Cause‐Specific Cumulative Incidence Estimation and the Fine and Gray Model Under Both Left Truncation and Right Censoring

Summary The standard estimator for the cause‐specific cumulative incidence function in a competing risks setting with left truncated and/or right censored data can be written in two alternative forms. One is a weighted empirical cumulative distribution function and the other a product‐limit estimator. This equivalence suggests an alternative view of the analysis of time‐to‐event data with left truncation and right censoring: individuals who are still at risk or experienced an earlier competing event receive weights from the censoring and truncation mechanisms. As a consequence, inference on the cumulative scale can be performed using weighted versions of standard procedures. This holds for estimation of the cause‐specific cumulative incidence function as well as for estimation of the regression parameters in the Fine and Gray proportional subdistribution hazards model. We show that, with the appropriate filtration, a martingale property holds that allows deriving asymptotic results for the proportional subdistribution hazards model in the same way as for the standard Cox proportional hazards model. Estimation of the cause‐specific cumulative incidence function and regression on the subdistribution hazard can be performed using standard software for survival analysis if the software allows for inclusion of time‐dependent weights. We show the implementation in the R statistical package. The proportional subdistribution hazards model is used to investigate the effect of calendar period as a deterministic external time varying covariate, which can be seen as a special case of left truncation, on AIDS related and non‐AIDS related cumulative mortality.     

Marginal Models for Clustered Time‐to‐Event Data with Competing Risks Using Pseudovalues

Summary Many time‐to‐event studies are complicated by the presence of competing risks and by nesting of individuals within a cluster, such as patients in the same center in a multicenter study. Several methods have been proposed for modeling the cumulative incidence function with independent observations. However, when subjects are clustered, one needs to account for the presence of a cluster effect either through frailty modeling of the hazard or subdistribution hazard, or by adjusting for the within‐cluster correlation in a marginal model. We propose a method for modeling the marginal cumulative incidence function directly. We compute leave‐one‐out pseudo‐observations from the cumulative incidence function at several time points. These are used in a generalized estimating equation to model the marginal cumulative incidence curve, and obtain consistent estimates of the model parameters. A sandwich variance estimator is derived to adjust for the within‐cluster correlation. The method is easy to implement using standard software once the pseudovalues are obtained, and is a generalization of several existing models. Simulation studies show that the method works well to adjust the SE for the within‐cluster correlation. We illustrate the method on a dataset looking at outcomes after bone marrow transplantation.     

Dynamic prediction of cumulative incidence functions by direct binomial regression

In recent years there have been a series of advances in the field of dynamic prediction. Among those is the development of methods for dynamic prediction of the cumulative incidence function in a competing risk setting. These models enable the predictions to be updated as time progresses and more information becomes available, for example when a patient comes back for a follow‐up visit after completing a year of treatment, the risk of death, and adverse events may have changed since treatment initiation. One approach to model the cumulative incidence function in competing risks is by direct binomial regression, where right censoring of the event times is handled by inverse probability of censoring weights. We extend the approach by combining it with landmarking to enable dynamic prediction of the cumulative incidence function. The proposed models are very flexible, as they allow the covariates to have complex time‐varying effects, and we illustrate how to investigate possible time‐varying structures using Wald tests. The models are fitted using generalized estimating equations. The method is applied to bone marrow transplant data and the performance is investigated in a simulation study.     

