A proportional hazards model for arbitrarily censored and truncated data

Turnbull (1976, Journal of Royal Statistical Society, Series B 38, 290-295) proposed a method for nonparametric estimation of the distribution function when the data are incomplete because of censoring and truncation. However, as noted by Frydman (1994, Journal of Royal Statistical Society, Series B 56, 71-74), Turnbull's method has to be modified to accommodate both truncation and censoring. This paper presents a detailed correction of Turnbull's method and an extension to the regression analysis: a method of fitting the proportional hazards model for arbitrarily censored and truncated data is-developed. The method allows partial testing for zero regression coefficients. The test-can be performed using the likelihood ratio test or the Wald test. The methodology is applied to estimate the distribution of the induction time of patients diagnosed with transfusion-associated AIDS and to estimate the distribution of time from diabetes onset to development of diabetic nephropathy for insulin-dependent diabetics.