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

A comparison of two frameworks for multi-state modelling, applied to outcomes after hospital admissions with COVID-19

We compare two multi-state modelling frameworks that can be used to represent dates of events following hospital admission for people infected during an epidemic. The methods are applied to data from people admitted to hospital with COVID-19, to estimate the probability of admission to intensive care unit, the probability of death in hospital for patients before and after intensive care unit admission, the lengths of stay in hospital, and how all these vary with age and gender. One modelling framework is based on defining transition-specific hazard functions for competing risks. A less commonly used framework defines partially-latent subpopulations who will experience each subsequent event, and uses a mixture model to estimate the probability that an individual will experience each event, and the distribution of the time to the event given that it occurs. We compare the advantages and disadvantages of these two frameworks, in the context of the COVID-19 example. The issues include the interpretation of the model parameters, the computational efficiency of estimating the quantities of interest, implementation in software and assessing goodness of fit. In the example, we find that some groups appear to be at very low risk of some events, in particular intensive care unit admission, and these are best represented by using 'cure-rate' models to define transition-specific hazards. We provide general-purpose software to implement all the models we describe in the flexsurv R package, which allows arbitrarily flexible distributions to be used to represent the cause-specific hazards or times to events.

Burden of Antimicrobial Resistance: Compared to What?

The increased focus on the public health burden of antimicrobial resistance (AMR) raises conceptual challenges, such as determining how much harm multidrug-resistant organisms do compared to what, or how to establish the burden. Here, we present a counterfactual framework and provide guidance to harmonize methodologies and optimize study quality. In AMR-burden studies, 2 counterfactual approaches have been applied: the harm of drug-resistant infections relative to the harm of the same drug-susceptible infections (the susceptible-infection counterfactual); and the total harm of drug-resistant infections relative to a situation where such infections were prevented (the no-infection counterfactual). We propose to use an intervention-based causal approach to determine the most appropriate counterfactual. We show that intervention scenarios, species of interest, and types of infections influence the choice of counterfactual. We recommend using purpose-designed cohort studies to apply this counterfactual framework, whereby the selection of cohorts (patients with drug-resistant, drug-susceptible infections, and those with no infection) should be based on matching on time to infection through exposure density sampling to avoid biased estimates. Application of survival methods is preferred, considering competing events. We conclude by advocating estimation of the burden of AMR by using the no-infection and susceptible-infection counterfactuals. The resulting numbers will provide policy-relevant information about the upper and lower bound of future interventions designed to control AMR. The counterfactuals should be applied in cohort studies, whereby selection of the unexposed cohorts should be based on exposure density sampling, applying methods avoiding time-dependent bias and confounding.

Causal inference in the face of competing events

Purpose of review

Epidemiologists frequently must handle competing events, which prevent the event of interest from occurring. We review considerations for handling competing events when interpreting results causally.

Recent findings

When interpreting statistical associations as causal effects, we recommend following a causal inference "roadmap" as one would in an analysis without competing events. There are, however, special considerations to be made for competing events when choosing the causal estimand that best answers the question of interest, selecting the statistical estimand (e.g. the cause-specific or subdistribution) that will target that causal estimand, and assessing whether causal identification conditions (e.g., conditional exchangeability, positivity, and consistency) have been sufficiently met.

Summary

When doing causal inference in the competing events setting, it is critical to first ascertain the relevant question and the causal estimand that best answers it, with the choice often being between estimands that do and do not eliminate competing events.

Nonparametric Assessment of Differences Between Competing Risk Hazard Ratios: Application to Racial Differences in Pediatric Chronic Kidney Disease Progression

Associations between an exposure and multiple competing events are typically described by cause-specific hazard ratios (csHR) or subdistribution hazard ratios (sHR). However, diagnostic tools to assess differences between them have not been described. Under the proportionality assumption for both, it can be shown mathematically that the sHR and csHR must be equal, so reporting different time-constant sHR and csHR implies non-proportionality for at least one. We propose a simple, intuitive approach using the ratio of sHR/csHR to nonparametrically compare these metrics. In general, for the non-null case, there must be at least one event type for which the sHR and csHR differ, and the proposed diagnostic will be useful to identify these cases. Furthermore, once standard methods are used to estimate the csHR, multiplying it with our nonparametric estimate for the sHR/csHR ratio will yield estimates of sHR which fulfill intrinsic linkages of the subhazards that separate analysis may violate. In addition, for non-null cases, at least one must be time dependent (i.e., non-proportional), and thus our tool serves as an indirect test of the proportionality assumption. We applied this proposed diagnostic tool to data from a cohort of children with congenital kidney disease to describe racial differences in the time to first dialysis or first transplant and extend methods to include adjustment for socioeconomic factors.

