Inference Progress in Missing Data Analysis from Independent to Longitudinal Setup

In the independent setup with multivariate responses, the data become incomplete when partial responses, such as responses on some variables as opposed to all variables, are available from some individuals. The main challenge here is obtaining valid inferences such as unbiased and consistent estimates of mean parameters of all response variables by using available responses. Typically, unbalanced correlation matrices are formed and moments or likelihood analysis based on the available responses are employed for such inferences. Various imputation techniques also have been used. In the longitudinal setup, when a univariate response is repeatedly collected from an individual, these repeated responses become correlated and the responses form a multivariate distribution. In this setup, it may happen that a portion of responses are not available from some individuals under study. These non-responses may be monotonic or intermittent. Also the response may be missing following a mechanism such as missing completely at random (MCAR), missing at random (MAR), or missing non-ignorably. In a longitudinal regression setup, the covariates may also be missing, but typically they are known for all time periods. Obtaining unbiased and consistent regression estimates specially when longitudinal responses are missing following MAR or ignorable mechanism becomes a challenge. This happens because one requires to accommodate both longitudinal correlations and missing mechanism to develop a proper inference tool. Over the last three decades some progress has been made toward this mainly by taking partial care of missing mechanism in developing estimation techniques. But overall, they fall short and may still produce biased and hence inconsistent estimates. The purpose of this paper is to outline these perspectives in a comprehensive manner so that real progress and challenges are understood in order to develop proper inference techniques.