Parametric mixture models to evaluate and summarize hazard ratios in the presence of competing risks with time-dependent hazards and delayed entry

In the analysis of survival data, there are often competing events that preclude an event of interest from occurring. Regression analysis with competing risks is typically undertaken using a cause-specific proportional hazards model. However, modern alternative methods exist for the analysis of the subdistribution hazard with a corresponding subdistribution proportional hazards model. In this paper, we introduce a flexible parametric mixture model as a unifying method to obtain estimates of the cause-specific and subdistribution hazards and hazard-ratio functions. We describe how these estimates can be summarized over time to give a single number comparable to the hazard ratio that is obtained from a corresponding cause-specific or subdistribution proportional hazards model. An application to the Women's Interagency HIV Study is provided to investigate injection drug use and the time to either the initiation of effective antiretroviral therapy, or clinical disease progression as a competing event.     

Sample size formula for proportional hazards modelling of competing risks

To test the effect of a therapeutic or prognostic factor on the occurrence of a particular cause of failure in the presence of other causes, the interest has shifted in some studies from the modelling of the cause-specific hazard to that of the subdistribution hazard. We present approximate sample size formulas for the proportional hazards modelling of competing risk subdistribution, considering either independent or correlated covariates. The validity of these approximate formulas is investigated through numerical simulations. Two illustrations are provided, a randomized clinical trial, and a prospective prognostic study.     

Modelling competing risks data with missing cause of failure

When competing risks data arise, information on the actual cause of failure for some subjects might be missing. Therefore, a cause-specific proportional hazards model together with multiple imputation (MI) methods have been used to analyze such data. Modelling the cumulative incidence function is also of interest, and thus we investigate the proportional subdistribution hazards model (Fine and Gray model) together with MI methods as a modelling approach for competing risks data with missing cause of failure. Possible strategies for analyzing such data include the complete case analysis as well as an analysis where the missing causes are classified as an additional failure type. These approaches, however, may produce misleading results in clinical settings. In the present work we investigate the bias of the parameter estimates when fitting the Fine and Gray model in the above modelling approaches. We also apply the MI method and evaluate its comparative performance under various missing data scenarios. Results from simulation experiments showed that there is substantial bias in the estimates when fitting the Fine and Gray model with naive techniques for missing data, under missing at random cause of failure. Compared to those techniques the MI-based method gave estimates with much smaller biases and coverage probabilities of 95 per cent confidence intervals closer to the nominal level. All three methods were also applied on real data modelling time to AIDS or non-AIDS cause of death in HIV-1 infected individuals.     

Estimating and testing for center effects in competing risks

The problems of fitting Gaussian frailties proportional hazards models for the subdistribution of a competing risk and of testing for center effects are considered. In the analysis of competing risks data, Fine and Gray proposed a proportional hazards model for the subdistribution to directly assess the effects of covariates on the marginal failure probabilities of a given failure cause. Katsahianbiet al. extended their model to clustered time to event data, by including random center effects or frailties in the subdistribution hazard. We first introduce an alternate estimation procedure to the one proposed by Katsahian et al. This alternate estimation method is based on the penalized partial likelihood approach often used in fitting Gaussian frailty proportional hazards models in the standard survival analysis context, and has the advantage of using standard survival analysis software. Second, four hypothesis tests for the presence of center effects are given and compared via Monte-Carlo simulations. Statistical and numerical considerations lead us to formulate pragmatic guidelines as to which of the four tests is preferable. We also illustrate the proposed methodology with registry data from bone marrow transplantation for acute myeloid leukemia (AML).     

A proportional hazards regression model for the subdistribution with right-censored and left-truncated competing risks data

With competing risks failure time data, one often needs to assess the covariate effects on the cumulative incidence probabilities. Fine and Gray proposed a proportional hazards regression model to directly model the subdistribution of a competing risk. They developed the estimating procedure for right-censored competing risks data, based on the inverse probability of censoring weighting. Right-censored and left-truncated competing risks data sometimes occur in biomedical researches. In this paper, we study the proportional hazards regression model for the subdistribution of a competing risk with right-censored and left-truncated data. We adopt a new weighting technique to estimate the parameters in this model. We have derived the large sample properties of the proposed estimators. To illustrate the application of the new method, we analyze the failure time data for children with acute leukemia. In this example, the failure times for children who had bone marrow transplants were left truncated.     

