Bayesian inference under cluster sampling with probability proportional to size.

Area of Science:

• Statistics
• Survey Methodology
• Bayesian Inference

Background:

• Cluster sampling is widely used in surveys, with inference typically relying on design-based methods.
• Existing methods face challenges with unknown cluster sizes in complex sampling designs.
• Bayesian approaches offer a flexible framework for handling such complexities.

Purpose of the Study:

• To develop a Bayesian framework for cluster sampling that accounts for design effects in outcome modeling.
• To propose methods for predicting unknown cluster sizes within a Bayesian framework.
• To integrate cluster size prediction with survey outcome modeling for enhanced inference.

Main Methods:

• A two-stage cluster sampling design was considered, with probability proportional to size cluster selection.
• Nonparametric and parametric Bayesian approaches were developed to predict unknown cluster sizes.
• Inference was performed simultaneously for cluster sizes and survey outcomes using the Stan inference engine.

Main Results:

• The integrated Bayesian approach demonstrated superior performance compared to classical methods.
• Significant efficiency gains were observed, particularly with informative cluster sampling and a limited number of selected clusters.
• The method was successfully applied to the Fragile Families and Child Wellbeing study.

Conclusions:

• The proposed Bayesian framework provides a robust and efficient method for cluster sampling inference.
• This approach effectively addresses the challenge of unknown cluster sizes in complex survey designs.
• The findings have implications for improving statistical inference in large-scale health and social surveys.