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In survival analysis with competing risks, the transformation model allows different functions between the outcome and explanatory variables. However, the model's prediction accuracy and the interpretation of parameters may be sensitive to the choice of link function. We review the practical implications of different link functions for regression of the absolute risk (or cumulative incidence) of an event. Specifically, we consider models in which the regression coefficients β have the following interpretation: The probability of dying from cause D during the next t years changes with a factor exp(β) for a one unit change of the corresponding predictor variable, given fixed values for the other predictor variables. The models have a direct interpretation for the predictive ability of the risk factors. We propose some tools to justify the models in comparison with traditional approaches that combine a series of cause-specific Cox regression models or use the Fine-Gray model. We illustrate the methods with the use of bone marrow transplant data. [ABSTRACT FROM AUTHOR]

Abstract Background In sub-Saharan Africa (SSA), malaria caused by Plasmodium falciparum has historically been a major contributor to morbidity and mortality. Recent reports indicate a pronounced decline in infection and disease rates which are commonly ascribed to large-scale bed net programmes and improved case management. However, the decline has also occurred in areas with limited or no intervention. The present study assessed temporal changes in Anopheline populations in two highly malaria-endemic communities of NE Tanzania during the period 1998-2009. Methods Between 1998 and 2001 (1st period) and between 2003 and 2009 (2nd period), mosquitoes were collected weekly in 50 households using CDC light traps. Data on rainfall were obtained from the nearby climate station and were used to analyze the association between monthly rainfall and malaria mosquito populations. Results The average number of Anopheles gambiae and Anopheles funestus per trap decreased by 76.8% and 55.3%, respectively over the 1st period, and by 99.7% and 99.8% over the 2nd period. During the last year of sampling (2009), the use of 2368 traps produced a total of only 14 Anopheline mosquitoes. With the exception of the decline in An. gambiae during the 1st period, the results did not reveal any statistical association between mean trend in monthly rainfall and declining malaria vector populations. Conclusion A longitudinal decline in the density of malaria mosquito vectors was seen during both study periods despite the absence of organized vector control. Part of the decline could be associated with changes in the pattern of monthly rainfall, but other factors may also contribute to the dramatic downward trend. A similar decline in malaria vector densities could contribute to the decrease in levels of malaria infection reported from many parts of SSA.

Abstract Background Malaria due to Plasmodium falciparum is the leading cause of morbidity and mortality in Tanzania. According to health statistics, malaria accounts for about 30% and 15% of hospital admissions and deaths, respectively. The risk of P. falciparum infection varies across the country. This study describes the spatial variation and socio-economic determinants of P. falciparum infection in northeastern Tanzania. Methods The study was conducted in 14 villages located in highland, lowland and urban areas of Korogwe district. Four cross-sectional malaria surveys involving individuals aged 0-19 years were conducted during short (Nov-Dec) and long (May-Jun) rainy seasons from November 2005 to June 2007. Household socio-economic status (SES) data were collected between Jan-April 2006 and household's geographical positions were collected using hand-held geographical positioning system (GPS) unit. The effects of risk factors were determined using generalized estimating equation and spatial risk of P. falciparum infection was modelled using a kernel (non-parametric) method. Results There was a significant spatial variation of P. falciparum infection, and urban areas were at lower risk. Adjusting for covariates, high risk of P. falciparum infection was identified in rural areas of lowland and highland. Bed net coverage levels were independently associated with reduced risk of P. falciparum by 19.1% (95%CI: 8.9-28.2, p < 0.001) and by 39.3% (95%CI: 28.9-48.2, p < 0.001) in households with low and high coverage, respectively, compared to those without bed nets. Households with moderate and lower SES had risk of infection higher than 60% compared to those with higher SES; while inhabitants of houses built of mud walls were at 15.5% (95%CI: 0.1 - 33.3, p < 0.048) higher risk compared to those living in houses built by bricks. Individuals in houses with thatched roof had an excess risk of 17.3% (95%CI: 4.1 - 32.2, p < 0.009) compared to those living in houses roofed with iron sheet. Conclusions There was high spatial variation of risk of P. falciparum infection and urban area was at the lowest risk. High bed net coverage, better SES and good housing were among the important risk factors associated with low risk of P. falciparum infection.

