Nested multiple imputation in large-scale assessments

Background: In order to measure the proficiency of person populations in various domains, large-scale assessments often use marginal maximum likelihood IRT models where person proficiency is modelled as a random variable. Thus, the model does not provide proficiency estimates for any single person. A popular approach to derive these proficiency estimates is the multiple imputation of plausible values (PV) to enable subsequent analyses on complete data sets. The main drawback is that all variables that are to be analyzed later have to be included in the imputation model to allow the distribution of plausible values to be conditional on these variables. These background variables (e.g., sex, age) have to be fully observed which is highly unlikely in practice. In several current large-scale assessment programs missing observations on background variables are dummy coded, and subsequently, dummy codes are used additionally in the PV imputation model. However, this approach is only appropriate for small proportions of missing data. Otherwise the resulting population scores may be biased. Methods: Alternatively, single imputation or multiple imputation methods can be used to account for missing values on background variables. With both imputation methods, the result is a two-step procedure in which the PV imputation is nested within the background variable imputation. In the single+multiple-imputation (SMI), each missing value on background variables is replaced by one value. In the multiple+multiple-imputation (MMI), each missing value is replaced by a set of imputed values. MMI is expected to outperform SMI as SMI ignores the uncertainty due to missing values in the background data. Results: In a simulation study, both methods yielded unbiased population estimates under most conditions. Still, the recovery proportion was slightly higher for the MMI method. Conclusions: The advantages of the MMI method are apparent for fairly high proportions of missing values in combination with fairly high dependency between the latent trait and the probability of missing data on background variables.