Optimal experimental designs for partial likelihood information

In a follow-up study the time-to-event may be censored either because of dropouts or the end of study is earlier. This situation is frequently modeled through a Cox-proportional hazards model including covariates, some of which are under the control of the experimenter. When the model is to be fitted to n observed times these are known and for each of them it is also known whether that time is censored or not. When the experiment is to be designed neither the observed times nor the information about whether a particular unit will be censored are known. For censoring some additional prior probability distribution has to be assumed. Thus, the design problem faces two sources of imprecision when the experiment is to be scheduled. On the one hand, the censored times are not known. On the other hand, there is uncertainty about occurrence of censoring. A prior probability distribution is needed for this. Moreover, the Cox partial likelihood instead of the full likelihood is usually considered for these models. A partial information matrix is built in this case and optimal designs are computed and compared with the optimal designs for the full likelihood information. The usual tools for computing optimal designs with full likelihood are no longer valid for partial information. Some general results are provided in order to deal with this new approach. An application to a simple case with two possible treatments is used to illustrate it. The partial information matrix depends on the parameters and therefore a sensitivity analysis is conducted in order to check the robustness of the designs for the choice of the nominal values of the parameters.