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Christopher Cox, Michael F. Schneider, and Alvaro Muñoz
Abstract In reliability theory, the lifetime remaining in a network of components after an initial run-in period is an important property of the system. Similarly, for medical interventions residual survival characterizes the subsequent experience of patients who survive beyond the beginning of follow-up. Here we show how quantiles of the residual survival distribution can be used to provide such a characterization. We first discuss properties of the residual quantile function and its close relationship to the hazard function. We then consider parametric estimation of the residual quantile function, focusing on the generalized gamma distribution.
Finally, we describe an application of quantiles of residual survival to help describe the effects at the population level of the introduction and sustained use of highly active antiretroviral therapy for the treatment of HIV/AIDS.
Introduction
In many applications, the comparison of two survival distributions is summarized by the estimation of a single relative hazard, under the standard proportional hazards assumption. We have previously argued [4], along with others, that this
C. Cox ()
Johns Hopkins Bloomberg School of Public Health, 615 N. Wolfe St E7642, Baltimore, MD 21205, USA
e-mail:[email protected] M.F. Schneider
Johns Hopkins Bloomberg School of Public Health, 615 N. Wolfe St E7012, Baltimore, MD 21205, USA
e-mail:[email protected] A. Muñoz
Johns Hopkins Bloomberg School of Public Health, 615 N. Wolfe St E7648, Baltimore, MD 21205, USA
e-mail:[email protected]
M.-L.T. Lee et al. (eds.), Risk Assessment and Evaluation of Predictions, Lecture Notes in Statistics 210, DOI 10.1007/978-1-4614-8981-8__6,
© Springer Science+Business Media New York 2013
87
assumption is frequently violated, and that estimation of selected quantiles of the two distributions and their comparison by relative times (relative quantiles) can be not only more appropriate but also more informative.
Patients returning for a follow-up visit after an initial diagnosis or treatment often want to know what to expect in the future. This information, contained in the conditional distribution of residual survival times, is an important metric for evaluating the long term effects of interventions. In reliability theory, mean residual life has been extensively studied, and is sometimes used instead of the hazard function to characterize families of survival distributions. Here, we argue that the use of quantiles of the residual survival distribution is a useful alternative.
We first discuss properties of the residual quantile function, including its close relationship to the hazard function. We show that, like mean residual life, the residual quantile function has the opposite shape from the hazard. More importantly, the residual quantiles provide useful information about residual survival that is not apparent in the behavior of the hazard. An advantage of the residual quantiles is that they are expressed in units of time, which is the natural metric of survival.
Using the generalized gamma (GG) family, which we have previously advocated as a platform for parametric survival analysis [4], we then discuss estimation of the residual quantile function from the parametric perspective. In this case, estimation of the residual quantile function for selected quantiles such as the median and quartiles is relatively straightforward, and can be accomplished using standard statistical software. Finally, using data from two multicenter cohort studies, we consider an application in which absolute and relative residual twenty-fifth percentiles are used to assess the effect at the population level of the introduction and continued use of highly active antiretroviral therapy (HAART) for the treatment of HIV/AIDS.