  Multiple imputation is a well-established general technique for analyzing
data with missing values. A convenient way to implement multiple imputation is
sequential regression multiple imputation (SRMI), also called chained equations
multiple imputation. In this approach, we impute missing values using
regression models for each variable, conditional on the other variables in the
data. This approach, however, assumes that the missingness mechanism is missing
at random, and it is not well-justified under not-at-random missingness without
additional modification. In this paper, we describe how we can generalize the
SRMI imputation procedure to handle not-at-random missingness (MNAR) in the
setting where missingness may depend on other variables that are also missing.
We provide algebraic justification for several generalizations of standard SRMI
using Taylor series and other approximations of the target imputation
distribution under MNAR. Resulting regression model approximations include
indicators for missingness, interactions, or other functions of the MNAR
missingness model and observed data. In a simulation study, we demonstrate that
the proposed SRMI modifications result in reduced bias in the final analysis
compared to standard SRMI, with an approximation strategy involving inclusion
of an offset in the imputation model performing the best overall. The method is
illustrated in a breast cancer study, where the goal is to estimate the
prevalence of a specific genetic pathogenic variant.
