Correlated individual frailty: an advantageous approach to survival analysis of bivariate data

Non-inferiority trials with time-to-event data: clarifying the impact of censoring

In non-inferiority (NI) trials with time-to-event data, different types and patterns of censoring may occur, but their impact on trial results is not entirely clear. We investigated the influence of informative and non-informative censoring by conducting extensive simulation studies under the assumption that the NI margin is defined as a maximum acceptable hazard ratio and scenarios typically observed in recent NI trials. We found that while non-informative censoring tends to only affect the power, informative censoring can impact the treatment effect estimates, type I error rate, and power. The magnitude of these effects depends on the between-group differences in the failure and informative censoring risks, as well as the correlation between censoring and failure times, among other factors. The adverse impact of informative censoring was generally decreased with larger NI margins.

A lean additive frailty model: With an application to clustering of melanoma in Norwegian families

Additive frailty models are used to model correlated survival data. However, the complexity of the models increases with cluster size to the extent that practical usage becomes increasingly challenging. We present a modification of the additive genetic gamma frailty (AGGF) model, the lean AGGF (L-AGGF) model, which alleviates some of these challenges by using a leaner additive decomposition of the frailty. The performances of the models were compared and evaluated in a simulation study. The L-AGGF model was used to analyze population-wide data on clustering of melanoma in 2 391 125 two-generational Norwegian families, 1960-2015. Using this model, we could analyze the complete data set, while the original model limited the analysis to a restricted data set (with cluster sizes≤ 7 $$ \le 7 $$ ). We found a substantial clustering of melanoma in Norwegian families and large heterogeneity in melanoma risk across the population, where 52% of the frailty was attributed to the 10% of the population at highest unobserved risk. Due to the improved scalability, the L-AGGF model enables a wider range of analyses of population-wide data compared to the AGGF model. Moreover, the methods outlined here make it possible to perform these analyses in a computationally efficient manner.

Efficient and accurate frailty model approach for genome-wide survival association analysis in large-scale biobanks

With decades of electronic health records linked to genetic data, large biobanks provide unprecedented opportunities for systematically understanding the genetics of the natural history of complex diseases. Genome-wide survival association analysis can identify genetic variants associated with ages of onset, disease progression and lifespan. We propose an efficient and accurate frailty model approach for genome-wide survival association analysis of censored time-to-event (TTE) phenotypes by accounting for both population structure and relatedness. Our method utilizes state-of-the-art optimization strategies to reduce the computational cost. The saddlepoint approximation is used to allow for analysis of heavily censored phenotypes (>90%) and low frequency variants (down to minor allele count 20). We demonstrate the performance of our method through extensive simulation studies and analysis of five TTE phenotypes, including lifespan, with heavy censoring rates (90.9% to 99.8%) on ~400,000 UK Biobank participants with white British ancestry and ~180,000 individuals in FinnGen. We further analyzed 871 TTE phenotypes in the UK Biobank and presented the genome-wide scale phenome-wide association results with the PheWeb browser.

Modeling Time to Blindness of Glaucoma Patients: A Case Study at Jimma University Medical Center

BACKGROUND

Glaucoma is a significant public health problem due to its substantial increase in the projected number of glaucoma cases. In Ethiopia, glaucoma accounts for 5.2% of irreversible blindness and is the fifth main cause of blindness. The main objective of this study was to modeling time to blindness of left and right eyes of glaucoma patients.

STUDY DESIGN

An institution-based retrospective cohort study.

METHODS

This study was conducted among 315 glaucoma patients admitted to the Ophthalmology Department of Jimma University Medical Center (JUMC), Southwest Ethiopia, from January 1, 2016, to August 30, 2020. Kaplan-Meier survival analysis and semiparametric and parametric copula models were applied to identify factors that affect time to the blindness in glaucoma patients and the dependence between time to the blindness of the left and right eyes, respectively. An Akaike information criterion (AIC) was used to select the best non-nested model.

RESULTS

In total, 211 (66.9%) out of 315 glaucoma patients were blind, whereas 104 (33.1%) patients were censored. The median time to the blindness of the left and right eyes was determined to be 12 months. The result suggested that the risk of the blindness in male patients was 1.005 (P = 0.01) times higher than that in female patients, and the risk of the blindness in patients who had early, moderate, and advanced glaucoma was estimated to be 0.582 (P = 0.002), 0.485 (P = 0.001) and 0.887 (P = 0.003) times less than that in the patients with absolute glaucoma, respectively.

CONCLUSIONS

Age, place of residence, gender, type of medication, diabetes disease, stage of glaucoma, duration of treatment, intraocular pressure (IOP), and cup-disk ratio were significantly associated with and affected by the time to the blindness of left and right eyes in glaucoma patients. Awareness should be given to the community to reduce the burden of glaucoma.