Abstract Background In Sub-Sahara Africa, malaria due to Plasmodium falciparum is the main cause of ill health. Evaluation of malaria interventions, such as drugs and vaccines depends on clinical definition of the disease, which is still a challenge due to lack of distinct malaria specific clinical features. Parasite threshold is used in definition of clinical malaria in evaluation of interventions. This however, is likely to be influenced by other factors such as transmission intensity as well as individual level of immunity against malaria. Methods This paper describes step function and dose response model with threshold parameter as a tool for estimation of parasite threshold for onset of malaria fever in highlands (low transmission) and lowlands (high transmission intensity) strata. These models were fitted using logistic regression stratified by strata and age groups (0-1, 2-3, 4-5, 6-9, and 10-19 years). Dose response model was further extended to fit all age groups combined in each stratum. Sub-sampling bootstrap was used to compute confidence intervals. Cross-sectional and passive case detection data from Korogwe district, north eastern Tanzania were used. Results Dose response model was better in the estimation of parasite thresholds. Parasite thresholds (scale = log parasite/μL) were high in lowlands than in highlands. In the lowlands, children in age group 4-5 years had the highest parasite threshold (8.73) while individuals aged 10-19 years had the lowest (6.81). In the highlands, children aged 0-1 years had the highest threshold (7.12) and those aged 10-19 years had the lowest (4.62). Regression analysis with all ages combined showed similar pattern of thresholds in both strata, whereby, in the lowlands the threshold was highest in age group 2-5 years and lowest in older individuals, while in the highlands was highest in age group 0-1 and decreased with increased age. The sensitivity of parasite threshold by age group ranged from 64%-74% in the lowlands and 67%-97% in the highlands; while specificity ranged between 67%-90% in the lowlands and 37%-73% in the highlands. Conclusion Dose response model with threshold parameter can be used to estimate parasite threshold associated with malaria fever onset. Parasite threshold were lower in older individuals and in low malaria transmission area.

Background: In the 1960s only 1/3 of children with soft-tissue sarcomas survived, however with improved treatments survival today has reached 70%. Given the previous poor survival and the rarity of soft-tissue sarcomas, the risk of somatic late effects in a large cohort of Nordic soft-tissue sarcoma survivors has not yet been assessed., Methods: In this population-based cohort study we identified 985 five-year soft-tissue sarcoma survivors in Nordic nationwide cancer registries and late effects in national hospital registries covering the period 1964-2012. Information on tumour site and radiotherapy was available for Danish and Finnish survivors ( N  = 531). Using disease-specific rates of first-time hospital contacts for somatic diseases in survivors and in 4,830 matched comparisons we calculated relative rates (RR) and rate differences (RD)., Results: Survivors had a RR of 1.5 (95% CI 1.4-1.7) and an absolute RD of 23.5 (17.7-29.2) for a first hospital contact per 1,000 person-years. The highest risks in both relative and absolute terms were of endocrine disorders (RR = 2.5; RD = 7.6), and diseases of the nervous system (RR = 1.9; RD = 6.6), digestive organs (RR = 1.7; RD = 5.4) and urinary system (RR = 1.7; RD = 5.6). By tumour site, excess risk was lower after extremity tumours. Irradiated survivors had a 2.6 (1.2-5.9) times higher risk than non-irradiated., Conclusions: Soft-tissue sarcoma survivors have an increased risk of somatic late effects in 5 out of 10 main diagnostic groups of diseases, and the risk remains increased up to 40 years after cancer diagnosis. Risks were slightly lower for those treated for tumours in the extremities, and radiotherapy increased the risk by more than two-fold.

We suggest a regression approach to estimate the excess cumulative incidence function (CIF) when matched data are available. In a competing risk setting, we define the excess risk as the difference between the CIF in the exposed group and the background CIF observed in the unexposed group. We show that the excess risk can be estimated through an extended binomial regression model that actively uses the matched structure of the data, avoiding further estimation of both the exposed and the unexposed CIFs. The method naturally deals with two time scales, age and time since exposure and simplifies how to deal with the left truncation on the age time-scale. The model makes it easy to predict individual excess risk scenarios and allows for a direct interpretation of the covariate effects on the cumulative incidence scale. After introducing the model and some theory to justify the approach, we show via simulations that our model works well in practice. We conclude by applying the excess risk model to data from the ALiCCS study to investigate the excess risk of late events in childhood cancer survivors., (© 2020 John Wiley & Sons, Ltd.)

We are interested in the estimation of average treatment effects based on right-censored data of an observational study. We focus on causal inference of differences between t-year absolute event risks in a situation with competing risks. We derive doubly robust estimation equations and implement estimators for the nuisance parameters based on working regression models for the outcome, censoring, and treatment distribution conditional on auxiliary baseline covariates. We use the functional delta method to show that these estimators are regular asymptotically linear estimators and estimate their variances based on estimates of their influence functions. In empirical studies, we assess the robustness of the estimators and the coverage of confidence intervals. The methods are further illustrated using data from a Danish registry study., (© 2020 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

We present an excess risk regression model for matched cohort data, where the occurrence of some events for individuals with a disease is compared to that of healthy controls that are matched at the onset-of-disease by various factors. By using the matched structure, we show how to estimate the excess risk and its dependence on covariates on both proportional and additive form. We remove the individual effects on background mortality related to matching factors by considering differences. The model handles two different time scales, namely attained age and follow-up time. First, we solve estimating equations for the non-parametric and parametric components of the excess risk model, providing large sample properties for the suggested estimators. Next, we report results from a simulation study. Lastly, we describe an application of the method on childhood cancer data, to study the excess risk of cardiovascular events in adults' life among childhood cancer survivors.

