Conditional-cumulant-of-exposure method in logistic missing covariate regression

We consider estimation in logistic regression where some covariate variable s may be missing at random. Satten and Kupper (1993, Journal of the America n Statistical Association 88, 200-208) proposed estimating odds ratio param eters using methods based on the probability of exposure. By approximating a partial likelihood, we extend their idea and propose a method that estima tes the cumulant-generating function of the missing covariate given observe d covariates and surrogates in the controls. Our proposed method first esti mates some lower order cumulants of the conditional distribution of the uno bserved data and then solves a resulting estimating equation for the logist ic regression parameter. A simple version of the proposed method is to repl ace a missing covariate by the summation of its conditional mean and condit ional variance given observed data in the controls. We note that orle impor tant property of the proposed method is that, when the validation is only o n controls, a class of inverse selection probability weighted semiparametri c estimators cannot be applied because selection probabilities on cases are zeros. The proposed estimator performs well unless the relative risk param eters are large, even though it is technically inconsistent. Small-sample s imulations are conducted. We illustrate the method by an example of real da ta analysis.