Nonproportional hazards and unobserved heterogeneity in clustered survival data: When can we tell the difference?

Individual frailty excess hazard models in cancer epidemiology

Unobserved individual heterogeneity is a common challenge in population cancer survival studies. This heterogeneity is usually associated with the combination of model misspecification and the failure to record truly relevant variables. We investigate the effects of unobserved individual heterogeneity in the context of excess hazard models, one of the main tools in cancer epidemiology. We propose an individual excess hazard frailty model to account for individual heterogeneity. This represents an extension of frailty modeling to the relative survival framework. In order to facilitate the inference on the parameters of the proposed model, we select frailty distributions which produce closed-form expressions of the marginal hazard and survival functions. The resulting model allows for an intuitive interpretation, in which the frailties induce a selection of the healthier individuals among survivors. We model the excess hazard using a flexible parametric model with a general hazard structure which facilitates the inclusion of time-dependent effects. We illustrate the performance of the proposed methodology through a simulation study. We present a real-data example using data from lung cancer patients diagnosed in England, and discuss the impact of not accounting for unobserved heterogeneity on the estimation of net survival. The methodology is implemented in the R package IFNS.

Intravenous edaravone treatment in ALS and survival: An exploratory, retrospective, administrative claims analysis

BACKGROUND

We aimed to evaluate overall survival in US patients with amyotrophic lateral sclerosis (ALS) treated with intravenous (IV) edaravone compared with those not treated with IV edaravone in a real-world setting.

METHODS

This exploratory retrospective comparative effectiveness observational analysis included patients with ALS who were enrolled in an administrative claims database from 8 August 2017 to 31 March 2020. Propensity score matching identified IV edaravone-treated patients (cases) and non-edaravone-treated patients (controls) matched for covariates: age, race, geographic region, sex, pre-index disease duration, insurance, history of cardiovascular disease, riluzole prescription, gastrostomy tube placement, artificial nutrition, noninvasive ventilation, and all-cause hospitalisation. For cases, the index date was the date of the first claim for IV edaravone. For controls, it was the date IV edaravone was available (8 August 2017). The effect of IV edaravone on all-cause mortality was estimated with shared frailty Cox regression analysis.

FINDINGS

318 cases were matched to 318 controls. In both groups, 208 patients (65.4%) had a history of riluzole prescription. As of 31 March 2021, there were 155 deaths (48.7%) among the cases and 196 among the controls (61.6%). Median overall survival time was 29.5 months with edaravone and 23.5 months without, respectively, and the risk of death was 27% lower in cases than in controls (HR, 0.73; 95% CI, 0.59-0.91; p=0.005).

INTERPRETATION

In this real-world analysis, IV edaravone treatment in a large predominantly riluzole-treated US cohort was associated with prolonged overall survival compared with not using IV edaravone. Data from adequately powered RCTs are needed to support this finding.

FUNDING

Funded by Mitsubishi Tanabe Pharma America.

Recurrent events analysis with piece-wise exponential additive mixed models

Recurrent events analysis plays an important role in many applications, including the study of chronic diseases or recurrence of infections. Historically, many models for recurrent events have been variants of the Cox model. In this article we introduce and describe the application of the piece-wise exponential Additive Mixed Model (PAMM) for recurrent events analysis and illustrate how PAMMs can be used to flexibly model the dependencies in recurrent events data. Simulations confirm that PAMMs provide unbiased estimates as well as equivalence to the Cox model when proportional hazards are assumed. Applications to recurrence of staphylococcus aureus and malaria in children illustrate the estimation of seasonality, bivariate non-linear effects, multiple timescales and relaxation of the proportional hazards assumption via time-varying effects. The R package pammtools is extended to facilitate estimation and visualization of PAMMs for recurrent events data.

Semiparametric marginal regression for clustered competing risks data with missing cause of failure

Clustered competing risks data are commonly encountered in multicenter studies. The analysis of such data is often complicated due to informative cluster size (ICS), a situation where the outcomes under study are associated with the size of the cluster. In addition, the cause of failure is frequently incompletely observed in real-world settings. To the best of our knowledge, there is no methodology for population-averaged analysis with clustered competing risks data with an ICS and missing causes of failure. To address this problem, we consider the semiparametric marginal proportional cause-specific hazards model and propose a maximum partial pseudolikelihood estimator under a missing at random assumption. To make the latter assumption more plausible in practice, we allow for auxiliary variables that may be related to the probability of missingness. The proposed method does not impose assumptions regarding the within-cluster dependence and allows for ICS. The asymptotic properties of the proposed estimators for both regression coefficients and infinite-dimensional parameters, such as the marginal cumulative incidence functions, are rigorously established. Simulation studies show that the proposed method performs well and that methods that ignore the within-cluster dependence and the ICS lead to invalid inferences. The proposed method is applied to competing risks data from a large multicenter HIV study in sub-Saharan Africa where a significant portion of causes of failure is missing.

