A Simple Test for the Absence of Covariate Dependence in Hazard Regression Models

This paper extends commonly used tests for equality of hazard rates in a two-sample or k-sample setup to a situation where the covariate under study is continuous. In other words, we test the hypothesis that the conditional hazard rate is the same for all covariate values, against the omnibus alternative as well as more specific alternatives, when the covariate is continuous. The tests developed are particularly useful for detecting trend in the underlying conditional hazard rates or changepoint trend alternatives. Asymptotic distribution of the test statistics are established and small sample properties of the tests are studied. An application to the e¤ect of aggregate Q on corporate failure in the UK shows evidence of trend in the covariate e¤ect, whereas a Cox regression model failed to detect evidence of any covariate effect. Finally, we discuss an important extension to testing for proportionality of hazards in the presence of individual level frailty with arbitrary distribution

Bayesian semiparametric inference for multivariate doubly-interval-censored data

Based on a data set obtained in a dental longitudinal study, conducted in Flanders (Belgium), the joint time to caries distribution of permanent first molars was modeled as a function of covariates. This involves an analysis of multivariate continuous doubly-interval-censored data since: (i) the emergence time of a tooth and the time it experiences caries were recorded yearly, and (ii) events on teeth of the same child are dependent. To model the joint distribution of the emergence times and the times to caries, we propose a dependent Bayesian semiparametric model. A major feature of the proposed approach is that survival curves can be estimated without imposing assumptions such as proportional hazards, additive hazards, proportional odds or accelerated failure time.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS368 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Simple Test for the Absence of Covariate Dependence in Hazard Regression Models

This paper extends commonly used tests for equality of hazard rates in a two-sample or k-sample setup to a situation where the covariate under study is continuous. In other words, we test the hypothesis that the conditional hazard rate is the same for all covariate values, against the omnibus alternative as well as more specific alternatives, when the covariate is continuous. The tests developed are particularly useful for detecting trend in the underlying conditional hazard rates or changepoint trend alternatives. Asymptotic distribution of the test statistics are established and small sample properties of the tests are studied. An application to the e¤ect of aggregate Q on corporate failure in the UK shows evidence of trend in the covariate e¤ect, whereas a Cox regression model failed to detect evidence of any covariate effect. Finally, we discuss an important extension to testing for proportionality of hazards in the presence of individual level frailty with arbitrary distribution.Covariate dependence; Continuous covariate; Two-sample tests; Trend tests; Proportional hazards; Frailty/ unobserved heterogeneity; Linear transformation model

The effect of behavioural changes over time on Cox proportional hazards estimates

This thesis considers bias when studying behaviours in a Cox proportional hazards model. In Cox proportional hazards regressions and cohort studies in general, measurements are often made during a limited period of time. Behaviours may, however, change rather dramatically over time, and if these changes are unknown, they will distort the results in the regression models. We study this problem in the context of the effects of smoking and physical activity on cardiovascular disease by simulating Cox proportional hazards models. Changes in behaviour are simulated with Markov chains in four scenarios. In each scenario we perform ten sets of simulations where each set has a different transition probability. The first scenario considers a dichotomous variable indicating physical inactivity. We find that an increasing probability of changing behaviour will eventually completely dilute the baseline estimates. In the second, third, and fourth scenario we instead look at a smoking status variable containing the categories smoker, ex-smoker, and non-smoker. The three-category variable was in the regressions decomposed into the two dichotomous variables Smoker and Ex-smoker. In the second scenario we only allow transitions from smoker to ex-smoker. That leads to the hazard ratio estimates of Smoker going towards the hazard ratio of Ex-smoker. In the third scenario transitions are also allowed from ex-smoker to smoker. This results in the hazard ratios of the two variables moving towards each other, as the transition probabilities become larger. Lastly, the fourth scenario have Markov chains where non-smokers are additionally allowed to transition to smokers. There we find that the hazard ratios of Smoker and Ex-smoker go towards 1.0 when the transition probability of going from non-smoker to smoker is large

Testing for Proportional Hazards with Unrestricted Univariate Unobserved Heterogeneity

We develop tests of the proportional hazards assumption, with respect to a continuous covariate, in the presence of unobserved heterogeneity with unknown distribution at the individual observation level. The proposed tests are specially powerful against ordered alternatives useful for modeling non-proportional hazards situations. By contrast to the case when the heterogeneity distribution is known up to …nite dimensional parameters, the null hypothesis for the current problem is similar to a test for absence of covariate dependence. However, the two testing problems di¤er in the nature of relevant alternative hypotheses. We develop tests for both the problems against ordered alternatives. Small sample performance and an application to real data highlight the usefulness of the framework and methodology

