Alleviating spatial confounding in frailty models.

The confounding between fixed effects and (spatial) random effects in a regression setup is termed spatial confounding. This topic continues to gain attention and has been studied extensively in recent years, given that failure to account for this may lead to a suboptimal inference. To mitigate this, a variety of projection-based approaches under the class of restricted spatial models are available in the context of generalized linear mixed models. However, these projection approaches cannot be directly extended to the spatial survival context via frailty models due to dimension incompatibility between the fixed and spatial random effects. In this work, we introduce a two-step approach to handle this, which involves (i) projecting the design matrix to the dimension of the spatial effect (via dimension reduction) and (ii) assuring that the random effect is orthogonal to this new design matrix (confounding alleviation). Under a fully Bayesian paradigm, we conduct fast estimation and inference using integrated nested Laplace approximation. Both simulation studies and application to a motivating data evaluating respiratory cancer survival in the US state of California reveal the advantages of our proposal in terms of model performance and confounding alleviation, compared to alternatives.

Assessing survival post-kidney transplantation in Australia: A multivariable prediction model.

AIM: Kidney transplantation remains the preferred standard of care for patients with kidney failure. Most patients do not access this treatment and wide variations exist in which patients access transplantation. We sought to develop a model to estimate post-kidney transplant survival to inform more accurate comparisons of access to kidney transplantation. METHODS: Development and validation of prediction models using demographic and clinical data from the Australia and New Zealand Dialysis and Transplant Registry. Adult deceased donor kidney only transplant recipients between 2000 and 2020 were included. Cox proportional hazards regression methods were used with a primary outcome of patient survival. Models were evaluated using Harrell's C-statistic for discrimination, and calibration plots, predicted survival probabilities and Akaike Information Criterion for goodness-of-fit. RESULTS: The model development and validation cohorts included 11 302 participants. Most participants were male (62.8%) and Caucasian (79.2%). Glomerulonephritis was the most common cause of kidney disease (45.6%). The final model included recipient, donor, and transplant related variables. The model had good discrimination (C-statistic, 0.72; 95% confidence interval (CI) 0.70-0.74 in the development cohort, 0.70; 95% CI 0.67-0.73 in the validation cohort and 0.72; 95% CI 0.69-0.75 in the temporal cohort) and was well calibrated. CONCLUSION: We developed a statistical model that predicts post-kidney transplant survival in Australian kidney failure patients. This model will aid in assessing the suitability of kidney transplantation for patients with kidney failure. Survival estimates can be used to make more informed comparisons of access to transplantation between units to better measure equity of access to organ transplantation.

Nonparametric scanning tests of homogeneity for hierarchical models with continuous covariates.

In many applications of hierarchical models, there is often interest in evaluating the inherent heterogeneity in view of observed data. When the underlying hypothesis involves parameters resting on the boundary of their support space such as variances and mixture proportions, it is a usual practice to entertain testing procedures that rely on common heterogeneity assumptions. Such procedures, albeit omnibus for general alternatives, may entail a substantial loss of power for specific alternatives such as heterogeneity varying with covariates. We introduce a novel and flexible approach that uses covariate information to improve the power to detect heterogeneity, without imposing unnecessary restrictions. With continuous covariates, the approach does not impose a regression model relating heterogeneity parameters to covariates or rely on arbitrary discretizations. Instead, a scanning approach requiring continuous dichotomizations of the covariates is proposed. Empirical processes resulting from these dichotomizations are then used to construct the test statistics, with limiting null distributions shown to be functionals of tight random processes. We illustrate our proposals and results on a popular class of two-component mixture models, followed by simulation studies and applications to two real datasets in cancer and caries research.

A flexible class of generalized joint frailty models for the analysis of survival endpoints.