A competing risks analysis of bloodstream infection after stem-cell transplantation using subdistribution hazards and cause-specific hazards

After peripheral blood stem-cell transplantation, patients treated for severe haematologic diseases enter a critical phase (neutropenia). Analysis of bloodstream infection during neutropenia has to account for competing risks. Separate Cox analyses of all cause-specific hazards are the standard technique of choice, but are hard to interpret when the overall effects of covariates on the cumulative incidence function (CIF) are of interest. Proportional subdistribution hazards modelling of the subdistribution of the CIF is establishing itself as an interpretation-friendly alternative. We apply both methods and discuss their relative merits.     

Sample size calculations in the presence of competing risks

Recently, with the growth of statistical developments for competing risks analysis, some methods have been proposed to compute sample size in this context. These methods differ from a modelling approach: one is based on the Cox regression model for the cause-specific hazard, while another relies on the Fine and Gray regression model for the subdistribution hazard of a competing risk. In this work, we compare these approaches, derive a new sample size for comparing cumulative incidence functions when the hazards are not proportional (either cause-specific or subdistribution) and give practical advices to choose the approach best suited for the study question.     

Simulating competing risks data in survival analysis

Competing risks analysis considers time‐to‐first‐event (‘survival time’) and the event type (‘cause’), possibly subject to right‐censoring. The cause‐, i.e. event‐specific hazards, completely determine the competing risk process, but simulation studies often fall back on the much criticized latent failure time model. Cause‐specific hazard‐driven simulation appears to be the exception; if done, usually only constant hazards are considered, which will be unrealistic in many medical situations. We explain simulating competing risks data based on possibly time‐dependent cause‐specific hazards. The simulation design is as easy as any other, relies on identifiable quantities only and adds to our understanding of the competing risks process. In addition, it immediately generalizes to more complex multistate models. We apply the proposed simulation design to computing the least false parameter of a misspecified proportional subdistribution hazard model, which is a research question of independent interest in competing risks. The simulation specifications have been motivated by data on infectious complications in stem‐cell transplanted patients, where results from cause‐specific hazards analyses were difficult to interpret in terms of cumulative event probabilities. The simulation illustrates that results from a misspecified proportional subdistribution hazard analysis can be interpreted as a time‐averaged effect on the cumulative event probability scale. Copyright © 2009 John Wiley & Sons, Ltd.     

Propensity-score matching with competing risks in survival analysis

Propensity-score matching is a popular analytic method to remove the effects of confounding due to measured baseline covariates when using observational data to estimate the effects of treatment. Time-to-event outcomes are common in medical research. Competing risks are outcomes whose occurrence precludes the occurrence of the primary time-to-event outcome of interest. All non-fatal outcomes and all cause-specific mortality outcomes are potentially subject to competing risks. There is a paucity of guidance on the conduct of propensity-score matching in the presence of competing risks. We describe how both relative and absolute measures of treatment effect can be obtained when using propensity-score matching with competing risks data. Estimates of the relative effect of treatment can be obtained by using cause-specific hazard models in the matched sample. Estimates of absolute treatment effects can be obtained by comparing cumulative incidence functions (CIFs) between matched treated and matched control subjects. We conducted a series of Monte Carlo simulations to compare the empirical type I error rate of different statistical methods for testing the equality of CIFs estimated in the matched sample. We also examined the performance of different methods to estimate the marginal subdistribution hazard ratio. We recommend that a marginal subdistribution hazard model that accounts for the within-pair clustering of outcomes be used to test the equality of CIFs and to estimate subdistribution hazard ratios. We illustrate the described methods by using data on patients discharged from hospital with acute myocardial infarction to estimate the effect of discharge prescribing of statins on cardiovascular death.     

Estimation with Cox models: cause-specific survival analysis with misclassified cause of failure

While epidemiologic and clinical research often aims to analyze predictors of specific endpoints, time-to-the-specific-event analysis can be hampered by problems with cause ascertainment. Under typical assumptions of competing risks analysis (and missing-data settings), we correct the cause-specific proportional hazards analysis when information on the reliability of diagnosis is available. Our method avoids bias in effect estimates at low cost in variance, thus offering a perspective for better-informed decision making. The ratio of different cause-specific hazards can be estimated flexibly for this purpose. It thus complements an all-cause analysis. In a sensitivity analysis, this approach can reveal the likely extent and direction of the bias of a standard cause-specific analysis when the diagnosis is suspect. These 2 uses are illustrated in a randomized vaccine trial and an epidemiologic cohort study, respectively.     

First‐event or marginal estimation of cause‐specific hazards for analysing correlated multivariate failure‐time data?

