Bayesian Selection Model with Shrinking Priors for Nonignorable Missingness

This study investigates the effectiveness of Bayesian variable selection (BVS) procedures in dealing with missing not at random (MNAR) data for identification in selection models. Three BVS-adapted selection models, namely Bayesian LASSO, horseshoe prior, and spike-and-slab prior, were compared, along with established missing data methods such as a model that assumes a missing at random (MAR) process and full-selection model. The results indicate that the spike-and-slab prior consistently outperformed other BVS methods in terms of accuracy and bias for various parameters, including slope estimates, residual variance, and intercept. When compared with the full-selection model, the spike-and-slab model exhibited superior performance across all parameters based on mean squared error (MSE) results.Although the MAR and spike-and-slab models showed comparable performance for slope estimates, the spike-and-slab model consistently outperformed the MAR model in estimating residual variance and intercept. This comparable performance is attributed to the bias-variance tradeoff. The MAR model, while biased, demonstrated efficiency by estimating fewer parameters than selection models and obtaining robust support from the observed data. On the other hand, the spike-and-slab model outperformed the full-selection model, even when the full-selection model aligned with the true data-generating model. The adaptation of BVS to selection models, particularly through the spike-and-slab method, yielded promising results with unbiased estimates under various conditions. However, it is important to acknowledge that this study represents an initial exploration of this subject, and its scope was inherently limited. Finally, the BVS adaptations to the selection model was illustrated with data from a clinical-trial study