This article focuses on shared frailty models for correlated failure times, as well as joint frailty models for the simultaneous analysis of recurrent events (eg, appearance of new cancerous lesions or hospital readmissions) and a major terminal event (typically, death). As extensions of the Cox model, these joint models usually assume a frailty proportional hazards model for each of the recurrent and terminal event processes. In order to extend these models beyond the proportional hazards assumption, our proposal is to replace these proportional hazards models with generalized survival models, for which the survival function is modeled as a linear predictor through a link function. Depending on the link function considered, these can be reduced to proportional hazards, proportional odds, additive hazards, or probit models. We first consider a fully parametric framework for the time and covariate effects. For proportional and additive hazards models, our approach also allows the use of smooth functions for baseline hazard functions and time-varying coefficients. The dependence between recurrent and terminal event processes is modeled by conditioning on a shared frailty acting differently on the two processes. Parameter estimates are provided using the maximum (penalized) likelihood method, implemented in the R package frailtypack (function GenfrailtyPenal). We perform simulation studies to assess the method, which is also illustrated on real datasets.

Including uncertainty of the expected mortality rates in the prediction of loss in life expectancy.

PURPOSE: This study introduces a novel method for estimating the variance of life expectancy since diagnosis (LEC) and loss in life expectancy (LLE) for cancer patients within a relative survival framework in situations where life tables based on the entire general population are not accessible. LEC and LLE are useful summary measures of survival in population-based cancer studies, but require information on the mortality in the general population. Our method addresses the challenge of incorporating the uncertainty of expected mortality rates when using a sample from the general population. METHODS: To illustrate the approach, we estimated LEC and LLE for patients diagnosed with colon and breast cancer in Sweden. General population mortality rates were based on a random sample drawn from comparators of a matched cohort. Flexible parametric survival models were used to model the mortality among cancer patients and the mortality in the random sample from the general population. Based on the models, LEC and LLE together with their variances were estimated. The results were compared with those obtained using fixed expected mortality rates. RESULTS: By accounting for the uncertainty of expected mortality rates, the proposed method ensures more accurate estimates of variances and, therefore, confidence intervals of LEC and LLE for cancer patients. This is particularly valuable for older patients and some cancer types, where underestimation of the variance can be substantial when the entire general population data are not accessible. CONCLUSION: The method can be implemented using existing software, making it accessible for use in various cancer studies. The provided example of Stata code further facilitates its adoption.

Five-year follow-up mortality prognostic index for colorectal patients.

PURPOSE: To identify 5-year survival prognostic variables in patients with colorectal cancer (CRC) and to propose a survival prognostic score that also takes into account changes over time in the patient's health-related quality of life (HRQoL) status. METHODS: Prospective observational cohort study of CRC patients. We collected data from their diagnosis, intervention, and at 1, 2, 3, and 5 years following the index intervention, also collecting HRQoL data using the EuroQol-5D-5L (EQ-5D-5L), European Organization for Research and Treatment of Cancer's Quality of Life Questionnaire-Core 30 (EORTC-QLQ-C30), and Hospital Anxiety and Depression Scale (HADS) questionnaires. Multivariate Cox proportional models were used. RESULTS: We found predictors of mortality over the 5-year follow-up to be being older; being male; having a higher TNM stage; having a higher lymph node ratio; having a result of CRC surgery classified as R1 or R2; invasion of neighboring organs; having a higher score on the Charlson comorbidity index; having an ASA IV; and having worse scores, worse quality of life, on the EORTC and EQ-5D questionnaires, as compared to those with higher scores in each of those questionnaires respectively. CONCLUSIONS: These results allow preventive and controlling measures to be established on long-term follow-up of these patients, based on a few easily measurable variables. IMPLICATIONS FOR CANCER SURVIVORS: Patients with colorectal cancer should be monitored more closely depending on the severity of their disease and comorbidities as well as the perceived health-related quality of life, and preventive measures should be established to prevent adverse outcomes and therefore to ensure that better treatment is received. TRIAL REGISTRATION: ClinicalTrials.gov identifier: NCT02488161.

Association between childhood adversities and premature and potentially avoidable mortality in adulthood: a population-based study.

