Regression modeling of competing risks data based on pseudovalues of the cumulative incidence function

Typically, regression models for competing risks outcomes are based on proportional hazards models for the crude hazard rates. These estimates often do not agree with impressions drawn from plots of cumulative incidence functions for each level of a risk factor. We present a technique which models the cumulative incidence functions directly. The method is based on the pseudovalues from a jackknife statistic constructed from the cumulative incidence curve. These pseudovalues are used in a generalized estimating equation to obtain estimates of model parameters. We study the properties of this estimator and apply the technique to a study of the effect of alternative donors on relapse for patients given a bone marrow transplant for leukemia.     

Semiparametric models for cumulative incidence functions

In analyses of time-to-failure data with competing risks, cumulative incidence functions may be used to estimate the time-dependent cumulative probability of failure due to specific causes. These functions are commonly estimated using nonparametric methods, but in cases where events due to the cause of primary interest are infrequent relative to other modes of failure, nonparametric methods may result in rather imprecise estimates for the corresponding subdistribution. In such cases, it may be possible to model the cause-specific hazard of primary interest parametrically, while accounting for the other modes of failure using nonparametric estimators. The cumulative incidence estimators so obtained are simple to compute and are considerably more efficient than the usual nonparametric estimator, particularly with regard to interpolation of cumulative incidence at early or intermediate time points within the range of data used to fit the function. More surprisingly, they are often nearly as efficient as fully parametric estimators. We illustrate the utility of this approach in the analysis of patients treated for early stage breast cancer.     

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.     

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.     

Estimation of separable direct and indirect effects in continuous time

Many research questions involve time-to-event outcomes that can be prevented from occurring due to competing events. In these settings, we must be careful about the causal interpretation of classical statistical estimands. In particular, estimands on the hazard scale, such as ratios of cause-specific or subdistribution hazards, are fundamentally hard to interpret causally. Estimands on the risk scale, such as contrasts of cumulative incidence functions, do have a clear causal interpretation, but they only capture the total effect of the treatment on the event of interest; that is, effects both through and outside of the competing event. To disentangle causal treatment effects on the event of interest and competing events, the separable direct and indirect effects were recently introduced. Here we provide new results on the estimation of direct and indirect separable effects in continuous time. In particular, we derive the nonparametric influence function in continuous time and use it to construct an estimator that has certain robustness properties. We also propose a simple estimator based on semiparametric models for the two cause-specific hazard functions. We describe the asymptotic properties of these estimators and present results from simulation studies, suggesting that the estimators behave satisfactorily in finite samples. Finally, we reanalyze the prostate cancer trial from Stensrud et al. (2020).     

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.     

Nonparametric estimation in a cure model with random cure times

Acute respiratory distress syndrome (ARDS) is a life-threatening acute condition that sometimes follows pneumonia or surgery. Patients who recover and leave the hospital are considered to have been cured at the time they leave the hospital. These data differ from typical data in which cure is a possibility: death times are not observed for patients who are cured and cure times are observed and vary among patients. Here we apply a competing risks model to these data and show it to be equivalent to a mixture model, the more common approach for cure data. Further, we derive an estimator for the variance of the cumulative incidence function from the competing risks model, and thus for the cure rate, based on elementary calculations. We compare our variance estimator to Gray's (1988, Annals of Statistics 16, 1140-1154) estimator, which is based on counting process theory. We find our estimator to be slightly more accurate in small samples. We apply these results to data from an ARDS clinical trial.     

