Generating Survival Times to Simulate Cox Proportional Hazards Models

This paper discusses techniques to generate survival times for simulation studies regarding Cox proportional hazards models. In linear regression models, the response variable is directly connected with the considered covariates, the regression coefficients and the simulated random errors. Thus, the response variable can be generated from the regression function, once the regression coefficients and the error distribution are specified. However, in the Cox model, which is formulated via the hazard function, the effect of the covariates have to be translated from the hazards to the survival times, because the usual software packages for estimation of Cox models require the individual survival time data. A general formula describing the relation between the hazard and the corresponding survival time of the Cox model is derived. It is shown how the exponential, the Weibull and the Gompertz distribution can be used to generate appropriate survival times for simulation studies. Additionally, the general relation between hazard and survival time can be used to develop own distributions for special situations and to handle flexibly parameterized proportional hazards models. The use of other distributions than the exponential distribution only is indispensable to investigate the characteristics of the Cox proportional hazards model, especially in non-standard situations, where the partial likelihood depends on the baseline hazard

Cox regression survival analysis with compositional covariates: application to modelling mortality risk from 24-h physical activity patterns

Survival analysis is commonly conducted in medical and public health research to assess the association of an exposure or intervention with a hard end outcome such as mortality. The Cox (proportional hazards) regression model is probably the most popular statistical tool used in this context. However, when the exposure includes compositional covariables (that is, variables representing a relative makeup such as a nutritional or physical activity behaviour composition), some basic assumptions of the Cox regression model and associated significance tests are violated. Compositional variables involve an intrinsic interplay between one another which precludes results and conclusions based on considering them in isolation as is ordinarily done. In this work, we introduce a formulation of the Cox regression model in terms of log-ratio coordinates which suitably deals with the constraints of compositional covariates, facilitates the use of common statistical inference methods, and allows for scientifically meaningful interpretations. We illustrate its practical application to a public health problem: the estimation of the mortality hazard associated with the composition of daily activity behaviour (physical activity, sitting time and sleep) using data from the U.S. National Health and Nutrition Examination Survey (NHANES)

Modeling Firm-Size Distribution Using Box-Cox Heteroscedastic Regression

Using the Box-Cox regression model with heteroscedasticity, we examine the size distribution of firms. Analyzing the data set of Portuguese manufacturing firms as in Machado and Mata (2000), we show that our approach compares favorably against the Box-Cox quantile regression method. In particular, we are able to answer the key questions addressed by Machado and Mata, with the additional advantage that our empirical quantile functions are monotonic. Furthermore, confidence intervals of the regression quantiles are easy to compute, and the estimation of the Box-Cox heteroscedastic regression model is straightforward.Box-Cox transformation, Firm-size distribution, Quantile regression.

A Note on Implementing Box-Cox Quantile Regression

The Box-Cox quantile regression model using the two stage method suggested by Chamberlain (1994) and Buchinsky (1995) provides a flexible and numerically attractive extension of linear quantile regression techniques. However, the objective function in stage two of the method may not exists. We suggest a simple modification of the estimator which is easy to implement. The modified estimator is still pn{consistent and we derive its asymptotic distribution. A simulation study confirms that the modified estimator works well in situations, where the original estimator is not well defined. --Box-Cox quantile regression,iterative estimator

Time-varying effects when analysing customer lifetime duration, application to the insurance market

The Cox model (Cox, 1972) is widely used in customer lifetime duration research, but it assumes that the regression coefficients are time invariant. In order to analyse the temporal covariate effects on the duration times, we propose to use an extended version of the Cox model where the parameters are allowed to vary over time. We apply this methodology to real insurance policy cancellation data and we conclude that the kind of contracts held by the customer and the concurrence of an external insurer in the cancellation influence the risk of the customer leaving the company, but the effect differs as time goes by.Cox model, customer lifetime.

An exact corrected log-likelihood function for Cox's proportional hazards model under measurement error and some extensions

This paper studies Cox`s proportional hazards model under covariate measurement error. Nakamura`s (1990) 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 (1992), Kong, Huang and Li (1998) and Kong and Gu (1999) are reestablished 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