Methods of competing risks flexible parametric modeling for estimation of the risk of the first disease among HIV infected men

Background

Patients infected with the Human Immunodeficiency Virus (HIV) are susceptible to many diseases. In these patients, the occurrence of one disease alters the chance of contracting another. Under such circumstances, methods for competing risks are required. Recently, competing risks analyses in the scope of flexible parametric models have risen to address this requirement. These lesser-known analyses have considerable advantages over conventional methods.

Methods

Using data from Multi Centre AIDS Cohort Study (MACS), this paper reviews and applies methods of competing risks flexible parametric models to analyze the risk of the first disease (AIDS or non-AIDS) among HIV-infected patients. We compared two alternative subdistribution hazard flexible parametric models (SDH_FPM1 and SDH_FPM2) with the Fine & Gray model. To make a complete inference, we performed cause-specific hazard flexible parametric models for each event separately as well.

Results

Both SDH_FPM1 and SDH_FPM2 provided consistent results regarding the magnitude of coefficients and risk estimations compared with estimations obtained from the Fine & Gray model, However, competing risks flexible parametric models provided more efficient and smoother estimations for the baseline risks of the first disease. We found that age at HIV diagnosis indirectly affected the risk of AIDS as the first event by increasing the number of patients who experience a non-AIDS disease prior to AIDS among > 40 years. Other significant covariates had direct effects on the risks of AIDS and non-AIDS.

Discussion

The choice of an appropriate model depends on the research goals and computational challenges. The SDH_FPM1 models each event separately and requires calculating censoring weights which is time-consuming. In contrast, SDH_FPM2 models all events simultaneously and is more appropriate for large datasets, however, when the focus is on one particular event SDH_FPM1 is more preferable.

The impact of BMI and smoking on risk of revision following knee and hip replacement surgery: evidence from routinely collected data

OBJECTIVE

The aim of this study was to assess the association of body mass index (BMI) and smoking with risk of revision following total knee replacement (TKR) and total hip replacement (THR).

DESIGN

Primary care data, from the Clinical Practice Research Datalink (CPRD), was linked to inpatient hospital records, from Hospital Episode Statistics Admitted Patient Care (HES APC), and covered 1997 to 2014. Parametric survival models, with BMI and smoking status included as explanatory variables, were estimated for 10-year risk of revision and mortality, and were extrapolated to estimate lifetime risk of revision.

FINDINGS

TKR and THR cohorts included 10,260 and 10,961 individuals, respectively. For a change in BMI from 25 to 35, the 10-year risk of revision is expected change from 4.6% (3.3-6.4%) to 3.7% (2.6-5.1%) for TKR and 3.7% (2.8-5.1%) to 4.0% (2.8-5.7%) for THR for an otherwise average patient profile. Meanwhile, changing from a non-smoker to a current smoker is expected to change the risk of revision from 4.1% (3.1-5.5%) to 2.8% (1.7-4.7%) for TKR and from 3.8% (2.8-5.3%) to 2.9% (1.9-4.7%) for THR for an otherwise average patient profile. Estimates of lifetime risk were also similar for different values of BMI or smoking status.

CONCLUSIONS

Obesity and smoking do not appear to have a meaningful impact on the risk of revision following TKR and THR.

Missingness in the Setting of Competing Risks: from missing values to missing potential outcomes

Purpose of review

The setting of competing risks in which there is an event that precludes the event of interest from occurring is prevalent in epidemiological research. Unless studying all-cause mortality, any study following up individuals is subject to having a competing risk should individuals die during time period that the study covers. While there are prior papers discussing the need for competing risk methods in epidemiologic research, we are not aware of any review that discusses issues of missing data in a competing risk setting.

Recent Findings

We provide an overview of causal inference in competing risks as potential outcomes are missing, provide some strategies in dealing with missing (or misclassified) event type, and missing covariate data in competing risks. The strategies presented are specifically focused on those that may easily be implemented in standard statistical packages. There is ongoing work in terms of causal analyses, dealing with missing event type information, and missing covariate values specific to competing risk analyses.

Summary

Competing events are common in epidemiologic research. While there has been a focus on why one should conduct a proper competing risk analysis, a perhaps unrecognized issue is in terms of missingness. Strategies exist to minimize the impact of missingness in analyses of competing risks.

