Modeling the cumulative incidence function of multivariate competing risks data allowing for within-cluster dependence of risk and timing.

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.

Excess cumulative incidence estimation for matched cohort survival studies.

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.

Risk of late health effects after soft-tissue sarcomas in childhood - a population-based cohort study within the Adult Life after Childhood Cancer in Scandinavia research programme.

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.

The Role of Comorbidity in Mortality After Hip Fracture: A Nationwide Norwegian Study of 38,126 Women With Hip Fracture Matched to a General-Population Comparison Cohort.

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.

Analysis of Generalized Semiparametric Regression Models for Cumulative Incidence Functions with Missing Covariates.

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.

Penalized estimation for competing risks regression with applications to high-dimensional covariates.

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.

Regression models for the restricted residual mean life for right-censored and left-truncated data.

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.

Incorporation of the time aspect into the liability-threshold model for case-control-family data.

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.

Cox regression with missing covariate data using a modified partial likelihood method.

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.

Estimating Twin Pair Concordance for Age of Onset.

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.

Measuring early or late dependence for bivariate lifetimes of twins.

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.

Checking Fine and Gray subdistribution hazards model with cumulative sums of residuals.

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.

Estimating twin concordance for bivariate competing risks twin data.

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.

Estimating heritability for cause specific mortality based on twin studies.

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.

On cross-odds ratio for multivariate competing risks data.

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.

Competing risks with missing covariates: effect of haplotypematch on hematopoietic cell transplant patients.

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.

Absolute risk regression for competing risks: interpretation, link functions, and prediction.

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 ß have the following interpretation: The probability of dying from cause D during the next t years changes with a factor exp(ß) 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.

Is the current decline in malaria burden in sub-Saharan Africa due to a decrease in vector population?

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.

Spatial variation and socio-economic determinants of Plasmodium falciparum infection in northeastern Tanzania.

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.