Comparing multilevel and fixed effect approaches in the generalized linear model setting

Abstract

We extend prior work comparing linear multilevel models (MLM) and fixed effect (FE) models to the generalized linear model (GLM) setting, where the coefficient on a treatment variable is of primary interest. This leads to three insights. (i) First, as in the linear setting, MLM can be thought of as a regularized form of FE (RegFE). This explains why group-level confounding can greatly bias MLM's treatment coefficient estimates. However, unlike the linear setting, there is not an exact equivalence between MLM and RegFE in GLMs. (ii) Second, we study a generalization of "bias-corrected MLM" (bcMLM) to the GLM setting, and a corresponding "bias-corrected RegFE" (bcRegFE). None of FE, bcMLM, or bcRegFE entirely solve MLM's bias problem in GLMs, but bcMLM and bcRegFE tend to show less bias than does FE. (iii) Third, as in the linear setting, MLM's default standard errors can misspecify the true intragroup dependence structure in the GLM setting, which can yield downwardly biased standard errors. A cluster bootstrap is a more agnostic alternative. We also consider a cluster-robust standard error for (bc)RegFE. Ultimately, for non-linear GLMs, we recommend bcMLM for estimating the treatment coefficient, and a cluster bootstrap for standard errors and confidence intervals. If a bootstrap is not computationally feasible, then we recommend bcRegFE with cluster-robust standard errors, or FE with cluster-robust standard errors when group sizes are larger.