A New Procedure to Test Mediation With Missing Data Through Nonparametric Bootstrapping and Multiple Imputation

Abstract

This article proposes a new procedure to test mediation with the presence of missing data by combining nonparametric bootstrapping with multiple imputation (MI). This procedure performs MI first and then bootstrapping for each imputed data set. The proposed procedure is more computationally efficient than the procedure that performs bootstrapping first and then MI for each bootstrap sample. The validity of the procedure is evaluated using a simulation study under different sample size, missing data mechanism, missing data proportion, and shape of distribution conditions. The result suggests that the proposed procedure performs comparably to the procedure that combines bootstrapping with full information maximum likelihood under most conditions. However, caution needs to be taken when using this procedure to handle missing not-at-random or nonnormal data.

Notes

^a BCCI = biased corrected confidence interval.

^b The computing times were recorded on a 2.3 GHz quad-core PC.

¹The sufficient number of imputations depends on the fraction of missing information (FMI) on the parameter(s) of interest and the criterion to be considered. Three criteria have been considered in the past: relative efficiency (the standard error should be close to the standard error with K = infinite), relative power (the power to detect an effect should be close to that from K = infinite), and reproducibility (repeat MI on the same data should reproduce the same results). A small number of imputations (e.g., K = 5) would be sufficient if the purpose is just to achieve efficient parameter estimates (Rubin, 1987). However, more imputations are needed to prevent power loss and Monte Carlo error. Graham et al. (2007) showed that 100 imputations yield power that is similar to that from FIML, even when FMI = .90. White et al. (2011) suggested a rule of thumb that considers repeatability. Their criterion says that K should be ≥ 100 × FMI. That is, if FMI = .50, then at least 50 imputed data sets should be obtained.

^a Averaged across sample sizes. MCAR = missing completely at random; B(FIML) = FIML nested within bootstrapping; MI(B) = bootstrapping nested within MI.

^a Averaged across sample sizes.

^a Averaged across sample size.

²To investigate the extent of this dependency, we computed the intraclass correlation for normal MAR data with 30% missing data and moderate mediation. The intraclass correlation coefficient was about .26, which suggests that the parameter estimates from the same imputed data set are correlated.