Inference from Sampling with Response Probabilities Estimated via Calibration

TL;DR

This paper compares calibration and maximum likelihood methods for estimating response probabilities in survey sampling, showing calibration's advantages in efficiency and robustness, and discussing practical estimation issues.

Contribution

It provides an asymptotic analysis of calibration-based estimators, demonstrating their efficiency and double robustness, and highlights practical challenges in response probability estimation.

Findings

Calibration estimators are asymptotically unbiased and more efficient than true response probability estimators.

Calibration-based estimators are doubly robust to model misspecification.

Problems like convergence and extreme weights are more common with calibration than maximum likelihood.

Abstract

A solution to control for nonresponse bias consists of multiplying the design weights of respondents by the inverse of estimated response probabilities to compensate for the nonrespondents. Maximum likelihood and calibration are two approaches that can be applied to obtain estimated response probabilities. We consider a common framework in which these approaches can be compared. We develop an asymptotic study of the behavior of the resulting estimator when calibration is applied. A logistic regression model for the response probabilities is postulated. Missing at random and unclustered data are supposed. Three main contributions of this work are: 1) we show that the estimators with the response probabilities estimated via calibration are asymptotically equivalent to unbiased estimators and that a gain in efficiency is obtained when estimating the response probabilities via calibration as compared to the estimator with the true response probabilities, 2) we show that the estimators with the response probabilities estimated via calibration are doubly robust to model misspecification and explain why double robustness is not guaranteed when maximum likelihood is applied, and 3) we discuss and illustrate problems related to response probabilities estimation, namely existence of a solution to the estimating equations, problems of convergence, and extreme weights. We explain and illustrate why the first aforementioned problem is more likely with calibration than with maximum likelihood estimation. We present the results of a simulation study in order to illustrate these elements.