  We propose a copula based method to handle missing values in multivariate
data of mixed types in multilevel data sets. Building upon the extended rank
likelihood of \cite{hoff2007extending} and the multinomial probit model, our
model is a latent variable model which is able to capture the relationship
among variables of different types as well as accounting for the clustering
structure. We fit the model by approximating the posterior distribution of the
parameters and the missing values through a Gibbs sampling scheme. We use the
multiple imputation procedure to incorporate the uncertainty due to missing
values in the analysis of the data. Our proposed method is evaluated through
simulations to compare it with several conventional methods of handling missing
data. We also apply our method to a data set from a cluster randomized
controlled trial of a multidisciplinary intervention in acute stroke units. We
conclude that our proposed copula based imputation model for mixed type
variables achieves reasonably good imputation accuracy and recovery of
parameters in some models of interest, and that adding random effects enhances
performance when the clustering effect is strong.