We propose to model the cause-specific cumulative incidence function of multivariate competing risks data using a random effects model that allows for within-cluster dependence of both risk and timing. The model contains parameters that makes it possible to assess how the two are connected, e.g. if high-risk is related to early onset. Under the proposed model, the cumulative incidences of all failure causes are modeled and all cause-specific and cross-cause associations specified. Consequently, left-truncation and right-censoring are easily dealt with. The proposed model is assessed using simulation studies and applied in analysis of Danish register-based family data on breast cancer., (© The Author 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com.)

Hip fracture patients often have comorbid conditions. We investigated whether the combination of comorbidity and hip fracture could explain the previously observed excess mortality among hip fracture patients as compared with the general population. Using a population-based matched study design with 38,126 Norwegian women who suffered a hip fracture during the period 2009-2015 and the same number of women in a matched comparison cohort, we matched participants on prefracture comorbidity, age, and education. We estimated relative survival and additive and multiplicative comorbidity-hip fracture interactions. An additive comorbidity-hip fracture interaction of 4 or 9 additional deaths per 100 patients, depending on Charlson Comorbidity Index (CCI) score, was observed 1 year after hip fracture. Among women with a CCI score of ≥3, 15 additional deaths per 100 patients were observed; of these, 9 deaths could be attributed to the interaction and 6 to the hip fracture per se. On the relative scale, we observed increasing heterogeneity in survival by comorbidity over time; survival was reduced by 39% after 6 years among patients with a CCI score of ≥3, while among women with no comorbidity, survival was reduced by 17% (hip fracture vs. no hip fracture). In summary, prefracture comorbidity was associated with short-term absolute excess mortality and long-term relative excess mortality.

The cumulative incidence function quantifies the probability of failure over time due to a specific cause for competing risks data. The generalized semiparametric regression models for the cumulative incidence functions with missing covariates are investigated. The effects of some covariates are modeled as non-parametric functions of time while others are modeled as parametric functions of time. Different link functions can be selected to add flexibility in modeling the cumulative incidence functions. The estimation procedures based on the direct binomial regression and the inverse probability weighting of complete cases are developed. This approach modifies the full data weighted least squares equations by weighting the contributions of observed members through the inverses of estimated sampling probabilities which depend on the censoring status and the event types among other subject characteristics. The asymptotic properties of the proposed estimators are established. The finite-sample performances of the proposed estimators and their relative efficiencies under different two-phase sampling designs are examined in simulations. The methods are applied to analyze data from the RV144 vaccine efficacy trial to investigate the associations of immune response biomarkers with the cumulative incidence of HIV-1 infection.

The hazard ratios resulting from a Cox's regression hazards model are hard to interpret and to be converted into prolonged survival time. As the main goal is often to study survival functions, there is increasing interest in summary measures based on the survival function that are easier to interpret than the hazard ratio; the residual mean time is an important example of those measures. However, because of the presence of right censoring, the tail of the survival distribution is often difficult to estimate correctly. Therefore, we consider the restricted residual mean time, which represents a partial area under the survival function, given any time horizon τ, and is interpreted as the residual life expectancy up to τ of a subject surviving up to time t. We present a class of regression models for this measure, based on weighted estimating equations and inverse probability of censoring weighted estimators to model potential right censoring. Furthermore, we show how to extend the models and the estimators to deal with delayed entries. We demonstrate that the restricted residual mean life estimator is equivalent to integrals of Kaplan-Meier estimates in the case of simple factor variables. Estimation performance is investigated by simulation studies. Using real data from Danish Monitoring Cardiovascular Risk Factor Surveys, we illustrate an application to additive regression models and discuss the general assumption of right censoring and left truncation being dependent on covariates. Copyright © 2017 John Wiley & Sons, Ltd., (Copyright © 2017 John Wiley & Sons, Ltd.)

Familial aggregation and the role of genetic and environmental factors can be investigated through family studies analysed using the liability-threshold model. The liability-threshold model ignores the timing of events including the age of disease onset and right censoring, which can lead to estimates that are difficult to interpret and are potentially biased. We incorporate the time aspect into the liability-threshold model for case-control-family data following the same approach that has been applied in the twin setting. Thus, the data are considered as arising from a competing risks setting and inverse probability of censoring weights are used to adjust for right censoring. In the case-control-family setting, recognising the existence of competing events is highly relevant to the sampling of control probands. Because of the presence of multiple family members who may be censored at different ages, the estimation of inverse probability of censoring weights is not as straightforward as in the twin setting but requires consideration. We propose to employ a composite likelihood conditioning on proband status that markedly simplifies adjustment for right censoring. We assess the proposed approach using simulation studies and apply it in the analysis of two Danish register-based case-control-family studies: one on cancer diagnosed in childhood and adolescence, and one on early-onset breast cancer. Copyright © 2017 John Wiley & Sons, Ltd., (Copyright © 2017 John Wiley & Sons, Ltd.)

