Multiple Imputation

Motivation

MI is most commonly used in conjunction with Bayesian imputation methods, in which samples drawn from the posterior distribution of the missing data given the observed data are used to fill in the missing values. However, as long as there is some element of randomness in the imputation process, one can imagine executing the process multiple times and storing the answers from each application (i.e. replication). The variance of a statistic of interest across these replications can then be calculated. This variance can be added to the “naive” estimate of variance (obtained by treating all imputed data as if they were observed) to produce a variance estimate for the statistic that reflects the uncertainty due to both sampling and imputation. That is the essence of multiple imputation.

Controversy

There is a long-standing heated debate within the community of survey research statisticians about the utility of MI for analyses unanticipated by the data publisher. It is easy to find examples where mechanical application of MI results in over- or undercorrection. Rubin has a theorem that identifies the class of [Page 490]imputation procedures that can be used in conjunction with MI to obtain asymptotically valid inferences for a given statistic. He labels such imputation procedures as “proper.” However, a series of debates in the 1990s, culminating in a trio of 1996 papers, demonstrated that proper imputation methods are difficult to construct. Moreover, an imputation procedure that is proper for one analysis might be improper for another analysis.

Basic Formulae

Suppose that the entire imputation process of choice is repeated m times and that all m imputed values are stored along with the reported data. Conceptually, the process produces m completed data sets representing m replicates of this process. If there were originally p columns with missing data, then there will be mp corresponding columns in the new multiply imputed dataset. The user then applies his or her full-sample analysis procedure of choice m times, once to each set of p columns. Suppose that is the point estimate of some parameter, θ based on the kth set of p columns. (The subscript, I, indicates employment of imputed data.) Also suppose that is the variance estimate for provided by the standard complex survey analysis software when applied to the kth set of p columns.

Assuming that the imputation method of choice has a stochastic component, such as imputation that is based on a linear regression model to predict imputed values from covariates, multiple imputations can be used to improve the point estimate and provide better leverage for variance estimation. Rubin's point estimate is and his variance estimate

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