Joint analysis of longitudinal measurements and spatially clustered competing risks HIV/AIDS data

The joint modeling of repeated measurements and time-to-event provides a general framework to describe better the link between the progression of disease through longitudinal measurements and time-to-event outcome. In the survival data, a sample of individuals is frequently grouped into clusters. In some applications, these clusters could be arranged spatially, for example, based on geographical regions. There are two benefits of considering spatial variation in these data, enhancing the efficiency and accuracy of the parameters estimations, and investigating the survivorship spatial pattern. On the other hand, in survival data, there is a situation that subjects are supposed to experience more than one type of event potentially, but the occurrence of one type of event prevents the occurrence of the others. In this article, we considered a Bayesian joint model of longitudinal and competing risks outcomes for spatially clustered HIV/AIDS data. The data were from a registry-based study carried in Hamadan Province, Iran, from December 1997 to June 2020. In this joint model, a linear mixed effects model was used for the longitudinal submodel and a cause-specific hazard model with spatial and spatial-risk random effects was used for the survival submodel. Also, a latent structure was defined by random effects to link both event times and longitudinal processes. We used a univariate intrinsic conditional autoregressive (ICAR) distribution and a multivariate ICAR distribution for modeling the areal spatial and spatial-risk random effects, respectively. The performance of our proposed model using simulation studies and analysis of HIV/AIDS data were assessed.

Evaluating Association Between Two Event Times with Observations Subject to Informative Censoring

This article is concerned with evaluating the association between two event times without specifying the joint distribution parametrically. This is particularly challenging when the observations on the event times are subject to informative censoring due to a terminating event such as death. There are few methods suitable for assessing covariate effects on association in this context. We link the joint distribution of the two event times and the informative censoring time using a nested copula function. We use flexible functional forms to specify the covariate effects on both the marginal and joint distributions. In a semiparametric model for the bivariate event time, we estimate simultaneously the association parameters, the marginal survival functions, and the covariate effects. A byproduct of the approach is a consistent estimator for the induced marginal survival function of each event time conditional on the covariates. We develop an easy-to-implement pseudolikelihood-based inference procedure, derive the asymptotic properties of the estimators, and conduct simulation studies to examine the finite-sample performance of the proposed approach. For illustration, we apply our method to analyze data from the breast cancer survivorship study that motivated this research. Supplementary materials for this article are available online.

The Value of the Illness-Death Model for Predicting Outcomes in Patients with Non‒Small Cell Lung Cancer

Purpose

The illness-death model (IDM) is a comprehensive approach to evaluate the relationship between relapse and death. This study aimed to illustrate the value of the IDM for identifying risk factors and evaluating predictive probabilities for relapse and death in patients with non–small cell lung cancer (NSCLC) in comparison with the disease-free survival (DFS) model.

Materials and Methods

We retrospectively analyzed 612 NSCLC patients who underwent a curative operation. Using the IDM, the risk factors and predictive probabilities for relapse, death without relapse, and death after relapse were simultaneously evaluated and compared to those obtained from a DFS model.

Results

The IDM provided more detailed risk factors according to the patient’s disease course, including relapse, death without relapse, and death after relapse, in patients with resected lung cancer. In the IDM, history of malignancy (other than lung cancer) was related to relapse and smoking history was associated with death without relapse; both were indistinguishable in the DFS model. In addition, the IDM was able to evaluate the predictive probability and risk factors for death after relapse; this information could not be obtained from the DFS model.

Conclusion

Compared to the DFS model, we found that the IDM provides more comprehensive information on transitions between states and disease stages and provides deeper insights with respect to understanding the disease process among lung cancer patients.

Estimating the correlation between semi-competing risk survival endpoints

Time-to-event, bivariate, semi-competing risk data occur when a terminal event can censor a non-terminal event, but not vice versa. There are potential correlations between these endpoints as they are measured on the same individual. However, traditional methods to estimate the correlations cannot be used directly due to the censoring of time-to-event endpoints. We develop methods using a copula-based approach to study the dependence structures between the two survival endpoints. We use a variety of copulas to estimate the correlation between endpoints and to acknowledge different dependence structures. The estimated association parameter in the copula function is transformed into Spearman's rank correlation coefficient. We conduct a simulation study to evaluate the estimation from the proposed models along with the effects of misspecification of the copula functions and survival distributions. The proposed methods are applied to two real-life data sets.

Impact of Immunosuppressive Agents on Clinical Manifestations and Outcome of Staphylococcus aureus Bloodstream Infection: A Propensity Score-Matched Analysis in 2 Large, Prospectively Evaluated Cohorts

BACKGROUND

Staphylococcus aureus bloodstream infection (SAB) is a common, life-threatening infection. The impact of immunosuppressive agents on the outcome of patients with SAB is incompletely understood.

