A Proportional Hazards Model for Arbitrarily Censored and Truncated Data

Abstract

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.

Multivariate survival analysis with doubly‐censored data: application to the assessment of Accutane treatment for fibrodysplasia ossificans progressiva

“…Several authors [10][11][12] have examined the estimation of semi-parametric proportional hazard models with doubly-censored or interval-censored data, but time-dependent covariates seem to present a special challenge in this context. Recent work by Goggins et al [13] applies the Cox model for an interval-censored time-dependent covariate to data which is exact or right-censored.…”

Nonparametric tests for left-truncated and interval-censored data

“…Without loss of generality, suppose the observed data are ordered according to L ij such that L i1 ≤ L i2 ≤ · · · ≤ L in i . Based on Turnbull [27] (also see Frydman [28] and Alioum and Commenges [29]), we consider nonparametric estimation of F i (t) using the n i independent pairs…”

Turnbull Estimator

“…The form of (1) indicates that L(F ) will be maximized when the values of F (x) are as large as possible for x ∈ R and as small as possible for x ∈ L subject to the constraint that F is the distribution function. Accord-…”

“…Other generalizations are given by De Gruttola et al [8], Sun [25], and Frydman [15]. The generalizations by Kim et al [19], Tu et al [26], and Alioum & Commenges [1] incorporate covariates.…”