Bivariate Survival Copula Analysis of Glaucoma Patients during Blindness: Glaucoma Cases at Alert Hospital in Addis Ababa City of Ethiopia

BACKGROUND

Glaucoma is a worldwide problem that causes vision loss and even blindness, with a prevalence rate ranging from 1.9% to 15%. In Ethiopia, glaucoma is the fifth cause of blindness. This study aimed to explore the dependence between blindness of the right and the left eyes of glaucoma patients and assess the effects of the covariates under the dependence structure.

STUDY DESIGN

A retrospective cohort study.

METHODS

The study population included the glaucoma patients at Alert hospital from January 1, 2018, to December 30, 2021. The copula model was used to estimate the time to the blindness of the right and the left eyes of the glaucoma patients by specifying the dependence between the event times.

RESULTS

Out of 537 glaucoma patients, 224 (41.71%) became blind at least in one eye during the follow-up period. The results of the Clayton copula model revealed that factors, such as age, residence, diabetes mellitus, stage of glaucoma, and hypertension are considered the most prognostic factors for blindness in glaucoma patients. The findings also revealed that there was a strong dependence between the time to the blindness of the right and the left eyes in the glaucoma patients (τ = 0.43).

CONCLUSION

Based on the obtained results, high age, urban residence, hypertension, diabetes mellitus, and higher stage of glaucoma were factors associated with time to the blindness in the glaucoma patients. There was also a dependence between the right and the left eyes of the glaucoma patients. The results revealed that the Clayton Archimedean copula model was the best statistical model for accurate description of glaucoma patients' datasets.

Heavy-tailed phase-type distributions: a unified approach

A phase-type distribution is the distribution of the time until absorption in a finite state-space time-homogeneous Markov jump process, with one absorbing state and the rest being transient. These distributions are mathematically tractable and conceptually attractive to model physical phenomena due to their interpretation in terms of a hidden Markov structure. Three recent extensions of regular phase-type distributions give rise to models which allow for heavy tails: discrete- or continuous-scaling; fractional-time semi-Markov extensions; and inhomogeneous time-change of the underlying Markov process. In this paper, we present a unifying theory for heavy-tailed phase-type distributions for which all three approaches are particular cases. Our main objective is to provide useful models for heavy-tailed phase-type distributions, but any other tail behavior is also captured by our specification. We provide relevant new examples and also show how existing approaches are naturally embedded. Subsequently, two multivariate extensions are presented, inspired by the univariate construction which can be considered as a matrix version of a frailty model. We provide fully explicit EM-algorithms for all models and illustrate them using synthetic and real-life data.

Response-adaptive treatment allocation for clinical studies with recurrent event and terminal event data

In long-term clinical studies, recurrent event data are frequently collected to contrast the efficacy of two different treatments. However, the recurrent event process can be stopped by a terminal event, such as death. For analyzing recurrent event and terminal event data, joint frailty modeling has recently received considerable attention because it makes it possible to study the joint evolution over time of both recurrent and terminal event processes and gives consistent and efficient parameters. For a two-arm clinical trial design based on these data sets, there has been limited research on investigating the balanced design, let alone adaptive treatment allocation. Although equal sample size allocation obtained for both treatments is intuitively first adopted in a trial design, if one treatment is expected to be superior, it may be desirable to allocate more subjects to the effective treatment. In this article, we calculate the required sample size based on restricted randomization and then propose a target response-adaptive randomization procedure for recurrent and terminal event outcomes based on the joint frailty model. A randomization procedure, the doubly adaptive biased coin design that targets some optimal allocations, is implemented. The proposed adaptive treatment allocation schemes have been shown to be capable of reducing the number of trial participants who receive inferior treatment while simultaneously reaching an optimal target, as well as retaining a comparable test power as compared to a restricted randomization design. Finally, two clinical studies, the COAPT trial and the A-HeFT trial, are used to illustrate the advantages of adopting the proposed procedure.

A tutorial on frailty models

The hazard function plays a central role in survival analysis. In a homogeneous population, the distribution of the time to event, described by the hazard, is the same for each individual. Heterogeneity in the distributions can be accounted for by including covariates in a model for the hazard, for instance a proportional hazards model. In this model, individuals with the same value of the covariates will have the same distribution. It is natural to think that not all covariates that are thought to influence the distribution of the survival outcome are included in the model. This implies that there is unobserved heterogeneity; individuals with the same value of the covariates may have different distributions. One way of accounting for this unobserved heterogeneity is to include random effects in the model. In the context of hazard models for time to event outcomes, such random effects are called frailties, and the resulting models are called frailty models. In this tutorial, we study frailty models for survival outcomes. We illustrate how frailties induce selection of healthier individuals among survivors, and show how shared frailties can be used to model positively dependent survival outcomes in clustered data. The Laplace transform of the frailty distribution plays a central role in relating the hazards, conditional on the frailty, to hazards and survival functions observed in a population. Available software, mainly in R, will be discussed, and the use of frailty models is illustrated in two different applications, one on center effects and the other on recurrent events.

