Baseline Hazard Rate Research Articles

Stochastic comparisons in frailty models

The frailty model in survival analysis accounts for unobserved heterogeneity between individuals by assuming that the hazard rate of an individual is the product of an individual specific quantity, called “frailty” and a baseline hazard rate. It is well known that the choice of the frailty distribution strongly affects the nonparametric estimate of the baseline hazard as well as that of the conditional probabilities. This paper reviews the basic concepts of a frailty model, presents various probability inequalities and other monotonicity results which may prove useful in choosing among alternative specifications. More specifically, our main result lies in seeing how well known stochastic orderings between distributions of two frailities translate into orderings between the corresponding survival functions. Some probabilistic aspects and implications of the models resulting from competing choices of the distributions of frailty or the baseline are compared.

Tumour Incidence, Prevalence and Lethality Estimation in the Absence of Cause-of-Death Information

Summary A Bayesian intensity model is presented for studying a bioassay problem involving interval-censored tumour onset times, and without discretization of times of death. Both tumour lethality and base-line hazard rates are estimated in the absence of cause-of-death information. Markov chain Monte Carlo methods are used in the numerical estimation, and sophisticated group updating algorithms are applied to achieve reasonable convergence properties. This method was tried on the rat tumorigenicity data that have previously been analysed by Ahn, Moon and Kodell, and our results seem to be more realistic.

An Exact Corrected Log‐Likelihood Function for Cox's Proportional Hazards Model under Measurement Error and Some Extensions

Abstract. This paper studies Cox's proportional hazards model under covariate measurement error. Nakamura's [Biometrika 77 (1990) 127] methodology of corrected log‐likelihood will be applied to the so‐called Breslow likelihood, which is, in the absence of measurement error, equivalent to partial likelihood. For a general error model with possibly heteroscedastic and non‐normal additive measurement error, corrected estimators of the regression parameter as well as of the baseline hazard rate are obtained. The estimators proposed by Nakamura [Biometrics 48 (1992) 829], Kong et al. [Scand. J. Statist. 25 (1998) 573] and Kong & Gu [Statistica Sinica 9 (1999) 953] are re‐established in the special cases considered there. This sheds new light on these estimators and justifies them as exact corrected score estimators. Finally, the method will be extended to some variants of the Cox model.

BITE: A Bayesian Intensity Estimator

BITE is a software package designed for the analysis of event history data using flexible hierarchical models and Bayesian inference, with a particular emphasis on the application of flexible intensities as a description of the distribution of lifetimes. BITE provides a framework for combining flexible baseline hazard rates and observed data into intensity processes. Inclusion of covariate information is possible, and data can be non-informatively and independently filtered, or censored. The model and the data are described by a command language and data are stored into text files. Markov chain Monte Carlo methods are used for numerical approximation of expectations with respect to the posterior. Output consists of (i) parameter values stored during simulations, (ii) estimated expectations of functionals of parameters, or (iii) graphs (created with Splus or R software packages) presenting point-wise expectations (and credibility intervals) of the baseline hazard rates.

A Novel Method For Deriving Intersubject Variability For Baseline Hazard Rates and Relative Risks For Secondary Vascular Events

Clinical Pharmacology & Therapeutics (2003) 73, P88–P88; doi:

Smooth Goodness-of-Fit Tests for the Baseline Hazard in Cox's Proportional Hazards Model

For the Cox proportional hazards model, the goodness-of-fit problem of testing whether the baseline hazard rate function λ(·) equals some specified hazard rate function λ0(·) in the presence of incomplete data is considered. Extending Neyman's idea, the goodness-of-fit tests are score tests obtained by embedding λ(·) in a larger family of hazard rate functions developed through smooth and possibly random transformations of λ0(·). But whereas their smooth alternatives were formulated through the density function, my formulation is through the hazard rate function, which is more natural in failure time models where model specification is now done through hazard rates. Furthermore, my smooth alternatives are allowed to be data driven or dynamic. Through this machinery, classes of omnibus, as well as directional goodness-of-fit tests are obtained. Moreover, this machinery provides a theoretically sound method for combining varied directional goodness-of-fit tests into omnibus tests. Some of the resulting tests are specialized to the randomly censored model. Finite and asymptotic properties of the resulting tests are presented. The asymptotic results shed light into properties of generalized residuals typically used in model validation and diagnostics. The theoretical results indicate the adjustments that need to be made to properly use these residuals for goodness of fit and model validation. Two medical datasets are reanalyzed using the proposed procedures.

Trends in the incidence rate and risk factors for breast cancer in Japan.

The incidence rate of breast cancer in Japan rose more than two-fold from 1959-60 to 1983-87. To assess to what extent this increase can be explained by changes in the prevalence of four major risk factors of breast cancer (i.e. age at menarche, age at first birth, age at menopause, and parity), we estimated the probability of developing breast cancer based on the joint distribution and relative risks of these four risk factors. The age-specific incidence rate during 1959-60 reported by the Miyagi Prefectural Cancer Registry was used to estimate the baseline hazard rate for women without the four risk factors in the same age group. Assuming that the baseline hazard rate is constant during all periods, we calculated the expected incidence rates during the periods of 1959-60, 1962-64, 1968-71, 1973-77, 1978-81, and 1983-87 for each age group. Large discrepancies were noted between the observed and expected incidence rates during 1983-87 in all age groups. The change in the joint distribution of the four risk factors accounted for less than 40% of the increase observed from 1959-60 to 1983-87, suggesting the effects of other powerful risk factors.