BACKGROUND: The association of childhood adversities with mortality has rarely been explored, and even less studied is the question of whether any excess mortality may be potentially preventable. This study examined the association between specific childhood adversities and premature and potentially avoidable mortality (PPAM) in adulthood in a representative sample of the general population. Also, we examined whether the associations were potentially mediated by various adult socioeconomic, psychosocial, and behavioral factors. METHODS: The study used data from the National Population Health Survey (NPHS-1994) linked to the Canadian Vital Statistics Database (CVSD 1994-2014) available from Statistics Canada. The NPHS interview retrospectively assessed childhood exposure to prolonged hospitalization, parental divorce, prolonged parental unemployment, prolonged trauma, parental problematic substance use, physical abuse, and being sent away from home for doing something wrong. An existing definition of PPAM, consisting of causes of death considered preventable or treatable before age 75, was used. Competing cause survival models were used to examine the associations of specific childhood adversities with PPAM in adulthood among respondents aged 18 to 74 years (rounded n = 11,035). RESULTS: During the 20-year follow-up, 5.4% of the sample died prematurely of a cause that was considered potentially avoidable. Childhood adversities had a differential effect on mortality. Physical abuse (age-adjusted sub-hazard ratio; SHR 1.44; 95% CI 1.03, 2.00) and being sent away from home (age-adjusted SHR 2.26; 95% CI 1.43,3.57) were significantly associated with PPAM. The associations were attenuated when adjusted for adulthood factors, namely smoking, poor perceived health, depression, low perceived social support, and low income, consistent with possible mediating effects. Other adversities under study were not associated with PPAM. CONCLUSION: The findings imply that the psychological sequelae of childhood physical abuse and being sent away from home and subsequent uptake of adverse health behavior may lead to increased risk of potentially avoidable mortality. The potential mediators identified offer directions for future research to perform causal mediation analyses with suitable data and identify interventions aimed at preventing premature mortality due to potentially avoidable causes. Other forms of adversities, mostly related to household dysfunction, may not be determinants of the distal health outcome of mortality.

Impact of socio-economic conditions and perinatal factors on risk of becoming a child looked after: a whole population cohort study using routinely collected data in Wales.

OBJECTIVES: Between 1997 and 2021, the number of children looked after (CLA) in Wales, UK, increased steadily, with stark inequalities. We aimed to assess how deprivation and maternal and child perinatal characteristics influence the risk of becoming CLA in Wales. STUDY DESIGN: We constructed a prospective longitudinal cohort of children born in Wales between April 2006 and March 2021 (n = 395,610) using linked administrative records. METHODS: Survival models examined the risk of CLA from birth by small-area deprivation and maternal and child perinatal characteristics. Population attributable fractions quantify the potential impact of action on modifiable risk factors. RESULTS: Children from the most deprived fifth of the population were 3.4 times more likely to enter care than those in the least deprived (demographic adjusted hazard ratios [aHRs] 3.40, 95% confidence interval [CI] 3.08, 3.74). Maternal mental health problems in pregnancy (fully aHR, 2.03, 95% CI 1.88, 2.19) and behavioural factors, such as smoking (aHR 2.46, 95% CI 2.34-2.60), alcohol problems (aHR 2.35, 95% CI 1.70-3.23) and substance use in pregnancy (aHR 5.72, 95% CI 5.03-6.51), as well as child congenital anomalies (aHR 1.46, 95% CI 1.16-1.84), low birth weight (aHR 1.28, 95% CI 1.17, 1.39) and preterm birth (aHR 1.16, 95% CI 1.06, 1.26), were associated with higher risk of CLA status. The risk of CLA in the population may be reduced by 35% (95% CI 0.33, 0.38) if children in the two most deprived fifths of the population experienced the conditions of those in the least deprived. CONCLUSIONS: Deprivation and perinatal maternal health are important modifiable risk factors for children becoming CLA. Our analysis provides insight into the mechanisms of intergenerational transfer of disadvantage in a vulnerable section of the child population and identifies targets for public health action.

Adaptive MCMC for Bayesian Variable Selection in Generalised Linear Models and Survival Models.