A general frailty model to accommodate individual heterogeneity in the acquisition of multiple infections: An application to bivariate current status data

The analysis of multivariate time-to-event (TTE) data can become complicated due to the presence of clustering, leading to dependence between multiple event times. For a long time, (conditional) frailty models and (marginal) copula models have been used to analyze clustered TTE data. In this article, we propose a general frailty model employing a copula function between the frailty terms to construct flexible (bivariate) frailty distributions with the application to current status data. The model has the advantage to impose a less restrictive correlation structure among latent frailty variables as compared to traditional frailty models. Specifically, our model uses a copula function to join the marginal distributions of the frailty vector. In this article, we considered different copula functions, and we relied on marginal gamma distributions due to their mathematical convenience. Based on a simulation study, our novel model outperformed the commonly used additive correlated gamma frailty model, especially in the case of a negative association between the frailties. At the end of the article, the new methodology is illustrated on real-life data applications entailing bivariate serological survey data.

Androgens In Men Study (AIMS): protocol for meta-analyses of individual participant data investigating associations of androgens with health outcomes in men

INTRODUCTION

This study aims to clarify the role(s) of endogenous sex hormones to influence health outcomes in men, specifically to define the associations of plasma testosterone with incidence of cardiovascular events, cancer, dementia and mortality risk, and to identify factors predicting testosterone concentrations. Data will be accrued from at least three Australian, two European and four North American population-based cohorts involving approximately 20 000 men.

METHODS AND ANALYSIS

Eligible studies include prospective cohort studies with baseline testosterone concentrations measured using mass spectrometry and 5 years of follow-up data on incident cardiovascular events, mortality, cancer diagnoses or deaths, new-onset dementia or decline in cognitive function recorded. Data for men, who were not taking androgens or drugs suppressing testosterone production, metabolism or action; and had no prior orchidectomy, are eligible. Systematic literature searches were conducted from 14 June 2019 to 31 December 2019, with no date range set for searches. Aggregate level data will be sought where individual participant data (IPD) are not available. One-stage IPD random-effects meta-analyses will be performed, using linear mixed models, generalised linear mixed models and either stratified or frailty-augmented Cox regression models. Heterogeneity in estimates from different studies will be quantified and bias investigated using funnel plots. Effect size estimates will be presented in forest plots and non-negligible heterogeneity and bias investigated using subgroup or meta-regression analyses.

ETHICS AND DISSEMINATION

Ethics approvals obtained for each of the participating cohorts state that participants have consented to have their data collected and used for research purposes. The Androgens In Men Study has been assessed as exempt from ethics review by the Human Ethics office at the University of Western Australia (file reference number RA/4/20/5014). Each of the component studies had obtained ethics approvals; please refer to respective component studies for details. Research findings will be disseminated to the scientific and broader community via the publication of four research articles, with each involving a separate set of IPD meta-analyses (articles will investigate different, distinct outcomes), at scientific conferences and meetings of relevant professional societies. Collaborating cohort studies will disseminate findings to study participants and local communities.

PROSPERO REGISTRATION NUMBER

CRD42019139668.

Fast Algorithms for Conducting Large-Scale GWAS of Age-at-Onset Traits Using Cox Mixed-Effects Models

Age-at-onset is one of the critical traits in cohort studies of age-related diseases. Large-scale genome-wide association studies (GWAS) of age-at-onset traits can provide more insights into genetic effects on disease progression and transitions between stages. Moreover, proportional hazards (or Cox) regression models can achieve higher statistical power in a cohort study than a case-control trait using logistic regression. Although mixed-effects models are widely used in GWAS to correct for sample dependence, application of Cox mixed-effects models (CMEMs) to large-scale GWAS is so far hindered by intractable computational cost. In this work, we propose COXMEG, an efficient R package for conducting GWAS of age-at-onset traits using CMEMs. COXMEG introduces fast estimation algorithms for general sparse relatedness matrices including, but not limited to, block-diagonal pedigree-based matrices. COXMEG also introduces a fast and powerful score test for dense relatedness matrices, accounting for both population stratification and family structure. In addition, COXMEG generalizes existing algorithms to support positive semidefinite relatedness matrices, which are common in twin and family studies. Our simulation studies suggest that COXMEG, depending on the structure of the relatedness matrix, is orders of magnitude computationally more efficient than coxme and coxph with frailty for GWAS. We found that using sparse approximation of relatedness matrices yielded highly comparable results in controlling false-positive rate and retaining statistical power for an ethnically homogeneous family-based sample. By applying COXMEG to a study of Alzheimer's disease (AD) with a Late-Onset Alzheimer's Disease Family Study from the National Institute on Aging sample comprising 3456 non-Hispanic whites and 287 African Americans, we identified the APOE ε4 variant with strong statistical power (P = 1e-101), far more significant than that reported in a previous study using a transformed variable and a marginal Cox model. Furthermore, we identified novel SNP rs36051450 (P = 2e-9) near GRAMD1B, the minor allele of which significantly reduced the hazards of AD in both genders. These results demonstrated that COXMEG greatly facilitates the application of CMEMs in GWAS of age-at-onset traits.

