RSURV: a function to perform relative survival analysis with S-PLUS or R

The performance of multiple imputation for missing covariate data within the context of regression relative survival analysis

Relative survival assesses the effects of prognostic factors on disease-specific mortality when the cause of death is uncertain or unavailable. It provides an estimate of patients' survival, allowing for the effects of other independent causes of death. Regression-based relative survival models are commonly used in population-based studies to model the effects of some prognostic factors and to estimate net survival. Most often, studies focus on routinely collected prognostic factors for which the proportion of missing values is usually low (around 5 per cent). However, in some cases, additional factors are collected with a greater proportion of missingness. In the present article, we systematically assess the performance of multiple imputation in regression analysis of relative survival through a series of simulation experiments. According to the assumptions concerning the missingness mechanism (completely at random, at random, and not at random) and the missingness pattern (monotone, non-monotone), several strategies were considered and compared: all cases analysis, complete cases analysis, missing data indicator analysis, and multiple imputation by chained equations (MICE) analysis. We showed that MICE performs well in estimating the hazard ratios and the baseline hazard function when the missing mechanism is missing at random (MAR) conditionally on the vital status. In the situations where the missing mechanism was not MAR conditionally on vital status, complete case behaves consistently. As illustration, we used data of the French Cancer Registries on relative survival of patients with colorectal cancer.

An approach to estimation in relative survival regression

The goal of relative survival methodology is to compare the survival experience of a cohort with that of the background population. Most often an additive excess hazard model is employed, which assumes that each person's hazard is a sum of 2 components--the population hazard obtained from life tables and an excess hazard attributable to the specific condition. Usually covariate effects on the excess hazard are assumed to have a proportional hazards structure with parametrically modelled baseline. In this paper, we introduce a new fitting procedure using the expectation-maximization algorithm, treating the cause of death as missing data. The method requires no assumptions about the baseline excess hazard thus reducing the risk of bias through misspecification. It accommodates the possibility of knowledge of cause of death for some patients, and as a side effect, the method yields an estimate of the ratio between the excess and the population hazard for each subject. More importantly, it estimates the baseline excess hazard flexibly with no additional degrees of freedom spent. Finally, it is a generalization of the Cox model, meaning that all the wealth of options in existing software for the Cox model can be used in relative survival. The method is applied to a data set on survival after myocardial infarction, where it shows how a particular form of the hazard function could be missed using the existing methods.

An overall strategy based on regression models to estimate relative survival and model the effects of prognostic factors in cancer survival studies

Relative survival provides a measure of the proportion of patients dying from the disease under study without requiring the knowledge of the cause of death. We propose an overall strategy based on regression models to estimate the relative survival and model the effects of potential prognostic factors. The baseline hazard was modelled until 10 years follow-up using parametric continuous functions. Six models including cubic regression splines were considered and the Akaike Information Criterion was used to select the final model. This approach yielded smooth and reliable estimates of mortality hazard and allowed us to deal with sparse data taking into account all the available information. Splines were also used to model simultaneously non-linear effects of continuous covariates and time-dependent hazard ratios. This led to a graphical representation of the hazard ratio that can be useful for clinical interpretation. Estimates of these models were obtained by likelihood maximization. We showed that these estimates could be also obtained using standard algorithms for Poisson regression.

Relative survival analysis in R

Relative survival techniques are used to compare the survival experience in a study cohort with the one expected should they follow the background population mortality rates. The techniques are especially useful when the cause-specific death information is not accurate or not available since they provide a measure of excess mortality in a group of patients with a certain disease. There are several approaches to modeling relative survival, but there is no widely used statistical package that would incorporate the relevant techniques. The existing software was mostly written by the authors of different methods, in different computer languages and with different requirements for the data input, which makes it almost impossible for a user to choose between available models. We describe our R package relsurv that provides functions for easy and flexible fitting of several relative survival regression models.

Revue comparative des modèles régressifs de survie brute et de survie relative

BACKGROUND

Statistical analysis of lifetime data is frequently used in the biomedical area. Our objective was to present a comparative review of the different regression models according to the survival concept (crude survival or relative survival) in order to express guidelines.

METHODS

From a methodological point of view, we compared a regressive crude survival model (Cox model) and regressive relative survival models for grouped data (Hakulinen and Tenkanen) and for individuals data (Esteve et al.). We illustrated our work with an analysis of survival data of 3,355 incident cases of breast cancer identified by a hospital registry. Comportment of the models was studied in situation where censors rates ranged from 31.7 to 96.5%.

RESULTS

Because relative survival analysis takes into account natural mortality, the risk of death was smaller for women older than 50 years than for women aged from 35 to 49; this was not demonstrated in the crude survival analysis (Cox). Estimations obtained from Cox model were more accurate than those obtained from both studied regressive relative survival models. Estimations obtained from Esteve et al. model were not very different from those obtained from Hakulinen and Tenkanen model and they were more accurate.

CONCLUSION

By definition, analysis of relative survival is more appropriate to estimate survival to a specific cause of death. It is preferable to use models based on individual estimation when data set is small or when the number of individual per strata is small.

Goodness of fit of relative survival models

Additive regression models are preferred over multiplicative models in the analysis of relative survival data. Such preferences are mainly grounded in practical experience with mostly cancer registries data, where the basic assumption of the additivity of hazards is more likely to be met. Also, the interpretation of coefficients is more meaningful in additive than in multiplicative models. Nonetheless, the question of goodness of fit of the assumed model must still be addressed, and while there is an abundance of methods to check the goodness of fit of multiplicative models, the respective arsenal for additive models is almost empty. We propose here a variety of procedures for testing the null hypothesis of a good fit. These are based on partial residuals defined similarly to Schoenfeld residuals familiar for Cox model diagnostics. The tests have appropriate sizes under the null hypothesis, and good power under different alternatives. We investigate their performance through simulations and apply the methods to data from a study into survival of colon cancer patients.

Analysis of time-dependent covariates in a regressive relative survival model

Relative survival is a method for assessing prognostic factors for disease-specific mortality. However, most relative survival models assume that the effect of covariate on disease-specific mortality is fixed-in-time, which may not hold in some studies and requires adapted modelling. We propose an extension of the Esteve et al. regressive relative survival model that uses the counting process approach to accommodate time-dependent effect of a predictor's on disease-specific mortality. This approach had shown its robustness, and the properties of the counting process give a simple and attractive computational solution to model time-dependent covariates. Our approach is illustrated with the data from the Stanford Heart Transplant Study and with data from a hospital-based study on invasive breast cancer. Advantages of modelling time-dependent covariates in relative survival analysis are discussed.