Envelope Methods with Ignorable Missing Data

TL;DR

This paper extends the envelope method for multivariate regression to handle missing data using an EM algorithm, improving efficiency and reducing bias, with proven asymptotic properties and demonstrated benefits in simulations and real data.

Contribution

It introduces a novel EM algorithm incorporating envelope structure for missing data, addressing bias and efficiency issues in multivariate regression.

Findings

The method is more efficient than standard EM.

It outperforms full data MLE in simulations.

Asymptotic properties are established under various distributions.

Abstract

Envelope method was recently proposed as a method to reduce the dimension of responses in multivariate regressions. However, when there exists missing data, the envelope method using the complete case observations may lead to biased and inefficient results. In this paper, we generalize the envelope estimation when the predictors and/or the responses are missing at random. Specifically, we incorporate the envelope structure in the expectation-maximization (EM) algorithm. As the parameters under the envelope method are not pointwise identifiable, the EM algorithm for the envelope method was not straightforward and requires a special decomposition. Our method is guaranteed to be more efficient, or at least as efficient as, the standard EM algorithm. Moreover, our method has the potential to outperform the full data MLE. We give asymptotic properties of our method under both normal and non-normal cases. The efficiency gain over the standard EM is confirmed in simulation studies and in an application to the Chronic Renal Insufficiency Cohort (CRIC) study.