A Frailty Mixture Model for Estimating Vaccine Efficacy

SUMMARY Vaccines can have heterogeneous effects on the host's immune response. Somevaccinated people may have a strong response and be highly protected against infectious challenge, but others may only denve varying levels of partial protection. We derive a statistical model for estimating vaccine efficacy that expresses the often unmeasured heterogeneous host response and other vaccine effects in terms of estimable parameters. In addition, the-model incorporates the infection process into the base-line hazard rate. The model falls into the general category of frailty models (with point mass at 0) employed in survival analysis. As an example, we estimate the efficacy of a measles vaccine from a measles outbreak in Burundi.

Power considerations for testing an interaction in a 2 × k factorial design with a failure time outcome

We examine the power and sample size requirements for testing an interaction in the situation of a 2 × k factorial design with time to failure as the outcome of interest. Using the distribution of a general test statistic, based on a weighted residual sum of squares for testing a general interaction in a 2 × k factorial experiment, we describe the relationship between the power of the test and the size of the sample. In a Simulation study, we evaluate the behavior of three commonly used estimators as methods for estimating the parameters of the test statistic. These are the Mantel-Haenszel (MH) method, a method (O/E) based on the ratio of the observed to expected number of events, and the maximum likelihood (MLE) method. We show that in most cases nominal test sizes and appropriate powers are attained using the MLE and MH methods, whereas the O/E method yielded test sizes and powers less than expected. With both large baseline hazard rates and large differences in relative hazard rates, the difference between the simulated and asymptotic powers for all three methods become larger; however, the size of this difference is small and unlikely to seriously affect the use of either the MLE or MH methods. The proposed methods could also be used to calculate the power and sample size for testing a treatment-covariate interaction in a stratified data analysis.

Nonparametric kernel estimation in counting processes with explanatory variables

SUMMARY Cox's proportional hazards model assumes that the duration hazard rate factors into a product of a baseline hazard rate and a nonnegative function of explanatory variables. An estimate of the baseline hazard rate, hence also of overall hazard rate, is proposed, based on a smoothing procedure using a kernel function. It is shown that the proposed estimator is uniformly consistent and converges weakly to a Gaussian process. Bandwidth selection issues are also discussed.

Econometric estimation of proportional hazard models

When analyzing duration data, covariates are typically assumed to modify hazard rates through the use of the proportional hazard model, in which the baseline hazard is multiplied by some function of the regressors and associated parameters, g(·), to determine individual hazard rates. The specification for g(·) which is in virtually universal use, g(·) = exp(·), may be overly restrictive. This paper considers one alternative—setting g(·) equal to the cumulative density function of the standard normal distribution. The relative performance of this functional form for g(·), compared to the exponential transformation, is shown to depend on the parametrization of the baseline hazard rate.

Local Hazard Models

"We introduce a class of local hazard models that maintain parametric assumptions locally rather than globally. These models allow estimation of a flexible baseline hazard rate and nonproportional covariate effects. Recently developed local likelihood methods, which generalize maximum likelihood methods, can be used to estimate these models. We illustrate these techniques by estimating two local hazard models of first marriage--a local exponential model and a local Gompertz model--from data in the June 1980 [U.S.] Current Population Survey. We also compare results of the local hazard models with those of Cox's model, an exponential model, a piecewise exponential model, and a piecewise Gompertz model. Estimates for the local exponential and local Gompertz models agree closely with nonparametric estimates for various subgroups. Moreover, results of the local models provide interesting substantive insights into the process of first marriage that are not easily obtained from global models."

Projecting individualized probabilities of developing breast cancer for white females who are being examined annually.

To assist in medical counseling, we present a method to estimate the chance that a woman with given age and risk factors will develop breast cancer over a specified interval. The risk factors used were age at menarche, age at first live birth, number of previous biopsies, and number of first-degree relatives with breast cancer. A model of relative risks for various combinations of these factors was developed from case-control data from the Breast Cancer Detection Demonstration Project (BCDDP). The model allowed for the fact that relative risks associated with previous breast biopsies were smaller for women aged 50 or more than for younger women. Thus, the proportional hazards models for those under age 50 and for those of age 50 or more. The baseline age-specific hazard rate, which is the rate for a patient without identified risk factors, is computed as the product of the observed age-specific composite hazard rate times the quantity 1 minus the attributable risk. We calculated individualized breast cancer probabilities from information on relative risks and the baseline hazard rate. These calculations take competing risks and the interval of risk into account. Our data were derived from women who participated in the BCDDP and who tended to return for periodic examinations. For this reason, the risk projections given are probably most reliable for counseling women who plan to be examined about once a year.

Survival Estimation Using Splines

A nonparametric maximum likelihood procedure is given for estimating the survivor function from right-censored data. It approximates the hazard rate by a simple function such as a spline, with different approximations yielding different estimators. A special case is that proposed by Nelson (1969, Journal of Quality Technology 1, 27-52) and Altshuler (1970, Mathematical Biosciences 6, 1-11). The estimators are uniformly consistent and have the same asymptotic weak convergence properties as the Kaplan-Meier (1958, Journal of the American Statistical Association 53, 457-481) estimator. However, in small and in heavily censored samples, the simplest spline estimators have uniformly smaller mean squared error than do the Kaplan-Meier and Nelson-Altshuler estimators. The procedure is extended to estimate the baseline hazard rate and regression coefficients in the Cox (1972, Journal of the Royal Statistical Society, Series B 34, 187-220) proportional hazards model and is illustrated using experimental carcinogenesis data.