Beyond the proportional frailty model: Bayesian estimation of individual heterogeneity on mortality parameters

Today, we know that demographic rates can be greatly influenced by differences among individuals in their capacity to survive and reproduce. These intrinsic differences, commonly known as individual heterogeneity, can rarely be measured and are thus treated as latent variables when modeling mortality. Finite mixture models and mixed effects models have been proposed as alternative approaches for inference on individual heterogeneity in mortality. However, in general models assume that individual heterogeneity influences mortality proportionally, which limits the possibility to test hypotheses on the effect of individual heterogeneity on other aspects of mortality such as ageing rates. Here, we propose a Bayesian model that builds upon the mixture models previously developed, but that facilitates making inferences on the effect of individual heterogeneity on mortality parameters other than the baseline mortality. As an illustration, we apply this framework to the Gompertz-Makeham mortality model, commonly used in human and wildlife studies, by assuming that the Gompertz rate parameter is affected by individual heterogeneity. We provide results of a simulation study where we show that the model appropriately retrieves the parameters used for simulation, even for low variances in the heterogeneous parameter. We then apply the model to a dataset on captive chimpanzees and on a cohort life table of 1751 Swedish men, and show how model selection against a null model (i.e., without heterogeneity) can be carried out.

Penalized survival models for the analysis of alternating recurrent event data

Recurrent event data are widely encountered in clinical and observational studies. Most methods for recurrent events treat the outcome as a point process and, as such, neglect any associated event duration. This generally leads to a less informative and potentially biased analysis. We propose a joint model for the recurrent event rate (of incidence) and duration. The two processes are linked through a bivariate normal frailty. For example, when the event is hospitalization, we can treat the time to admission and length-of-stay as two alternating recurrent events. In our method, the regression parameters are estimated through a penalized partial likelihood, and the variance-covariance matrix of the frailty is estimated through a recursive estimating formula. Moreover, we develop a likelihood ratio test to assess the dependence between the incidence and duration processes. Simulation results demonstrate that our method provides accurate parameter estimation, with a relatively fast computation time. We illustrate the methods through an analysis of hospitalizations among end-stage renal disease patients.

Nonproportional hazards and unobserved heterogeneity in clustered survival data: When can we tell the difference?

Multivariate survival data are frequently encountered in biomedical applications in the form of clustered failures (or recurrent events data). A popular way of analyzing such data is by using shared frailty models, which assume that the proportional hazards assumption holds conditional on an unobserved cluster‐specific random effect. Such models are often incorporated in more complicated joint models in survival analysis.

If the random effect distribution has finite expectation, then the conditional proportional hazards assumption does not carry over to the marginal models. It has been shown that, for univariate data, this makes it impossible to distinguish between the presence of unobserved heterogeneity (eg, due to missing covariates) and marginal nonproportional hazards. We show that time‐dependent covariate effects may falsely appear as evidence in favor of a frailty model also in the case of clustered failures or recurrent events data, when the cluster size or number of recurrent events is small. When true unobserved heterogeneity is present, the presence of nonproportional hazards leads to overestimating the frailty effect. We show that this phenomenon is somewhat mitigated as the cluster size grows.

We carry out a simulation study to assess the behavior of test statistics and estimators for frailty models in such contexts. The gamma, inverse Gaussian, and positive stable shared frailty models are contrasted using a novel software implementation for estimating semiparametric shared frailty models. Two main questions are addressed in the contexts of clustered failures and recurrent events: whether covariates with a time‐dependent effect may appear as indication of unobserved heterogeneity and whether the additional presence of unobserved heterogeneity can be detected in this case. Finally, the practical implications are illustrated in a real‐world data analysis example.