Developing an efficient computational scheme for high-dimensional Bayesian variable selection in generalised linear models and survival models has always been a challenging problem due to the absence of closed-form solutions to the marginal likelihood. The Reversible Jump Markov Chain Monte Carlo (RJMCMC) approach can be employed to jointly sample models and coefficients, but the effective design of the trans-dimensional jumps of RJMCMC can be challenging, making it hard to implement. Alternatively, the marginal likelihood can be derived conditional on latent variables using a data-augmentation scheme (e.g., Pólya-gamma data augmentation for logistic regression) or using other estimation methods. However, suitable data-augmentation schemes are not available for every generalised linear model and survival model, and estimating the marginal likelihood using a Laplace approximation or a correlated pseudo-marginal method can be computationally expensive. In this paper, three main contributions are presented. Firstly, we present an extended Point-wise implementation of Adaptive Random Neighbourhood Informed proposal (PARNI) to efficiently sample models directly from the marginal posterior distributions of generalised linear models and survival models. Secondly, in light of the recently proposed approximate Laplace approximation, we describe an efficient and accurate estimation method for marginal likelihood that involves adaptive parameters. Additionally, we describe a new method to adapt the algorithmic tuning parameters of the PARNI proposal by replacing Rao-Blackwellised estimates with the combination of a warm-start estimate and the ergodic average. We present numerous numerical results from simulated data and eight high-dimensional genetic mapping data-sets to showcase the efficiency of the novel PARNI proposal compared with the baseline add-delete-swap proposal.

How Does the Statutory Retirement Age Affect Older workers' Employment in Relation to Individual and Work-Related Factors?

Policies aimed at increasing employment among older people often focus on the statutory retirement age. Taking into account the characteristics of workers and work-related factors, we examine the impact of reaching the statutory retirement age on continuing employment. In addition to the use of survival trees, we propose a novel method to predict the probability of staying in employment based on an ensemble of survival trees. We focus on Poland as an example of a European country with a particularly low share of older workers in the labor force. Moreover, reform was carried out in Poland in 2017, lowering the previously raised pension eligibility age. Like other EU countries, pension eligibility in Poland starts after reaching the statutory retirement age. Our results suggest that the timing of retirement is determined by the statutory retirement age to a limited extent compared to other factors. In the case of women, a match of education and occupation, the employment sector, and holding a managerial position had a greater impact on continuing employment than reaching retirement age. In the case of men, the type of job contract had the greatest impact on continuing employment. Our findings indicate that the policies and initiatives aimed at extending working life should pay more attention to work-related factors and gender differences in employment.

A multistate competing risks framework for preconception prediction of pregnancy outcomes.

BACKGROUND: Preconception pregnancy risk profiles-characterizing the likelihood that a pregnancy attempt results in a full-term birth, preterm birth, clinical pregnancy loss, or failure to conceive-can provide critical information during the early stages of a pregnancy attempt, when obstetricians are best positioned to intervene to improve the chances of successful conception and full-term live birth. Yet the task of constructing and validating risk assessment tools for this earlier intervention window is complicated by several statistical features: the final outcome of the pregnancy attempt is multinomial in nature, and it summarizes the results of two intermediate stages, conception and gestation, whose outcomes are subject to competing risks, measured on different time scales, and governed by different biological processes. In light of this complexity, existing pregnancy risk assessment tools largely focus on predicting a single adverse pregnancy outcome, and make these predictions at some later, post-conception time point. METHODS: We reframe the individual pregnancy attempt as a multistate model comprised of two nested multinomial prediction tasks: one corresponding to conception and the other to the subsequent outcome of that pregnancy. We discuss the estimation of this model in the presence of multiple stages of outcome missingness and then introduce an inverse-probability-weighted Hypervolume Under the Manifold statistic to validate the resulting multivariate risk scores. Finally, we use data from the Effects of Aspirin in Gestation and Reproduction (EAGeR) trial to illustrate how this multistate competing risks framework might be utilized in practice to construct and validate a preconception pregnancy risk assessment tool. RESULTS: In the EAGeR study population, the resulting risk profiles are able to meaningfully discriminate between the four pregnancy attempt outcomes of interest and represent a significant improvement over classification by random chance. CONCLUSIONS: As illustrated in our analysis of the EAGeR data, our proposed prediction framework expands the pregnancy risk assessment task in two key ways-by considering a broader array of pregnancy outcomes and by providing the predictions at an earlier, preconception intervention window-providing obstetricians and their patients with more information and opportunities to successfully guide pregnancy attempts.

Assessing the impact of including variation in general population mortality on standard errors of relative survival and loss in life expectancy.

