Bayesian models for subgroup analysis in clinical trials

Effects of IV fluid restriction according to site-specific intensity of standard fluid treatment-protocol

BACKGROUND

Variation in usual practice in fluid trials assessing lower versus higher volumes may affect overall comparisons. To address this, we will evaluate the effects of heterogeneity in treatment intensity in the Conservative versus Liberal Approach to Fluid Therapy of Septic Shock in Intensive Care trial. This will reflect the effects of differences in site-specific intensities of standard fluid treatment due to local practice preferences while considering participant characteristics.

METHODS

We will assess the effects of heterogeneity in treatment intensity across one primary (all-cause mortality) and three secondary outcomes (serious adverse events or reactions, days alive without life support and days alive out of hospital) after 90 days. We will classify sites based on the site-specific intensity of standard fluid treatment, defined as the mean differences in observed versus predicted intravenous fluid volumes in the first 24 h in the standard-fluid group while accounting for differences in participant characteristics. Predictions will be made using a machine learning model including 22 baseline predictors using the extreme gradient boosting algorithm. Subsequently, sites will be grouped into fluid treatment intensity subgroups containing at least 100 participants each. Subgroups differences will be assessed using hierarchical Bayesian regression models with weakly informative priors. We will present the full posterior distributions of relative (risk ratios and ratios of means) and absolute differences (risk differences and mean differences) in each subgroup.

DISCUSSION

This study will provide data on the effects of heterogeneity in treatment intensity while accounting for patient characteristics in critically ill adult patients with septic shock.

REGISTRATIONS

The European Clinical Trials Database (EudraCT): 2018-000404-42, ClinicalTrials. gov: NCT03668236.

Modern approaches for evaluating treatment effect heterogeneity from clinical trials and observational data

In this paper, we review recent advances in statistical methods for the evaluation of the heterogeneity of treatment effects (HTE), including subgroup identification and estimation of individualized treatment regimens, from randomized clinical trials and observational studies. We identify several types of approaches using the features introduced in Lipkovich et al (Stat Med 2017;36: 136-196) that distinguish the recommended principled methods from basic methods for HTE evaluation that typically rely on rules of thumb and general guidelines (the methods are often referred to as common practices). We discuss the advantages and disadvantages of various principled methods as well as common measures for evaluating their performance. We use simulated data and a case study based on a historical clinical trial to illustrate several new approaches to HTE evaluation.

Bayesian Hierarchical Models for Subgroup Analysis

In conventional subgroup analyses, subgroup treatment effects are estimated using data from each subgroup separately without considering data from other subgroups in the same study. The subgroup treatment effects estimated this way may be heterogenous with high variability due to small sample sizes in some subgroups and much different from the treatment effect in the overall population. A Bayesian hierarchical model (BHM) can be used to derive more precise, and less heterogenous estimates of subgroup treatment effects that are closer to the treatment effect in the overall population. BHM assumes exchangeability in treatment effect across subgroups after adjusting for effect modifiers and other relevant covariates. In this article, we will discuss the technical details for applying one-way and multi-way BHM using summary-level statistics, and patient-level data for subgroup analysis. Four case studies based on four new drug applications are used to illustrate the application of these models in subgroup analyses for continuous, dichotomous, time-to-event, and count endpoints.

Bayesian Multivariate Logistic Regression for Superiority and Inferiority Decision-Making under Observable Treatment Heterogeneity

The effects of treatments may differ between persons with different characteristics. Addressing such treatment heterogeneity is crucial to investigate whether patients with specific characteristics are likely to benefit from a new treatment. The current paper presents a novel Bayesian method for superiority decision-making in the context of randomized controlled trials with multivariate binary responses and heterogeneous treatment effects. The framework is based on three elements: a) Bayesian multivariate logistic regression analysis with a Pólya-Gamma expansion; b) a transformation procedure to transfer obtained regression coefficients to a more intuitive multivariate probability scale (i.e., success probabilities and the differences between them); and c) a compatible decision procedure for treatment comparison with prespecified decision error rates. Procedures for a priori sample size estimation under a non-informative prior distribution are included. A numerical evaluation demonstrated that decisions based on a priori sample size estimation resulted in anticipated error rates among the trial population as well as subpopulations. Further, average and conditional treatment effect parameters could be estimated unbiasedly when the sample was large enough. Illustration with the International Stroke Trial dataset revealed a trend toward heterogeneous effects among stroke patients: Something that would have remained undetected when analyses were limited to average treatment effects.