Investigating hospital heterogeneity with a competing risks frailty model

Survival analysis is used in the medical field to identify the effect of predictive variables on time to a specific event. Generally, not all variation of survival time can be explained by observed covariates. The effect of unobserved variables on the risk of a patient is called frailty. In multicenter studies, the unobserved center effect can induce frailty on its patients, which can lead to selection bias over time when ignored. For this reason, it is common practice in multicenter studies to include a random frailty term modeling center effect. In a more complex event structure, more than one type of event is possible. Independent frailty variables representing center effect can be incorporated in the model for each competing event. However, in the medical context, events representing disease progression are likely related and correlation is missed when assuming frailties to be independent. In this work, an additive gamma frailty model to account for correlation between frailties in a competing risks model is proposed, to model frailties at center level. Correlation indicates a common center effect on both events and measures how closely the risks are related. Estimation of the model using the expectation‐maximization algorithm is illustrated. The model is applied to a data set from a multicenter clinical trial on breast cancer from the European Organisation for Research and Treatment of Cancer (EORTC trial 10854). Hospitals are compared by employing empirical Bayes estimates methodology together with corresponding confidence intervals.

Correlated gamma frailty models for bivariate survival time data

Frailty models have been developed to quantify both heterogeneity as well as association in multivariate time-to-event data. In recent years, numerous shared and correlated frailty models have been proposed in the survival literature allowing for different association structures and frailty distributions. A bivariate correlated gamma frailty model with an additive decomposition of the frailty variables into a sum of independent gamma components was introduced before. Although this model has a very convenient closed-form representation for the bivariate survival function, the correlation among event- or subject-specific frailties is bounded above which becomes a severe limitation when the values of the two frailty variances differ substantially. In this article, we review existing correlated gamma frailty models and propose novel ones based on bivariate gamma frailty distributions. Such models are found to be useful for the analysis of bivariate survival time data regardless of the censoring type involved. The frailty methodology was applied to right-censored and left-truncated Danish twins mortality data and serological survey current status data on varicella zoster virus and parvovirus B19 infections in Belgium. From our analyses, it has been shown that fitting more flexible correlated gamma frailty models in terms of the imposed association and correlation structure outperforms existing frailty models including the one with an additive decomposition.

Early-life mortality clustering in families: A literature review

Research on early-life mortality in contemporary and historical populations has shown that infant and child mortality tend to cluster in a limited number of high-mortality families, a phenomenon known as 'mortality clustering'. This paper is the first to review the literature on the role of the family in early-life mortality. Contemporary results, methodological and theoretical shortfalls, recent developments, and opportunities for future research are all discussed in this review. Four methodological approaches are distinguished: those based on sibling deaths, mother heterogeneity, thresholds, and excess deaths in populations. It has become clear from research to date that the death of an older child harms the survival chances of younger children in that family, and that fertility behaviour, earlier stillbirths, remarriages, and socio-economic status all explain mortality clustering to some extent.

Modelling time varying heterogeneity in recurrent infection processes: an application to serological data

Frailty models are often used in survival analysis to model multivariate time‐to‐event data. In infectious disease epidemiology, frailty models have been proposed to model heterogeneity in the acquisition of infection and to accommodate association in the occurrence of multiple types of infection. Although traditional frailty models rely on the assumption of lifelong immunity after recovery, refinements have been made to account for reinfections with the same pathogen. Recently, Abrams and Hens quantified the effect of misspecifying the underlying infection process on the basic and effective reproduction number in the context of bivariate current status data on parvovirus B19 and varicella zoster virus. Furthermore, Farrington, Unkel and their co‐workers introduced and applied time varying shared frailty models to paired bivariate serological data. In this paper, we consider an extension of the proposed frailty methodology by Abrams and Hens to account for age‐dependence in individual heterogeneity through the use of age‐dependent shared and correlated gamma frailty models. The methodology is illustrated by using two data applications.

How the effects of aging and stresses of life are integrated in mortality rates: insights for genetic studies of human health and longevity

Increasing proportions of elderly individuals in developed countries combined with substantial increases in related medical expenditures make the improvement of the health of the elderly a high priority today. If the process of aging by individuals is a major cause of age related health declines then postponing aging could be an efficient strategy for improving the health of the elderly. Implementing this strategy requires a better understanding of genetic and non-genetic connections among aging, health, and longevity. We review progress and problems in research areas whose development may contribute to analyses of such connections. These include genetic studies of human aging and longevity, the heterogeneity of populations with respect to their susceptibility to disease and death, forces that shape age patterns of human mortality, secular trends in mortality decline, and integrative mortality modeling using longitudinal data. The dynamic involvement of genetic factors in (i) morbidity/mortality risks, (ii) responses to stresses of life, (iii) multi-morbidities of many elderly individuals, (iv) trade-offs for diseases, (v) genetic heterogeneity, and (vi) other relevant aging-related health declines, underscores the need for a comprehensive, integrated approach to analyze the genetic connections for all of the above aspects of aging-related changes. The dynamic relationships among aging, health, and longevity traits would be better understood if one linked several research fields within one conceptual framework that allowed for efficient analyses of available longitudinal data using the wealth of available knowledge about aging, health, and longevity already accumulated in the research field.

