[Paper Review] Nonparametric Pattern-Mixture Models for Inference with Missing Data

Yen‐Chi Chen, Mauricio Sadinle|arXiv (Cornell University)|Apr 24, 2019

Bayesian Methods and Mixture Models52 references4 citations

TL;DR

This paper proposes a nonparametric pattern-mixture model for inference with missing data, using a surrogate empirical distribution with kernel smoothing and Monte Carlo sampling under donor-based identifying restrictions. It enables valid point estimation and bootstrap confidence intervals for functionals like ATEs, even under nonmonotone missingness, with theoretical guarantees and robustness to misspecified assumptions.

ABSTRACT

Pattern-mixture models provide a transparent approach for handling missing data, where the full-data distribution is factorized in a way that explicitly shows the parts that can be estimated from observed data alone, and the parts that require identifying restrictions. We introduce a nonparametric estimator of the full-data distribution based on the pattern-mixture model factorization. Our approach uses the empirical observed-data distribution and augments it with a nonparametric estimator of the missing-data distributions under a given identifying restriction. Our results apply to a large class of donor-based identifying restrictions that encompasses commonly used ones and can handle both monotone and nonmonotone missingness. We propose a Monte Carlo procedure to derive point estimates of functionals of interest, and the bootstrap to construct confidence intervals.

Motivation & Objective

• To develop a nonparametric framework for pattern-mixture models that handles both monotone and nonmonotone missingness.
• To unify and generalize donor-based identifying restrictions for use in nonparametric settings.
• To provide a computationally feasible method for estimating functionals of interest under missing data, such as average treatment effects.
• To establish theoretical validity of Monte Carlo estimation and bootstrap confidence intervals under nonparametric estimation.
• To enable sensitivity analysis by allowing multiple identifying restrictions to be evaluated under a single consistent framework.

Proposed method

• The method uses the empirical distribution of observed data as a base and nonparametrically estimates missing-data distributions using conditional kernel-density estimators.
• It introduces a surrogate observed-data distribution based on kernel smoothing to enable estimation under donor-based identifying restrictions.
• A Monte Carlo procedure is used to simulate from the estimated full-data distribution, enabling numerical computation of functionals like ATEs.
• The bootstrap is applied to construct valid 95% confidence intervals for functionals, with theoretical justification provided via Efron’s bootstrap diagram.
• The approach is designed to be robust: inferences depend only on observed data and a specified identifying restriction, even if the restriction is misspecified.
• Convergence rates and asymptotic normality are derived under regularity conditions, supporting inferential validity.

Experimental results

Research questions

• RQ1 How can nonparametric inference be conducted under pattern-mixture models when missingness is nonmonotone?
• RQ2 Can a unified framework for donor-based identification be extended to handle nonmonotone missing data?
• RQ3 What is the theoretical validity of Monte Carlo approximation for functionals under nonparametric pattern-mixture models?
• RQ4 How can bootstrap-based confidence intervals be constructed and justified under this nonparametric framework?
• RQ5 To what extent are inferences robust to misspecification of identifying restrictions?

Key findings

• The proposed Monte Carlo method for estimating functionals is theoretically valid, with convergence rates and asymptotic normality established under regularity conditions.
• The bootstrap procedure for constructing confidence intervals is valid, as demonstrated through theoretical proof and Efron’s bootstrap diagram.
• In a clinical trial on schizophrenia, all 95% bootstrap confidence intervals for the ATE comparing new vs. standard treatment at week 8 fell below zero, indicating strong evidence for superiority of the new treatment across all missing-data assumptions.
• The method successfully handles nonmonotone missingness, extending the applicability of pattern-mixture models beyond longitudinal monotone settings.
• The approach allows for sensitivity analysis by evaluating the same functional under multiple identifying restrictions (e.g., AC, 3NC, NC), with consistent conclusions across assumptions.
• Theoretical results show that inferences depend only on observed data and the specified identifying restriction, ensuring transparency and interpretability.