Tests for multivariate recurrent events in the presence of a terminal event

In studies involving diseases associated with high rates of mortality, trials are frequently conducted to evaluate the effects of therapeutic interventions on recurrent event processes terminated by death. In this setting, cumulative mean functions form a natural basis for inference for questions of a health economic nature, and Ghosh and Lin (2000) recently proposed a relevant class of test statistics. Trials of patients with cancer metastatic to bone, however, involve multiple types of skeletal complications, each of which may be repeatedly experienced by patients over their lifetime. Traditionally the distinction between the various types of events is ignored and univariate analyses are conducted based on a composite recurrent event. However, when the events have different impacts on patients' quality of life, or when they incur different costs, it can be important to gain insight into the relative frequency of the specific types of events and treatment effects thereon. This may be achieved by conducting separate marginal analyses with each analysis focusing on one type of recurrent event. Global inferences regarding treatment benefit can then be achieved by carrying out multiplicity adjusted marginal tests, more formal multiple testing procedures, or by constructing global test statistics. We describe methods for testing for differences in mean functions between treatment groups which accommodate the fact that each particular event process is ultimately terminated by death. The methods are illustrated by application to a motivating study designed to examine the effect of bisphosphonate therapy on the incidence of skeletal complications among patients with breast cancer metastatic to bone. We find that there is a consistent trend towards a reduction in the cumulative mean for all four types of skeletal complications with bisphosphonate therapy; there is a significant reduction in the need for radiation therapy for the treatment of bone. The global test suggests that bisphosphonate therapy significantly reduces the overall number of skeletal complications.     

Multi-lineage potential of human mesenchymal stem cells following clonal expansion

Bone marrow contains mesenchymal cells that can be isolated and grown in vitro. Using appropriate treatment protocols such cultures can be induced to differentiate to yield osteoblasts, adipocytes, and chondrocytes. However, previous experiments had not addressed the question whether single pluripotent stem cells exist and can give rise to these different cell lineages or whether bone marrow mesenchymal cell preparations represent a mixture of committed precursors. We have used human adult bone marrow-derived mesenchymal cells obtained from iliac crest biopsies to demonstrate clonal outgrowth after limiting dilution and we show that some clones can be expanded over more than 20 cumulative population doublings and differentiated to osteoblasts, adipocytes, and chondrocytes. Our data provide direct experimental evidence that cultures of bone marrow-derived mesenchymal cells contain individual cells that fulfil two essential stem cell criteria: (i) extensive self-renewal capacity and (ii) multi-lineage potential.     

Time-dependent ROC curves for censored survival data and a diagnostic marker

ROC curves are a popular method for displaying sensitivity and specificity of a continuous diagnostic marker, X, for a binary disease variable, D. However, many disease outcomes are time dependent, D(t), and ROC curves that vary as a function of time may be more appropriate. A common example of a time-dependent variable is vital status, where D(t) = 1 if a patient has died prior to time t and zero otherwise. We propose summarizing the discrimination potential of a marker X, measured at baseline (t = 0), by calculating ROC curves for cumulative disease or death incidence by time t, which we denote as ROC(t). A typical complexity with survival data is that observations may be censored. Two ROC curve estimators are proposed that can accommodate censored data. A simple estimator is based on using the Kaplan-Meier estimator for each possible subset X > c. However, this estimator does not guarantee the necessary condition that sensitivity and specificity are monotone in X. An alternative estimator that does guarantee monotonicity is based on a nearest neighbor estimator for the bivariate distribution function of (X, T), where T represents survival time (Akritas, M. J., 1994, Annals of Statistics 22, 1299-1327). We present an example where ROC(t) is used to compare a standard and a modified flow cytometry measurement for predicting survival after detection of breast cancer and an example where the ROC(t) curve displays the impact of modifying eligibility criteria for sample size and power in HIV prevention trials.     

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.     

Impact of and Correction for Outcome Misclassification in Cumulative Incidence Estimation

Cohort studies and clinical trials may involve multiple events. When occurrence of one of these events prevents the observance of another, the situation is called “competing risks”. A useful measure in such studies is the cumulative incidence of an event, which is useful in evaluating interventions or assessing disease prognosis. When outcomes in such studies are subject to misclassification, the resulting cumulative incidence estimates may be biased. In this work, we study the mechanism of bias in cumulative incidence estimation due to outcome misclassification. We show that even moderate levels of misclassification can lead to seriously biased estimates in a frequently unpredictable manner. We propose an easy to use estimator for correcting this bias that is uniformly consistent. Extensive simulations suggest that this method leads to unbiased estimates in practical settings. The proposed method is useful, both in settings where misclassification probabilities are known by historical data or can be estimated by other means, and for performing sensitivity analyses when the misclassification probabilities are not precisely known.     