Using Bayesian statistics in confirmatory clinical trials in the regulatory setting: a tutorial review

Bayesian statistics plays a pivotal role in advancing medical science by enabling healthcare companies, regulators, and stakeholders to assess the safety and efficacy of new treatments, interventions, and medical procedures. The Bayesian framework offers a unique advantage over the classical framework, especially when incorporating prior information into a new trial with quality external data, such as historical data or another source of co-data. In recent years, there has been a significant increase in regulatory submissions using Bayesian statistics due to its flexibility and ability to provide valuable insights for decision-making, addressing the modern complexity of clinical trials where frequentist trials are inadequate. For regulatory submissions, companies often need to consider the frequentist operating characteristics of the Bayesian analysis strategy, regardless of the design complexity. In particular, the focus is on the frequentist type I error rate and power for all realistic alternatives. This tutorial review aims to provide a comprehensive overview of the use of Bayesian statistics in sample size determination, control of type I error rate, multiplicity adjustments, external data borrowing, etc., in the regulatory environment of clinical trials. Fundamental concepts of Bayesian sample size determination and illustrative examples are provided to serve as a valuable resource for researchers, clinicians, and statisticians seeking to develop more complex and innovative designs.

Differential Treatment Effects of Subgroup Analyses in Phase 3 Oncology Trials From 2004 to 2020

Importance

Subgroup analyses are often performed in oncology to investigate differential treatment effects and may even constitute the basis for regulatory approvals. Current understanding of the features, results, and quality of subgroup analyses is limited.

Objective

To evaluate forest plot interpretability and credibility of differential treatment effect claims among oncology trials.

Design, Setting, and Participants

This cross-sectional study included randomized phase 3 clinical oncology trials published prior to 2021. Trials were screened from ClinicalTrials.gov.

Main Outcomes and Measures

Missing visual elements in forest plots were defined as a missing point estimate or use of a linear x-axis scale for hazard and odds ratios. Multiplicity of testing control was recorded. Differential treatment effect claims were rated using the Instrument for Assessing the Credibility of Effect Modification Analyses. Linear and logistic regressions evaluated associations with outcomes.

Results

Among 785 trials, 379 studies (48%) enrolling 331 653 patients reported a subgroup analysis. The forest plots of 43% of trials (156 of 363) were missing visual elements impeding interpretability. While 4148 subgroup effects were evaluated, only 1 trial (0.3%) controlled for multiple testing. On average, trials that did not meet the primary end point conducted 2 more subgroup effect tests compared with trials meeting the primary end point (95% CI, 0.59-3.43 tests; P = .006). A total of 101 differential treatment effects were claimed across 15% of trials (55 of 379). Interaction testing was missing in 53% of trials (29 of 55) claiming differential treatment effects. Trials not meeting the primary end point were associated with greater odds of no interaction testing (odds ratio, 4.47; 95% CI, 1.42-15.55, P = .01). The credibility of differential treatment effect claims was rated as low or very low in 93% of cases (94 of 101).

Conclusions and Relevance

In this cross-sectional study of phase 3 oncology trials, nearly half of trials presented a subgroup analysis in their primary publication. However, forest plots of these subgroup analyses largely lacked essential features for interpretation, and most differential treatment effect claims were not supported. Oncology subgroup analyses should be interpreted with caution, and improvements to the quality of subgroup analyses are needed.

Comparing Bayesian hierarchical meta-regression methods and evaluating the influence of priors for evaluations of surrogate endpoints on heterogeneous collections of clinical trials

BACKGROUND

Surrogate endpoints, such as those of interest in chronic kidney disease (CKD), are often evaluated using Bayesian meta-regression. Trials used for the analysis can evaluate a variety of interventions for different sub-classifications of disease, which can introduce two additional goals in the analysis. The first is to infer the quality of the surrogate within specific trial subgroups defined by disease or intervention classes. The second is to generate more targeted subgroup-specific predictions of treatment effects on the clinical endpoint.

METHODS

Using real data from a collection of CKD trials and a simulation study, we contrasted surrogate endpoint evaluations under different hierarchical Bayesian approaches. Each approach we considered induces different assumptions regarding the relatedness (exchangeability) of trials within and between subgroups. These include partial-pooling approaches, which allow subgroup-specific meta-regressions and, yet, facilitate data adaptive information sharing across subgroups to potentially improve inferential precision. Because partial-pooling models come with additional parameters relative to a standard approach assuming one meta-regression for the entire set of studies, we performed analyses to understand the impact of the parameterization and priors with the overall goals of comparing precision in estimates of subgroup-specific meta-regression parameters and predictive performance.

