Standard procedures for drawing inferences from complex samples do not apply when the variable of interest θ cannot be observed directly, but must be inferred from the values of secondary random variables that depend on θ stochastically. Examples are proficiency variables in item response models and class memberships in latent class models. Rubin's “multiple imputation” techniques yield approximations of sample statistics that would have been obtained, had θ been observable, and associated variance estimates that account for uncertainty due to both the sampling of respondents and the latent nature of θ. The approach is illustrated with data from the National Assessment for Educational Progress.

Rubin's “multiple imputation” approach to missing data creates synthetic data sets, in which each missing variable is replaced by a draw from its predictive distribution, conditional on the observed data. By construction, analyses of such filled-in data sets as if the imputations were true values have the correct expectations for population parameters. In a recent paper, Mislevy showed how this approach can be applied to estimate the distributions of latent variables from complex samples. Multiple imputations for a latent variable bear a surface similarity to classical “multiple indicators” of a latent variable, as might be addressed in structural equation modelling or hierarchical modelling of successive stages of random sampling. This note demonstrates with a simple example why analyzing “multiple imputations” as if they were “multiple indicators” does not generally yield correct results; they must instead be analyzed by means concordant with their construction.