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.

翻译：我们将先前在线性多层模型（MLM）与固定效应（FE）模型比较的研究扩展至广义线性模型（GLM）设定，其中处理变量的系数是主要关注对象。这引出了三个关键发现。（i）首先，与线性设定类似，MLM可被视为FE的正则化形式。这解释了为何在存在组层面混杂因素时，MLM的处理系数估计可能呈现较大偏差。然而，与线性设定不同，在GLM中MLM与正则化FE的系数估计并不存在精确等价关系。（ii）其次，我们研究了“偏差校正MLM”（bcMLM）在GLM设定中的推广形式。无论是FE还是bcMLM都未能完全解决GLM中MLM的偏差问题，但bcMLM通常表现出比FE更小的偏差。（iii）第三，也是最后一点，正如在线性设定中那样，在GLM设定下MLM的默认标准误可能错误设定组内真实依赖结构，从而导致标准误的低估。聚类自助法是一种更具不可知论特征的替代方案。最终，对于非线性GLM，我们建议采用bcMLM进行处理系数估计，并使用聚类自助法计算标准误和置信区间。若自助法在计算上不可行，则推荐采用具有聚类稳健标准误的FE模型。