RESULTS

In the analyses considered, partial-pooling approaches to surrogate endpoint evaluation improved accuracy of estimation of subgroup-specific meta-regression parameters relative to fitting separate models within subgroups. A random rather than fixed effects approach led to reduced bias in estimation of meta-regression parameters and in prediction in subgroups where the surrogate was strong. Finally, we found that subgroup-specific meta-regression posteriors were robust to use of constrained priors under the partial-pooling approach, and that use of constrained priors could facilitate more precise prediction for clinical effects in trials of a subgroup not available for the initial surrogacy evaluation.

CONCLUSION

Partial-pooling modeling strategies should be considered for surrogate endpoint evaluation on collections of heterogeneous studies. Fitting these models comes with additional complexity related to choosing priors. Constrained priors should be considered when using partial-pooling models when the goal is to predict the treatment effect on the clinical endpoint.

Quantifying and reducing inequity in average treatment effect estimation

BACKGROUND

Across studies of average treatment effects, some population subgroups consistently have lower representation than others which can lead to discrepancies in how well results generalize.

METHODS

We develop a framework for quantifying inequity due to systemic disparities in sample representation and a method for mitigation during data analysis. Assuming subgroup treatment effects are exchangeable, an unbiased sample average treatment effect estimator will have lower mean-squared error, on average across studies, for subgroups with less representation when treatment effects vary. We present a method for estimating average treatment effects in representation-adjusted samples which enables subgroups to optimally leverage information from the full sample rather than only their own subgroup's data. Two approaches for specifying representation adjustment are offered-one minimizes average mean-squared error for each subgroup separately and the other balances minimization of mean-squared error and equal representation. We conduct simulation studies to compare the performance of the proposed estimators to several subgroup-specific estimators.

RESULTS

We find that the proposed estimators generally provide lower mean squared error, particularly for smaller subgroups, relative to the other estimators. As a case study, we apply this method to a subgroup analysis from a published study.

CONCLUSIONS

We recommend the use of the proposed estimators to mitigate the impact of disparities in representation, though structural change is ultimately needed.

Change plane model averaging for subgroup identification

Central to personalized medicine and tailored therapies is discovering the subpopulations that account for treatment effect heterogeneity and are likely to benefit more from given interventions. In this article, we introduce a change plane model averaging method to identify subgroups characterized by linear combinations of predictive variables and multiple cut-offs. We first fit a sequence of statistical models, each incorporating the thresholding effect of one particular covariate. The estimation of submodels is accomplished through an iterative integration of a change point detection method and numerical optimization algorithms. A frequentist model averaging approach is then employed to linearly combine the submodels with optimal weights. Our approach can deal with high-dimensional settings involving enormous potential grouping variables by adopting the sparsity-inducing penalties. Simulation studies are conducted to investigate the prediction and subgrouping performance of the proposed method, with a comparison to various competing subgroup detection methods. Our method is applied to a dataset from a warfarin pharmacogenetics study, producing some new findings.

Comprehensive analysis and prediction of long-term durability of factor IX activity following etranacogene dezaparvovec gene therapy in the treatment of hemophilia B

OBJECTIVE

Congenital hemophilia B is a rare bleeding disorder caused by defects in the gene encoding factor IX (FIX) leading to coagulation deficiency. Recurrent bleeds may cause chronic pain, disability, and reduced quality of life. Phase 2 b and 3 single-arm, open-label, single-dose trials assessing etranacogene dezaparvovec gene therapy for hemophilia B have demonstrated sustained FIX activity levels over observed periods, but long-term durability of the treatment effect has not been established. Using statistical modeling, we estimate long-term durability of FIX activity levels after receiving etranacogene dezaparvovec.

METHODS

Participants from Phase 2 b (N = 3; NCT03489291) and 3 studies (N = 52; NCT03569891) were included. Two participants who did not respond to treatment were excluded. FIX activity was assessed by one-stage activated partial thromboplastin time assay. FIX activity levels at Month 6 post-treatment were considered baseline. Bayesian and Frequentist linear mixed models predicted FIX activity levels up to 25.5 years at an individual and population level with pre-treatment adeno-associated virus 5 (AAV5) neutralizing antibody (NAb) status as primary covariate.

RESULTS

Bayesian and Frequentist linear mixed models predicted no more than 6/55 (10.91%) observed participants would have FIX activity levels <2% up to 25.5 years post-infusion. Bayesian model-based predictions of future participants suggest >80% would be free from prophylactic FIX replacement products 25.5 years post-infusion. Both models predicted FIX activity levels were not significantly influenced by pre-treatment AAV5 NAb status.

