A dynamic approach for reconstructing missing longitudinal data using the linear increments model

Abstract

Missing observations are commonplace in longitudinal data. We discuss how to model and analyze such data in a dynamic framework, that is, taking into consideration the time structure of the process and the influence of the past on the present and future responses. An autoregressive model is used as a special case of the linear increments model defined by Farewell (2006. Linear models for censored data, [PhD Thesis]. Lancaster University) and Diggle and others (2007. Analysis of longitudinal data with drop-out: objectives, assumptions and a proposal. Journal of the Royal Statistical Society, Series C (Applied Statistics, 56, 499-550). We wish to reconstruct responses for missing data and discuss the required assumptions needed for both monotone and nonmonotone missingness. The computational procedures suggested are very simple and easily applicable. They can also be used to estimate causal effects in the presence of time-dependent confounding. There are also connections to methods from survival analysis: The Aalen-Johansen estimator for the transition matrix of a Markov chain turns out to be a special case. Analysis of quality of life data from a cancer clinical trial is analyzed and presented. Some simulations are given in the supplementary material available at Biostatistics online.

Fig. 1.

A Markov model for generating longitudinal data. This consists of both observed states (1–3) and unobserved states (4–6). Subjects in the unobserved states constitute the missing ones. Possible transitions between various states in the model are indicated by arrows.

Fig. 2.

Observed and estimated mean scores of item 30 (quality of life), as functions of weeks since treatment onset in the 2 treatment arms. The estimates are computed by the compensator and the imputation approach.

Fig. 3.

The empirical variance of the mean score estimates of item 30 (quality of life), based on 1000 bootstrap samples, as functions of weeks since treatment onset.

Fig. 4.

Least squares estimates of the effects of baseline covariates on the increment score of item 30 (quality of life). The 95% confidence intervals are computed by bootstrap.

Fig. 5.

Least squares estimates of the effects of previous responses on the increment score of item 30 (quality of life). The 95% confidence intervals are computed by bootstrap.