Bias Due to Confounders for the Exposure-Competing Risk Relationship

BACKGROUND

Epidemiologic studies that aim to estimate a causal effect of an exposure on a particular event of interest may be complicated by the existence of competing events that preclude the occurrence of the primary event. Recently, many articles have been published in the epidemiologic literature demonstrating the need for appropriate models to accommodate competing risks when they are present. However, there has been little attention to variable selection for confounder control in competing risk analyses.

METHODS

We employ simulation to demonstrate the bias in two variable selection strategies include covariates that are associated with the exposure and (1) which change the cause-specific hazard of any of the outcomes; or (2) which change the cause-specific hazard of the specific event of interest.

RESULTS

We demonstrated minimal to no bias in estimators adjusted for confounders of exposure and either the event of interest or the competing event, but bias of varying magnitude in almost all estimators adjusted only for confounders of exposure and the primary outcome.

DISCUSSION

When estimating causal effects for which there are competing risks, the analysis should control for confounders of both the exposure-primary outcome effect and of the exposure-competing outcome effect.

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.

Smooth semi-nonparametric (SNP) estimation of the cumulative incidence function

This paper presents a novel approach to estimation of the cumulative incidence function in the presence of competing risks. The underlying statistical model is specified via a mixture factorization of the joint distribution of the event type and the time to the event. The time to event distributions conditional on the event type are modeled using smooth semi‐nonparametric densities. One strength of this approach is that it can handle arbitrary censoring and truncation while relying on mild parametric assumptions. A stepwise forward algorithm for model estimation and adaptive selection of smooth semi‐nonparametric polynomial degrees is presented, implemented in the statistical software R, evaluated in a sequence of simulation studies, and applied to data from a clinical trial in cryptococcal meningitis. The simulations demonstrate that the proposed method frequently outperforms both parametric and nonparametric alternatives. They also support the use of ‘ad hoc’ asymptotic inference to derive confidence intervals. An extension to regression modeling is also presented, and its potential and challenges are discussed. © 2017 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.

Recursive Partitioning Method on Competing Risk Outcomes

In some cancer clinical studies, researchers have interests to explore the risk factors associated with competing risk outcomes such as recurrence-free survival. We develop a novel recursive partitioning framework on competing risk data for both prognostic and predictive model constructions. We define specific splitting rules, pruning algorithm, and final tree selection algorithm for the competing risk tree models. This methodology is quite flexible that it can corporate both semiparametric method using Cox proportional hazards model and parametric competing risk model. Both prognostic and predictive tree models are developed to adjust for potential confounding factors. Extensive simulations show that our methods have well-controlled type I error and robust power performance. Finally, we apply both Cox proportional hazards model and flexible parametric model for prognostic tree development on a retrospective clinical study on oropharyngeal cancer patients.

Are all biases missing data problems?

Estimating causal effects is a frequent goal of epidemiologic studies. Traditionally, there have been three established systematic threats to consistent estimation of causal effects. These three threats are bias due to confounders, selection, and measurement error. Confounding, selection, and measurement bias have typically been characterized as distinct types of biases. However, each of these biases can also be characterized as missing data problems that can be addressed with missing data solutions. Here we describe how the aforementioned systematic threats arise from missing data as well as review methods and their related assumptions for reducing each bias type. We also link the assumptions made by the reviewed methods to the missing completely at random (MCAR) and missing at random (MAR) assumptions made in the missing data framework that allow for valid inferences to be made based on the observed, incomplete data.

Survival analysis in hematologic malignancies: recommendations for clinicians

The widespread availability of statistical packages has undoubtedly helped hematologists worldwide in the analysis of their data, but has also led to the inappropriate use of statistical methods. In this article, we review some basic concepts of survival analysis and also make recommendations about how and when to perform each particular test using SPSS, Stata and R. In particular, we describe a simple way of defining cut-off points for continuous variables and the appropriate and inappropriate uses of the Kaplan-Meier method and Cox proportional hazard regression models. We also provide practical advice on how to check the proportional hazards assumption and briefly review the role of relative survival and multiple imputation.

Hospital-acquired Clostridium difficile infections: estimating all-cause mortality and length of stay

BACKGROUND

Clostridium difficile is a health care-associated infection of increasing importance. The purpose of this study was to estimate the time until death from any cause and time until release among patients with C. difficile, comparing the burden of those in the intensive care unit (ICU) with those in the general hospital population.