High-dimensional regression has become an increasingly important topic for many research fields. For example, biomedical research generates an increasing amount of data to characterize patients' bio-profiles (e.g. from a genomic high-throughput assay). The increasing complexity in the characterization of patients' bio-profiles is added to the complexity related to the prolonged follow-up of patients with the registration of the occurrence of possible adverse events. This information may offer useful insight into disease dynamics and in identifying subset of patients with worse prognosis and better response to the therapy. Although in the last years the number of contributions for coping with high and ultra-high-dimensional data in standard survival analysis have increased (Witten and Tibshirani, 2010. Survival analysis with high-dimensional covariates. Statistical Methods in Medical Research 19: (1), 29-51), the research regarding competing risks is less developed (Binder and others, 2009. Boosting for high-dimensional time-to-event data with competing risks. Bioinformatics 25: (7), 890-896). The aim of this work is to consider how to do penalized regression in the presence of competing events. The direct binomial regression model of Scheike and others (2008. Predicting cumulative incidence probability by direct binomial regression. Biometrika 95: (1), 205-220) is reformulated in a penalized framework to possibly fit a sparse regression model. The developed approach is easily implementable using existing high-performance software to do penalized regression. Results from simulation studies are presented together with an application to genomic data when the endpoint is progression-free survival. An R function is provided to perform regularized competing risks regression according to the binomial model in the package timereg (Scheike and Martinussen, 2006. Dynamic Regression models for survival data New York: Springer), available through CRAN., (© The Author 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com.)

Missing covariate values is a common problem in survival analysis. In this paper we propose a novel method for the Cox regression model that is close to maximum likelihood but avoids the use of the EM-algorithm. It exploits that the observed hazard function is multiplicative in the baseline hazard function with the idea being to profile out this function before carrying out the estimation of the parameter of interest. In this step one uses a Breslow type estimator to estimate the cumulative baseline hazard function. We focus on the situation where the observed covariates are categorical which allows us to calculate estimators without having to assume anything about the distribution of the covariates. We show that the proposed estimator is consistent and asymptotically normal, and derive a consistent estimator of the variance-covariance matrix that does not involve any choice of a perturbation parameter. Moderate sample size performance of the estimators is investigated via simulation and by application to a real data example.

With competing risks data, one often needs to assess the treatment and covariate effects on the cumulative incidence function. Fine and Gray proposed a proportional hazards regression model for the subdistribution of a competing risk with the assumption that the censoring distribution and the covariates are independent. Covariate-dependent censoring sometimes occurs in medical studies. In this paper, we study the proportional hazards regression model for the subdistribution of a competing risk with proper adjustments for covariate-dependent censoring. We consider a covariate-adjusted weight function by fitting the Cox model for the censoring distribution and using the predictive probability for each individual. Our simulation study shows that the covariate-adjusted weight estimator is basically unbiased when the censoring time depends on the covariates, and the covariate-adjusted weight approach works well for the variance estimator as well. We illustrate our methods with bone marrow transplant data from the Center for International Blood and Marrow Transplant Research (CIBMTR). Here cancer relapse and death in complete remission are two competing risks.

Twin and family data provide a key source for evaluating inheritance of specific diseases. A standard analysis of such data typically involves the computation of prevalences and different concordance measures such as the casewise concordance, that is the probability that one twin has the disease given that the co-twin has the disease. Most diseases have a varying age-of-onset that will lead to age-specific prevalence. Typically, this aspect is not considered, and this may lead to severe bias as well as make it very unclear exactly what population quantities that we are estimating. In addition, one will typically need to deal with censoring in the data, that is the fact that we for some subjects only know that they are alive at a specific age without having the disease. These subjects needs to be considered age specifically, and clearly if they are young there is still a risk that they will develop the disease. The aim of this contribution is to show that the standard casewise concordance and standard prevalence estimators do not work in general for age-of-onset data. We show how one can in fact do something easy and simple even with censored data. The key is to take age into account when analysing such data.

We consider data from the Danish twin registry and aim to study in detail how lifetimes for twin-pairs are correlated. We consider models where we specify the marginals using a regression structure, here Cox's regression model or the additive hazards model. The best known such model is the Clayton-Oakes model. This model can be extended in several directions. One extension is to allow the dependence parameter to depend on covariates. Another extension is to model dependence via piecewise constant cross-hazard ratio models. We show how both these models can be implemented for large sample data, and suggest a computational solution for obtaining standard errors for such models for large registry data. In addition we consider alternative models that have some computational advantages and with different dependence parameters based on odds ratios of the survival function using the Plackett distribution. We also suggest a way of assessing how and if the dependence is changing over time, by considering either truncated or right-censored versions of the data to measure late or early dependence. This can be used for formally testing if the dependence is constant, or decreasing/increasing. The proposed procedures are applied to Danish twin data to describe dependence in the lifetimes of the twins. Here we show that the early deaths are more correlated than the later deaths, and by comparing MZ and DZ associations we suggest that early deaths might be more driven by genetic factors. This conclusion requires models that are able to look at more local dependence measures. We further show that the dependence differs for MZ and DZ twins and appears to be the same for males and females, and that there are indications that the dependence increases over calendar time.