Peripheral deletion after bone marrow transplantation with costimulatory blockade has features of both activation-induced cell death and passive cell death

Two major pathways of death of previously activated T cells have been described: activation-induced cell death can be triggered by restimulating activated T cells with high concentrations of Ag, is Fas-dependent, is not influenced by proteins of the Bcl family, and is blocked by cyclosporin A; in contrast, passive cell death is induced by the withdrawal of growth factors and activation stimuli, is Fas-independent, and is blocked by Bcl family proteins. We examined the role of these two forms of cell death in the peripheral deletion of donor-reactive host T cells after allogeneic bone marrow transplantation and costimulatory blockade with anti-CD154 plus CTLA4Ig in two murine models. The substantial decline in donor-reactive CD4 cells seen in wild-type recipients 1 wk after bone marrow transplantation with costimulatory blockade was largely inhibited in Fas-deficient recipients and in Bcl-x(L)-transgenic recipients. We observed these effects both in a model involving low-dose total body irradiation and a conventional dose of bone marrow, and in a radiation-free regimen using high-dose bone marrow transplantation. Furthermore, cyclosporin A did not completely block the deletion of donor-reactive CD4(+) T cells in recipients of bone marrow transplantation with costimulatory blockade. Thus, the deletion of donor-reactive T cells occurring early after bone marrow transplantation with costimulatory blockade has features of both activation-induced cell death and passive cell death. Furthermore, these in vivo data demonstrate for the first time the significance of in vitro results indicating that proteins of the Bcl family can prevent Fas-mediated apoptosis under certain circumstances.     

Comparing treatments in the presence of crossing survival curves: an application to bone marrow transplantation

Summary .   In some clinical studies comparing treatments in terms of their survival curves, researchers may anticipate that the survival curves will cross at some point, leading to interest in a long-term survival comparison. However, simple comparison of the survival curves at a fixed point may be inefficient, and use of a weighted log-rank test may be overly sensitive to early differences in survival. We formulate the problem as one of testing for differences in survival curves after a prespecified time point, and propose a variety of techniques for testing this hypothesis. We study these methods using simulation and illustrate them on a study comparing survival for autologous and allogeneic bone marrow transplants.     

Up-regulation of alpha2,6 sialylation during myeloid maturation: a potential role in myeloid cell release from the bone marrow

The mechanisms by which mature myeloid cells are released from the bone marrow into the peripheral blood are not clearly understood. Glycosylation is likely to play an important role, as has been shown in the homing of lymphocytes to lymph nodes and of neutrophils to inflamed endothelia. Cell surface sialylation is an important component of many cellular adhesive interactions, both as ligand-promoting interactions, as occurs in selectin and sialoadhesin-mediated adhesion, and for reducing cell adhesion as in some cancer cells. We have studied the expression of cell surface alpha2,6-linked sialic acid in the maturation of normal bone marrow myeloid cells, the expression of alpha2,6-sialyltransferase mRNA, and the role of sialylation in the adherence of myeloid cells to bone marrow stroma. Our data show that there is a dramatic increase in cell surface alpha2,6-sialylation during the late stage of maturation. This up-regulation is restricted to specific glycoproteins including CD11b and CD18. It is associated with a relative increase in the level of alpha2,6-sialyltransferase mRNA compared with alpha2,3-sialyltransferase mRNA. The changes in mature bone marrow myeloid cells are associated with reduced cell binding to fibronectin and cultured bone marrow stroma. Our data strongly suggest that alpha2,6-sialylation may be important in the interaction between maturing myeloid cells and bone marrow stroma and may govern the release of cells from the bone marrow into the peripheral blood.     