Has long become longer or short become shorter? Evidence from a censored quantile regression analysis of the changes in the distribution of U.S. unemployment duration

There is conflicting evidence regarding the recent evolution of unemployment duration in the U.S. In this study we rely on censored quantile regression methods to analyze the changes in the US unemployment duration distribution. We employed the decomposition method proposed by Machado and Mata (2003) to disentangle the contribution of the changes generated by the covariate distribution and by the conditional distribution and adapted it to a duration analysis framework.The data used in this inquiry are taken from the nationally representative Displaced Worker Survey of 1988 and 1998. We provide evidence that the unemployment duration distribution shifted leftward. The main driving force behind that shift was the sharp leftward move in the unemployment rate distribution. This force was partially counteracted by the ageing of the displaced population, the striking absence of impact from being displaced via a plant shutdown, and the higher sensitivity of unemployment duration to unemployment ratesQuantile Regression, Duration Analysis, Unemployment Duration, Counterfactual Decomposition

Bayesian Inference for Multivariate Survival Data with a Cure Fraction

AbstractWe develop Bayesian methods for right censored multivariate failure time data for populations with a cure fraction. We propose a new model, called the multivariate cure rate model, and provide a natural motivation and interpretation of it. To create the correlation structure between the failure times, we introduce a frailty term, which is assumed to have a positive stable distribution. The resulting correlation structure induced by the frailty term is quite appealing and leads to a nice characterization of the association between the failure times. Several novel properties of the model are derived. First, conditional on the frailty term, it is shown that the model has a proportional hazards structure with the covariates depending naturally on the cure rate. Second, we establish mathematical relationships between the marginal survivor functions of the multivariate cure rate model and the more standard mixture model for modelling cure rates. With the introduction of latent variables, we show that the new model is computationally appealing, and novel computational Markov chain Monte Carlo (MCMC) methods are developed to sample from the posterior distribution of the parameters. Specifically, we propose a modified version of the collapsed Gibbs technique (J. S. Liu, 1994, J. Amer. Statist. Assoc.89, 958–966) to sample from the posterior distribution. This development will lead to an efficient Gibbs sampling procedure, which would otherwise be extremely difficult. We characterize the propriety of the joint posterior distribution of the parameters using a class of noninformative improper priors. A real dataset from a melanoma clinical trial is presented to illustrate the methodology

Estimating the availability of hydraulic drive systems operating under different functional profiles through simulation

Hydraulic drive systems are widely used in a variety of industrial applications where high torque and low speed rotational power are required. The advantages include maximum torque from zero speed, continuously variable speed within wide limits, high reliability and insensitivity to shock loads. A drive system consists of a hydraulic circuit, electric motors, hydraulic pumps, hydraulic motors and auxiliary components. The stress on the components, and hence wear and failure rate, varies with the torque and speed output by the drive. The reliability of a hydraulic drive system of a particular design can therefore vary significantly between installations operating in applications with different functional requirements. Predicting the availability of a drive system in a particular application is useful for several purposes such as optimising the system design and estimating support costs. This paper describes a simulation model, developed to estimate the availability of a hydraulic drive system in a given functional profile, consisting of output torque and speed time phase requirements. It outputs statistics on system availability and component failure rates. As an example, the simulation model is used to compare these statistics for a drive design operating under two distinct operational profiles

Bootstrap applications in proportional hazards models

Experiments in which the measured responses are times until events occur are common in a variety of fields. When only one response is measured on each subject, the proportional hazards model of Cox (1972) is often used to assess the effects of one or more explanatory variables on the event times. Two new resampling plans are introduced for bootstrapping estimators from this model when explanatory variables are fixed by design. One method resamples from the Uniform (0,1) distribution of the probability integral transformation corresponding to the conditional failure time distribution, and it is easily adapted to a wide variety of censoring schemes. The other method is an analog to the residual-resampling method for regression introduced by Efron (1979), and it admits random censoring from a class of distributions which includes the Koziol-Green model;Multivariate extensions of resampling methods are developed for situations where multiple event times are monitored on individual subjects. Marginal models are fit using an independence working model approach. Resampling procedures are then applied to the joint distribution of the multiple responses or residuals to make bias corrections to the parameter estimates, estimate covariance matrices, and construct confidence intervals. Simulation studies indicate that each of the proposed methods provides substantial improvements in mean squared errors over existing techniques for estimation of model parameters. The proposed methods also provide better estimates of standard errors and more reliable confidence intervals for model parameters than existing methods which rely largely on asymptotic approximations. These methods are demonstrated through applications to data sets available in the literature