Making relative survival analysis relatively easy

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Abstract

In survival analysis we are interested in time from the beginning of an observation until certain event (death, relapse, etc.). We assume that the final event is well defined, so that we are never in doubt whether the final event has occurred or not. In practice this is not always true. If we are interested in cause-specific deaths, then it may sometimes be difficult or even impossible to establish the cause of death, or there may be different causes of death, making it impossible to assign death to just one cause. Suicides of terminal cancer patients are a typical example. In such cases, standard survival techniques cannot be used for estimation of mortality due to a certain cause. The cure to the problem are relative survival techniques which compare the survival experience in a study cohort to the one expected should they follow the background population mortality rates. This enables the estimation of the proportion of deaths due to a certain cause. In this paper, we briefly review some of the techniques to model relative survival, and outline a new fitting method for the additive model, which solves the problem of dependency of the parameter estimation on the assumption about the baseline excess hazard. We then direct the reader's attention to our R package relsurv that provides functions for easy and flexible fitting of all the commonly used relative survival regression models. The basic features of the package have been described in detail elsewhere, but here we additionally explain the usage of the new fitting method and the interface for using population mortality data freely available on the Internet. The combination of the package and the data sets provides a powerful informational tool in the hands of a skilled statistician/informatician.

Section snippets

Motivation

If a person with an incurable disease commits suicide, the cause of death written in the death certificate will be suicide. And if there were many such cases, the mortality statistics would show much lower proportion of deaths due to the disease in question than it really should. And while suicides are just an obvious, more or less hypothetical, example, it is less well known that it is often difficult or even impossible to select among different possible causes of death or assign a certain

Relative survival

The cumulative relative survival function is defined [2] asr(t)=SO(t)SP(t),where SO(t) denotes observed survival and SP(t) stands for population or expected survival, which is estimated on the basis of population mortality tables. Obviously, r(t) can be any non-negative number, although the methods are most often applied to data where r(t) is less than 1. Correct calculation of the expected survival is not a straightforward task, and it is now generally accepted that the method of Hakulinen [3]

Estimation

Fitting any model to the transformed data presents no new problems, and as mentioned fitting the multiplicative model is also straightforward. Additive model on the other hand has its own peculiarities. The problem is that the existing methods require baseline excess hazard in (5) to be specified. This is usually done by assuming the baseline excess hazard to be constant in prespecified intervals. Noncritical usage of such an assumption can lead to problems. First, if the assumption is wrong,

Population tables available on the internet

Population tables are an indispensable part of any relative survival analysis. They can be obtained from the national statistical offices, but usually come in quite different formats depending on the national mortality and census data organization. Some work might then be needed for transforming such tables into a format required by R programs. Fortunately, web sites now exist that provide tables for various countries in a uniform format. One such site is the human mortality database (HMD, //www.mortality.org

The R package relsurv

In previous sections, various relative survival methods and their accompanying problems have been presented. The relsurv package, that can be downloaded from CRAN [5], provides a software tool that simplifies their use. While the core functions are described in [13], the package has now been enriched with new methods and several functions that make it more user friendly. Two important additions, the EM-based estimation and the usage of the population tables, available on the Internet, are

Examples

In this section we present three examples of the usage of the package, with emphasis on the usage of population tables. We first explain how to create figures from Section 3.3, and then present two examples of relative survival analysis.

Discussion

Relative survival analysis offers answers to questions that cannot be answered with standard analysis. For now, the methods are used extensively in cancer registries, but are almost unknown to other medical professionals. For example, no textbook on survival analysis has a chapter, or even a section, devoted to the topic. This is partly due to the fact that relative survival analysis is inseparably tied to the population data, which is seen as an obstacle to its usage. With ever greater

Maja Pohar has graduated from mathematics at the University of Ljubljana and is now a final year Ph.D. student of statistics at the same university. She is the author of six SCI papers and has written papers on relative survival in the Journal of the Royal Statistical Society—Series C, Statistics in Medicine, and Computer Methods and Programs in Biomedicine. She is the author of the R package relsurv that is included in CRAN and provides functions for regression in relative survival.

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Maja Pohar has graduated from mathematics at the University of Ljubljana and is now a final year Ph.D. student of statistics at the same university. She is the author of six SCI papers and has written papers on relative survival in the Journal of the Royal Statistical Society—Series C, Statistics in Medicine, and Computer Methods and Programs in Biomedicine. She is the author of the R package relsurv that is included in CRAN and provides functions for regression in relative survival.

Janez Stare is a full profesor at the Faculty of Medicine at the University of Ljubjana and the head of the Department of Biomedical Informatics.

He is the author of more than 25 SCI papers and has recently written papers on relative survival in the Journal of the Royal Statistical Society—Series C, Statistics in Medicine, and Computer Methods and Programs in Biomedicine.

He has been an invited speaker on the topic of relative survival at the Annual Conference of the International Society for Clinical Biostatistics in Leiden, 2004, and at the ROeS Seminar in Graz, 2005.

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