Flexible modelling of simultaneously interval censored and truncated time-to-event data

This paper deals with the analysis of data from a HET-CAM(VT) experiment. From a statistical perspective, such data yield many challenges. First of all, the data are typically time-to-event like data, which are at the same time interval censored and right truncated. In addition, one has to cope with overdispersion as well as clustering. Traditional analysis approaches ignore overdispersion and clustering and summarize the data into a continuous score that can be analysed using simple linear models. In this paper, a novel combined frailty model is developed that simultaneously captures all of the aforementioned statistical challenges posed by the data.

Dynamic frailty models based on compound birth-death processes

Frailty models are used in survival analysis to model unobserved heterogeneity. They accommodate such heterogeneity by the inclusion of a random term, the frailty, which is assumed to multiply the hazard of a subject (individual frailty) or the hazards of all subjects in a cluster (shared frailty). Typically, the frailty term is assumed to be constant over time. This is a restrictive assumption and extensions to allow for time-varying or dynamic frailties are of interest. In this paper, we extend the auto-correlated frailty models of Henderson and Shimakura and of Fiocco, Putter and van Houwelingen, developed for longitudinal count data and discrete survival data, to continuous survival data. We present a rigorous construction of the frailty processes in continuous time based on compound birth-death processes. When the frailty processes are used as mixtures in models for survival data, we derive the marginal hazards and survival functions and the marginal bivariate survival functions and cross-ratio function. We derive distributional properties of the processes, conditional on observed data, and show how to obtain the maximum likelihood estimators of the parameters of the model using a (stochastic) expectation-maximization algorithm. The methods are applied to a publicly available data set.

Statistical challenges of high-dimensional methylation data

With the fast growing field of epigenetics comes the need to better understand the intricacies of DNA methylation data analysis. High-throughput profiling using techniques, such as Illumina's BeadArray assay, enable the quantitative assessment of methylation. Challenges arise from the fact that resulting methylation levels (so-called beta values) are proportions between 0 and 1, often from an asymmetric, bimodal distribution with peaks close to 0 and 1. Therefore, the majority of standard statistical approaches do not apply. The logit transformation into so-called M-values is a common approach to circumvent this problem and aims to allow the use of common statistical methods. However, it can be observed that the transformation from beta to M-values does not necessarily result in an approximately homoscedastic distribution. Often, bimodality, asymmetry and heteroscedasticity are conserved even after transformation. We give an overview and discussion of methods suggested in the recent years that attempt to address the characteristics of methylation data in univariate screening settings. In order to identify 'differential' methylation with respect to covariates of interest while adjusting for confounders, we compare parametric methods, such as linear and beta regression, and nonparametric methods, such as rank-based regression. Our goal is to sensitise researchers to the challenges and issues that arise from this type of data as well as to present possible solutions.

Modeling individual heterogeneity in the acquisition of recurrent infections: an application to parvovirus B19

In recent years, it has been shown that individual heterogeneity in the acquisition of infectious diseases has a large impact on the estimation of important epidemiological parameters such as the (basic) reproduction number. Therefore, frailty modeling has become increasingly popular in infectious disease epidemiology. However, so far, using frailty models, it was assumed infections confer lifelong immunity after recovery, an assumption which is untenable for non-immunizing infections. Our work concentrates on refining the existing frailty models to encompass complexities of waning immunity and consequently recurrent infections while accounting for individual heterogeneity. Univariate and shared gamma frailty models, frequently used in practice, and correlated gamma frailty models that have proven to be a valuable alternative are considered. We show that incorrectly assuming lifelong immunity when applying frailty models introduces substantial bias in the estimation of both the baseline hazard and the frailty parameters, and consequently of the basic and effective reproduction number. We illustrate our work using cross-sectional serological data on parvovirus B19 (PVB19) from Belgium for which the link with varicella zoster virus is exploited.

A joint frailty model to estimate the recurrence process and the disease-specific mortality process without needing the cause of death

In chronic diseases, such as cancer, recurrent events (such as relapses) are commonly observed; these could be interrupted by death. With such data, a joint analysis of recurrence and mortality processes is usually conducted with a frailty parameter shared by both processes. We examined a joint modeling of these processes considering death under two aspects: 'death due to the disease under study' and 'death due to other causes', which enables estimating the disease-specific mortality hazard. The excess hazard model was used to overcome the difficulties in determining the causes of deaths (unavailability or unreliability); this model allows estimating the disease-specific mortality hazard without needing the cause of death but using the mortality hazards observed in the general population. We propose an approach to model jointly recurrence and disease-specific mortality processes within a parametric framework. A correlation between the two processes is taken into account through a shared frailty parameter. This approach allows estimating unbiased covariate effects on the hazards of recurrence and disease-specific mortality. The performance of the approach was evaluated by simulations with different scenarios. The method is illustrated by an analysis of a population-based dataset on colon cancer with observations of colon cancer recurrences and deaths. The benefits of the new approach are highlighted by comparison with the 'classical' joint model of recurrence and overall mortality. Moreover, we assessed the goodness of fit of the proposed model. Comparisons between the conditional hazard and the marginal hazard of the disease-specific mortality are shown, and differences in interpretation are discussed.

