proposed model may include random coefficients. We compare inferences under different approaches in estimation of a contextual effects model using data from two national surveys of high school achievement.