Flexible Bayesian excess hazard models using low-rank thin plate splines

Abstract

Excess hazard models became the preferred modelling tool in population-based cancer survival research. In this setting, the model is commonly formulated as the additive decomposition of the overall hazard into two components: the excess hazard due to the cancer of interest and the population hazard due to all other causes of death. We introduce a flexible Bayesian regression model for the log-excess hazard where the baseline log-excess hazard and any non-linear effects of covariates are modelled using low-rank thin plate splines. Using this type of splines will ensure that the log-likelihood function retains tractability not requiring numerical integration. We demonstrate how to derive posterior distributions for the excess hazard and for net survival, a population-level measure of cancer survival that can be derived from excess hazard models. We illustrate the proposed model using survival data for patients diagnosed with colon cancer during 2009 in London, England.

Variation in colon cancer survival for patients living and receiving care in London, 2006–2013: does where you live matter?

“…Maps were created using the software ArcGIS V.10.5. 20 In order to investigate the variability in cancer survival at CCG and hospital levels, net survival (survival from the cancer) and excess hazard of death (hazard due to the cancer) were estimated using flexible Bayesian excess hazard models proposed by Quaresma et al 21 Separate models were fitted for men and women, adjusting for age at diagnosis, deprivation category and stage at diagnosis. To accommodate the hierarchical structure of the data (ie, that patients within a given CCG of residence or hospital of cancer care are likely to share some characteristics), the original model by Quaresma et al 21 was extended with the inclusion of a pair of random effects for CCG and hospital.…”

“…20 In order to investigate the variability in cancer survival at CCG and hospital levels, net survival (survival from the cancer) and excess hazard of death (hazard due to the cancer) were estimated using flexible Bayesian excess hazard models proposed by Quaresma et al 21 Separate models were fitted for men and women, adjusting for age at diagnosis, deprivation category and stage at diagnosis. To accommodate the hierarchical structure of the data (ie, that patients within a given CCG of residence or hospital of cancer care are likely to share some characteristics), the original model by Quaresma et al 21 was extended with the inclusion of a pair of random effects for CCG and hospital. To isolate the excess (cancer-related) hazards of death, the hazards of death from other causes were obtained for each patient with cancer from English life tables defined for each calendar year in 2006-2014 and stratified by single year of age, sex, deprivation category and region of residence.…”

A Unifying Framework for Flexible Excess Hazard Modelling with Applications in Cancer Epidemiology

“…Building on Marra and Radice [2020], we present a flexible methodology that is capable of handling simultaneously all types of censoring as well as left-truncation, while accounting for the excess hazard. Often only right-censoring and potentially left-truncation is allowed [e.g., Fauvernier et al, 2019, Quaresma et al, 2019, thus accounting for any type of censoring broadens the applicability of our framework. Further, a variety of covariate effects, including time-dependent effects, can be flexibly estimated via additive predictors with several types of smoothers.…”

“…The excess hazard function is typically modelled using the available patient characteristics, denoted by x which can, for instance, incorporate continuous and categorical variables in our framework. Several approaches for estimating the excess hazard have been explored in the literature, such as non‐parametric methods, which aim at estimating the cumulative excess hazard (Perme et al, 2012) and the net survival (Pavlič & Pohar‐Perme, 2019; Pohar‐Perme et al, 2009, 2016), parametric methods based on flexibly modelling the baseline excess hazard or cumulative hazard using splines (Charvat et al, 2016; Cramb et al, 2016; Fauvernier et al, 2019; Lambert & Royston, 2009; Quaresma et al, 2019) and modelling the baseline excess hazard function using flexible parametric distributions (Rubio et al, 2019). Most approaches assume a proportional hazards (PH) structure (with the option of adding time‐dependent effects as originally proposed by Cox (1972), which is a convenient way of bypassing the proportionality assumed by the PH setting), with the exception of Rubio et al (2019), who adopt a general hazard structure that contains the PH, accelerated hazards and the accelerated failure time (AFT) models as particular cases.…”

“…Often only right censoring and potentially left truncation is allowed (e.g. Fauvernier et al, 2019; Quaresma et al, 2019), thus accounting for any type of censoring broadens the applicability of our framework. Furthermore, a variety of covariate effects, including time‐dependent effects, can be flexibly estimated via additive predictors with several types of smoothers.…”