In the analysis of multivariate failure-time data, the effect of a treatment or an exposure on the hazard of each failure type is sometimes evaluated using only the information on the first event that occurs in every individual, ignoring all events that follow. A Cox proportional hazards model may be fitted to such data, yielding a cause-specific hazard ratio (HR) estimate of the exposure for each failure type conditional on surviving all other failure types. However, such an estimate would not fully utilize all the available information on event times. Alternatively, a marginal approach may be implemented to model the time distribution of each failure type beyond the subject's first failure to (any) second and later failures. We investigate the performance of these two approaches by simulating positive and negative correlated event times from exponential distributions. Surprisingly, our results suggest that the first-event-only method (when multiple failures are possible) performs as well as the marginal method in most practical situations. Generally, for a modest sample size of 400, it is possible to achieve at least 85 per cent coverage of the true marginal HR with the first-event method. Although the coverage is poor for a correlation of 0.7 and beyond, such a high correlation between competing event times may be biologically rather implausible.     

Estimation of the adjusted cause-specific cumulative probability using flexible regression models for the cause-specific hazards

In competing risks setting, we account for death according to a specific cause and the quantities of interest are usually the cause-specific hazards (CSHs) and the cause-specific cumulative probabilities. A cause-specific cumulative probability can be obtained with a combination of the CSHs or via the subdistribution hazard. Here, we modeled the CSH with flexible hazard-based regression models using B-splines for the baseline hazard and time-dependent (TD) effects. We derived the variance of the cause-specific cumulative probabilities at the population level using the multivariate delta method and showed how we could easily quantify the impact of a covariate on the cumulative probability scale using covariate-adjusted cause-specific cumulative probabilities and their difference. We conducted a simulation study to evaluate the performance of this approach in its ability to estimate the cumulative probabilities using different functions for the cause-specific log baseline hazard and with or without a TD effect. In the scenario with TD effect, we tested both well-specified and misspecified models. We showed that the flexible regression models perform nearly as well as the nonparametric method, if we allow enough flexibility for the baseline hazards. Moreover, neglecting the TD effect hardly affects the cumulative probabilities estimates of the whole population but impacts them in the various subgroups. We illustrated our approach using data from people diagnosed with monoclonal gammopathy of undetermined significance and provided the R-code to derive those quantities, as an extension of the R-package mexhaz .     

Patient death as a censoring event or competing risk event in models of nursing home placement

Participant death is often observed in studies that examine predictors of events, such as hospitalization or institutionalization, in older adult populations. The Cox proportional hazards modeling of the target event, whereby death is treated as a censoring event, is the standard analysis in this competing risks situation. However, the assumption of noninformative censoring applied to a frequently occurring competing event like death may be invalid and complicate interpretation in terms of the probability of the event. Multiple cause‐specific hazard (CSH) models can be estimated, but ambiguities may arise when interpreting covariate effects across multiple CSH models and in terms of the cumulative incidence function (CIF). Alternatively, one can model the proportional hazards of the subdistribution of the CIF and evaluate the covariate effects on the CIF directly. We examine and compare these two approaches with nursing home (NH) placement data from a randomized controlled trial of a counseling and support intervention for spouse‐caregivers of patients with Alzheimer's disease. CSHs for NH placement (where death is treated as a censoring event) and death (where NH placement is treated as a censoring event) and subdistribution hazards of the CIF for NH placement are modeled separately. In the presence of multiple covariates, the intervention effect is significant in both approaches, but the interpretation of the covariate effects requires joint evaluation of all estimated models. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Goodness‐of‐fit test for proportional subdistribution hazards model

This paper concerns using modified weighted Schoenfeld residuals to test the proportionality of subdistribution hazards for the Fine–Gray model, similar to the tests proposed by Grambsch and Therneau for independently censored data. We develop a score test for the time‐varying coefficients based on the modified Schoenfeld residuals derived assuming a certain form of non‐proportionality. The methods perform well in simulations and a real data analysis of breast cancer data, where the treatment effect exhibits non‐proportional hazards. Copyright © 2013 John Wiley & Sons, Ltd.     

Competing risks analysis of patients with osteosarcoma: a comparison of four different approaches

In failure time studies involving a chronic disease such as cancer, several competing causes of mortality may be operating. Commonly, the conventional statistical technique of Kaplan-Meier, which is only meaningfully interpreted by assuming independence of failure types and the censoring mechanism, is employed in clinical research involving competing risks data. Some authors have advocated the use of a cause-specific cumulative incidence function which takes into account the existence of other events within a competing risks framework, without making any assumption about independence. Lunn and McNeil have proposed an approach based on an extension of the Cox proportional hazards regression, which enables direct comparisons between failure types. We have extended this approach to estimate cause-specific cumulative incidence. As it is often not easy to follow competing risks methodology in the literature, this paper sets out systematically the assumptions made and the steps taken to implement four different methods of analysing competing risks data using cumulative incidence rates or the Kaplan-Meier estimates of cause-specific failure probabilities. The data obtained from a randomized trial of patients with osteosarcoma were used to compare these four approaches. As illustrated using the osteosarcoma data, the estimates of the classical Kaplan-Meier methods have larger numerical values than the cause-specific cumulative incidence. On the other hand, estimates of the cause-specific cumulative incidence rates from the conventional method and the modified Cox method are highly comparable.     

