Modelling disease-specific mortality using an extended multi-state model

1. Project description with basic information regarding (co)financing

In the field of survival analysis, one usually examines the occurrence of a primary event of interest through time for a given sample of individuals. In the medical domain, it is common to analyse the death rate for a group of patients that are followed through time. Thus, by taking death to be the primary event of interest, one may consider the patients’ overall survival probability and understand the lethality of the disease in question.

An extension of the basic survival scenario is to consider multiple events (other than the primary event – death). The multi-state model is a standard tool for investigating multiple events that may occur through time. Examples of additional events that are commonly considered in practice are disease recurrence, progression, additional treatment, adverse event, recovery, transplantation etc. Hence, multi-state models provide a framework for simultaneously analysing competing events and sequences of events. Furthermore, these models help to study the impact of intermediate events on the prognosis of patients and estimate separate death probabilities (with and without an intermediate event) at different time points. The multi-state model theory has been well-developed [Putter2007, Andersen2002] and complemented with various software implementations available for use such as [Mstate2011], hence allowing for diverse clinical applications to be performed in practice.

A common theme in survival analysis is to consider the cause of death. More specifically, apart from observing death, the goal can be to distinguish between deaths due to the studied disease (excess mortality) and deaths due to other causes (population mortality). The field of relative survival deals with this question in the case when the cause of death is not given in the data. The basic relative survival idea is to assume that the overall mortality hazard can be written as a sum of an excess (disease-specific) and population (other-cause) hazard. This allows for estimating the proportion of deaths attributed to the disease itself and due to other causes. Such analysis is useful in cases when patients with a good prognosis are considered (i.e., their probability of dying due to other causes is non-negligible). In practice, this commonly occurs when observing an older patient population over a longer time period (where the risk of dying due to other causes is high and may considerably contribute to the overall proportion of deaths).

In practice, if the cause of death is provided in the data, a multi-state model may be used where different causes of death are considered as separate events. However, there are many cases where cause of death is not provided [Mariotto2014], is unreliable [Begg2002], or cannot be unequivocally attributed to the disease or other causes. Relative survival tackles this problem by using external population mortality tables from which information on other-cause (population) mortality is obtained. Thus, by merging the observed data with external mortality tables one may extract cause of death information from the data even though the cause of death is not explicitly given. There have been numerous advances in the field of relative survival in the past decades [PoharPerme2012] which has greatly increased its usage.

The goal of this project is to focus on cause-specific mortality in a general multi-state model using relative survival. A recent paper [Manevski2022] has laid the foundations of how the two approaches can be merged. The article has proposed to split all mortality in disease and non-disease-related mortality, with and without intermediate events, in datasets where cause of death is not known. This project makes one step forward and focuses on regression modelling. This enables to understand the relationship between the patients' survival and a given set of (confounding) covariates. Regression modelling is a standard step of any statistical analysis (like the usage of the Cox proportional-hazards model in survival analysis). The goal of this project is to provide the necessary tools for performing regression modelling for an extended multi-state model using relative survival.

The project is solely funded by ARIS, Z3-50124.

2. Team members of the project with links to the SICRIS system

This is a Postdoctoral Research Project, thus Damjan Manevski is the only project member.

https://cris.cobiss.net/ecris/si/sl/researcher/48273

3. Project phases and description of their realization

Apart from establishing the connection between multi-state models and relative survival, our recent paper [Manevski2022] considers non-parametric estimation. In general, non-parametric estimators provide a basic understanding of the data in question. The next step is to fully establish the regression framework which is covered in this project. The following areas are addressed:

a)       The measures of interest. The first step is to introduce an extended multi-state model framework and consider the measures with suitable theoretical and applicative properties. Commonly, one considers transition hazards which in this case allows establishing the link between multi-state models and relative survival, and transition probabilities which provide appealing clinical interpretation of the results. 

b)      Regression modelling. The goal is to provide a regression framework to be used in the extended multi-state model. In survival analysis, the most common regression approach is the semi-parametric Cox model which considers the effects of covariates on the hazard function. Cox models have been standardly used for modelling separate transitions in a multi-state model. We see how this model can be used in the case when cause-specific mortality is considered, meaning that our interest shifts to the effect of covariates on the excess (disease-specific) hazard. Cox-type models have been already proposed for modelling the excess hazard. However, such proposals have been given in the basic relative survival setting, whereas additional challenges arise in the multi-state context, such as the late entry of patients and estimating the regression coefficients and baseline hazards. Furthermore, providing predictions for future patients is an important component of any regression model and this is considered as well. Hence, a comprehensive regression framework covering all these points is a focal point of the project. 

c)       Variance estimation. Once the covariate effects (regression coefficients) and predictions are estimated from the regression model, the next sensible step is to evaluate the uncertainty of these estimates, i.e. estimate the variance. Two different approaches for variance estimation are explored: the first through exact variance estimators, and the second using resampling methods such as bootstrap.

d)      Practical applications. It is our belief that any theoretical statistical work should be complemented with practical medical applications. Thus, the benefit of the developed methodology is illustrated using real-life clinical data.

e)      Software development. The key to using any new statistical methodology is to provide a free user-friendly software implementation which is available to the general public. This toolbox is implemented in the R programming language and based on already available R packages for multi-state models [Mstate2011] and relative survival [Relsurv2018].

4. Literature:

[Putter2007] Putter H, Fiocco M and Geskus RB. Tutorial in biostatistics: competing risks and multi-state models. Statistics in Medicine 2007; 26:2389–2430.

[Andersen2002] Andersen PK and Keiding N. Multi-state models for event history analysis. Statistical Methods in Medical Research 2002; 11:91–115.

[Mstate2011] de Wreede LC, Fiocco M and Putter H. The mstate package for estimation and prediction in non- and semi-parametric multi-state and competing risks models. Computer Methods and Programs in Biomedicine 2010; 99(3): 261–274.

[Mariotto2014] Mariotto AB, Noone AM, Howlader N et al. Cancer Survival: An Overview of Measures, Uses, and Interpretation. Journal of the National Cancer Institute Monographs 2014; 2014(49): 145–186.

[Begg2002] Begg CB and Schrag D. Attribution of deaths following cancer treatment. Journal of the National Cancer Institute 2002; 94(14):1044–1045.

[PoharPerme2012] Pohar Perme M, Stare J, Esteve J. On estimation in relative survival. Biometrics. 2012;68(1):113-20.

[Manevski2022] Manevski D, Putter H, Pohar Perme M, Bonneville EF, Schetelig J, de Wreede LC. Integrating relative survival in multi-state models—a non-parametric approach. Statistical Methods in Medical Research. 2022 Jun;31(6):997-1012.

[Relsurv2018] Pohar Perme M, Pavlič K. Nonparametric Relative Survival Analysis with the R Package relsurv. Journal of Statistical Software, Articles. 2018;87(8):1-27.