BACKGROUND: A relative survival approach is often used in population-based cancer studies, where other cause (or expected) mortality is assumed to be the same as the mortality in the general population, given a specific covariate pattern. The population mortality is assumed to be known (fixed), i.e. measured without uncertainty. This could have implications for the estimated standard errors (SE) of any measures obtained within a relative survival framework, such as relative survival (RS) ratios and the loss in life expectancy (LLE). We evaluated the existing approach to estimate SE of RS and the LLE in comparison to if uncertainty in the population mortality was taken into account. METHODS: The uncertainty from the population mortality was incorporated using parametric bootstrap approach. The analysis was performed with different levels of stratification and sizes of the general population used for creating expected mortality rates. Using these expected mortality rates, SEs of 5-year RS and the LLE for colon cancer patients in Sweden were estimated. RESULTS: Ignoring uncertainty in the general population mortality rates had negligible (less than 1%) impact on the SEs of 5-year RS and LLE, when the expected mortality rates were based on the whole general population, i.e. all people living in a country or region. However, the smaller population used for creating the expected mortality rates, the larger impact. For a general population reduced to 0.05% of the original size and stratified by age, sex, year and region, the relative precision for 5-year RS was 41% for males diagnosed at age 85. For the LLE the impact was more substantial with a relative precision of 1286%. The relative precision for marginal estimates of 5-year RS was 3% and 30% and for the LLE 22% and 313% when the general population was reduced to 0.5% and 0.05% of the original size, respectively. CONCLUSIONS: When the general population mortality rates are based on the whole population, the uncertainty in the estimates of the expected measures can be ignored. However, when based on a smaller population, this uncertainty should be taken into account, otherwise SEs may be too small, particularly for marginal values, and, therefore, confidence intervals too narrow.

Structured expert elicitation to inform long-term survival extrapolations using alternative parametric distributions: a case study of CAR T therapy for relapsed/ refractory multiple myeloma.

BACKGROUND: Our aim was to extend traditional parametric models used to extrapolate survival in cost-effectiveness analyses (CEAs) by integrating individual-level patient data (IPD) from a clinical trial with estimates from experts regarding long-term survival. This was illustrated using a case study evaluating survival of patients with triple-class exposed relapsed/refractory multiple myeloma treated with the chimeric antigen receptor (CAR) T cell therapy idecabtagene vicleucel (ide-cel, bb2121) in KarMMa (a phase 2, single-arm trial). METHODS: The distribution of patients expected to be alive at 3, 5, and 10 years given the observed survival from KarMMa (13.3 months of follow-up) was elicited from 6 experts using the SHeffield ELicitation Framework. Quantities of interest were elicited from each expert individually, which informed the consensus elicitation including all experts. Estimates for each time point were assumed to follow a truncated normal distribution. These distributions were incorporated into survival models, which constrained the expected survival based on standard survival distributions informed by IPD from KarMMa. RESULTS: Models for ide-cel that combined KarMMa data with expert opinion were more consistent in terms of survival as well as mean survival at 10 years (survival point estimates under different parametric models were 29-33% at 3 years, 5-17% at 5 years, and 0-6% at 10 years) versus models with KarMMa data alone (11-39% at 3 years, 0-25% at 5 years, and 0-11% at 10 years). CONCLUSION: This case study demonstrates a transparent approach to integrate IPD from trials with expert opinion using traditional parametric distributions to ensure long-term survival extrapolations are clinically plausible.

Modelling multiple time-scales with flexible parametric survival models.

BACKGROUND: There are situations when we need to model multiple time-scales in survival analysis. A usual approach in this setting would involve fitting Cox or Poisson models to a time-split dataset. However, this leads to large datasets and can be computationally intensive when model fitting, especially if interest lies in displaying how the estimated hazard rate or survival change along multiple time-scales continuously. METHODS: We propose to use flexible parametric survival models on the log hazard scale as an alternative method when modelling data with multiple time-scales. By choosing one of the time-scales as reference, and rewriting other time-scales as a function of this reference time-scale, users can avoid time-splitting of the data. RESULT: Through case-studies we demonstrate the usefulness of this method and provide examples of graphical representations of estimated hazard rates and survival proportions. The model gives nearly identical results to using a Poisson model, without requiring time-splitting. CONCLUSION: Flexible parametric survival models are a powerful tool for modelling multiple time-scales. This method does not require splitting the data into small time-intervals, and therefore saves time, helps avoid technological limitations and reduces room for error.

Generating high-fidelity synthetic time-to-event datasets to improve data transparency and accessibility.