CONCLUSIONS

People with hemophilia B receiving etranacogene dezaparvovec would likely achieve durable FIX activity levels and remain free of prophylactic FIX replacement products for up to 25.5 years following single administration. The long-term factor IX durability predictions are based on statistical methods and results in vivo may differ.

Bayesian sparse modeling to identify high-risk subgroups in meta-analysis of safety data

Meta-analysis allows researchers to combine evidence from multiple studies, making it a powerful tool for synthesizing information on the safety profiles of new medical interventions. There is a critical need to identify subgroups at high risk of experiencing treatment-related toxicities. However, this remains quite challenging from a statistical perspective as there are a variety of clinical risk factors that may be relevant for different types of adverse events, and adverse events of interest may be rare or incompletely reported. We frame this challenge as a variable selection problem and propose a Bayesian hierarchical model which incorporates a horseshoe prior on the interaction terms to identify high-risk groups. Our proposed model is motivated by a meta-analysis of adverse events in cancer immunotherapy, and our results uncover key factors driving the risk of specific types of treatment-related adverse events.

Addressing patient heterogeneity in disease predictive model development

This paper addresses patient heterogeneity associated with prediction problems in biomedical applications. We propose a systematic hypothesis testing approach to determine the existence of patient subgroup structure and the number of subgroups in patient population if subgroups exist. A mixture of generalized linear models is considered to model the relationship between the disease outcome and patient characteristics and clinical factors, including targeted biomarker profiles. We construct a test statistic based on expectation maximization (EM) algorithm and derive its asymptotic distribution under the null hypothesis. An important computational advantage of the test is that the involved parameter estimates under the complex alternative hypothesis can be obtained through a small number of EM iterations, rather than optimizing the objective function. We demonstrate the finite sample performance of the proposed test in terms of type-I error rate and power, using extensive simulation studies. The applicability of the proposed method is illustrated through an application to a multicenter prostate cancer study.

Conservative vs. liberal fluid therapy in septic shock - Protocol for secondary Bayesian analyses of the CLASSIC trial

BACKGROUND

Clinical equipoise exists regarding intravenous (IV) fluid volumes in sepsis. The Conservative vs. Liberal Approach to fluid therapy of Septic Shock in Intensive Care (CLASSIC) trial investigates the effect of restricted vs. standard IV fluid therapy in 1554 adult intensive care unit patients with septic shock.

METHODS

This protocol describes secondary Bayesian analyses of the primary outcome (90-day all-cause mortality) and three secondary outcomes at day 90. We will analyse all binary outcomes with adjusted Bayesian logistic regressions and present results as conditional relative risks and risk differences with 95% credibility intervals (CrIs). The secondary count outcome will be analysed using adjusted Bayesian linear regression with results summarised as conditional mean differences and ratios of means with 95% Crls. We will use weakly informative priors for the primary analyses, and sceptical and evidence-based priors in the sensitivity analyses. Exact probabilities will be presented for any benefit/harm, clinically important benefit/harm and no clinically important difference. We will assess whether heterogeneity of treatment effects on mortality is present using Bayesian hierarchical models in subgroups and on the continuous scale using models with interactions according to five baseline variables assessing the overall severity of illness and the degree of circulatory and renal impairment.

DISCUSSION

The outlined analyses will supplement the primary analysis of the CLASSIC trial by describing probabilities of beneficial and harmful effects and evaluating heterogeneity of treatment effects in a framework that may be easier to interpret for researchers and clinicians.

Borrowing information across patient subgroups in clinical trials, with application to a paediatric trial

Background

Clinical trial investigators may need to evaluate treatment effects in a specific subgroup (or subgroups) of participants in addition to reporting results of the entire study population. Such subgroups lack power to detect a treatment effect, but there may be strong justification for borrowing information from a larger patient group within the same trial, while allowing for differences between populations. Our aim was to develop methods for eliciting expert opinions about differences in treatment effect between patient populations, and to incorporate these opinions into a Bayesian analysis.

Methods

We used an interaction parameter to model the relationship between underlying treatment effects in two subgroups. Elicitation was used to obtain clinical opinions on the likely values of the interaction parameter, since this parameter is poorly informed by the data. Feedback was provided to experts to communicate how uncertainty about the interaction parameter corresponds with relative weights allocated to subgroups in the Bayesian analysis. The impact on the planned analysis was then determined.

