Various process regression methods have been developed over the last 10-15 years. In the setting of our interest, the response is a binary indicator process representing the joint event of being alive and occupying a specific state. The process is indexed by time (e.g., time since diagnosis) and observed continuously. Data of this sort occur frequently in the study of chronic disease. A semiparametric multiplicative model is assumed for probability of being alive and in the (presumably transient) state of interest. Under the proposed methods, the regression parameter is estimated through a procedure which does not require estimating the baseline probability. Unlike the majority of process regression methods, the proposed methods accommodate multiple sources of censoring. We show that the regression parameter estimator is asymptotically normal, and that the baseline probability function estimator converges to a Gaussian process. Simulations demonstrate that our estimators have satisfactory finite sample performance. We apply our methods to end-stage liver disease (ESLD) data obtained from a national registry.