Estimation in the progressive illness‐death model: A nonexhaustive review

Multistate models can be successfully used for describing complex event history data, for example, describing stages in the disease progression of a patient. The so‐called “illness‐death” model plays a central role in the theory and practice of these models. Many time‐to‐event datasets from medical studies with multiple end points can be reduced to this generic structure. In these models one important goal is the modeling of transition rates but biomedical researchers are also interested in reporting interpretable results in a simple and summarized manner. These include estimates of predictive probabilities, such as the transition probabilities, occupation probabilities, cumulative incidence functions, and the sojourn time distributions. We will give a review of some of the available methods for estimating such quantities in the progressive illness‐death model conditionally (or not) on covariate measures. For some of these quantities estimators based on subsampling are employed. Subsampling, also referred to as landmarking, leads to small sample sizes and usually to heavily censored data leading to estimators with higher variability. To overcome this issue estimators based on a preliminary estimation (presmoothing) of the probability of censoring may be used. Among these, the presmoothed estimators for the cumulative incidences are new. We also introduce feasible estimation methods for the cumulative incidence function conditionally on covariate measures. The proposed methods are illustrated using real data. A comparative simulation study of several estimation approaches is performed and existing software in the form of R packages is discussed.     

Omnibus tests for comparison of competing risks with adjustment for covariate effects

This article develops omnibus tests for comparing cause-specific hazard rates and cumulative incidence functions at specified covariate levels. Confidence bands for the difference and the ratio of two conditional cumulative incidence functions are also constructed. The omnibus test is formulated in terms of a test process given by a weighted difference of estimates of cumulative cause-specific hazard rates under Cox proportional hazards models. A simulation procedure is devised for sampling from the null distribution of the test process, leading to graphical and numerical technques for detecting significant differences in the risks. The approach is applied to a cohort study of type-specific HIV infection rates.     

Estimating the cumulative incidence of SARS-CoV-2 with imperfect serological tests: Exploiting cutoff-free approaches

Large-scale serological testing in the population is essential to determine the true extent of the current SARS-CoV-2 pandemic. Serological tests measure antibody responses against pathogens and use predefined cutoff levels that dichotomize the quantitative test measures into sero-positives and negatives and use this as a proxy for past infection. With the imperfect assays that are currently available to test for past SARS-CoV-2 infection, the fraction of seropositive individuals in serosurveys is a biased estimator of the cumulative incidence and is usually corrected to account for the sensitivity and specificity. Here we use an inference method—referred to as mixture-model approach—for the estimation of the cumulative incidence that does not require to define cutoffs by integrating the quantitative test measures directly into the statistical inference procedure. We confirm that the mixture model outperforms the methods based on cutoffs, leading to less bias and error in estimates of the cumulative incidence. We illustrate how the mixture model can be used to optimize the design of serosurveys with imperfect serological tests. We also provide guidance on the number of control and case sera that are required to quantify the test’s ambiguity sufficiently to enable the reliable estimation of the cumulative incidence. Lastly, we show how this approach can be used to estimate the cumulative incidence of classes of infections with an unknown distribution of quantitative test measures. This is a very promising application of the mixture-model approach that could identify the elusive fraction of asymptomatic SARS-CoV-2 infections. An R-package implementing the inference methods used in this paper is provided. Our study advocates using serological tests without cutoffs, especially if they are used to determine parameters characterizing populations rather than individuals. This approach circumvents some of the shortcomings of cutoff-based methods at exactly the low cumulative incidence levels and test accuracies that we are currently facing in SARS-CoV-2 serosurveys.     

A Note on Variance Estimation of the Aalen–Johansen Estimator of the Cumulative Incidence Function in Competing Risks,with a View towards Left‐Truncated Data

The Aalen–Johansen estimator is the standard nonparametric estimator of the cumulative incidence function in competing risks. Estimating its variance in small samples has attracted some interest recently, together with a critique of the usual martingale‐based estimators. We show that the preferred estimator equals a Greenwood‐type estimator that has been derived as a recursion formula using counting processes and martingales in a more general multistate framework. We also extend previous simulation studies on estimating the variance of the Aalen–Johansen estimator in small samples to left‐truncated observation schemes, which may conveniently be handled within the counting processes framework. This investigation is motivated by a real data example on spontaneous abortion in pregnancies exposed to coumarin derivatives, where both competing risks and left‐truncation have recently been shown to be crucial methodological issues (Meister and Schaefer (2008), Reproductive Toxicology 26 , 31–35). Multistate‐type software and data are available online to perform the analyses. The Greenwood‐type estimator is recommended for use in practice.     