METHODS

Data from 2 large prospective, international, multicenter cohort studies (Invasive Staphylococcus aureus Infections Cohort [INSTINCT] and International Staphylococcus aureus Collaboration [ISAC]) between 2006 and 2015 were analyzed. Patients receiving immunosuppressive agents were identified and a 1:1 propensity score-matched analysis was performed to adjust for baseline characteristics of patients. Overall survival and time to SAB-related late complications (SAB relapse, infective endocarditis, osteomyelitis, or other deep-seated manifestations) were analyzed by Cox regression and competing risk analyses, respectively. This approach was then repeated for specific immunosuppressive agents (corticosteroid monotherapy and immunosuppressive agents other than steroids [IMOTS]).

RESULTS

Of 3188 analyzed patients, 309 were receiving immunosuppressive treatment according to our definitions and were matched to 309 nonimmunosuppressed patients. After propensity score matching, baseline characteristics were well balanced. In the Cox regression analysis, we observed no significant difference in survival between the 2 groups (death during follow-up: 105/309 [33.9%] immunosuppressed vs 94/309 [30.4%] nonimmunosuppressed; hazard ratio [HR], 1.20 [95% confidence interval {CI}, .84-1.71]). Competing risk analysis showed a cause-specific HR of 1.81 (95% CI, .85-3.87) for SAB-related late complications in patients receiving immunosuppressive agents. The cause-specific HR was higher in patients taking IMOTS (3.69 [95% CI, 1.41-9.68]).

CONCLUSIONS

Immunosuppressive agents were not associated with an overall higher mortality. The risk for SAB-related late complications in patients receiving specific immunosuppressive agents such as IMOTS warrants further investigations.

Statistical models for composite endpoints of death and non-fatal events: a review

The proper analysis of composite endpoints consisting of both death and non-fatal events is an intriguing and sometimes contentious topic. The current practice of analyzing time to the first event often draws criticisms for ignoring the unequal importance between component events and for leaving recurrent-event data unused. Novel methods that address these limitations have recently been proposed. To compare the novel versus traditional approaches, we review three typical models for composite endpoints based on time to the first event, composite event process, and pairwise hierarchical comparisons. The pros and cons of these models are discussed with reference to the relevant regulatory guidelines, such as the recently released ICH-E9(R1) Addendum "Estimands and Sensitivity Analysis in Clinical Trials". We also discuss the impact of censoring when the model assumptions are violated and explore sensitivity analysis strategies. Simulation studies are conducted to assess the performance of the reviewed methods under different settings. As demonstration, we use publicly available R-packages to analyze real data from a major cardiovascular trial.

Correlation-based joint feature screening for semi-competing risks outcomes with application to breast cancer data

Ultrahigh-dimensional gene features are often collected in modern cancer studies in which the number of gene features p is extremely larger than sample size n. While gene expression patterns have been shown to be related to patients' survival in microarray-based gene expression studies, one has to deal with the challenges of ultrahigh-dimensional genetic predictors for survival predicting and genetic understanding of the disease in precision medicine. The problem becomes more complicated when two types of survival endpoints, distant metastasis-free survival and overall survival, are of interest in the study and outcome data can be subject to semi-competing risks due to the fact that distant metastasis-free survival is possibly censored by overall survival but not vice versa. Our focus in this paper is to extract important features, which have great impacts on both distant metastasis-free survival and overall survival jointly, from massive gene expression data in the semi-competing risks setting. We propose a model-free screening method based on the ranking of the correlation between gene features and the joint survival function of two endpoints. The method accounts for the relationship between two endpoints in a simply defined utility measure that is easy to understand and calculate. We show its favorable theoretical properties such as the sure screening and ranking consistency, and evaluate its finite sample performance through extensive simulation studies. Finally, an application to classifying breast cancer data clearly demonstrates the utility of the proposed method in practice.

Causal mediation of semicompeting risks

The semi-competing risks problem arises when one is interested in the effect of an exposure or treatment on both intermediate (e.g., having cancer) and primary events (e.g., death) where the intermediate event may be censored by the primary event, but not vice versa. Here we propose a nonparametric approach casting the semi-competing risks problem in the framework of causal mediation modeling. We set up a mediation model with the intermediate and primary events, respectively as the mediator and the outcome, and define an indirect effect as the effect of the exposure on the primary event mediated by the intermediate event and a direct effect as that not mediated by the intermediate event. A nonparametric estimator with time-varying weights is proposed for direct and indirect effects where the counting process at time t of the primary event N 2 n 1 ( t ) and its compensator A n 1 ( t ) are both defined conditional on the status of the intermediate event right before t, N 1 ( t - ) = n 1 . We show that N 2 n 1 ( t ) - A n 1 ( t ) is a zero-mean martingale. Based on this, we further establish theoretical properties for the proposed estimators. Simulation studies are presented to illustrate the finite sample performance of the proposed method. Its advantage in causal interpretation over existing methods is also demonstrated in a hepatitis study.