BACKGROUND: A lack of available data and statistical code being published alongside journal articles provides a significant barrier to open scientific discourse, and reproducibility of research. Information governance restrictions inhibit the active dissemination of individual level data to accompany published manuscripts. Realistic, high-fidelity time-to-event synthetic data can aid in the acceleration of methodological developments in survival analysis and beyond by enabling researchers to access and test published methods using data similar to that which they were developed on. METHODS: We present methods to accurately emulate the covariate patterns and survival times found in real-world datasets using synthetic data techniques, without compromising patient privacy. We model the joint covariate distribution of the original data using covariate specific sequential conditional regression models, then fit a complex flexible parametric survival model from which to generate survival times conditional on individual covariate patterns. We recreate the administrative censoring mechanism using the last observed follow-up date information from the initial dataset. Metrics for evaluating the accuracy of the synthetic data, and the non-identifiability of individuals from the original dataset, are presented. RESULTS: We successfully create a synthetic version of an example colon cancer dataset consisting of 9064 patients which aims to show good similarity to both covariate distributions and survival times from the original data, without containing any exact information from the original data, therefore allowing them to be published openly alongside research. CONCLUSIONS: We evaluate the effectiveness of the methods for constructing synthetic data, as well as providing evidence that there is minimal risk that a given patient from the original data could be identified from their individual unique patient information. Synthetic datasets using this methodology could be made available alongside published research without breaching data privacy protocols, and allow for data and code to be made available alongside methodological or applied manuscripts to greatly improve the transparency and accessibility of medical research.

Health care providers' effect on long-term mortality after the first-ever stroke: application of shared frailty survival models.

BACKGROUND: Stroke is a global public health challenge. Frailty models can detect and consider the effects of the unknown factors influencing survival along with other known factors. This study aims to evaluate health care providers' effect, along with the demographic and clinical factors, on the stroke patients' survival by using the shared frailty survival models. METHODS: In the 2-year follow-up, a total of 1036 patients with first-ever stroke were recruited from 2013 up to 2015 with census sampling method from two hospitals of Iran, as the health care providers. For model selection, we fitted parametric and semiparametric survival models with parametric shared frailty and used the goodness of fit criteria to compare the models. RESULT: The median follow-up was 730 days. The rate of mortality was 38% during the follow-up period. The Weibull model with gamma frailty had a better fit than the other survival models. The significant variables from the Weibull model were NIHSS score as the stroke severity (score < 5: reference category; scores 5-19: HR = 2.99, p value < 0.001; score ≥ 20: HR = 5.66, p value < 0.001) and age (HR = 1.03, p value < 0.001). Even with the incorporation of the demographic and clinical factors in the survival model, the effect of health care providers as the shared frailty effect was significant (p < 0.001). CONCLUSIONS: Despite considering the known demographic and clinical prognostic factors, health care providers' effect on the patients' survival after stroke was still significant. This may be due to the existing difference between two hospitals in facilities, management, coordination, and efficiency of treatment.

Accounting for delayed entry into observational studies and clinical trials: length-biased sampling and restricted mean survival time.

Individuals in many observational studies and clinical trials for chronic diseases are enrolled well after onset or diagnosis of their disease. Times to events of interest after enrollment are therefore residual or left-truncated event times. Individuals entering the studies have disease that has advanced to varying extents. Moreover, enrollment usually entails probability sampling of the study population. Finally, event times over a short to moderate time horizon are often of interest in these investigations, rather than more speculative and remote happenings that lie beyond the study period. This research report looks at the issue of delayed entry into these kinds of studies and trials. Time to event for an individual is modelled as a first hitting time of an event threshold by a latent disease process, which is taken to be a Wiener process. It is emphasized that recruitment into these studies often involves length-biased sampling. The requisite mathematics for this kind of sampling and delayed entry are presented, including explicit formulas needed for estimation and inference. Restricted mean survival time (RMST) is taken as the clinically relevant outcome measure. Exact parametric formulas for this measure are derived and presented. The results are extended to settings that involve study covariates using threshold regression methods. Methods adapted for clinical trials are presented. An extensive case illustration for a clinical trial setting is then presented to demonstrate the methods, the interpretation of results, and the harvesting of useful insights. The closing discussion covers a number of important issues and concepts.

