Identifiability of Normal and Normal Mixture Models With Nonignorable Missing Data

TL;DR

This paper investigates the conditions under which normal and mixture models are identifiable when data are missing in a nonignorable way, providing theoretical results, conditions, and practical illustrations.

Contribution

It establishes identifiability results for normal and mixture models with nonignorable missing data, including conditions for Logistic and Probit mechanisms.

Findings

Normal distribution is identifiable under monotone missing mechanisms.

Identifiability extends to normal and t mixture models with non-monotone missing data.

Logistic missing mechanisms are less identifiable than Probit mechanisms.

Abstract

Missing data problems arise in many applied research studies. They may jeopardize statistical inference of the model of interest, if the missing mechanism is nonignorable, that is, the missing mechanism depends on the missing values themselves even conditional on the observed data. With a nonignorable missing mechanism, the model of interest is often not identifiable without imposing further assumptions. We find that even if the missing mechanism has a known parametric form, the model is not identifiable without specifying a parametric outcome distribution. Although it is fundamental for valid statistical inference, identifiability under nonignorable missing mechanisms is not established for many commonly-used models. In this paper, we first demonstrate identifiability of the normal distribution under monotone missing mechanisms. We then extend it to the normal mixture and $t$ mixture models with non-monotone missing mechanisms. We discover that models under the Logistic missing mechanism are less identifiable than those under the Probit missing mechanism. We give necessary and sufficient conditions for identifiability of models under the Logistic missing mechanism, which sometimes can be checked in real data analysis. We illustrate our methods using a series of simulations, and apply them to a real-life dataset.