Parametric models for accelerated and long-term survival: a comment on proportional hazards

The Cox proportional hazards model (CPH) is routinely used in clinical trials, but it may encounter serious difficulties with departures from the proportional hazards assumption, even when the departures are not readily detected by commonly used diagnostics. We consider the Gamel-Boag (GB) model, a log-normal model for accelerated failure in which a proportion of subjects are long-term survivors. When the CPH model is fit to simulated data generated from this model, the results can range from gross overstatement of the effect size, to a situation where increasing follow-up may cause a decline in power. We implement a fitting algorithm for the GB model that permits separate covariate effects on the rapidity of early failure and the fraction of long-term survivors. When effects are detected by both the CPH and GB methods, the attribution of the effect to long-term or short-term survival may change the interpretation of the data. We believe these examples motivate more frequent use of parametric survival models in conjunction with the semi-parametric Cox proportional hazards model.     

A competing risks analysis should report results on all cause-specific hazards and cumulative incidence functions

Competing risks endpoints are frequently encountered in hematopoietic stem cell transplantation where patients are exposed to relapse and treatment-related mortality. Both cause-specific hazards and direct models for the cumulative incidence functions have been used for analyzing such competing risks endpoints. For both approaches, the popular models are of a proportional hazards type. Such models have been used for studying prognostic factors in acute and chronic leukemias.We argue that a complete understanding of the event dynamics requires that both hazards and cumulative incidence be analyzed side by side, and that this is generally the most rigorous scientific approach to analyzing competing risks data. That is, understanding the effects of covariates on cause-specific hazards and cumulative incidence functions go hand in hand. A case study illustrates our proposal.     

Incorporating prognostic factors into causal estimators: A comparison of methods for randomised controlled trials with a time‐to‐event outcome

In randomised controlled trials, the effect of treatment on those who comply with allocation to active treatment can be estimated by comparing their outcome to those in the comparison group who would have complied with active treatment had they been allocated to it. We compare three estimators of the causal effect of treatment on compliers when this is a parameter in a proportional hazards model and quantify the bias due to omitting baseline prognostic factors. Causal estimates are found directly by maximising a novel partial likelihood; based on a structural proportional hazards model; and based on a ‘corrected dataset’ derived after fitting a rank‐preserving structural failure time model. Where necessary, we extend these methods to incorporate baseline covariates. Comparisons use simulated data and a real data example. Analysing the simulated data, we found that all three methods are accurate when an important covariate was included in the proportional hazards model (maximum bias 5.4%). However, failure to adjust for this prognostic factor meant that causal treatment effects were underestimated (maximum bias 11.4%), because estimators were based on a misspecified marginal proportional hazards model. Analysing the real data example, we found that adjusting causal estimators is important to correct for residual imbalances in prognostic factors present between trial arms after randomisation. Our results show that methods of estimating causal treatment effects for time‐to‐event outcomes should be extended to incorporate covariates, thus providing an informative compliment to the corresponding intention‐to‐treat analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

Semiparametric accelerated failure time cure rate mixture models with competing risks

Modern medical treatments have substantially improved survival rates for many chronic diseases and have generated considerable interest in developing cure fraction models for survival data with a non‐ignorable cured proportion. Statistical analysis of such data may be further complicated by competing risks that involve multiple types of endpoints. Regression analysis of competing risks is typically undertaken via a proportional hazards model adapted on cause‐specific hazard or subdistribution hazard. In this article, we propose an alternative approach that treats competing events as distinct outcomes in a mixture. We consider semiparametric accelerated failure time models for the cause‐conditional survival function that are combined through a multinomial logistic model within the cure‐mixture modeling framework. The cure‐mixture approach to competing risks provides a means to determine the overall effect of a treatment and insights into how this treatment modifies the components of the mixture in the presence of a cure fraction. The regression and nonparametric parameters are estimated by a nonparametric kernel‐based maximum likelihood estimation method. Variance estimation is achieved through resampling methods for the kernel‐smoothed likelihood function. Simulation studies show that the procedures work well in practical settings. Application to a sarcoma study demonstrates the use of the proposed method for competing risk data with a cure fraction.