Robustness of a multivariate normal approximation for imputation of incomplete binary data

Multiple imputation has become easier to perform with the advent of several software packages that provide imputations under a multivariate normal model, but imputation of missing binary data remains an important practical problem. Here, we explore three alternative methods for converting a multivariate normal imputed value into a binary imputed value: (1) simple rounding of the imputed value to the nearer of 0 or 1, (2) a Bernoulli draw based on a 'coin flip' where an imputed value between 0 and 1 is treated as the probability of drawing a 1, and (3) an adaptive rounding scheme where the cut-off value for determining whether to round to 0 or 1 is based on a normal approximation to the binomial distribution, making use of the marginal proportions of 0's and 1's on the variable. We perform simulation studies on a data set of 206,802 respondents to the California Healthy Kids Survey, where the fully observed data on 198,262 individuals defines the population, from which we repeatedly draw samples with missing data, impute, calculate statistics and confidence intervals, and compare bias and coverage against the true values. Frequently, we found satisfactory bias and coverage properties, suggesting that approaches such as these that are based on statistical approximations are preferable in applied research to either avoiding settings where missing data occur or relying on complete-case analyses. Considering both the occurrence and extent of deficits in coverage, we found that adaptive rounding provided the best performance.