Tsallis Entropy for Loss Models and Survival Models Involving Truncated and Censored Random Variables.

The aim of this paper consists in developing an entropy-based approach to risk assessment for actuarial models involving truncated and censored random variables by using the Tsallis entropy measure. The effect of some partial insurance models, such as inflation, truncation and censoring from above and truncation and censoring from below upon the entropy of losses is investigated in this framework. Analytic expressions for the per-payment and per-loss entropies are obtained, and the relationship between these entropies are studied. The Tsallis entropy of losses of the right-truncated loss random variable corresponding to the per-loss risk model with a deductible d and a policy limit u is computed for the exponential, Weibull, χ2 or Gamma distribution. In this context, the properties of the resulting entropies, such as the residual loss entropy and the past loss entropy, are studied as a result of using a deductible and a policy limit, respectively. Relationships between these entropy measures are derived, and the combined effect of a deductible and a policy limit is also analyzed. By investigating residual and past entropies for survival models, the entropies of losses corresponding to the proportional hazard and proportional reversed hazard models are derived. The Tsallis entropy approach for actuarial models involving truncated and censored random variables is new and more realistic, since it allows a greater degree of flexibility and improves the modeling accuracy.

The Extrapolation Performance of Survival Models for Data With a Cure Fraction: A Simulation Study.

OBJECTIVES: Curative treatments can result in complex hazard functions. The use of standard survival models may result in poor extrapolations. Several models for data which may have a cure fraction are available, but comparisons of their extrapolation performance are lacking. A simulation study was performed to assess the performance of models with and without a cure fraction when fit to data with a cure fraction. METHODS: Data were simulated from a Weibull cure model, with 9 scenarios corresponding to different lengths of follow-up and sample sizes. Cure and noncure versions of standard parametric, Royston-Parmar, and dynamic survival models were considered along with noncure fractional polynomial and generalized additive models. The mean-squared error and bias in estimates of the hazard function were estimated. RESULTS: With the shortest follow-up, none of the cure models provided good extrapolations. Performance improved with increasing follow-up, except for the misspecified standard parametric cure model (lognormal). The performance of the flexible cure models was similar to that of the correctly specified cure model. Accurate estimates of the cured fraction were not necessary for accurate hazard estimates. Models without a cure fraction provided markedly worse extrapolations. CONCLUSIONS: For curative treatments, failure to model the cured fraction can lead to very poor extrapolations. Cure models provide improved extrapolations, but with immature data there may be insufficient evidence to choose between cure and noncure models, emphasizing the importance of clinical knowledge for model choice. Dynamic cure fraction models were robust to model misspecification, but standard parametric cure models were not.

Systemic inflammation markers and cancer incidence in the UK Biobank.

Systemic inflammation markers have been linked to increased cancer risk and mortality in a number of studies. However, few studies have estimated pre-diagnostic associations of systemic inflammation markers and cancer risk. Such markers could serve as biomarkers of cancer risk and aid in earlier identification of the disease. This study estimated associations between pre-diagnostic systemic inflammation markers and cancer risk in the prospective UK Biobank cohort of approximately 440,000 participants recruited between 2006 and 2010. We assessed associations between four immune-related markers based on blood cell counts: systemic immune-inflammation index (SII), neutrophil-to-lymphocyte ratio (NLR), platelet-to-lymphocyte ratio (PLR), lymphocyte-to-monocyte ratio (LMR), and risk for 17 cancer sites by estimating hazard ratios (HR) using flexible parametric survival models. We observed positive associations with risk for seven out of 17 cancers with SII, NLR, PLR, and negative associations with LMR. The strongest associations were observed for SII for colorectal and lung cancer risk, with associations increasing in magnitude for cases diagnosed within one year of recruitment. For instance, the HR for colorectal cancer per standard deviation increment in SII was estimated at 1.09 (95% CI 1.02-1.16) in blood drawn five years prior to diagnosis and 1.50 (95% CI 1.24-1.80) in blood drawn one month prior to diagnosis. We observed associations between systemic inflammation markers and risk for several cancers. The increase in risk the last year prior to diagnosis may reflect a systemic immune response to an already present, yet clinically undetected cancer. Blood cell ratios could serve as biomarkers of cancer incidence risk with potential for early identification of disease in the last year prior to clinical diagnosis.