Recently, Fine and Gray (J Am Stat Assoc 94:496-509, 1999) proposed a semi-parametric proportional regression model for the subdistribution hazard function which has been used extensively for analyzing competing risks data. However, failure of model adequacy could lead to severe bias in parameter estimation, and only a limited contribution has been made to check the model assumptions. In this paper, we present a class of analytical methods and graphical approaches for checking the assumptions of Fine and Gray's model. The proposed goodness-of-fit test procedures are based on the cumulative sums of residuals, which validate the model in three aspects: (1) proportionality of hazard ratio, (2) the linear functional form and (3) the link function. For each assumption testing, we provide a p-values and a visualized plot against the null hypothesis using a simulation-based approach. We also consider an omnibus test for overall evaluation against any model misspecification. The proposed tests perform well in simulation studies and are illustrated with two real data examples.

There has been considerable interest in studying the magnitude and type of inheritance of specific diseases. This is typically derived from family or twin studies, where the basic idea is to compare the correlation for different pairs that share different amount of genes. We here consider data from the Danish twin registry and discuss how to define heritability for cancer occurrence. The key point is that this should be done taking censoring as well as competing risks due to e.g.  death into account. We describe the dependence between twins on the probability scale and show that various models can be used to achieve sensible estimates of the dependence within monozygotic and dizygotic twin pairs that may vary over time. These dependence measures can subsequently be decomposed into a genetic and environmental component using random effects models. We here present several novel models that in essence describe the association in terms of the concordance probability, i.e., the probability that both twins experience the event, in the competing risks setting. We also discuss how to deal with the left truncation present in the Nordic twin registries, due to sampling only of twin pairs where both twins are alive at the initiation of the registries.

For twin time-to-event data, we consider different concordance probabilities, such as the casewise concordance that are routinely computed as a measure of the lifetime dependence/correlation for specific diseases. The concordance probability here is the probability that both twins have experienced the event of interest. Under the assumption that both twins are censored at the same time, we show how to estimate this probability in the presence of right censoring, and as a consequence, we can then estimate the casewise twin concordance. In addition, we can model the magnitude of within pair dependence over time, and covariates may be further influential on the marginal risk and dependence structure. We establish the estimators large sample properties and suggest various tests, for example, for inferring familial influence. The method is demonstrated and motivated by specific twin data on cancer events with the competing risk death. We thus aim to quantify the degree of dependence through the casewise concordance function and show a significant genetic component., (Copyright © 2013 John Wiley & Sons, Ltd.)

In this paper we consider a problem from hematopoietic cell transplant (HCT) studies where there is interest on assessing the effect of haplotype match for donor and patient on the cumulative incidence function for a right censored competing risks data. For the HCT study, donor's and patient's genotype are fully observed and matched but their haplotypes are missing. In this paper we describe how to deal with missing covariates of each individual for competing risks data. We suggest a procedure for estimating the cumulative incidence functions for a flexible class of regression models when there are missing data, and establish the large sample properties. Small sample properties are investigated using simulations in a setting that mimics the motivating haplotype matching problem. The proposed approach is then applied to the HCT study.

The cross-odds ratio is defined as the ratio of the conditional odds of the occurrence of one cause-specific event for one subject given the occurrence of the same or a different cause-specific event for another subject in the same cluster over the unconditional odds of occurrence of the cause-specific event. It is a measure of the association between the correlated cause-specific failure times within a cluster. The joint cumulative incidence function can be expressed as a function of the marginal cumulative incidence functions and the cross-odds ratio. Assuming that the marginal cumulative incidence functions follow a generalized semiparametric model, this paper studies the parametric regression modeling of the cross-odds ratio. A set of estimating equations are proposed for the unknown parameters and the asymptotic properties of the estimators are explored. Non-parametric estimation of the cross-odds ratio is also discussed. The proposed procedures are applied to the Danish twin data to model the associations between twins in their times to natural menopause and to investigate whether the association differs among monozygotic and dizygotic twins and how these associations have changed over time.

An additive hazards model may be used to quantify the effect of genetic and environmental predictors on flowering of sugar beet plants recorded as data-grouped time-to-event data. Estimated predictor effects have an intuitive interpretation rooted in the underlying time dynamics of the flowering process. However, agricultural experiments are often designed using several plots containing a large number of plants that are subsequently being monitored. In this article, we consider an additive hazards model with an additional plot structure induced by latent shared frailty variables. This approach enables us to derive a method to assess the quality of predictors in terms of how much plot variation they explain. We apply the method to a large data set exploring flowering of sugar beet and conclude that the genetic predictor biotype, which has a strong effect, also explains a substantial amount of the plot variation. The method is also applied to a data set from medical research concerning days to virus positivity of serum samples in AIDS patients., (© 2011, The International Biometric Society.)

