Parametric fractional imputation for mixed models with nonignorable missing data

Inference in the presence of non-ignorable missing data is a widely encountered and difficult problem in statistics. Imputation is often used to facilitate parameter estimation, which allows one to use the complete sample estimators on the imputed data set. We develop a parametric fractional imputation (PFI) method proposed by Kim (2011), which simplifies the computation associated with the EM algorithm for maximum likelihood estimation with missing data. We first consider the problem of parameter estimation for linear mixed models with non-ignorable missing values, which assumes that missingness depends on the missing values only through the random effects, leading to shared parameter models (Follmann and Wu, 1995). In the M-step, the restricted or adjusted profiled maximum likelihood method is used to reduce the bias of maximum likelihood estimation of the variance components. Results from a limited simulation study are presented to compare the proposed method with the existing methods, which demonstrates that imputation can significantly reduce the non-response bias and the idea of adjusted profiled maximum likelihood works nicely in PFI for the bias correction in estimating the variance components. Variance estimation is also discussed. We next extend PFI to generalized linear mixed model and the flexibility of this method is illustrated by analyzing the infamous salamander mating data (McCullagh and Nelder, 1989).