Constrained parametric model for simultaneous inference of two cumulative incidence functions

We propose a parametric regression model for the cumulative incidence functions (CIFs) commonly used for competing risks data. The model adopts a modified logistic model as the baseline CIF and a generalized odds‐rate model for covariate effects, and it explicitly takes into account the constraint that a subject with any given prognostic factors should eventually fail from one of the causes such that the asymptotes of the CIFs should add up to one. This constraint intrinsically holds in a nonparametric analysis without covariates, but is easily overlooked in a semiparametric or parametric regression setting. We hence model the CIF from the primary cause assuming the generalized odds‐rate transformation and the modified logistic function as the baseline CIF. Under the additivity constraint, the covariate effects on the competing cause are modeled by a function of the asymptote of the baseline distribution and the covariate effects on the primary cause. The inference procedure is straightforward by using the standard maximum likelihood theory. We demonstrate desirable finite‐sample performance of our model by simulation studies in comparison with existing methods. Its practical utility is illustrated in an analysis of a breast cancer dataset to assess the treatment effect of tamoxifen, adjusting for age and initial pathological tumor size, on breast cancer recurrence that is subject to dependent censoring by second primary cancers and deaths.     

Parametric regression on cumulative incidence function

We propose parametric regression analysis of cumulative incidence function with competing risks data. A simple form of Gompertz distribution is used for the improper baseline subdistribution of the event of interest. Maximum likelihood inferences on regression parameters and associated cumulative incidence function are developed for parametric models, including a flexible generalized odds rate model. Estimation of the long-term proportion of patients with cause-specific events is straightforward in the parametric setting. Simple goodness-of-fit tests are discussed for evaluating a fixed odds rate assumption. The parametric regression methods are compared with an existing semiparametric regression analysis on a breast cancer data set where the cumulative incidence of recurrence is of interest. The results demonstrate that the likelihood-based parametric analyses for the cumulative incidence function are a practically useful alternative to the semiparametric analyses.     

Nonparametric association analysis of bivariate left‐truncated competing risks data

We develop time‐varying association analyses for onset ages of two lung infections to address the statistical challenges in utilizing registry data where onset ages are left‐truncated by ages of entry and competing‐risk censored by deaths. Two types of association estimators are proposed based on conditional cause‐specific hazard function and cumulative incidence function that are adapted from unconditional quantities to handle left truncation. Asymptotic properties of the estimators are established by using the empirical process techniques. Our simulation study shows that the estimators perform well with moderate sample sizes. We apply our methods to the Cystic Fibrosis Foundation Registry data to study the relationship between onset ages of Pseudomonas aeruginosa and Staphylococcus aureus infections.     

Correcting for variable age of onset in the estimation of familial relative risk when there is a secular trend in incidence of disease

We discuss the effects that a secular trend in incidence would have on estimation of familial relative risk (ratio of observed to expected cumulative incidence among relatives of index cases). For example, when age-specific incidence rates of a condition have increased during the lifetimes of relatives among whom relative risk is to be estimated, familial relative risk will be biased downward if cross-sectional, age-specific incidence data are used to estimate expected cumulative incidence among relatives. The stronger the trend and the older the ages of unaffected relatives, the greater the bias will be. Incorporating different age-specific incidence curves for different birth cohorts into the analysis is an approach we suggest for correcting the bias.     

Confidence bands for cumulative incidence curves under the additive risk model

In the context of competing risks, the cumulative incidence function is often used to summarize the cause-specific failure-time data. As an alternative to the proportional hazards model, the additive risk model is used to investigate covariate effects by specifying that the subject-specific hazard function is the sum of a baseline hazard function and a regression function of covariates. Based on such a formulation, we present an approach to constructing simultaneous confidence intervals for the cause-specific cumulative incidence function of patients with given risk factors. A melanoma data set is used for the purpose of illustration.     

