School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, 14155-6455, Iran.
Stat Med. 2018 Nov 10;37(25):3616-3636. doi: 10.1002/sim.7829. Epub 2018 Jun 5.
The sensitivity of Bayesian inferences to non-ignorability is an important issue which should be carefully handled when analyzing incomplete data sets. Generally, sensitivity analysis quantifies the effect that non-ignorability parameter variations have on model outputs or inferences. This sensitivity can be achieved locally around the ignorable model. Previously, some local sensitivity measures to assess the impact of non-ignorable coarsening on Bayesian inferences have been established based on the first-order derivation of the posterior expectations. This may not be adequate to show potential sensitivity when there is a considerable amount of curvature around the ignorable model estimate. Specifically, it becomes more important when the posterior expectation is U-shaped near the ignorable estimate so that the first-order sensitivity index is approximately zero even if the posterior mean might be highly curved around the ignorable model and hence sensitive to the ignorability assumption. In this paper, we present a method for determining the second-order sensitivity to non-ignorability of Bayesian inferences locally around the ignorable model in GLMs which perform equally well when the impact of non-ignorability is locally linear. Calculation of the proposed second-order sensitivity index only requires some posterior covariances of the simple ignorable model and is conducted efficiently and with minimal computational overhead compared with the first-order sensitivity index. To show the need for the second-order sensitivity index as a more precise screening tool, some simulation studies are conducted. Also, the approach is applied to analyze a real data example with CD4 cell counts as an incomplete response variable.
贝叶斯推断对不可忽略性的敏感性是分析不完全数据集时需要谨慎处理的一个重要问题。通常，敏感性分析量化了不可忽略性参数变化对模型输出或推断的影响。这种敏感性可以在可忽略模型的局部范围内实现。以前，已经基于后验期望的一阶导数，建立了一些局部敏感性度量方法来评估不可忽略性粗化对贝叶斯推断的影响。当可忽略模型估计周围存在相当大的曲率时，这可能不足以显示潜在的敏感性。具体来说，当后验期望在可忽略估计值附近呈 U 形时，情况变得更加重要，即使后验均值可能在可忽略模型周围呈高度弯曲，并且对不可忽略性假设敏感，一阶敏感性指数也接近零。在本文中，我们提出了一种用于确定 GLM 中可忽略模型局部范围内贝叶斯推断对不可忽略性的二阶敏感性的方法，当不可忽略性的影响在局部线性时，该方法同样有效。拟议的二阶敏感性指数的计算仅需要一些简单可忽略模型的后验协方差，并且与一阶敏感性指数相比，计算效率高，计算开销小。为了表明二阶敏感性指数作为更精确的筛选工具的必要性，进行了一些模拟研究。此外，该方法还应用于分析 CD4 细胞计数作为不完全响应变量的真实数据示例。