Quantile Regression for Survival Data

Quantile regression offers a useful alternative strategy for analyzing survival data. Compared to traditional survival analysis methods, quantile regression allows for comprehensive and flexible evaluations of covariate effects on a survival outcome of interest, while providing simple physical interpretations on the time scale. Moreover, many quantile regression methods enjoy easy and stable computation. These appealing features make quantile regression a valuable practical tool for delivering in-depth analyses of survival data. In this paper, I review a comprehensive set of statistical methods for performing quantile regression with different types of survival data. This review covers various survival scenarios, including randomly censored data, data subject to left truncation or censoring, competing risks and semi-competing risks data, and recurrent events data. Two real examples are presented to illustrate the utility of quantile regression for practical survival data analyses.

Fitting a shared frailty illness-death model to left-truncated semi-competing risks data to examine the impact of education level on incident dementia

BACKGROUND

Semi-competing risks arise when interest lies in the time-to-event for some non-terminal event, the observation of which is subject to some terminal event. One approach to assessing the impact of covariates on semi-competing risks data is through the illness-death model with shared frailty, where hazard regression models are used to model the effect of covariates on the endpoints. The shared frailty term, which can be viewed as an individual-specific random effect, acknowledges dependence between the events that is not accounted for by covariates. Although methods exist for fitting such a model to right-censored semi-competing risks data, there is currently a gap in the literature for fitting such models when a flexible baseline hazard specification is desired and the data are left-truncated, for example when time is on the age scale. We provide a modeling framework and openly available code for implementation.

METHODS

We specified the model and the likelihood function that accounts for left-truncated data, and provided an approach to estimation and inference via maximum likelihood. Our model was fully parametric, specifying baseline hazards via Weibull or B-splines. Using simulated data we examined the operating characteristics of the implementation in terms of bias and coverage. We applied our methods to a dataset of 33,117 Kaiser Permanente Northern California members aged 65 or older examining the relationship between educational level (categorized as: high school or less; trade school, some college or college graduate; post-graduate) and incident dementia and death.

RESULTS

A simulation study showed that our implementation provided regression parameter estimates with negligible bias and good coverage. In our data application, we found higher levels of education are associated with a lower risk of incident dementia, after adjusting for sex and race/ethnicity.

CONCLUSIONS

As illustrated by our analysis of Kaiser data, our proposed modeling framework allows the analyst to assess the impact of covariates on semi-competing risks data, such as incident dementia and death, while accounting for dependence between the outcomes when data are left-truncated, as is common in studies of aging and dementia.

Tailored optimal posttreatment surveillance for cancer recurrence

A substantial rise in the number of cancer survivors has led to urgent management questions regarding effective posttreatment imaging-based surveillance strategies for cancer recurrence. Current surveillance guidelines provided by a number of professional societies all warn against overly aggressive surveillance, especially for low-risk patients, but all fail to provide more specific directions to accommodate underlying heterogeneity of cancer recurrence. Therefore it is imperative to develop data-driven strategies that can tailor the surveillance schedules to recurrence risk in this era of stricter insurance regulations, provider shortages, and rising costs of health care. Due to a lack of statistical methods for optimizing surveillance scheduling in presence of competing risks, we propose a general approach that uses an intuitive loss function for optimization of early detection of recurrence before death. The proposed strategies can tailor to patient risks of recurrence, in terms of both intensity and amount of surveillance. Using general three-state Markov models, our method is flexible and includes earlier works as special cases. We illustrate our method in both simulation studies and an application to breast cancer surveillance.

Time from diagnosis to 2nd treatment is a promising surrogate for overall survival in patients with advanced stage follicular lymphoma

It is difficult to demonstrate an overall survival (OS) benefit in trials of immediate therapy vs observation in follicular lymphoma (FL). Time to 2nd treatment (TT2T) may be a preferred endpoint. We identified 584 consecutive patients at our institution with advanced stage FL grade 1-3 A for whom intention was observation (n = 248) or therapy (n = 338). Median time to 1st treatment (TT1T), TT2T, and OS were estimated (subdistribution function). Modified Kendall's tau (mKτ) was used to assess correlation between survival endpoints. Among initially observed patients, median TT1T was 3.3 years, TT2T was 12.1 years, 10-year treatment-free survival was 23%, and 10-year OS was 82%. TT2T was strongly correlated with OS following initial observation (mKτ 0.46, p = .004) or therapy (mKτ 0.53, p < .0001), while duration of observation was not. TT2T is a potential surrogate for OS. Given the outstanding survival in this population, early intervention trials should focus on identifying high risk patients.