In this paper we consider a problem from bone marrow transplant (BMT) studies where there is interest on assessing the effect of haplotype match for donor and patient on the overall survival. The BMT study we consider is based on donors and patients that are genotype matched, and this therefore leads to a missing data problem. We show how Aalen's additive risk model can be applied in this setting with the benefit that the time-varying haplo-match effect can be easily studied. This problem has not been considered before, and the standard approach where one would use the EM-algorithm cannot be applied for this model because the likelihood is hard to evaluate without additional assumptions. We suggest an approach based on multivariate estimating equations that are solved using a recursive structure. This approach leads to an estimator where the large sample properties can be developed using product-integration theory. Small sample properties are investigated using simulations in a setting that mimics the motivating haplo-match problem.

In this paper we describe flexible competing risks regression models using the comp.risk() function available in the timereg package for R based on Scheike et al. (2008). Regression models are specified for the transition probabilities, that is the cumulative incidence in the competing risks setting. The model contains the Fine and Gray (1999) model as a special case. This can be used to do goodness-of-fit test for the subdistribution hazards' proportionality assumption (Scheike and Zhang 2008). The program can also construct confidence bands for predicted cumulative incidence curves.We apply the methods to data on follicular cell lymphoma from Pintilie (2007), where the competing risks are disease relapse and death without relapse. There is important non-proportionality present in the data, and it is demonstrated how one can analyze these data using the flexible regression models.

Genetic association studies often investigate the effect of haplotypes on an outcome of interest. Haplotypes are not observed directly, and this complicates the inclusion of such effects in survival models. We describe a new estimating equations approach for Cox's regression model to assess haplotype effects for survival data. These estimating equations are simple to implement and avoid the use of the EM algorithm, which may be slow in the context of the semiparametric Cox model with incomplete covariate information. These estimating equations also lead to easily computable, direct estimators of standard errors, and thus overcome some of the difficulty in obtaining variance estimators based on the EM algorithm in this setting. We also develop an easily implemented goodness-of-fit procedure for Cox's regression model including haplotype effects. Finally, we apply the procedures presented in this article to investigate possible haplotype effects of the PAF-receptor on cardiovascular events in patients with coronary artery disease, and compare our results to those based on the EM algorithm., (© 2009, The International Biometric Society.)

We propose a semiparametric random effects model for multivariate competing risks data when the failures of a particular type are of interest. Under this model, the marginal cumulative incidence functions follow a generalized semiparametric additive model. The associations between the cause-specific failure times can be studied through dependence parameters of copula functions that are allowed to depend on cluster-level covariates. A cross-odds ratio-type measure is proposed to describe the associations between cause-specific failure times, and its relationship to the dependence parameters is explored. We develop a two-stage estimation procedure where the marginal models are estimated in the first stage and the dependence parameters are estimated in the second stage. The large sample properties of the proposed estimators are derived. The proposed procedures are applied to Danish twin data to model the cumulative incidence for the age of natural menopause and to investigate the association in the onset of natural menopause between monozygotic and dizygotic twins.

Regression analysis of survival data, and more generally event history data, is typically based on Cox's regression model. We here review some recent methodology, focusing on the limitations of Cox's regression model. The key limitation is that the model is not well suited to represent time-varying effects. We start by considering classical and also more recent goodness-of-fit procedures for the Cox model that will reveal when the Cox model does not capture important aspects of the data, such as time-varying effects. We present recent regression models that are able to deal with and describe such time-varying effects. The introduced models are all applied to data on breast cancer from the Norwegian cancer registry, and these analyses clearly reveal the shortcomings of Cox's regression model and the need for other supplementary analyses with models such as those we present here.

This paper considers estimation and prediction in the Aalen additive hazards model in the case where the covariate vector is high-dimensional such as gene expression measurements. Some form of dimension reduction of the covariate space is needed to obtain useful statistical analyses. We study the partial least squares regression method. It turns out that it is naturally adapted to this setting via the so-called Krylov sequence. The resulting PLS estimator is shown to be consistent provided that the number of terms included is taken to be equal to the number of relevant components in the regression model. A standard PLS algorithm can also be constructed, but it turns out that the resulting predictor can only be related to the original covariates via time-dependent coefficients. The methods are applied to a breast cancer data set with gene expression recordings and to the well known primary biliary cirrhosis clinical data.