Results

The methods were applied to an ongoing non-inferiority trial designed to compare antiretroviral therapy regimens in 707 children living with HIV and weighing ≥ 14 kg, with an additional group of 85 younger children weighing < 14 kg in whom the treatment effect will be estimated separately. Expert clinical opinion was elicited and demonstrated that substantial borrowing is supported. Clinical experts chose on average to allocate a relative weight of 78% (reduced from 90% based on sample size) to data from children weighing ≥ 14 kg in a Bayesian analysis of the children weighing < 14 kg. The total effective sample size in the Bayesian analysis was 386 children, providing 84% predictive power to exclude a difference of more than 10% between arms, whereas the 85 younger children weighing < 14 kg provided only 20% power in a standalone frequentist analysis.

Conclusions

Borrowing information from a larger subgroup or subgroups can facilitate estimation of treatment effects in small subgroups within a clinical trial, leading to improved power and precision. Informative prior distributions for interaction parameters are required to inform the degree of borrowing and can be informed by expert opinion. We demonstrated accessible methods for obtaining opinions.

Survival with Cemiplimab in Recurrent Cervical Cancer

BACKGROUND

Patients with recurrent cervical cancer have a poor prognosis. Cemiplimab, the fully human programmed cell death 1 (PD-1)-blocking antibody approved to treat lung and skin cancers, has been shown to have preliminary clinical activity in this population.

METHODS

In this phase 3 trial, we enrolled patients who had disease progression after first-line platinum-containing chemotherapy, regardless of their programmed cell death ligand 1 (PD-L1) status. Women were randomly assigned (1:1) to receive cemiplimab (350 mg every 3 weeks) or the investigator's choice of single-agent chemotherapy. The primary end point was overall survival. Progression-free survival and safety were also assessed.

RESULTS

A total of 608 women were enrolled (304 in each group). In the overall trial population, median overall survival was longer in the cemiplimab group than in the chemotherapy group (12.0 months vs. 8.5 months; hazard ratio for death, 0.69; 95% confidence interval [CI], 0.56 to 0.84; two-sided P<0.001). The overall survival benefit was consistent in both histologic subgroups (squamous-cell carcinoma and adenocarcinoma [including adenosquamous carcinoma]). Progression-free survival was also longer in the cemiplimab group than in the chemotherapy group in the overall population (hazard ratio for disease progression or death, 0.75; 95% CI, 0.63 to 0.89; two-sided P<0.001). In the overall population, an objective response occurred in 16.4% (95% CI, 12.5 to 21.1) of the patients in the cemiplimab group, as compared with 6.3% (95% CI, 3.8 to 9.6) in the chemotherapy group. An objective response occurred in 18% (95% CI, 11 to 28) of the cemiplimab-treated patients with PD-L1 expression greater than or equal to 1% and in 11% (95% CI, 4 to 25) of those with PD-L1 expression of less than 1%. Overall, grade 3 or higher adverse events occurred in 45.0% of the patients who received cemiplimab and in 53.4% of those who received chemotherapy.

CONCLUSIONS

Survival was significantly longer with cemiplimab than with single-agent chemotherapy among patients with recurrent cervical cancer after first-line platinum-containing chemotherapy. (Funded by Regeneron Pharmaceuticals and Sanofi; EMPOWER-Cervical 1/GOG-3016/ENGOT-cx9 ClinicalTrials.gov number, NCT03257267.).

A shrinkage estimator for subgroup analysis without the exchangeability assumption

Shrinkage estimators for exploratory subgroup analyses are intuitively appealing and can greatly improve estimation over standard analysis approaches; however, adoption of these estimators has been limited by reliance on the exchangeability assumption. This paper describes a new shrinkage estimator that does not rely on this assumption. Rather than assuming that treatment effect sizes within subgroups are randomly distributed around an overall mean, this new estimator assumes that the difference between the effect sizes in any given pair of subgroups is randomly distributed around zero. The estimator is illustrated using data from a clinical trial in which the treatment effect size in one region was substantially different from the sizes in other regions. Simulation results show that the estimator has properties that are comparable to or superior to a standard shrinkage estimator when exchangeability is assumed, while allowing the flexibility to handle situations where exchangeability cannot be assumed.

Outcomes of Participants With Diabetes in the ISCHEMIA Trials

BACKGROUND

Among patients with diabetes and chronic coronary disease, it is unclear if invasive management improves outcomes when added to medical therapy.

METHODS

The ISCHEMIA (International Study of Comparative Health Effectiveness with Medical and Invasive Approaches) trials (ie, ISCHEMIA and ISCHEMIA-Chronic Kidney Disease) randomized chronic coronary disease patients to an invasive (medical therapy + angiography and revascularization if feasible) or a conservative approach (medical therapy alone with revascularization if medical therapy failed). Cohorts were combined after no trial-specific effects were observed. Diabetes was defined by history, hemoglobin A1c ≥6.5%, or use of glucose-lowering medication. The primary outcome was all-cause death or myocardial infarction (MI). Heterogeneity of effect of invasive management on death or MI was evaluated using a Bayesian approach to protect against random high or low estimates of treatment effect for patients with versus without diabetes and for diabetes subgroups of clinical (female sex and insulin use) and anatomic features (coronary artery disease severity or left ventricular function).

