Baseline Cumulative Hazard Function Research Articles

Efficient estimation of regression coefficients and baseline hazard under proportionality of conditional hazards

Efficient estimation of the regression parameter and the cumulative baseline hazard function under proportionality assumptions of the conditional hazards is considered in this paper. The cases of complete and incomplete information on the censoring indicators are studied. In both cases, a convenient reparametrization of the likelihood for a single observation places this problem in the setting investigated by Ritov and Wellner (1988). Their results on information bounds for the regression parameter and the cumulative baseline hazard function of the Cox model for the uncensored case apply here. The asymptotic properties of proposed estimators are derived and the estimators are shown to be asymptotically efficient.

Smoothed Cox regression

Nonparametric regression was shown by Beran and McKeague and Utikal to provide a flexible method for analysis of censored failure times and more general counting processes models in the presence of covariates. We discuss application of kernel smoothing towards estimation in a generalized Cox regression model with baseline intensity dependent on a covariate. Under regularity conditions we show that estimates of the regression parameters are asymptotically normal at rate root-n, and we also discuss estimation of the baseline cumulative hazard function and related parameters.

Regression on hazard ratios and cross ratios in multivariate failure time analysis

SUMMARY Cox model marginal survivor function and pairwise correlation models are specified for a multivariate failure time vector. The corresponding mean and covariance structure for the cumulative baseline hazard variates and standard baseline hazard function estimators are used to develop joint estimating equations for hazard ratio and correlation parameters, in the absence of censorship. Semiparametric models for pairwise survivor functions are required to generalise these equations to allow arbitrary right censorship. Under Clayton model bivariate distributions the resulting equations lead to joint estimators of hazard ratio and cross ratio parameters, and to inferences with useful and ready interpretation. For example, these estimates yield summary measures of pairwise dependency that have been adjusted for covariate effects on marginal hazard rates. Solutions to the proposed estimating equations are shown to be quite generally consistent and asymptotically normally distributed. Moderate sample size properties are examined in simulation studies, and illustrations are provided.

Efficient estimation for the proportional hazards model with interval censoring

The maximum likelihood estimator (MLE) for the proportional hazards model with case 1 interval censored data is studied. It is shown that the MLE for the regression parameter is asymptotically normal with $\sqrt{n}$ convergence rate and achieves the information bound, even though the MLE for the baseline cumulative hazard function only converges at $n^{1/3}$ rate. Estimation of the asymptotic variance matrix for the MLE of the regression parameter is also considered. To prove our main results, we also establish a general theorem showing that the MLE of the finite-dimensional parameter in a class of semiparametric models is asymptotically efficient even though the MLE of the infinite-dimensional parameter converges at a rate slower than $\sqrt{n}$. The results are illustrated by applying them to a data set from a tumorigenicity study.

Semiparametric Analysis of General Additive-Multiplicative Hazard Models for Counting Processes

The additive-multiplicative hazard model specifies that the hazard function for the counting process associated with a multidimensional covariate process $Z = (W^T, X^T)^T$ takes the form of $\lambda(t\mid Z) = g\{\beta^T_0 W(t)\} + \lambda_0(t)h\{\gamma^T_0X(t)\}$, where $\theta_0 = (\beta^T_0, \gamma^T_0)^T$ is a vector of unknown regression parameters, $g$ and $h$ are known link functions and $\lambda_0$ is an unspecified hazard function. In this paper, we develop a class of simple estimating functions for $\theta_0$, which contains the partial likelihood score function in the special case of proportional hazards models. The resulting estimators are shown to be consistent and asymptotically normal under appropriate regularity conditions. Weak convergence of the Aalen-Breslow type estimators for the cumulative baseline hazard function $\Lambda_0(t) = \int^t_0\lambda_0(u) du$ is also established. Furthermore, we construct adaptive estimators for $\theta_0$ and $\Lambda_0$ that achieve the (semiparametric) information bounds. Finally, a real example is provided along with some simulation results.

Martingale-based residuals for survival models

SUMMARY Graphical methods based on the analysis of residuals are considered for the setting of the highly-used Cox (1972) regression model and for the Andersen-Gill (1982) generalization of that model. We start with a class of martingale-based residuals as proposed by Barlow & Prentice (1988). These residuals and/or their transforms are useful for investigating the functional form of a covariate, the proportional hazards assumption, the leverage of each subject upon the estimates of 13, and the lack of model fit to a given subject. 1 1. Model Consider a set of n independent subjects such that the counting process Ni {Ni(t), t } O} for the ith subject in the set indicates the number of observed events experienced over time t. The sample paths of the Ni are step functions with jumps of size +1 and with Ni(0) =0. We assume that the intensity function for Ni(t) is given by Yi(t)dA{t, Zi(t)} = Yi(t) eP'Z(t) dAO(t), (1) where Yi(t) is a 0-1 process indicating whether the ith subject is a risk at time t, 13 is a vector of regression coefficients, Zi(t) is a p dimensional vector of covariate processes, and Ao is the baseline cumulative hazard function. Several familar survival models fit into this framework. The Andersen & Gill (1982) generalization of the Cox (1972) model arises when AO(t) is completely unspecified. The further restriction that Yi(t) = 1 until the first event or censoring, and is 0 thereafter yields the Cox model. The parametric form AO(t) = t yields a Poisson model, or an exponential if restricted to a single event per subject, and AO(t) = tP a Weibull model. Our attention will focus primarily on the Andersen-Gill and Cox models; however, the