Multiple imputation of missing covariates with non-linear effects and interactions: an evaluation of statistical methods

Abstract

Background: Multiple imputation is often used for missing data. When a model contains as covariates more than one function of a variable, it is not obvious how best to impute missing values in these covariates. Consider a regression with outcome Y and covariates X and X2. In 'passive imputation' a value X* is imputed for X and then X2 is imputed as (X*)2. A recent proposal is to treat X2 as 'just another variable' (JAV) and impute X and X2 under multivariate normality.

Methods: We use simulation to investigate the performance of three methods that can easily be implemented in standard software: 1) linear regression of X on Y to impute X then passive imputation of X2; 2) the same regression but with predictive mean matching (PMM); and 3) JAV. We also investigate the performance of analogous methods when the analysis involves an interaction, and study the theoretical properties of JAV. The application of the methods when complete or incomplete confounders are also present is illustrated using data from the EPIC Study.

Results: JAV gives consistent estimation when the analysis is linear regression with a quadratic or interaction term and X is missing completely at random. When X is missing at random, JAV may be biased, but this bias is generally less than for passive imputation and PMM. Coverage for JAV was usually good when bias was small. However, in some scenarios with a more pronounced quadratic effect, bias was large and coverage poor. When the analysis was logistic regression, JAV's performance was sometimes very poor. PMM generally improved on passive imputation, in terms of bias and coverage, but did not eliminate the bias.

Conclusions: Given the current state of available software, JAV is the best of a set of imperfect imputation methods for linear regression with a quadratic or interaction effect, but should not be used for logistic regression.

Figures

Figure 1

Typical datasets for normally or log-normally distributed X (each with mean 2 and variance 1), normally distributed Y with mean 2X + X2 or (X - 2)2, and R2 = 0.1, 0.5 or 0.8. Dotted line shows expected value of Y given X.

Figure 2

Log plasma vitamin C and log dietary vitamin C in 15415 individuals for whom both variables are observed.