A phase I-II design based on periodic and continuous monitoring of disease status and the times to toxicity and death

A Bayesian phase I-II dose-finding design is presented for a clinical trial with four coprimary outcomes that reflect the actual clinical observation process. During a prespecified fixed follow-up period, the times to disease progression, toxicity, and death are monitored continuously, and an ordinal disease status variable, including progressive disease (PD) as one level, is evaluated repeatedly by scheduled imaging. We assume a proportional hazards model with piecewise constant baseline hazard for each continuous variable and a longitudinal multinomial probit model for the ordinal disease status process and include multivariate patient frailties to induce association among the outcomes. A finite partition of the nonfatal outcome combinations during the follow-up period is constructed, and the utility of each set in the partition is elicited. Posterior mean utility is used to optimize each patient's dose, subject to a safety rule excluding doses with an unacceptably high rate of PD, severe toxicity, or death. A simulation study shows that, compared with the proposed design, a simpler design based on commonly used efficacy and toxicity outcomes obtained by combining the four variables described above performs poorly and has substantially smaller probabilities of correctly choosing truly optimal doses and excluding truly unsafe doses.

Multiple event times in the presence of informative censoring: modeling and analysis by copulas

Motivated by a breast cancer research program, this paper is concerned with the joint survivor function of multiple event times when their observations are subject to informative censoring caused by a terminating event. We formulate the correlation of the multiple event times together with the time to the terminating event by an Archimedean copula to account for the informative censoring. Adapting the widely used two-stage procedure under a copula model, we propose an easy-to-implement pseudo-likelihood based procedure for estimating the model parameters. The approach yields a new estimator for the marginal distribution of a single event time with semicompeting-risks data. We conduct both asymptotics and simulation studies to examine the proposed approach in consistency, efficiency, and robustness. Data from the breast cancer program are employed to illustrate this research.

On shared gamma-frailty conditional Markov model for semicompeting risks data

Semicompeting risks data are a mixture of competing risks data and progressive state data. This type of data occurs when a nonterminal event is subject to truncation by a well-defined terminal event, but not vice versa. The shared gamma-frailty conditional Markov model (GFCMM) has been used to analyze semicompeting risks data because of its flexibility. There are two versions of this model: the restricted and the unrestricted model. Maximum likelihood estimation methodology has been proposed in the literature. However, we found through numerical experiments that the unrestricted model sometimes yields nonparametrically biased estimation. In this article, we provide a practical guideline for using the GFCMM in the analysis of semicompeting risk data that includes: (a) a score test to assess if the restricted model, which does not exhibit estimation problems, is reasonable under a proportional hazards assumption, and (b) a graphical illustration to justify whether the unrestricted model yields nonparametric estimation with substantial bias for cases where the test provides a statistical significant result against the restricted model. This guideline was applied to the Indianapolis-Ibadan Dementia Project data as an illustration to explore how dementia occurrence changes mortality risk.

Estimation and inference for semi-competing risks based on data from a nested case-control study

In semi-competing risks, the occurrence of some non-terminal event is subject to a terminal event, usually death. While existing methods for semi-competing risks data analysis assume complete information on all relevant covariates, data on at least one covariate are often not readily available in practice. In this setting, for standard univariate time-to-event analyses, researchers may choose from several strategies for sub-sampling patients on whom to collect complete data, including the nested case-control study design. Here, we consider a semi-competing risks analysis through the reuse of data from an existing nested case-control study for which risk sets were formed based on either the non-terminal or the terminal event. Additionally, we introduce the supplemented nested case-control design in which detailed data are collected on additional events of the other type. We propose estimation with respect to a frailty illness-death model through maximum weighted likelihood, specifying the baseline hazard functions either parametrically or semi-parametrically via B-splines. Two standard error estimators are proposed: (i) a computationally simple sandwich estimator and (ii) an estimator based on a perturbation resampling procedure. We derive the asymptotic properties of the proposed methods and evaluate their small-sample properties via simulation. The designs/methods are illustrated with an investigation of risk factors for acute graft-versus-host disease among N = 8838 patients undergoing hematopoietic stem cell transplantation, for which death is a significant competing risk.

