Comparing multilevel and fixed effect approaches in the generalized linear model setting
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广义线性模型中多层模型与固定效应方法的比较
- 研究问题
- 在广义线性模型中估计处理变量系数时，多层模型与固定效应方法如何受到组层混杂影响，其标准误如何处理组内依赖？
- 方法与思路
- 作者比较多层模型、正则化固定效应及其偏差校正版，讨论非线性下的不完全等价，并考察聚类自助法和聚类稳健标准误。
- 材料陈述的贡献
- 摘要称，各方法均未彻底解决偏差问题，但两种校正版通常偏差较小；作者据此提出系数估计与不确定性评估的组合建议。
- 阅读关注点
- 应核对偏差校正的适用条件、组大小与组层混杂设计、聚类自助法实施，以及课程效应应用所需的因果识别依据。
- 阅读边界
- 所给材料为 arXiv 预印本，不能确认同行评审或期刊接收。摘要中的方法建议具有条件性，未说明具体识别假设和偏差数值。
原文摘要
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, fit through (approximate) maximum likelihood estimation, 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. We apply our recommendations to estimate the effects of taking a mathematics course in twelfth grade for students in the Los Angeles Unified School District. Our mlmfe package easily implements our recommendations in the R programming language.