A hierarchical frailty model for familial testicular germ-cell tumors

Using a 2-level hierarchical frailty model, we analyzed population-wide data on testicular germ-cell tumor (TGCT) status in 1,135,320 two-generational Norwegian families to examine the risk of TGCT in family members of patients. Follow-up extended from 1954 (cases) or 1960 (unaffected persons) to 2008. The first-level frailty variable was compound Poisson-distributed. The underlying Poisson parameter was randomized to model the frailty variation between families and was decomposed additively to characterize the correlation structure within a family. The frailty relative risk (FRR) for a son, given a diseased father, was 4.03 (95% confidence interval (CI): 3.12, 5.19), with a borderline significantly higher FRR for nonseminoma than for seminoma (P = 0.06). Given 1 affected brother, the lifetime FRR was 5.88 (95% CI: 4.70, 7.36), with no difference between subtypes. Given 2 affected brothers, the FRR was 21.71 (95% CI: 8.93, 52.76). These estimates decreased with the number of additional healthy brothers. The estimated FRRs support previous findings. However, the present hierarchical frailty approach allows for a very precise definition of familial risk. These FRRs, estimated according to numbers of affected/nonaffected family members, provide new insight into familial TGCT. Furthermore, new light is shed on the different familial risks of seminoma and nonseminoma.

The Cox shared frailty model with log-skew-normal frailties

Clustered survival data arise when groups of failure times share a common ingredient; typically, they refer to the same individual or individuals with a common factor. When the association between failure times within the same cluster is of interest, statistical methods called frailty models have been used. The frailty is an unobserved random component which affects the risk level, changing from cluster to cluster but shared by all observations within the same cluster. Various probability distributions have been proposed for the frailty term, with special emphasis on the gamma and log-normal distribution. Since adequate modelling of the frailty distribution is essential to properly investigate the dependence structure, we introduce a new class of frailty models with a flexible distribution form. Specifically, we adopt the skew-normal distribution for the log-transformed frailty, leading to an extension of the log-normal model. After presenting the methodology connected to this choice, we illustrate it with a case study of multiple myeloma patients with autologous stem cells transplantation.

The Heritability of CHD Mortality in Danish Twins After Controlling for Smoking and BMI

Cause-specific mortality data on Danish monozygotic and dizygotic twins are used to analyze heritability estimates of susceptibility to coronary heart disease (CHD) after controlling for smoking and Body Mass Index (BMI). The sample includes 1209 like-sexed twin pairs born between 1890 and 1920, where both individuals were still alive in 1966. The participants completed a questionnaire in 1966 which included questions on smoking, height and weight. The analysis was conducted with both sexes pooled due to the relatively small number of twin pairs. Follow-up was conducted from January 1, 1966 to December 31, 1993. The correlated gammafrailty model with observed covariates was used for the genetic analysis of frailty to account for censoring and truncation in the lifetime data. During the follow-up, 1437 deaths occurred, including 435 deaths due to CHD. Proportions of variance of frailty attributable to genetic and environmental factors were analyzed using the structural equation model approach. Different standard biometric models are fitted to the data to evaluate the magnitude and nature of genetic and environmental factors on mortality. Using the best-fitting model without covariates, heritability of frailty to CHD was found to be 0.45 (0.11). This result changes only slightly to 0.55 (0.13) in a DE model after controlling for smoking and BMI. This analysis underlines the existence of a substantial genetic influence on individual frailty associated with mortality caused by CHD.

The Heritability of Mortality Due to Heart Diseases: A Correlated Frailty Model Applied to Danish Twins