RESULTS

Of 5900 participants with complete baseline data, the median age was 64 years (interquartile range, 57-70), 24% were female, and the median estimated glomerular filtration was 80 mL·min-1·1.73-2 (interquartile range, 64-95). Among the 2553 (43%) of participants with diabetes, the median percent hemoglobin A1c was 7% (interquartile range, 7-8), and 30% were insulin-treated. Participants with diabetes had a 49% increased hazard of death or MI (hazard ratio, 1.49 [95% CI, 1.31-1.70]; P<0.001). At median 3.1-year follow-up the adjusted event-free survival was 0.54 (95% bootstrapped CI, 0.48-0.60) and 0.66 (95% bootstrapped CI, 0.61-0.71) for patients with diabetes versus without diabetes, respectively, with a 12% (95% bootstrapped CI, 4%-20%) absolute decrease in event-free survival among participants with diabetes. Female and male patients with insulin-treated diabetes had an adjusted event-free survival of 0.52 (95% bootstrapped CI, 0.42-0.56) and 0.49 (95% bootstrapped CI, 0.42-0.56), respectively. There was no difference in death or MI between strategies for patients with diabetes versus without diabetes, or for clinical (female sex or insulin use) or anatomic features (coronary artery disease severity or left ventricular function) of patients with diabetes.

CONCLUSIONS

Despite higher risk for death or MI, chronic coronary disease patients with diabetes did not derive incremental benefit from routine invasive management compared with initial medical therapy alone. Registration: URL: https://www.clinicaltrials.gov; Unique identifier: NCT01471522.

Set-regression with applications to subgroup analysis

Regression is a commonly used statistical model. It is the conditional mean of the response given covariates μ ( x ) = E ( Y | X = x ) . However, in some practical problems, the interest is the conditional mean of the response given the covariates belonging to some set A. Notably, in precision medicine and subgroup analysis in clinical trials, the aim is to identify subjects who benefit the most from the treatment, or identify an optimal set in the covariate space which manifests treatment favoritism if a subject's covariates fall in this set and the subject is classified to the favorable treatment subgroup. Existing methods for subgroup analysis achieve this indirectly by using classical regression. This motivates us to develop a new type of regression: set-regression, defined as μ ( A ) = E ( Y | X ∈ A ) which directly addresses the subgroup analysis problem. This extends not only the classical regression model but also improves recursive partitioning and support vector machine approaches, and is particularly suitable for objectives involving optimization of the regression over sets, such as subgroup analysis. We show that the new versatile set-regression identifies the subgroup with increased accuracy. It is easy to use. Simulation studies also show superior performance of the proposed method in finite samples.

Improving the Estimation of Subgroup Effects for Clinical Trial Participants with Multimorbidity by Incorporating Drug Class-Level Information in Bayesian Hierarchical Models: A Simulation Study

Background

There is limited guidance for using common drug therapies in the context of multimorbidity. In part, this is because their effectiveness for patients with specific comorbidities cannot easily be established using subgroup analyses in clinical trials. Here, we use simulations to explore the feasibility and implications of concurrently estimating effects of related drug treatments in patients with multimorbidity by partially pooling subgroup efficacy estimates across trials.

Methods

We performed simulations based on the characteristics of 161 real clinical trials of noninsulin glucose-lowering drugs for diabetes, estimating subgroup effects for patients with a hypothetical comorbidity across related trials in different scenarios using Bayesian hierarchical generalized linear models. We structured models according to an established ontology—the World Health Organization Anatomic Chemical Therapeutic Classifications—allowing us to nest all trials within drugs and all drugs within anatomic chemical therapeutic classes, with effects partially pooled at each level of the hierarchy. In a range of scenarios, we compared the performance of this model to random effects meta-analyses of all drugs individually.

Results

Hierarchical, ontology-based Bayesian models were unbiased and accurately recovered simulated comorbidity-drug interactions. Compared with single-drug meta-analyses, they offered a relative increase in precision of up to 250% in some scenarios because of information sharing across the hierarchy. Because of the relative precision of the approaches, a large proportion of small subgroup effects was detectable only using the hierarchical model.

Conclusions

By assuming that similar drugs may have similar subgroup effects, Bayesian hierarchical models based on structures defined by existing ontologies can be used to improve the precision of treatment efficacy estimates in patients with multimorbidity, with potential implications for clinical decision making.

