In many biomedical and epidemiological studies, data are often clustered due to longitudinal follow up or repeated sampling. While in some clustered data the cluster size is pre-determined, in others it may be correlated with the outcome of subunits, resulting in informative cluster size. When the cluster size is informative, standard statistical procedures that ignore cluster size may produce biased estimates. One attractive framework for modeling data with informative cluster size is the joint modeling approach in which a common set of random effects are shared by both the outcome and cluster size models. In addition to making distributional assumptions on the shared random effects, the joint modeling approach needs to specify the cluster size model. Questions arise as to whether the joint modeling approach is robust to misspecification of the cluster size model. In this paper, we studied both asymptotic and finite-sample characteristics of the maximum likelihood estimators in joint models when the cluster size model is misspecified. We found that using an incorrect distribution for the cluster size may induce small to moderate biases, while using a misspecified functional form for the shared random parameter in the cluster size model results in nearly unbiased estimation of outcome model parameters. We also found that there is little efficiency loss under this model misspecification. A developmental toxicity study was used to motivate the research and to demonstrate the findings. Copyright (c) 2011 John Wiley & Sons, Ltd.
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机译：在许多生物医学和流行病学研究中，由于纵向随访或重复采样，数据经常被聚类。在某些聚类数据中，聚簇大小是预先确定的，而在另一些聚类数据中，聚簇大小可能与亚基的结果相关，从而导致信息量大。当聚类大小有用时，忽略聚类大小的标准统计程序可能会产生有偏差的估计。一种具有吸引力的集群大小的有吸引力的数据建模框架是联合建模方法，其中结果和集群大小模型共享一组通用的随机效应。除了对共享随机效应进行分布假设外，联合建模方法还需要指定集群大小模型。联合建模方法是否对簇大小模型的错误指定具有鲁棒性。在本文中，我们研究了联合模型中最大可能似然估计的渐近和有限样本特征，当簇大小模型未正确指定时。我们发现，对于簇大小使用不正确的分布可能会引起较小到中等的偏差，而对于簇大小模型中的共享随机参数使用错误指定的函数形式会导致结果模型参数的近似无偏估计。我们还发现，在这种模型错误指定下，效率损失很小。发育毒性研究被用来激励研究并证明发现。版权所有（c）2011 John Wiley＆Sons，Ltd.
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