A causal framework for classical statistical estimands in failure-time settings with competing events

In failure-time settings, a competing event is any event that makes it impossible for the event of interest to occur. For example, cardiovascular disease death is a competing event for prostate cancer death because an individual cannot die of prostate cancer once he has died of cardiovascular disease. Various statistical estimands have been defined as possible targets of inference in the classical competing risks literature. Many reviews have described these statistical estimands and their estimating procedures with recommendations about their use. However, this previous work has not used a formal framework for characterizing causal effects and their identifying conditions, which makes it difficult to interpret effect estimates and assess recommendations regarding analytic choices. Here we use a counterfactual framework to explicitly define each of these classical estimands. We clarify that, depending on whether competing events are defined as censoring events, contrasts of risks can define a total effect of the treatment on the event of interest or a direct effect of the treatment on the event of interest not mediated by the competing event. In contrast, regardless of whether competing events are defined as censoring events, counterfactual hazard contrasts cannot generally be interpreted as causal effects. We illustrate how identifying assumptions for all of these counterfactual estimands can be represented in causal diagrams, in which competing events are depicted as time-varying covariates. We present an application of these ideas to data from a randomized trial designed to estimate the effect of estrogen therapy on prostate cancer mortality.

Association measures for clustered competing risks

We propose a semiparameteric model for multivariate clustered competing risks data when the cause-specific failure times and the occurrence of competing risk events among subjects within the same cluster are of interest. The cause-specific hazard functions are assumed to follow Cox proportional hazard models, and the associations between failure times given the same or different cause events and the associations between occurrences of competing risk events within the same cluster are investigated through copula models. A cross-odds ratio measure is explored under our proposed models. Two-stage estimation procedure is proposed in which the marginal models are estimated in the first stage, and the dependence parameters are estimated via an expectation-maximization algorithm in the second stage. The proposed estimators are shown to yield consistent and asymptotically normal under mild regularity conditions. Simulation studies are conducted to assess finite sample performance of the proposed method. The proposed technique is demonstrated through an application to a multicenter Bone Marrow transplantation dataset.

Frailty modelling approaches for semi-competing risks data

In the semi-competing risks situation where only a terminal event censors a non-terminal event, observed event times can be correlated. Recently, frailty models with an arbitrary baseline hazard have been studied for the analysis of such semi-competing risks data. However, their maximum likelihood estimator can be substantially biased in the finite samples. In this paper, we propose effective modifications to reduce such bias using the hierarchical likelihood. We also investigate the relationship between marginal and hierarchical likelihood approaches. Simulation results are provided to validate performance of the proposed method. The proposed method is illustrated through analysis of semi-competing risks data from a breast cancer study.

Time-to-event analysis when the event is defined on a finite time interval

Acute graft-versus-host disease (GVHD) is a frequent complication following hematopoietic cell transplantation (HCT). Research on risk factors for acute GVHD has tended to ignore two important clinical issues. First, post-transplant mortality is high. In our motivating data, 100-day post-HCT mortality was 15.4%. Second, acute GVHD in its classic form is only diagnosed within 100 days of the transplant; beyond 100 days, a patient may be diagnosed with late onset acute or chronic GVHD. Standard modeling of time-to-event outcomes, however, generally conceive of patients being able to experience the event at any point on the time scale. In this paper, we propose a novel multi-state model that simultaneously: (i) accounts for mortality through joint modeling of acute GVHD and death, and (ii) explicitly acknowledges the finite time interval during which the event of interest can take place. The observed data likelihood is derived, with estimation and inference via maximum likelihood. Additionally, we provide methods for estimating the absolute risk of acute GVHD and death simultaneously. The proposed framework is compared via comprehensive simulations to a number of alternative approaches that each acknowledge some but not all aspects of acute GVHD, and illustrated with an analysis of HCT data that motivated this work.

Prioritized concordance index for hierarchical survival outcomes

We propose an extension of Harrell's concordance (C) index to evaluate the prognostic utility of biomarkers for diseases with multiple measurable outcomes that can be prioritized. Our prioritized concordance index measures the probability that, given a random subject pair, the subject with the worst disease status as of a time τ has the higher predicted risk. Our prioritized concordance index uses the same approach as the win ratio, by basing generalized pairwise comparisons on the most severe or clinically important comparable outcome. We use an inverse probability weighting technique to correct for study-specific censoring. Asymptotic properties are derived using U-statistic properties. We apply the prioritized concordance index to two types of disease processes with a rare primary outcome and a more common secondary outcome. Our simulation studies show that when a predictor is predictive of both outcomes, the new concordance index can gain efficiency and power in identifying true prognostic variables compared to using the primary outcome alone. Using the prioritized concordance index, we examine whether novel clinical measures can be useful in predicting risk of type II diabetes in patients with impaired glucose resistance whose disease status can also regress to normal glucose resistance. We also examine the discrimination ability of four published risk models among ever smokers at risk of lung cancer incidence and subsequent death.

