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.