| The case cohort sampling design is a popular method used to evaluate possible relationships between exposure and disease observed in a cohort followed over time. One advantage of the case-cohort sampling scheme is that data need not be collected on the entire cohort; such a procedure would be costly and difficult. This savings can only be gained at the price of some loss of efficiency relative to the full cohort design. This comparison also raises the question of how efficient the case cohort design is in an absolute sense, that is, does the usual maximum partial likelihood estimator achieve the theoretical limit? Using the concepts of Hellinger differentiability to compute the proper score functions for both the parametric and the non-parametric parts of the model, the effective score for the parametric part of interest can be obtained by determining the component of the parametric score orthogonal to the space generated by the infinite dimensional nuisance parameter. By this technique, asymptotic variance lower bounds for estimation of the parameter theta can be calculated in both the full cohort model, with general relative risk function r(thetaz), and for the case-cohort sampling design in the Cox model with relative risk ethetaz , where z is the scalar covariate value of an individual included in the cohort. We show that the maximum partial likelihood estimator for the full cohort model is efficient while the maximum pseudolikelihood estimator for the case-cohort sampling design is inefficient. A further discussion on the asymptotic efficiency of the case-cohort design is also provided. |