Comparison of statistical methods for the analysis of recurrent adverse events in the presence of non-proportional hazards and unobserved heterogeneity: a simulation study

Background

In preventive drug trials such as intermittent preventive treatment for malaria prevention during pregnancy (IPTp), where there is repeated treatment administration, recurrence of adverse events (AEs) is expected. Challenges in modelling the risk of the AEs include accounting for time-to-AE and within-patient-correlation, beyond the conventional methods. The correlation comes from two sources; (a) individual patient unobserved heterogeneity (i.e. frailty) and (b) the dependence between AEs characterised by time-dependent treatment effects. Potential AE-dependence can be modelled via time-dependent treatment effects, event-specific baseline and event-specific random effect, while heterogeneity can be modelled via subject-specific random effect. Methods that can improve the estimation of both the unobserved heterogeneity and treatment effects can be useful in understanding the evolution of risk of AEs, especially in preventive trials where time-dependent treatment effect is expected.

Methods

Using both a simulation study and the Chloroquine for Malaria in Pregnancy (NCT01443130) trial data to demonstrate the application of the models, we investigated whether the lognormal shared frailty models with restricted cubic splines and non-proportional hazards (LSF-NPH) assumption can improve estimates for both frailty variance and treatment effect compared to the conventional inverse Gaussian shared frailty model with proportional hazard (ISF-PH), in the presence of time-dependent treatment effects and unobserved patient heterogeneity. We assessed the bias, precision gain and coverage probability of 95% confidence interval of the frailty variance estimates for the models under varying known unobserved heterogeneity, sample sizes and time-dependent effects.

Results

The ISF-PH model provided a better coverage probability of 95% confidence interval, less bias and less precise frailty variance estimates compared to the LSF-NPH models. The LSF-NPH models yielded unbiased hazard ratio estimates at the expense of imprecision and high mean square error compared to the ISF-PH model.

Conclusion

The choice of the shared frailty model for the recurrent AEs analysis should be driven by the study objective. Using the LSF-NPH models is appropriate if unbiased hazard ratio estimation is of primary interest in the presence of time-dependent treatment effects. However, ISF-PH model is appropriate if unbiased frailty variance estimation is of primary interest.

Trial registration

ClinicalTrials.gov; NCT01443130

A comparison of semiparametric approaches to evaluate composite endpoints in heart failure trials

In heart failure (HF) trials efficacy is usually assessed by a composite endpoint including cardiovascular death (CVD) and heart failure hospitalizations (HFHs), which has traditionally been evaluated with a time-to-first-event analysis based on a Cox model. As a considerable fraction of events is ignored that way, methods for recurrent events were suggested, among others the semiparametric proportional rates models by Lin, Wei, Yang, and Ying (LWYY model) and Mao and Lin (Mao-Lin model). In our work we apply least false parameter theory to explain the behavior of the composite treatment effect estimates resulting from the Cox model, the LWYY model, and the Mao-Lin model in clinically relevant scenarios parameterized through joint frailty models. These account for both different treatment effects on the two outcomes (CVD, HFHs) and the positive correlation between their risk rates. For the important setting of beneficial outcome-specific treatment effects we show that the correlation results in composite treatment effect estimates, which are decreasing with trial duration. The estimate from the Cox model is affected more by the attenuation than the estimates from the recurrent event models, which both demonstrate very similar behavior. Since the Mao-Lin model turns out to be less sensitive to harmful effects on mortality, we conclude that, among the three investigated approaches, the LWYY model is the most appropriate one for the composite endpoint in HF trials. Our investigations are motivated and compared with empirical results from the PARADIGM-HF trial (ClinicalTrials.gov identifier: NCT01035255), a large multicenter trial including 8399 chronic HF patients.

Age Differences in Work-Disability Duration Across Canada: Examining Variations by Follow-Up Time and Context

Purpose This study aimed to understand age differences in wage-replacement duration by focusing on variations in the relationship across different periods of follow-up time. Methods We used administrative claims data provided by six workers' compensation systems in Canada. Included were time-loss claims for workers aged 15-80 years with a work-related injury/illness during the 2011 to 2015 period (N = 751,679 claims). Data were coded for comparability across cohorts. Survival analysis examined age-related differences in the hazard of transitioning off (versus remaining on) disability benefits, allowing for relaxed proportionality constraints on the hazard rates over time. Differences were examined on the absolute (hazard difference) and relative (hazard ratios [HR]) scales. Results Older age groups had a lower likelihood of transitioning off wage-replacement benefits compared to younger age groups in the overall models (e.g., 55-64 vs. 15-24 years: HR 0.62). However, absolute and relative differences in age-specific hazard rates varied as a function of follow-up time. The greatest age-related differences were observed at earlier event times and were attenuated towards a null difference across later follow-up event times. Conclusions Our study provides new insight into the workplace injury/illness claim and recovery processes and suggests that older age is not always strongly associated with worse disability duration outcomes. The use of data from multiple jurisdictions lends external validity to our findings and demonstrates the utility of using cross-jurisdictional data extracts. Future work should examine the social and contextual determinants that operate during various recovery phases, and how these factors interact with age.

