Defining, evaluating, and removing bias induced by linear imputation in longitudinal clinical trials with MNAR missing data.

Missing not at random (MNAR) post-dropout missing data from a longitudinal clinical trial result in the collection of "biased data," which leads to biased estimators and tests of corrupted hypotheses. In a full rank linear model analysis the model equation, E[Y] = Xß, leads to the definition of the primary parameter ß = (X'X)(-1)X'E[Y], and the definition of linear secondary parameters of the form θ = Lß = L(X'X)(-1)X'E[Y], including, for example, a parameter representing a "treatment effect." These parameters depend explicitly on E[Y], which raises the questions: What is E[Y] when some elements of the incomplete random vector Y are not observed and MNAR, or when such a Y is "completed" via imputation? We develop a rigorous, readily interpretable definition of E[Y] in this context that leads directly to definitions of ß, Bias(ß) = E[ß] - ß, Bias(θ) = E[θ] - Lß, and the extent of hypothesis corruption. These definitions provide a basis for evaluating, comparing, and removing biases induced by various linear imputation methods for MNAR incomplete data from longitudinal clinical trials. Linear imputation methods use earlier data from a subject to impute values for post-dropout missing values and include "Last Observation Carried Forward" (LOCF) and "Baseline Observation Carried Forward" (BOCF), among others. We illustrate the methods of evaluating, comparing, and removing biases and the effects of testing corresponding corrupted hypotheses via a hypothetical but very realistic longitudinal analgesic clinical trial.