Dynamic survival analysis: Modelling the hazard function via ordinary differential equations

The hazard function represents one of the main quantities of interest in the analysis of survival data. We propose a general approach for parametrically modelling the dynamics of the hazard function using systems of autonomous ordinary differential equations (ODEs). This modelling approach can be used to provide qualitative and quantitative analyses of the evolution of the hazard function over time. Our proposal capitalises on the extensive literature on ODEs which, in particular, allows for establishing basic rules or laws on the dynamics of the hazard function via the use of autonomous ODEs. We show how to implement the proposed modelling framework in cases where there is an analytic solution to the system of ODEs or where an ODE solver is required to obtain a numerical solution. We focus on the use of a Bayesian modelling approach, but the proposed methodology can also be coupled with maximum likelihood estimation. A simulation study is presented to illustrate the performance of these models and the interplay of sample size and censoring. Two case studies using real data are presented to illustrate the use of the proposed approach and to highlight the interpretability of the corresponding models. We conclude with a discussion on potential extensions of our work and strategies to include covariates into our framework. Although we focus on examples of Medical Statistics, the proposed framework is applicable in any context where the interest lies in estimating and interpreting the dynamics of the hazard function.

How group structure impacts the numbers at risk for coronary artery disease: polygenic risk scores and nongenetic risk factors in the UK Biobank cohort

The UK Biobank (UKB) is a large cohort study that recruited over 500,000 British participants aged 40-69 in 2006-2010 at 22 assessment centers from across the United Kingdom. Self-reported health outcomes and hospital admission data are 2 types of records that include participants' disease status. Coronary artery disease (CAD) is the most common cause of death in the UKB cohort. After distinguishing between prevalence and incidence CAD events for all UKB participants, we identified geographical variations in age-standardized rates of CAD between assessment centers. Significant distributional differences were found between the pooled cohort equation scores of UKB participants from England and Scotland using the Mann-Whitney test. Polygenic risk scores of UKB participants from England and Scotland and from different assessment centers differed significantly using permutation tests. Our aim was to discriminate between assessment centers with different disease rates by collecting data on disease-related risk factors. However, relying solely on individual-level predictions and averaging them to obtain group-level predictions proved ineffective, particularly due to the presence of correlated covariates resulting from participation bias. By using the Mundlak model, which estimates a random effects regression by including the group means of the independent variables in the model, we effectively addressed these issues. In addition, we designed a simulation experiment to demonstrate the functionality of the Mundlak model. Our findings have applications in public health funding and strategy, as our approach can be used to predict case rates in the future, as both population structure and lifestyle changes are uncertain.

An index of cancer survival to measure progress in cancer control: A tutorial

BACKGROUND

Cancer survival is a key component to assess the overall effectiveness of healthcare systems in their cancer management efforts. A key supporting tool for planning and decision making was introduced with the development of an index of cancer survival that summarises survival for all adults and cancer types into one single estimate, but the implementation details have not been previously described.

METHODS

We detail the construction of the index, including the structure, the calculation of 'sex-age-cancer' specific weights and our proposed modelling strategy to estimate net survival. We provide some practical recommendations through an illustration using a synthetic dataset ('Replica') that we generated for this purpose. An example of R code usage to estimate the index using our approach is provided.

RESULTS

The 'Replica' contains 500 000 artificial cancer records that mimic a cohort of adult cancer patients diagnosed with cancer in England between 1980 and 2004. Using this dataset, we estimated an index of cancer survival at one, five, and ten years after diagnosis for five selected periods of diagnosis, and provide an example of interpretation of these results.

DISCUSSION

We propose a flexible penalised regression modelling strategy to estimate the index's 'sex-age-cancer' specific cancer survival components that minimises the estimation challenge of these components. This tutorial will support researchers in constructing an index of cancer survival for their own setting, facilitating the enrichment of existing toolkits of cancer indicators to more effectively measure progress against cancer in their respective regions/countries.

Extended excess hazard models for spatially dependent survival data

Relative survival represents the preferred framework for the analysis of population cancer survival data. The aim is to model the survival probability associated with cancer in the absence of information about the cause of death. Recent data linkage developments have allowed for incorporating the place of residence into the population cancer databases; however, modeling this spatial information has received little attention in the relative survival setting. We propose a flexible parametric class of spatial excess hazard models (along with inference tools), named "Relative Survival Spatial General Hazard," that allows for the inclusion of fixed and spatial effects in both time-level and hazard-level components. We illustrate the performance of the proposed model using an extensive simulation study, and provide guidelines about the interplay of sample size, censoring, and model misspecification. We present a case study using real data from colon cancer patients in England. This case study illustrates how a spatial model can be used to identify geographical areas with low cancer survival, as well as how to summarize such a model through marginal survival quantities and spatial effects.

On near-redundancy and identifiability of parametric hazard regression models under censoring

We study parametric inference on a rich class of hazard regression models in the presence of right-censoring. Previous literature has reported some inferential challenges, such as multimodal or flat likelihood surfaces, in this class of models for some particular data sets. We formalize the study of these inferential problems by linking them to the concepts of near-redundancy and practical nonidentifiability of parameters. We show that the maximum likelihood estimators of the parameters in this class of models are consistent and asymptotically normal. Thus, the inferential problems in this class of models are related to the finite-sample scenario, where it is difficult to distinguish between the fitted model and a nested nonidentifiable (i.e., parameter-redundant) model. We propose a method for detecting near-redundancy, based on distances between probability distributions. We also employ methods used in other areas for detecting practical nonidentifiability and near-redundancy, including the inspection of the profile likelihood function and the Hessian method. For cases where inferential problems are detected, we discuss alternatives such as using model selection tools to identify simpler models that do not exhibit these inferential problems, increasing the sample size, or extending the follow-up time. We illustrate the performance of the proposed methods through a simulation study. Our simulation study reveals a link between the presence of near-redundancy and practical nonidentifiability. Two illustrative applications using real data, with and without inferential problems, are presented.

Individual frailty excess hazard models in cancer epidemiology

Unobserved individual heterogeneity is a common challenge in population cancer survival studies. This heterogeneity is usually associated with the combination of model misspecification and the failure to record truly relevant variables. We investigate the effects of unobserved individual heterogeneity in the context of excess hazard models, one of the main tools in cancer epidemiology. We propose an individual excess hazard frailty model to account for individual heterogeneity. This represents an extension of frailty modeling to the relative survival framework. In order to facilitate the inference on the parameters of the proposed model, we select frailty distributions which produce closed-form expressions of the marginal hazard and survival functions. The resulting model allows for an intuitive interpretation, in which the frailties induce a selection of the healthier individuals among survivors. We model the excess hazard using a flexible parametric model with a general hazard structure which facilitates the inclusion of time-dependent effects. We illustrate the performance of the proposed methodology through a simulation study. We present a real-data example using data from lung cancer patients diagnosed in England, and discuss the impact of not accounting for unobserved heterogeneity on the estimation of net survival. The methodology is implemented in the R package IFNS.