METHODS

A parametric mixture model was used to estimate event times, as well as the case-fatality ratio in ICU and non-ICU patients within a cohort of 609 adult incident cases of C. difficile in the Southeastern United States between 1 July 2009 and 31 December 2010.

RESULTS

ICU patients had twice the median time to death (relative time = 1.97 [95% confidence interval (CI) = 0.96-4.01]) and nearly twice the median time to release (1.88 [1.40-2.51]) compared with non-ICU patients. ICU patients also experienced 3.4 times the odds of mortality (95% CI = 1.8-6.2). Cause-specific competing risks analysis underestimated the relative survival time until death (0.65 [0.36-1.17]) compared with the mixture model.

CONCLUSIONS

Patients with C. difficile in the ICU experienced higher mortality and longer lengths of stay within the hospital. ICU patients with C. difficile infection represent a population in need of particular attention, both to prevent adverse patient outcomes and to minimize transmission of C. difficile to other hospitalized patients.

Competing risks mixture model for traffic incident duration prediction

Traffic incident duration is known to result from a combination of multiple factors, including covariates such as spatial and temporal characteristics, traffic conditions, and existence of secondary accidents but also the clearance method itself. In this paper, a competing risks mixture model is used to investigate the influence of clearance methods and various covariates on the duration of traffic incidents and predict traffic incident duration. The proposed mixture model considers the uncertainty in any of five clearance methods that occurred. The probability of the clearance method is specified in the mixture by using a multinomial logistic model. Three candidate distributions, namely, generalized gamma, Weibull, and log-logistic are tested to determine the most appropriate probability density function of the parametric survival analysis model. The unobserved heterogeneity is also incorporated into the mixture model in a way that allows parameters to vary across observations based on the three candidate distributions. The methods are illustrated with incident data from Singaporean expressways from January 2010 to December 2011. Regression analysis reveals that the probability of different clearance methods and the duration of traffic incidents are both significantly affected by various factors, such as traffic conditions and incident characteristics. Results show that the proposed mixture model is better than the traditional accelerated failure time model, and it predicts traffic incident duration with reasonable accuracy, as shown by the mean average percent error.

Determining the length of posttherapeutic follow-up for cancer patients using competing risks modeling

BACKGROUND/OBJECTIVE

After a curative treatment for cancer, patients enter into a posttherapeutic surveillance phase. This phase aims to detect relapses as soon as possible to improve the outcome. Mould and others predicted with a simple formula, using a parametric mixture cure model, how long early-stage breast cancer patients should be followed after treatment. However, patients in posttherapeutic surveillance phase are at risk of different events types with different responses according to their prognostic factors and different probabilities to be cured. This paper presents an adaptation of the method proposed by Mould and others, taking into account competing risks. Our loss function estimates, when follow-up is stopped at a given time, the proportion of patients who will fail after this time and who could have been treated successfully.

METHOD

We use the direct approach for cumulative incidence modeling in the presence of competing risks with an improper Gompertz probability distribution as proposed by Jeong and Fine. Prognostic factors can be taken into account, leading to a proportional hazards model. In a second step, the estimates of the Gompertz model are combined with the probability for a patient to be treated successfully in case of relapse for each event type. The method is applied to 2 examples, a numeric fictive example and a real data set on soft tissue sarcoma.

RESULTS

and

CONCLUSION

The model presented is a good tool for decision making to determine the total length of posttherapeutic surveillance. It can be applied to all cancers regardless of the localizations.

Applying competing risks regression models: an overview

In many clinical research applications the time to occurrence of one event of interest, that may be obscured by another--so called competing--event, is investigated. Specific interventions can only have an effect on the endpoint they address or research questions might focus on risk factors for a certain outcome. Different approaches for the analysis of time-to-event data in the presence of competing risks were introduced in the last decades including some new methodologies, which are not yet frequently used in the analysis of competing risks data. Cause-specific hazard regression, subdistribution hazard regression, mixture models, vertical modelling and the analysis of time-to-event data based on pseudo-observations are described in this article and are applied to a dataset of a cohort study intended to establish risk stratification for cardiac death after myocardial infarction. Data analysts are encouraged to use the appropriate methods for their specific research questions by comparing different regression approaches in the competing risks setting regarding assumptions, methodology and interpretation of the results. Notes on application of the mentioned methods using the statistical software R are presented and extensions to the presented standard methods proposed in statistical literature are mentioned.