Relative survival analysis in R
Introduction
Survival analysis is concerned with studying the random variable T, representing the time between entry to a study and some event of interest (e.g. death, recurrence of disease …). Instead of the cumulative distribution function , we usually estimate the cumulative survival function , which is defined as . In the case of the final event being death, is the probability of being alive at time t. When the final event of interest is death due to a certain disease, but the causes of death are not known, it is not possible to directly estimate the proportion of dead due to the disease in question. We then resort to the methods of relative survival. The cumulative relative survival function is defined [1] aswhere denotes observed survival and stands for population or expected survival, which is estimated on the basis of population life tables. Obviously, can be any non-negative number, although the methods are most often applied to data where is less than 1.
Several approaches to modelling relative survival exist, but all of the existing programmes (for example: Surv3 [2], SAS macros and Stata functions [3], RSurvR function [4]) focus on only one of the models and use specifically organised general population tables, making it difficult for the users to compare different methods.
We present three R [5] functions organised as a package called relsurv that largely simplify the usage of relative survival regression models. All the functions require the same basic organisation of the data and can be used with any format of the population mortality data.
Section 2 briefly describes the three most commonly used regression approaches and gives an outline of the theory for the five fitting methods used in the relsurv functions. Section 3 describes the functions, their arguments, the preparation of the data and the returned values. The usage is further explained through an example in Section 4. In case the user does not yet have the relevant population mortality tables organised in the required ratetableformat, the appendix provides some quick help.
Section snippets
Relative survival regression models
The relative survival literature most frequently refers to the additive models, dominating especially in cancer research. The main assumption is that the hazard of every individual can be split into two additive components:
where is the hazard every patient takes because of his age, sex and cohort year (or any other combination of covariates included in the population mortality data). denotes the excess hazard, specific for the disease in question and is used
The relsurv package
The core of the package are three functions that fit the models described in the previous section:
- •
rsadd fits an additive model. As described in Section 2, different methods of estimation exist, and the user can choose among them through the methodargument. The default is the Estève method (iii) which is specified by ”max.lik”, the other two options are method=”glm.bin”and method=”glm.poi” for models described in (i) and (ii) of Section 2. When using one of the glm methods, the observed and
Example
To illustrate the usage of the program, we use a subset of data from the study of survival of patients after acute myocardial infarction that is included in the package as the file rdata. The data were collected in the study carried out at the University Clinical Center in Ljubljana and contain 1040 patients diagnosed between 1982 and 1986 and followed up until 1997. During this time 547 deaths occurred and as the causes of death are not given, this is a good example of the need of the relative
To conclude
The usage of any statistical method depends mainly on the availability of adequate software, and relative survival techniques are no exception. For now, the choice of the model, and the method to fit it, were mostly influenced by software at hand. Different methods were programmed in different languages, and their usage was hindered by considerable effort needed to learn the special requirements of such programs. We believe that by incorporating all the different methods into an R package, now
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