A tutorial on frailty models

The hazard function plays a central role in survival analysis. In a homogeneous population, the distribution of the time to event, described by the hazard, is the same for each individual. Heterogeneity in the distributions can be accounted for by including covariates in a model for the hazard, for instance a proportional hazards model. In this model, individuals with the same value of the covariates will have the same distribution. It is natural to think that not all covariates that are thought to influence the distribution of the survival outcome are included in the model. This implies that there is unobserved heterogeneity; individuals with the same value of the covariates may have different distributions. One way of accounting for this unobserved heterogeneity is to include random effects in the model. In the context of hazard models for time to event outcomes, such random effects are called frailties, and the resulting models are called frailty models. In this tutorial, we study frailty models for survival outcomes. We illustrate how frailties induce selection of healthier individuals among survivors, and show how shared frailties can be used to model positively dependent survival outcomes in clustered data. The Laplace transform of the frailty distribution plays a central role in relating the hazards, conditional on the frailty, to hazards and survival functions observed in a population. Available software, mainly in R, will be discussed, and the use of frailty models is illustrated in two different applications, one on center effects and the other on recurrent events.

The role of maternal health care services as predictors of time to modern contraceptive use after childbirth in Northwest Ethiopia: Application of the shared frailty survival analysis

Introduction

The first year after birth is an ideal time to offer contraception services, as many women have many opportunities to be in contact with the health care system. Nevertheless, a large number of postpartum women in developing countries do not use the service owing to the interplay of factors operating at various stages. Therefore, this study aimed to assess predictors of modern contraceptive use in the extended postpartum period.

Methods

A community based retrospective cross-sectional study was done among 1281 women who gave birth within 12 months preceding the survey. Kaplan-Meier plots and log rank tests were used to explore the rate of modern contraceptive use. The Weibull regression survival model with multivariate frailty was employed to identify the predictors of time to contraception.

Results

Of the respondents, 59.1% (95% CI: 56.8%–62.2%) had started using modern contraceptive methods within 12 months after birth. By the second month after birth, only 11.1 percent of the women surveyed started to use a contraceptive method, which increased steadily to 25.9%, 37.7%, and 59.5% at 6, 9, and 12 months, respectively. The most preferred contraceptive method was injectable (71.5%), followed by implants (21.5%). Women’s education (aHR = 1.29; 95%CI: 1.02, 1.66), four or more antenatal care (aHR  =  1.59; 95% CI: 1.22, 2.06), early initiation of antenatal care (aHR = 2.03; 95% CI: 1.28, 3.21), and early postnatal checkup (aHR  =  1.39; 95% CI: 1.12, 1.73) were statistically significant predictors of earlier initiation of modern contraceptive methods.

Conclusions

A substantial proportion of women did not use modern contraceptive methods in the first year after birth. Maternal services were found to be the sole predictors in postpartum contraceptive use. Findings suggest the importance of linking postpartum family planning along the continuum of care. The observed heterogeneity at cluster level also urges the need of disaggregating data for decision-making.

Nonproportional hazards and unobserved heterogeneity in clustered survival data: When can we tell the difference?

Multivariate survival data are frequently encountered in biomedical applications in the form of clustered failures (or recurrent events data). A popular way of analyzing such data is by using shared frailty models, which assume that the proportional hazards assumption holds conditional on an unobserved cluster‐specific random effect. Such models are often incorporated in more complicated joint models in survival analysis.

If the random effect distribution has finite expectation, then the conditional proportional hazards assumption does not carry over to the marginal models. It has been shown that, for univariate data, this makes it impossible to distinguish between the presence of unobserved heterogeneity (eg, due to missing covariates) and marginal nonproportional hazards. We show that time‐dependent covariate effects may falsely appear as evidence in favor of a frailty model also in the case of clustered failures or recurrent events data, when the cluster size or number of recurrent events is small. When true unobserved heterogeneity is present, the presence of nonproportional hazards leads to overestimating the frailty effect. We show that this phenomenon is somewhat mitigated as the cluster size grows.

We carry out a simulation study to assess the behavior of test statistics and estimators for frailty models in such contexts. The gamma, inverse Gaussian, and positive stable shared frailty models are contrasted using a novel software implementation for estimating semiparametric shared frailty models. Two main questions are addressed in the contexts of clustered failures and recurrent events: whether covariates with a time‐dependent effect may appear as indication of unobserved heterogeneity and whether the additional presence of unobserved heterogeneity can be detected in this case. Finally, the practical implications are illustrated in a real‐world data analysis example.