Is convalescent plasma futile in COVID-19? A Bayesian re-analysis of the RECOVERY randomized controlled trial

BACKGROUND

Randomized trials are generally performed from a frequentist perspective, which can conflate absence of evidence with evidence of absence. The RECOVERY trial evaluated convalescent plasma for patients hospitalized with coronavirus disease 2019 (COVID-19) and concluded that there was no evidence of an effect. Re-analysis from a Bayesian perspective is warranted.

METHODS

Outcome data were extracted from the RECOVERY trial by serostatus and time of presentation. A Bayesian re-analysis with a wide variety of priors (vague, optimistic, sceptical, and pessimistic) was performed, calculating the posterior probability for: any benefit, an absolute risk difference of 0.5% (small benefit, number needed to treat 200), and an absolute risk difference of one percentage point (modest benefit, number needed to treat 100).

RESULTS

Across all patients, when analysed with a vague prior, the likelihood of any benefit or a modest benefit with convalescent plasma was estimated to be 64% and 18%, respectively. The estimated chance of any benefit was 95% if presenting within 7 days of symptoms, or 17% if presenting after this. In patients without a detectable antibody response at presentation, the chance of any benefit was 85%. However, it was only 20% in patients with a detectable antibody response at presentation.

CONCLUSIONS

Bayesian re-analysis suggests that convalescent plasma reduces mortality by at least one percentage point among the 39% of patients who present within 7 days of symptoms, and that there is a 67% chance of the same mortality reduction in the 38% who are seronegative at the time of presentation. This is in contrast to the results in people who already have antibodies when they present. This biologically plausible finding bears witness to the advantage of Bayesian analyses over misuse of hypothesis tests to inform decisions.

Study specific prediction intervals for random-effects meta-analysis: A tutorial: Prediction intervals in meta-analysis

The pooled estimate of the average effect is of primary interest when fitting the random-effects model for meta-analysis. However estimates of study specific effects, for example those displayed on forest plots, are also often of interest. In this tutorial, we present the case, with the accompanying statistical theory, for estimating the study specific true effects using so called 'empirical Bayes estimates' or 'Best Unbiased Linear Predictions' under the random-effects model. These estimates can be accompanied by prediction intervals that indicate a plausible range of study specific true effects. We coalesce and elucidate the available literature and illustrate the methodology using two published meta-analyses as examples. We also perform a simulation study that reveals that coverage probability of study specific prediction intervals are substantially too low if the between-study variance is small but not negligible. Researchers need to be aware of this defect when interpreting prediction intervals. We also show how empirical Bayes estimates, accompanied with study specific prediction intervals, can embellish forest plots. We hope that this tutorial will serve to provide a clear theoretical underpinning for this methodology and encourage its widespread adoption.

Subgroup Analyses in Oncology Trials: Regulatory Considerations and Case Examples

Subgroup analyses are assessments of treatment effects based on certain patient characteristics out of the total study population and are important for interpretation of pivotal oncology trials. However, appropriate use of subgroup analyses results for regulatory decision-making and product labeling is challenging. Typically, drugs approved by the FDA are indicated for use in the total patient population studied; however, there are examples of restriction to a subgroup of patients despite positive study results in the entire study population and also extension of an indication to the entire study population despite positive results appearing primarily in one or more subgroups. In this article, we summarize key issues related to subgroup analyses in the benefit-risk assessment of cancer drugs and provide case examples to illustrate approaches that the FDA Oncology Center of Excellence has taken when considering the appropriate patient population for cancer drug approval. In general, if a subgroup is of interest, the subgroup analysis should be hypothesis-driven and have adequate sample size to demonstrate evidence of a treatment effect. In addition to statistical efficacy considerations, the decision on what subgroups to include in labeling relies on the pathophysiology of the disease, mechanistic justification, safety data, and external information available. The oncology drug review takes the totality of the data into consideration during the decision-making process to ensure the indication granted and product labeling appropriately reflect the scientific evidence to support patient population for whom the drug is safe and effective.

A Bayesian Decision-Theoretic Design for Simultaneous Biomarker-Based Subgroup Selection and Efficacy Evaluation

The success of drug development of targeted therapy often hinges on an appropriate selection of the sensitive patient population, mostly based on patients' biomarker levels. At the planning stage of a phase II study, although a potential biomarker may have been identified, a threshold value for defining sensitive patient population is often unavailable for adopting many existing biomarker-guided designs. To address this issue, we propose a two-stage design that allows for simultaneous biomarker threshold selection and efficacy evaluation while accommodating situations where the drug is efficacious in the entire patient population. The design uses a Bayesian decision-theoretic approach and incorporates the benefit and cost considerations of the study into a utility function. The operating characteristics of the proposed design under different scenarios are investigated via simulations. We also provide a discussion on the choice of the benefit and cost parameters in practice.