Joint regression analysis for survival data in the presence of two sets of semi-competing risks

In many clinical trials, multiple time-to-event endpoints including the primary endpoint (e.g., time to death) and secondary endpoints (e.g., progression-related endpoints) are commonly used to determine treatment efficacy. These endpoints are often biologically related. This work is motivated by a study of bone marrow transplant (BMT) for leukemia patients, who may experience the acute graft-versus-host disease (GVHD), relapse of leukemia, and death after an allogeneic BMT. The acute GVHD is associated with the relapse free survival, and both the acute GVHD and relapse of leukemia are intermediate nonterminal events subject to dependent censoring by the informative terminal event death, but not vice versa, giving rise to survival data that are subject to two sets of semi-competing risks. It is important to assess the impacts of prognostic factors on these three time-to-event endpoints. We propose a novel statistical approach that jointly models such data via a pair of copulas to account for multiple dependence structures, while the marginal distribution of each endpoint is formulated by a Cox proportional hazards model. We develop an estimation procedure based on pseudo-likelihood and carry out simulation studies to examine the performance of the proposed method in finite samples. The practical utility of the proposed method is further illustrated with data from the motivating example.

A flexible and robust method for assessing conditional association and conditional concordance

When analyzing bivariate outcome data, it is often of scientific interest to measure and estimate the association between the bivariate outcomes. In the presence of influential covariates for one or both of the outcomes, conditional association measures can quantify the strength of association without the disturbance of the marginal covariate effects, to provide cleaner and less-confounded insights into the bivariate association. In this work, we propose estimation and inferential procedures for assessing the conditional Kendall's tau coefficient given the covariates, by adopting the quantile regression and quantile copula framework to handle marginal covariate effects. The proposed method can flexibly accommodate right censoring and be readily applied to bivariate survival data. It also facilitates an estimator of the conditional concordance measure, namely, a conditional C index, where the unconditional C index is commonly used to assess the predictive capacity for survival outcomes. The proposed method is flexible and robust and can be easily implemented using standard software. The method performed satisfactorily in extensive simulation studies with and without censoring. Application of our methods to two real-life data examples demonstrates their desirable practical utility.

Semiparametric Model for Bivariate Survival Data Subject to Biased Sampling

To better understand the relationship between patient characteristics and their residual survival after an intermediate event such as the local cancer recurrence, it is of interest to identify patients with the intermediate event and then analyze their residual survival data. One challenge in analyzing such data is that the observed residual survival times tend to be longer than those in the target population, since patients who die before experiencing the intermediate event are excluded from the identified cohort. We propose to jointly model the ordered bivariate survival data using a copula model and appropriately adjusting for the sampling bias. We develop an estimating procedure to simultaneously estimate the parameters for the marginal survival functions and the association parameter in the copula model, and use a two-stage expectation-maximization algorithm. Using empirical process theory, we prove that the estimators have strong consistency and asymptotic normality. We conduct simulations studies to evaluate the finite sample performance of the proposed method. We apply the proposed method to two cohort studies to evaluate the association between patient characteristics and residual survival.

Multivariate two-sample permutation tests for trials with multiple time-to-event outcomes

Clinical trials involving multiple time-to-event outcomes are increasingly common. In this paper, permutation tests for testing for group differences in multivariate time-to-event data are proposed. Unlike other two-sample tests for multivariate survival data, the proposed tests attain the nominal type I error rate. A simulation study shows that the proposed tests outperform their competitors when the degree of censored observations is sufficiently high. When the degree of censoring is low, it is seen that naive tests such as Hotelling's T² outperform tests tailored to survival data. Computational and practical aspects of the proposed tests are discussed, and their use is illustrated by analyses of three publicly available datasets. Implementations of the proposed tests are available in an accompanying R package.

Additive-multiplicative hazards regression models for interval-censored semi-competing risks data with missing intermediate events

BACKGROUND

In clinical trials and survival analysis, participants may be excluded from the study due to withdrawal, which is often referred to as lost-to-follow-up (LTF). It is natural to argue that a disease would be censored due to death; however, when an LTF is present it is not guaranteed that the disease has been censored. This makes it important to consider both cases; the disease is censored or not censored. We also note that the illness process can be censored by LTF. We will consider a multi-state model in which LTF is not regarded as censoring but as a non-fatal event.

METHODS

We propose a multi-state model for analyzing semi-competing risks data with interval-censored or missing intermediate events. More precisely, we employ the additive and multiplicative hazards model with log-normal frailty and construct the conditional likelihood to estimate the transition intensities among states in the multi-state model. Marginalization of the full likelihood is accomplished using adaptive importance sampling, and the optimal solution of the regression parameters is achieved through the iterative quasi-Newton algorithm.