In this paper we consider different approaches for estimation and assessment of covariate effects for the cumulative incidence curve in the competing risks model. The classic approach is to model all cause-specific hazards and then estimate the cumulative incidence curve based on these cause-specific hazards. Another recent approach is to directly model the cumulative incidence by a proportional model (Fine and Gray, J Am Stat Assoc 94:496-509, 1999), and then obtain direct estimates of how covariates influences the cumulative incidence curve. We consider a simple and flexible class of regression models that is easy to fit and contains the Fine-Gray model as a special case. One advantage of this approach is that our regression modeling allows for non-proportional hazards. This leads to a new simple goodness-of-fit procedure for the proportional subdistribution hazards assumption that is very easy to use. The test is constructive in the sense that it shows exactly where non-proportionality is present. We illustrate our methods to a bone marrow transplant data from the Center for International Blood and Marrow Transplant Research (CIBMTR). Through this data example we demonstrate the use of the flexible regression models to analyze competing risks data when non-proportionality is present in the data.

A natural way of modelling relative survival through regression analysis is to assume an additive form between the expected population hazard and the excess hazard due to the presence of an additional cause of mortality. Within this context, the existing approaches in the parametric, semiparametric and non-parametric setting are compared and discussed. We study the additive excess hazards models, where the excess hazard is on additive form. This makes it possible to assess the importance of time-varying effects for regression models in the relative survival framework. We show how recent developments can be used to make inferential statements about the non-parametric version of the model. This makes it possible to test the key hypothesis that an excess risk effect is time varying in contrast to being constant over time. In case some covariate effects are constant, we show how the semiparametric additive risk model can be considered in the excess risk setting, providing a better and more useful summary of the data. Estimators have explicit form and inference based on a resampling scheme is presented for both the non-parametric and semiparametric models. We also describe a new suggestion for goodness of fit of relative survival models, which consists on statistical and graphical tests based on cumulative martingale residuals. This is illustrated on the semiparametric model with proportional excess hazards. We analyze data from the TRACE study using different approaches and show the need for more flexible models in relative survival., (2008 John Wiley & Sons, Ltd)

A frequent occurrence in medical research is that a patient is subject to different causes of failure, where each cause is known as a competing risk. The cumulative incidence curve is a proper summary curve, showing the cumulative failure rates over time due to a particular cause. A common question in medical research is to assess the covariate effects on a cumulative incidence function. The standard approach is to construct regression models for all cause-specific hazard rate functions and then model a covariate-adjusted cumulative incidence curve as a function of all cause-specific hazards for a given set of covariates. New methods have been proposed in recent years, emphasizing direct assessment of covariate effects on cumulative incidence function. Fine and Gray proposed modeling the effects of covariates on a subdistribution hazard function. A different approach is to directly model a covariate-adjusted cumulative incidence function, including a pseudovalue approach by Andersen and Klein and a direct binomial regression by Scheike, Zhang and Gerds. In this paper, we review the standard and new regression methods for modeling a cumulative incidence function, and give the sources of computer packages/programs that implement these regression models. A real bone marrow transplant data set is analyzed to illustrate various regression methods.

The Cox hazards model (Cox, 1972, Journal of the Royal Statistical Society, Series B34, 187-220) for survival data is routinely used in many applied fields, sometimes, however, with too little emphasis on the fit of the model. A useful alternative to the Cox model is the Aalen additive hazards model (Aalen, 1980, in Lecture Notes in Statistics-2, 1-25) that can easily accommodate time changing covariate effects. It is of interest to decide which of the two models that are most appropriate to apply in a given application. This is a nontrivial problem as these two classes of models are nonnested except only for special cases. In this article we explore the Mizon-Richard encompassing test for this particular problem. It turns out that it corresponds to fitting of the Aalen model to the martingale residuals obtained from the Cox regression analysis. We also consider a variant of this method, which relates to the proportional excess model (Martinussen and Scheike, 2002, Biometrika 89, 283-298). Large sample properties of the suggested methods under the two rival models are derived. The finite-sample properties of the proposed procedures are assessed through a simulation study. The methods are further applied to the well-known primary biliary cirrhosis data set.

Background: Trends in biologic fertility are elusive. Possible negative trends in male reproductive health are still debated, and their effect on human fertility might be negligible. Time-to-pregnancy (TTP) is a functional measure of couple fecundability., Methods: We analyzed data on TTP among 832,000 primiparous women 20 years of age and older in the nationwide Swedish Medical Birth Registry from 1983 through 2002. This age restriction led to an exclusion of 10% of primiparous pregnancies. Subfertility (TTP > or =1 year) was analyzed as a function of maternal age, calendar time at initiation of attempt, and birth cohort-taking into account the truncation problems that are inherent in birth-based retrospective sampling., Results: Subfertility generally decreased over successive birth cohorts. When studied as a period effect, a transient increase in subfertility was seen in the early 1990s. Subfertility increased with age, except that for women in their late 1930s, an apparent decrease was observed, particularly among the early cohorts., Conclusion: We found decreasing subfertility over time. We speculate that these patterns might be related to a Sweden-specific decrease over time in sexually transmitted diseases, to changes in sexual behavior induced by socioeconomic conditions, or to broader biologic or educational trends.