Data of the Danish Twin Registry on monozygotic and dizygotic twins are used to analyse genetic and environmental influences on susceptibility to heart diseases for males and females, respectively. The sample includes 7955 like-sexed twin pairs born between 1870 and 1930. Follow-up was from 1 January 1943 to 31 December 1993 which results in truncation (twin pairs were included in the study if both individuals were still alive at the beginning of the follow-up) and censoring (nearly 40% of the study population was still alive at the end of the follow-up). We use the correlated gamma-frailty model for the genetic analysis of frailty to account for this censoring and truncation. During the follow-up 9370 deaths occurred, 3393 deaths were due to heart diseases in general, including 2476 deaths due to coronary heart disease (CHD). Proportions of variance of frailty attributable to genetic and environmental factors were analyzed using the structural equation model approach. Different standard biometric models are fitted to the data to evaluate the magnitude and nature of genetic and environmental factors on mortality. Using the best fitting model heritability of frailty (liability to death) was found to be 0.55 (0.07) and 0.53 (0.11) with respect to heart diseases and CHD, respectively, for males and 0.52 (0.10) and 0.58 (0.14) for females in a parametric analysis. A semi-parametric analysis shows very similar results. These analyses may indicate the existence of a strong genetic influence on individual frailty associated with mortality caused by heart diseases and CHD in both, males and females. The nature of genetic influences on frailty with respect to heart diseases and CHD is probably additive. No evidence for dominance and shared environment was found.

A frailty model for (interval) censored family survival data, applied to the age at onset of non-physical problems

Family survival data can be used to estimate the degree of genetic and environmental contributions to the age at onset of a disease or of a specific event in life. The data can be modeled with a correlated frailty model in which the frailty variable accounts for the degree of kinship within the family. The heritability (degree of heredity) of the age at a specific event in life (or the onset of a disease) is usually defined as the proportion of variance of the survival age that is associated with genetic effects. If the survival age is (interval) censored, heritability as usually defined cannot be estimated. Instead, it is defined as the proportion of variance of the frailty associated with genetic effects. In this paper we describe a correlated frailty model to estimate the heritability and the degree of environmental effects on the age at which individuals contact a social worker for the first time and to test whether there is a difference between the survival functions of this age for twins and non-twins.

A bivariate survival model with compound Poisson frailty

A correlated frailty model is suggested for analysis of bivariate time-to-event data. The model is an extension of the correlated power variance function (PVF) frailty model (correlated three-parameter frailty model) (J. Epidemiol. Biostat. 1999; 4:53-60). It is based on a bivariate extension of the compound Poisson frailty model in univariate survival analysis (Ann. Appl. Probab. 1992; 4:951-972). It allows for a non-susceptible fraction (of zero frailty) in the population, overcoming the common assumption in survival analysis that all individuals are susceptible to the event under study. The model contains the correlated gamma frailty model and the correlated inverse Gaussian frailty model as special cases. A maximum likelihood estimation procedure for the parameters is presented and its properties are studied in a small simulation study. This model is applied to breast cancer incidence data of Swedish twins. The proportion of women susceptible to breast cancer is estimated to be 15 per cent.

Detecting genes contributing to longevity using twin data

Searching for genes contributing to longevity is a typical task in association analysis. A number of methods can be used for finding this association - from the simplest method based on the technique of contingency tables to more complex algorithms involving demographic data, which allow us to estimate the genotype-specific hazard functions. The independence of individuals is the common assumption in all these methods. At the same time, data on related individuals such as twins are often used in genetic studies. This paper proposes an extension of the relative risk model to encompass twin data. We estimate the power and also discuss what happens if we treat the twin data using the univariate model.

Robust age at onset linkage analysis in nuclear families

OBJECTIVE

Standard methods for linkage analysis ignore the phenotype of the parents when they are not genotyped. However, this information can be useful for gene mapping. In this paper we propose methods for age at onset genetic linkage analysis in sibling pairs, taking into account parental age at onset.

METHODS

Two new score statistics are derived, one from an additive gamma frailty model and one from a log-normal frailty model. The score statistics are classical non-parametric linkage (NPL) statistics weighted by a function of the age at onset of the four family members. The weight depends on information from registries (age-specific incidences) and family studies (sib-sib and father-mother correlation).

RESULTS

In order to investigate how age at onset of sibs and their parents affect the information for linkage analysis the weight functions were studied for rare and common disease models, realistic models for breast cancer and human lifespan. We studied the performance of the weighted NPL methods by simulations. As illustration, the score statistics were applied to the GAW12 data. The results show that it is useful to include parental age at onset information in genetic linkage analysis.

Score test for age at onset genetic linkage analysis in selected sibling pairs

A new score statistic is derived, which uses information from registries (age-specific incidences) and family studies (sib-sib marginal correlation) to weight affected sibling pairs according to their age at onset. Age at onset of sibling pairs is modelled by a gamma frailty model. From this model we derive a bivariate survival function, which depends on the marginal survival and on the marginal correlation. The score statistic for linkage is a classical nonparametric linkage (NPL) statistic where the identical by descent sharing is weighted by a particular function of the age at onset data. Since the statistic is based on survival models, it can also be applied to discordant and healthy sibling pairs. Simulation studies show that the proposed method is robust and more powerful than standard NPL methods. As illustration we apply the new score statistic to data from a breast cancer study.