Nonparametric estimation of the cumulative incidences of competing risks under double truncation

Registry data typically report incident cases within a certain calendar time interval. Such interval sampling induces double truncation on the incidence times, which may result in an observational bias. In this paper, we introduce nonparametric estimation for the cumulative incidences of competing risks when the incidence time is doubly truncated. Two different estimators are proposed depending on whether the truncation limits are independent of the competing events or not. The asymptotic properties of the estimators are established, and their finite sample performance is investigated through simulations. For illustration purposes, the estimators are applied to childhood cancer registry data, where the target population is peculiarly defined conditional on future cancer development. Then, in our application, the cumulative incidences inform on the distribution by age of the different types of cancer.     

Apparent Incidence of Helicobacter pylori in Adulthood: To What Extent Do New Infections Reflect Misclassification?

Background: Helicobacter pylori infection is a key risk factor for a variety of gastrointestinal diseases. About half of the world population is infected. Most infections are acquired early in childhood, but the occurrence of new infections among adults has also been suggested. Methods: We review epidemiological studies providing estimates of incidence of H. pylori infection among adults and evaluate to what extent incidence estimates might have been affected by measurement error of infection status. Results: Thirty‐two studies could be included in the review. Annual incidence was lower than 1.0 % in 17 studies; no correlation between length of follow‐up and cumulative incidence was observed. Apparent cumulative incidences of the magnitudes observed in most studies would be expected, because of less than perfect sensitivity and specificity of the diagnostic tests, even in the absence of any true new infections. Conclusion/Impact: Apparent incidence rates of H. pylori infection among adults in Western populations should be interpreted with utmost caution.     

Identifying skeletal-related events for prostate cancer patients in routinely collected hospital data

BackgroundNon-osteoporotic skeletal-related events (SREs) are clinically important markers of disease progression in prostate cancer. We developed and validated an approach to identify SREs in men with prostate cancer using routinely-collected data.MethodsPatients diagnosed with prostate cancer between January 2010 and December 2013 were identified in the National Prostate Cancer Audit, based on English cancer registry data. A coding framework was developed based on diagnostic and procedure codes in linked national administrative hospital and routinely-collected radiotherapy data to identify SREs occurring before December 2015. Two coding definitions of SREs were assessed based on whether the SRE codes were paired with a bone metastasis code (‘specific definition’) or used in isolation (‘sensitive definition’). We explored the validity of both definitions by comparing the cumulative incidence of SREs from time of diagnosis according to prostate cancer stage at diagnosis with death as a competing risk.ResultsWe identified 40,063, 25,234 and 13,968 patients diagnosed with localised, locally advanced and metastatic disease, respectively. Using the specific definition, we found that the 5-year cumulative incidence of SREs was 1.0 % in patients with localised disease, 6.0 % in patients with locally advanced disease, and 42.3 % in patients with metastatic disease. Using the sensitive definition, the corresponding cumulative incidence figures were 9.0 %, 14.9 %, and 44.4 %, respectively.ConclusionThe comparison of the cumulative incidence of SREs identified in routinely collected hospital data, based on a specific coding definition in patients diagnosed with different prostate cancer stage, supports their validity as a clinically important marker of cancer progression.     

Multiple imputation methods for inference on cumulative incidence with missing cause of failure

Analysis of cumulative incidence (sometimes called absolute risk or crude risk) can be difficult if the cause of failure is missing for some subjects. Assuming missingness is random conditional on the observed data, we develop asymptotic theory for multiple imputation methods to estimate cumulative incidence. Covariates affect cause-specific hazards in our model, and we assume that separate proportional hazards models hold for each cause-specific hazard. Simulation studies show that procedures based on asymptotic theory have near nominal operating characteristics in cohorts of 200 and 400 subjects, both for cumulative incidence and for prediction error. The methods are illustrated with data on survival after breast cancer, obtained from the National Surgical Adjuvant Breast and Bowel Project (NSABP).     

