Second-order local sensitivity to non-ignorability in Bayesian inferences

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.