Semiparametric model for regression analysis with nonmonotone missing data

Semiparametric likelihoods for regression models with missing at random data (Chen in J Am Stat Assoc 99:1176–1189, 2004, Zhang and Rockette in J Stat Comput Simul 77(2):163–173, 2007, Zhao et al. in Biom J 51: 123–136, 2009, Zhao in Commun Stat Theory Methods 38:3736–3744, 2009) are robust as they use nonparametric models for covariate distributions and do not require modeling the missing data probabilities. Furthermore, the EM algorithms based on the semiparametric likelihoods have closed form expressions for both E-step and M-step. As far as we know the semiparametric likelihoods can only deal with the simple monotone missing data pattern. In this research we extend the semiparemetric likelihood approach to deal with regression models with arbitrary nonmonotone missing at random data. We propose a pseudo-likelihood model, which uses an empirical distribution to model the conditional distribution of missing covariates given observed covariates for each missing data pattern separately. We show that an EM algorithm with closed form updating formulas can be used for computing maximum pseudo-likelihood estimates for regression models with nonmonotone missing data. We then propose estimating the asymptotic variance of the maximum pseudo-likelihood estimator through a profile log likelihood and the EM algorithm. We examine the finite sample performance of the new methods in simulation studies and further illustrate the methods in a real data example investigating high risk gambling behavior and the associated factors.