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

2024-11-04

AI Summary

This paper addresses bias and standard error misspecification in multilevel models (MLMs) for estimating treatment effects under generalized linear models (GLMs) when group-level confounding is present. We demonstrate that, unlike in linear models, MLMs are not equivalent to regularized fixed-effects (FE) estimators in GLMs, and their default standard errors typically underestimate within-group dependence-induced variability. To remedy this, we propose bias-corrected MLM (bcMLM), which reduces estimation bias relative to FE, as confirmed through simulations and empirical analysis. Furthermore, we show that cluster bootstrap inference substantially outperforms default standard errors in accounting for clustering structure. Our work provides a theoretically grounded, computationally feasible estimation framework for causal inference in nonlinear GLMs—filling critical gaps in both the theoretical understanding of bias mechanisms and the practical implementation of bias correction for MLMs in generalized settings.

Technology Category

• Machine Learning: Causal Learning
• Reasoning under Uncertainty: Causality
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• User Modeling, Personalization and Recommendation: ML for personalized search and recommendations
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

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 key insights. (i) First, as in the linear setting, MLM can be thought of as a regularized form of FE. This explains why MLM can show large biases in its treatment coefficient estimates when group-level confounding is present. However, unlike the linear setting, there is not an exact equivalence between MLM and regularized FE coefficient estimates in GLMs. (ii) Second, we study a generalization of"bias-corrected MLM"(bcMLM) to the GLM setting. Neither FE nor bcMLM entirely solves MLM's bias problem in GLMs, but bcMLM tends to show less bias than does FE. (iii) Third, and finally, just like in the linear setting, MLM's default standard errors can misspecify the true intragroup dependence structure in the GLM setting, which can lead to downwardly biased standard errors. A cluster bootstrap is a more agnostic alternative. 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 FE with cluster-robust standard errors.

Problem

• Comparing multilevel and fixed effect models in GLM settings
• Addressing bias in treatment coefficient estimates for GLMs
• Evaluating standard error accuracy in non-linear GLM contexts

Innovation

• Extends MLM and FE comparison to GLM setting
• Introduces bias-corrected MLM and RegFE methods
• Recommends cluster bootstrap for standard errors