Spleen and bone marrow megakaryocytes as targets for inhaled vanadium

An increased incidence in ischemic and thromboembolic events in the population of cities with rising air suspended particle pollution has suggested the interaction of some of the components of these particles in the coagulation system. A previous report from our laboratory identified thrombocytosis as a consequence of the subacute and chronic inhalation of vanadium. With this preceding information we decided to evaluate the effects of this element in the spleen and bone marrow in a mouse experimental model. CD-1 male mice inhaled V2O5 0.02 M for one hour twice a week for twelve weeks. The spleen and bone marrow were processed for light microscopy. The increase in quantity and size of megakaryocytes (MKs) in the exposed group in both organs was striking. Also, modifications in the cytoplasm, granule content and nuclear ultrastructure were evident. Our results indicate the influence of vanadium on megakaryopoyesis, an effect which could be the onset of the thrombocytosis previously reported by our group. The modifications in MKs described here suggest that inhaled vanadium could induce megakaryocytic proliferation, which may result in increased production of platelets and increased risk for thromboembolic events.     

Therapy-related myelodysplastic syndrome presenting as fulminant heart failure secondary to myeloid sarcoma

Rapidly progressive heart failure is commonly caused by an extensive myocardial infarction, a mechanical complication of infarction, myocarditis, or acute valvular insufficiency. We present an unusual case that was caused by a diffuse infiltration of the myocardium with leukemic cells (myeloid sarcoma). The patient presented with episodic shortness of breath, he was anemic and thrombocytopenic, and his bone marrow biopsy revealed myelodysplastic syndrome from treatment for oligodendroglioma. His clinical course was characterized by a chronic leak of cardiac enzymes, a new right bundle branch block, and a large pericardial effusion causing tamponade and death from fulminant heart failure and ventricular arrhythmias within 2 weeks. At autopsy, the heart was massively infiltrated with myeloblasts and other immature myeloid cells. There was no evidence of acute leukemia in the bone marrow or peripheral blood. Cardiac infiltration in a patient with myelodysplastic syndrome is extremely rare, especially in the absence of bone marrow involvement by blasts. The recognition of this entity is becoming increasingly important as the incidence of cardiac myeloid sarcoma may be on the rise as the number of patients receiving chemotherapy increases.     

Estimation and interpretation of incidence rate difference for recurrent events when the estimation model is misspecified

Recurrent events data are common in experimental and observational studies. It is often of interest to estimate the effect of an intervention on the incidence rate of the recurrent events. The incidence rate difference is a useful measure of intervention effect. A weighted least squares estimator of the incidence rate difference for recurrent events was recently proposed for an additive rate model in which both the baseline incidence rate and the covariate effects were constant over time. In this article, we relax this model assumption and examine the properties of the estimator under the additive and multiplicative rate models assumption in which the baseline incidence rate and covariate effects may vary over time. We show analytically and numerically that the estimator gives an appropriate summary measure of the time‐varying covariate effects. In particular, when the underlying covariate effects are additive and time‐varying, the estimator consistently estimates the weighted average of the covariate effects over time. When the underlying covariate effects are multiplicative and time‐varying, and if there is only one binary covariate indicating the intervention status, the estimator consistently estimates the weighted average of the underlying incidence rate difference between the intervention and control groups over time. We illustrate the method with data from a randomized vaccine trial.     

Viral abrogation of stem cell transplantation tolerance causes graft rejection and host death by different mechanisms

Tolerance-based stem cell transplantation using sublethal conditioning is being considered for the treatment of human disease, but safety and efficacy remain to be established. We have shown that mouse bone marrow recipients treated with sublethal irradiation plus transient blockade of the CD40-CD154 costimulatory pathway develop permanent hematopoietic chimerism across allogeneic barriers. We now report that infection with lymphocytic choriomeningitis virus at the time of transplantation prevented engraftment of allogeneic, but not syngeneic, bone marrow in similarly treated mice. Infected allograft recipients also failed to clear the virus and died. Postmortem study revealed hypoplastic bone marrow and spleens. The cause of death was virus-induced IFN-alphabeta. The rejection of allogeneic bone marrow was mediated by a radioresistant CD8(+)TCR-alphabeta(+)NK1.1(-) T cell population. We conclude that a noncytopathic viral infection at the time of transplantation can prevent engraftment of allogeneic bone marrow and result in the death of sublethally irradiated mice treated with costimulation blockade. Clinical application of stem cell transplantation protocols based on costimulation blockade and tolerance induction may require patient isolation to facilitate the procedure and to protect recipients.