RESULTS

Simulation is performed to investigate the finite-sample performance of the proposed estimation method in terms of the relative bias and coverage probability of the regression parameters. The proposed estimators turned out to be robust to misspecifications of the frailty distribution. PAQUID data have been analyzed and yielded somewhat prominent results.

CONCLUSIONS

We propose a multi-state model for semi-competing risks data for which there exists information on fatal events, but information on non-fatal events may not be available due to lost to follow-up. Simulation results show that the coverage probabilities of the regression parameters are close to a nominal level of 0.95 in most cases. Regarding the analysis of real data, the risk of transition from a healthy state to dementia is higher for women; however, the risk of death after being diagnosed with dementia is higher for men.

Dynamic prediction of repeated events data based on landmarking model: application to colorectal liver metastases data

BACKGROUND

In some clinical situations, patients experience repeated events of the same type. Among these, cancer recurrences can result in terminal events such as death. Therefore, here we dynamically predicted the risks of repeated and terminal events given longitudinal histories observed before prediction time using dynamic pseudo-observations (DPOs) in a landmarking model.

METHODS

The proposed DPOs were calculated using Aalen-Johansen estimator for the event processes described in the multi-state model. Furthermore, in the absence of a terminal event, a more convenient approach without matrix operation was described using the ordering of repeated events. Finally, generalized estimating equations were used to calculate probabilities of repeated and terminal events, which were treated as multinomial outcomes.

RESULTS

Simulation studies were conducted to assess bias and investigate the efficiency of the proposed DPOs in a finite sample. Little bias was detected in DPOs even under relatively heavy censoring, and the method was applied to data from patients with colorectal liver metastases.

CONCLUSIONS

The proposed method enabled intuitive interpretations of terminal event settings.

Optimizing natural killer cell doses for heterogeneous cancer patients on the basis of multiple event times

A sequentially adaptive Bayesian design is presented for a clinical trial of cord blood derived natural killer cells to treat severe hematologic malignancies. Given six prognostic subgroups defined by disease type and severity, the goal is to optimize cell dose in each subgroup. The trial has five co-primary outcomes, the times to severe toxicity, cytokine release syndrome, disease progression or response, and death. The design assumes a multivariate Weibull regression model, with marginals depending on dose, subgroup, and patient frailties that induce association among the event times. Utilities of all possible combinations of the nonfatal outcomes over the first 100 days following cell infusion are elicited, with posterior mean utility used as a criterion to optimize dose. For each subgroup, the design stops accrual to doses having an unacceptably high death rate, and at the end of the trial selects the optimal safe dose. A simulation study is presented to validate the design's safety, ability to identify optimal doses, and robustness, and to compare it to a simplified design that ignores patient heterogeneity.

compound.Cox: Univariate feature selection and compound covariate for predicting survival

BACKGROUND AND OBJECTIVE

Univariate feature selection is one of the simplest and most commonly used techniques to develop a multigene predictor for survival. Presently, there is no software tailored to perform univariate feature selection and predictor construction.

METHODS

We develop the compound.Cox R package that implements univariate significance tests (via the Wald tests or score tests) for feature selection. We provide a cross-validation algorithm to measure predictive capability of selected genes and a permutation algorithm to assess the false discovery rate. We also provide three algorithms for constructing a multigene predictor (compound covariate, compound shrinkage, and copula-based methods), which are tailored to the subset of genes obtained from univariate feature selection. We demonstrate our package using survival data on the lung cancer patients. We examine the predictive capability of the developed algorithms by the lung cancer data and simulated data.

RESULTS

The developed R package, compound.Cox, is available on the CRAN repository. The statistical tools in compound.Cox allow researchers to determine an optimal significance level of the tests, thus providing researchers an optimal subset of genes for prediction. The package also allows researchers to compute the false discovery rate and various prediction algorithms.

Modeling Data With Semicompeting Risks: An Application to Chronic Kidney Disease in Colombia

In this paper, the structure of semicompeting risks data, dened by Fine, Jiang & Chappell (2001), is studied. Two events are of interest: a nonterminal and a terminal event, the last one, can censor the non-terminal event, but not vice versa. Due to the possible dependence between the times until the occurrence of such events, two approaches are evaluated: modelling the bivariate survival function through Archimedean copulas and a shared frailty model. A simulation is conducted to examine its performance and both approaches are applied to a real data set of patients with chronic kidney disease (CKD).