We consider tied survival data based on Cox proportional regression model. The standard approaches are the Breslow and Efron approximations and various so called exact methods. All these methods lead to biased estimates when the true underlying model is in fact a Cox model. In this paper we review the methods and suggest a new method based on the missing-data principle using EM-algorithm that leads to a score equation that can be solved directly. This score has mean zero. We also show that all the considered methods have the same asymptotic properties and that there is no loss of asymptotic efficiency when the tie sizes are bounded or even converge to infinity at a given rate. A simulation study is conducted to compare the finite sample properties of the methods.

I suggest an extension of the semiparametric transformation model that specifies a time-varying regression structure for the transformation, and thus allows time-varying structure in the data. Special cases include a stratified version of the usual semiparametric transformation model. The model can be thought of as specifying a first order Taylor expansion of a completely flexible baseline. Large sample properties are derived and estimators of the asymptotic variances of the regression coefficients are given. The method is illustrated by a worked example and a small simulation study. A goodness of fit procedure for testing if the regression effects lead to a satisfactory fit is also suggested.

Time-to-pregnancy(TTP), the duration that a couple waits from initiating attempts to conceive until conception occurs, is regarded as one of the direct measures of natural fecundity. Statistical tools for designing and analysing TTP studies belong to the general area of survival analysis, but several special features have been developed: it is customary to work in discrete time, and random heterogeneity between couples has always played a prominent role. This review works on this background with focus on how to perform valid analyses, under various prospective, retrospective and cross-sectional sampling frames. We illustrate using examples from our own experience.

Nested case-control sampling is designed to reduce the costs of large cohort studies. It is important to estimate the parameters of interest as efficiently as possible. We present a new maximum likelihood estimator (MLE) for nested case-control sampling in the context of Cox's proportional hazards model. The MLE is computed by the EM-algorithm, which is easy to implement in the proportional hazards setting. Standard errors are estimated by a numerical profile likelihood approach based on EM aided differentiation. The work was motivated by a nested case-control study that hypothesized that insulin-like growth factor I was associated with ischemic heart disease. The study was based on a population of 3784 Danes and 231 cases of ischemic heart disease where controls were matched on age and gender. We illustrate the use of the MLE for these data and show how the maximum likelihood framework can be used to obtain information additional to the relative risk estimates of covariates.

Cox's regression model is the standard regression tool for survival analysis in most applications. Often, however, the model only provides a rough summary of the effect of some covariates. Therefore, if the aim is to give a detailed description of covariate effects and to consequently calculate predicted probabilities, more flexible models are needed. In another article, Scheike and Zhang (2002, Scandinavian Journal of Statistics 29, 75-88), we suggested a flexible extension of Cox's regression model, which aimed at extending the Cox model only for those covariates where additional flexibility are needed. One important advantage of the suggested approach is that even though covariates are allowed a nonparametric effect, the hassle and difficulty of finding smoothing parameters are not needed. We show how the extended model also leads to simple formulae for predicted probabilities and their standard errors, for example, in the competing risk framework.

We use the additive risk model of Aalen (Aalen, 1980) as a model for the rate of a counting process. Rather than specifying the intensity, that is the instantaneous probability of an event conditional on the entire history of the relevant covariates and counting processes, we present a model for the rate function, i.e., the instantaneous probability of an event conditional on only a selected set of covariates. When the rate function for the counting process is of Aalen form we show that the usual Aalen estimator can be used and gives almost unbiased estimates. The usual martingale based variance estimator is incorrect and an alternative estimator should be used. We also consider the semi-parametric version of the Aalen model as a rate model (McKeague and Sasieni, 1994) and show that the standard errors that are computed based on an assumption of intensities are incorrect and give a different estimator. Finally, we introduce and implement a test-statistic for the hypothesis of a time-constant effect in both the non-parametric and semi-parametric model. A small simulation study was performed to evaluate the performance of the new estimator of the standard error.

Time to pregnancy, the number of menstrual cycles it takes a couple to conceive, and various covariates have been collected among couples ultimately achieving conception. To assess the influence of the covariates, we constructed a discrete survival model that allows time-dependent covariates. A random effect was included to account for unobserved heterogeneity. The collected waiting times are obtained through retrospective ascertainment and are analyzed as truncated data. Maximum likelihood estimation was implemented by Fisher scoring through iteratively reweighted least squares.

The nonparametric maximum likelihood estimator for current-status data has been known for at least 40 years, but only recently have the mathematical-statistical properties been clarified. This note provides a case study in the important and often studied context of estimating age-specific immunization intensities from a seroprevalence survey. Fully parametric and spline-based alternatives (also based on continuous-time models) are given. The basic reproduction number R0 exemplifies estimation of a functional. The limitations implied by the necessarily rather restrictive epidemiological assumptions are briefly discussed.