Updating latent class imputations with external auxiliary variables

Latent class models are often used to assign values to categorical variables that cannot be measured directly. This "imputed" latent variable is then used in further analyses with auxiliary variables. The relationship between the imputed latent variable and auxiliary variables can only be correctly estimated if these auxiliary variables are included in the latent class model. Otherwise, point estimates will be biased. We develop a method that correctly estimates the relationship between an imputed latent variable and external auxiliary variables, by updating the latent variable imputations to be conditional on the external auxiliary variables using a combination of multiple imputation of latent classes and the so-called three-step approach. In contrast with existing "one-step" and "three-step" approaches, our method allows the resulting imputations to be analyzed using the familiar methods favored by substantive researchers.