Bayesian Case Influence Measures for Statistical Models with Missing Data.

Area of Science:

• Statistics
• Computational Statistics

Background:

• Identifying influential observations is crucial in statistical modeling, especially with missing data.
• Bayesian case influence measures, such as φ-divergence and Cook's distances, are vital but computationally intensive for models with missing data.

Purpose of the Study:

• To derive and evaluate simple first-order approximations for three Bayesian case influence measures.
• To assess the utility of these approximations in identifying influential observations within statistical models featuring missing data.

Main Methods:

• Utilized Laplace approximation formula to derive first-order approximations for Bayesian case influence measures.
• Employed Markov chain Monte Carlo (MCMC) samples from the posterior distribution of the full data for computations.
• Applied approximations to identify influential sets in various statistical models, including longitudinal and latent variable models.

Main Results:

• Developed computationally efficient approximations for φ-divergence, Cook's posterior mode distance, and Cook's posterior mean distance.
• Demonstrated that these approximations can be effectively computed using standard MCMC output.
• Validated the methodology through analyses of simulated data and a real-world AIDS dataset.

Conclusions:

• The proposed first-order approximations offer a computationally feasible alternative for identifying influential observations in statistical models with missing data.
• These approximations facilitate the practical application of Bayesian influence diagnostics in complex missing data scenarios.
• The methodology is broadly applicable to various statistical models, enhancing robust data analysis.