Semiparametric regression analysis of interval-censored data with informative dropout

Interval-censored data arise when the event time of interest can only be ascertained through periodic examinations. In medical studies, subjects may not complete the examination schedule for reasons related to the event of interest. In this article, we develop a semiparametric approach to adjust for such informative dropout in regression analysis of interval-censored data. Specifically, we propose a broad class of joint models, under which the event time of interest follows a transformation model with a random effect and the dropout time follows a different transformation model but with the same random effect. We consider nonparametric maximum likelihood estimation and develop an EM algorithm that involves simple and stable calculations. We prove that the resulting estimators of the regression parameters are consistent, asymptotically normal, and asymptotically efficient with a covariance matrix that can be consistently estimated through profile likelihood. In addition, we show how to consistently estimate the survival function when dropout represents voluntary withdrawal and the cumulative incidence function when dropout is an unavoidable terminal event. Furthermore, we assess the performance of the proposed numerical and inferential procedures through extensive simulation studies. Finally, we provide an application to data on the incidence of diabetes from a major epidemiological cohort study.

Properties of composite time to first event versus joint marginal analyses of multiple outcomes

Many clinical studies (eg, cardiovascular outcome trials) investigate the effect of an intervention on multiple event time outcomes. The most common method of analysis is a so-called "composite" analysis of a composite outcome defined as the time to the first component event. Other approaches have been proposed, including the win ratio (or win difference) for ordered outcomes and the application of the Wei-Lachin test. Herein, we assess the influence of the marginal and joint distributions of the component events, and their correlation structures, on the operating characteristics of these methods for the analysis of multiple events. The operating characteristics are investigated using a bivariate exponential model with a shared frailty, under which these properties are obtained in closed form. While the composite time-to-first-event analysis provides an unbiased test of the hypothesis of equality of the distribution of the time to first event, we show that it can provide a biased test of the joint null hypothesis of equal marginal hazards when the correlation of event times differs between groups. The same applies to the win ratio. However, the operating characteristics of the Wei-Lachin or other tests of the joint equality of the marginal hazards are unaffected. Furthermore, when the correlations are equal, the Wei-Lachin test is more powerful to detect a difference in marginal hazards than the composite analysis test. Careful consideration of the properties of the various methods for analysis of composite outcome measures are in order before adopting one as primary analysis in a clinical study.

Estimating cross quantile residual ratio with left-truncated semi-competing risks data

A semi-competing risks setting often arises in biomedical studies, involving both a nonterminal event and a terminal event. Cross quantile residual ratio (Yang and Peng in Biometrics 72:770-779, 2016) offers a flexible and robust perspective to study the dependency between the nonterminal and the terminal events which can shed useful scientific insight. In this paper, we propose a new nonparametric estimator of this dependence measure with left truncated semi-competing risks data. The new estimator overcomes the limitation of the existing estimator that is resulted from demanding a strong assumption on the truncation mechanism. We establish the asymptotic properties of the proposed estimator and develop inference procedures accordingly. Simulation studies suggest good finite-sample performance of the proposed method. Our proposal is illustrated via an application to Denmark diabetes registry data.

Semiparametric Regression Analysis of Multiple Right- and Interval-Censored Events

Health sciences research often involves both right- and interval-censored events because the occurrence of a symptomatic disease can only be observed up to the end of follow-up, while the occurrence of an asymptomatic disease can only be detected through periodic examinations. We formulate the effects of potentially time-dependent covariates on the joint distribution of multiple right- and interval-censored events through semiparametric proportional hazards models with random effects that capture the dependence both within and between the two types of events. We consider nonparametric maximum likelihood estimation and develop a simple and stable EM algorithm for computation. We show that the resulting estimators are consistent and the parametric components are asymptotically normal and efficient with a covariance matrix that can be consistently estimated by profile likelihood or nonparametric bootstrap. In addition, we leverage the joint modelling to provide dynamic prediction of disease incidence based on the evolving event history. Furthermore, we assess the performance of the proposed methods through extensive simulation studies. Finally, we provide an application to a major epidemiological cohort study. Supplementary materials for this article are available online.

Methods for the analysis of multiple endpoints in small populations: A review

While current guidelines generally recommend single endpoints for primary analyses of confirmatory clinical trials, it is recognized that certain settings require inference on multiple endpoints for comprehensive conclusions on treatment effects. Furthermore, combining treatment effect estimates from several outcome measures can increase the statistical power of tests. Such an efficient use of resources is of special relevance for trials in small populations. This paper reviews approaches based on a combination of test statistics or measurements across endpoints as well as multiple testing procedures that allow for confirmatory conclusions on individual endpoints. We especially focus on feasibility in trials with small sample sizes and do not solely rely on asymptotic considerations. A systematic literature search in the Scopus database, supplemented by a manual search, was performed to identify research papers on analysis methods for multiple endpoints with relevance to small populations. The identified methods were grouped into approaches that combine endpoints into a single measure to increase the power of statistical tests and methods to investigate differential treatment effects in several individual endpoints by multiple testing.