Methods of survival analysis for long-term follow-up studies are illustrated by a study of mortality in 3878 breast cancer patients in Edinburgh followed for up to 20 years. The problems of life tables, advantages of hazard plots and difficulties in statistical modelling are demonstrated by studying the relationship between survival and both clinical stage and initial menopausal status at diagnosis. To assess the 'curability' of breast cancer, mortality by year of follow-up is compared with expected mortality using Scottish age-specific death rates. Techniques for analysing such relative survival data include age-corrected life tables, ratio of observed to expected deaths and excess death rates. Finally, an additive hazard model is developed to incorporate covariates in the analysis of relative survival and curability.
"In fact, in many applications, monotone departures from the PH model may be more reasonable even from a theoretical point of view. Examples include medical applications where one expects the prognostic relevance of some covariates to decay, or even disappear, in the long run (Pocock et al., 1982; Therneau and Grambsch, 2000). Similar decline in covariate e¤ects are observed in economic studies on the e¤ect of bene…ts on unemployment duration (Narendranathan and Stewart, 1993) and on the e¤ect of macroeconomic conditions on …rm exits (Bhattacharjee et al., 2007). "
[Show abstract][Hide abstract] ABSTRACT: Several omnibus tests of the proportional hazards assumption have been proposed in the literature. In the two-sample case, tests have also been developed against ordered alternatives like monotone hazard ratio and monotone ratio of cumulative hazards. Here we propose a natural extension of these partial orders to the case of continuous and potentially time varying covariates, and develop tests for the proportional hazards assumption against such ordered alternatives. The work is motivated by applications in biomedicine and economics where covariate effects often decay over lifetime. The proposed tests do not make restrictive assumptions on the underlying regression model, and are applicable in the presence of time varying covariates, multiple covariates and frailty. Small sample performance and an application to real data highlight the use of the framework and methodology to identify and model the nature of departures from proportionality.
Journal of Statistical Planning and Inference 01/2011; 141(1-141):243-261. DOI:10.1016/j.jspi.2010.06.012 · 0.68 Impact Factor
"For an introduction to mixture models in survival see Farewell  and Mc Lachlan and Mc Giffin  (see also  and ). Pack and Morgan  use a similar mixture model for the analysis of quantal assay data and Pocock et al.  use it for the analysis of curability of breast cancer. A review of survival analysis using gene expression levels as covariates is given in . "
[Show abstract][Hide abstract] ABSTRACT: A two-component parametric mixture is proposed to model survival after an invasive treatment, when patients may experience different hazards regimes: a risk of early mortality directly related to the treatment and/or the treated condition, and a risk of late death influenced by several exogenous factors. The parametric mixture is based on Weibull distributions for both components. Different sets of covariates can affect the Weibull scale parameters and the probability of belonging to one of the two latent classes. A logarithmic function is used to link explanatory variables to scale parameters while a logistic link is assumed for the probability of the latent classes. Inference about unknown parameters is developed in a Bayesian framework: point and interval estimates are based on posterior distributions, whereas the Schwarz criterion is used for testing hypotheses. The advantages of the approach are illustrated by analyzing data from an aorta aneurysm study.
"A third possibility could be that marketing variables act on the baseline hazard in an additive manner (instead of a proportional manner as in the PHM, and a shape-shifting manner as in the AFTM). Such an assumption yields a purchase-timing model called the Additive Risk Model 1 (ARM henceforth ), first proposed by Aalen (1980) and shown to be preferable to the PHM in several biological and epidemiology problems (Breslow and Day 1980, Pocock et al. 1982, Buckley 1984, Pierce and Preston 1984, Breslow 1986, Thomas 1986). The ARM also handles time-varying covariates and nests traditional purchase-timing models—such as NBD, Erlang-2, etc.—as special cases. "
[Show abstract][Hide abstract] ABSTRACT: This paper proposes the (ARM), first used by Aalen (1980), to explain households' interpurchase times. Unlike the Proportional Hazard Model (PHM), first proposed by Cox (1972), the ARM incorporates the effects of covariates on the individual hazard function in an (as opposed to ) manner. While a large number of previous studies on interpurchase timing have dealt with the question of correctly specifying the parametric distribution for interpurchase times, no study has explicitly investigated the question of correctly specifying the effects of covariates in the model. This study looks at this issue. We propose an ARM that is suitable for purchase-timing data, and compare its empirical performance to that of the PHM and the Accelerated Failure Time Model (AFTM) using scanner panel data on laundry detergents, paper towels, and toilet tissue. We find that the ARM not only estimates and validates the observed interpurchase times better than existing models, but also recovers a time-varying price elasticity and shows a high degree of robustness in the estimated covariate effects to alternative parametric specifications of the baseline hazard. The estimates of covariate parameters under the PHM, on the other hand, are highly sensitive to alternative parametric specifications of the baseline hazard.
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