A closer examination of three small-sample approximations to the multiple-imputation degrees of freedom, erratum

Abstract

Incomplete data is a common complication in applied research. In this study, we use simulation to compare two approaches to the multiple imputation of a continuous predictor: multiple imputation through chained equations and multivariate normal imputation. This study extends earlier work by being the first to 1) compare the small-sample approximations to the multiple-imputation degrees of freedom proposed by Barnard and Rubin (1999, Biometrika 86: 948- 955); Lipsitz, Parzen, and Zhao (2002, Journal of Statistical Computation and Simulation 72: 309-318); and Reiter (2007, Biometrika 94: 502-508) and 2) ask if the sampling distribution of the t statistics is in fact a Student's t distribution with the specified degrees of freedom. In addition to varying the imputation method, we varied the number of imputa- tions (m =5 , 10, 20, 100) that were averaged over 500,000 replications to obtain the combined estimates and standard errors for a linear model that regressed the log price of a home on its age (years) and size (square feet) in a sample of 25 ob- servations. Six age values were randomly set equal to missing for each replication. As assessed by the absolute percentage and relative percentage bias, the two ap- proaches performed similarly. The absolute bias of the regression coefficients for age and size was roughly −0.1% across the levels of m for both approaches; the ab- solute bias for the constant was 0.6% for the chained-equations approach and 1.0% for the multivariate normal model. The absolute biases of the standard errors for age, size, and the constant were 0.2%, 0.3%, and 1.2%, respectively. In general, the relative percentage bias was slightly smaller for the chained-equations approach. Graphical and numerical inspection of the empirical sampling distributions for the three t statistics suggested that the area from the shoulder to the tail was reasonably well approximated by a t distribution and that the small-sample ap- proximations to the multiple-imputation degrees of freedom proposed by Barnard and Rubin and by Reiter performed satisfactorily.