Reduced rank proportional hazards model for competing risks

Competing events concerning individual subjects are of interest in many medical studies. For example, leukemia-free patients surviving a bone marrow transplant are at risk of developing acute or chronic graft-versus-host disease, or they might develop infections. In this situation, competing risks models provide a natural framework to describe the disease. When incorporating covariates influencing the transition intensities, an obvious approach is to use Cox's proportional hazards model for each of the transitions separately. A practical problem then is how to deal with the abundance of regression parameters. Our objective is to describe the competing risks model in fewer parameters, both in order to avoid imprecise estimation in transitions with rare events and in order to facilitate interpretation of these estimates. Suppose that the regression parameters are gathered into a p x K matrix B, with p and K as the number of covariates and transitions, respectively. We propose the use of reduced rank models, where B is required to be of lower rank R, smaller than both p and K. One way to achieve this is to write B = A Gamma(inverted perpendicular) with A and Gamma matrices of dimensions p x R and K x R, respectively. We shall outline an algorithm to obtain estimates and their standard errors in a reduced rank proportional hazards model for competing risks and illustrate the approach on a competing risks model applied to 8966 leukemia patients from the European Group for Blood and Marrow Transplantation.