On the relationship between cumulative correlation coefficients and the quality of crystallographic data sets

In 2012, Karplus and Diederichs demonstrated that the Pearson correlation coefficient CC_1/2 is a far better indicator of the quality and resolution of crystallographic data sets than more traditional measures like merging R‐factor or signal‐to‐noise ratio. More specifically, they proposed that CC_1/2 be computed for data sets in thin shells of increasing resolution so that the resolution dependence of that quantity can be examined. Recently, however, the CC_1/2 values of entire data sets, i.e., cumulative correlation coefficients, have been used as a measure of data quality. Here, we show that the difference in cumulative CC_1/2 value between a data set that has been accurately measured and a data set that has not is likely to be small. Furthermore, structures obtained by molecular replacement from poorly measured data sets are likely to suffer from extreme model bias.     

Smoothed quantile regression analysis of competing risks

Censored quantile regression models, which offer great flexibility in assessing covariate effects on event times, have attracted considerable research interest. In this study, we consider flexible estimation and inference procedures for competing risks quantile regression, which not only provides meaningful interpretations by using cumulative incidence quantiles but also extends the conventional accelerated failure time model by relaxing some of the stringent model assumptions, such as global linearity and unconditional independence. Current method for censored quantile regressions often involves the minimization of the L₁‐type convex function or solving the nonsmoothed estimating equations. This approach could lead to multiple roots in practical settings, particularly with multiple covariates. Moreover, variance estimation involves an unknown error distribution and most methods rely on computationally intensive resampling techniques such as bootstrapping. We consider the induced smoothing procedure for censored quantile regressions to the competing risks setting. The proposed procedure permits the fast and accurate computation of quantile regression parameter estimates and standard variances by using conventional numerical methods such as the Newton–Raphson algorithm. Numerical studies show that the proposed estimators perform well and the resulting inference is reliable in practical settings. The method is finally applied to data from a soft tissue sarcoma study.     

germinator: a software package for high‐throughput scoring and curve fitting of Arabidopsis seed germination

Over the past few decades seed physiology research has contributed to many important scientific discoveries and has provided valuable tools for the production of high quality seeds. An important instrument for this type of research is the accurate quantification of germination; however gathering cumulative germination data is a very laborious task that is often prohibitive to the execution of large experiments. In this paper we present the germinator package: a simple, highly cost‐efficient and flexible procedure for high‐throughput automatic scoring and evaluation of germination that can be implemented without the use of complex robotics. The germinator package contains three modules: (i) design of experimental setup with various options to replicate and randomize samples; (ii) automatic scoring of germination based on the color contrast between the protruding radicle and seed coat on a single image; and (iii) curve fitting of cumulative germination data and the extraction, recap and visualization of the various germination parameters. The curve‐fitting module enables analysis of general cumulative germination data and can be used for all plant species. We show that the automatic scoring system works for Arabidopsis thaliana and Brassica spp. seeds, but is likely to be applicable to other species, as well. In this paper we show the accuracy, reproducibility and flexibility of the germinator package. We have successfully applied it to evaluate natural variation for salt tolerance in a large population of recombinant inbred lines and were able to identify several quantitative trait loci for salt tolerance. germinator is a low‐cost package that allows the monitoring of several thousands of germination tests, several times a day by a single person.     

A unique gender difference in early onset melanoma implies that in addition to ultraviolet light exposure other causative factors are important

Using US SEER17 Registry data, age‐specific melanoma incidence rates were calculated and comparisons were made between males and females. Relative Risk (RR) for males and females in each age group was computed and compared with that from Nordic Cancer Registry data set and to that for non‐melanoma skin cancer (NMSC). For age groups 44 and younger, females showed higher incidence rates, with a peak difference at age 20–24 (RR = 2.01, 95% CI = 1.21–3.33). Males exhibited higher incidence rates after age 44. The same bimodal gender difference was confirmed by the Nordic Cancer Registry data set, but it was not observed for NMSC, which is known to be strongly associated with cumulative exposure to solar UV radiation. We conclude that exposure to solar ultraviolet (UV) radiation is the major causative factor for melanoma at older age (>44 yr), but that other factors may play a role in early onset melanomas, particularly in females.