Bayesian Decision-Theoretic Methods for Survival Data using Stochastic Optimization

We introduce a principled method for Bayesian subgroup analysis. The approach is based on casting subgroup analysis as a Bayesian decision problem. The two main innovations are: (1) the explicit consideration of a "subgroup report," comprising multiple subpopulations; and (2) adapting an inhomogeneous Markov chain Monte Carlo simulation scheme to implement stochastic optimization. The latter makes the search for "subgroup reports" practically feasible.

Rivaroxaban and Aspirin in Patients With Symptomatic Lower Extremity Peripheral Artery Disease: A Subanalysis of the COMPASS Randomized Clinical Trial

Importance

Patients with symptomatic lower extremity peripheral artery disease (LE-PAD) experience an increased risk of major vascular events. There is limited information on what clinical features of symptomatic LE-PAD prognosticate major vascular events and whether patients at high risk have a greater absolute benefit from low-dose rivaroxaban and aspirin.

Objective

To quantify the risk of major vascular events and investigate the response to treatment with low-dose rivaroxaban and aspirin among patients with symptomatic LE-PAD based on clinical presentation and comorbidities.

Design, Setting, and Participants

This is a subanalysis of a previously reported subgroup of patients with symptomatic LE-PAD who were enrolled in a large, double-blind, placebo-controlled randomized clinical trial (Cardiovascular Outcomes for People Using Anticoagulation Strategies [COMPASS]) in 602 centers in 33 countries from March 2013 to January 2020. Data analysis was completed from May 2016 to June 2020.

Interventions

A combination of low-dose rivaroxaban and aspirin compared with aspirin alone.

Main Outcomes and Measures

Thirty-month incidence risk of myocardial infarction, stroke and cardiovascular death (MACE), major adverse limb events (MALE) including major vascular amputation, and bleeding.

Results

The COMPASS trial enrolled 4129 patients with symptomatic LE-PAD (mean [SD] age, 66.8 [8.8] years; 2932 men [71.0%]). The 30-month Kaplan-Meier incidence risk of MACE or MALE, including major amputation, was 22.6% in those with prior amputation (this outcome was observed in 54 patients), 17.6% (n = 15) in those with Fontaine III or IV symptoms, and 11.8% (n = 142) in those with previous peripheral artery revascularization, classifying these features as high-risk limb presentations. The 30-month incidence risk of MACE or MALE, including major amputation, was 14.1% (n = 118) in those with kidney dysfunction, 13.5% (n = 67) in those with heart failure, 13.4% (n = 199) in those with diabetes, and 12.8% (n = 222) in those with polyvascular disease, classifying these features as high-risk comorbidities. Among patients with either high-risk limb presentations or high-risk comorbidities, treatment with rivaroxaban and aspirin compared with aspirin alone was associated with an estimated 4.2% (95% CI, 1.9%-6.2%) absolute risk reduction for MACE or MALE, including major amputation, at 30 months. Although the estimated absolute risk increase of major bleeding was higher with rivaroxaban and aspirin in combination than aspirin alone (2.0% [95% CI, 0.5%-3.9%]) for patients with either high-risk limb presentation or high-risk comorbidity, the estimated absolute risk increase of fatal or critical organ bleeding was low in this high-risk group (0.4% [95% CI, 0.2%-1.8%]), such that the net clinical benefit was estimated to be 3.2% (95% CI, 0.6%-5.3%).

Conclusions and Relevance

Patients with LE-PAD with high-risk limb presentations or high-risk comorbidities had a high incidence of major vascular events. For these patients, treatment with rivaroxaban and aspirin in combination compared with aspirin alone led to a large absolute reduction in vascular risk.

Flexible Bayesian subgroup analysis in early and confirmatory trials

Subgroup analysis is one of the most important issues in clinical trials. In confirmatory trials, it is critical to investigate consistency of the treatment effect across subgroups, which could potentially result in incorrect scientific conclusion or regulatory decision. There are many challenges and methodological complications of interpreting subgroup results beyond the regulatory setting. For the early phase or proof of concept trials, particularly in basket trials, it is also important to have reliable estimation of subgroup treatment effect in order to guide the next phase go/no-go decision making when large biases can be introduced due to small sample size or random variability. In this paper, we review several recent methods that have been proposed for subgroup analysis in the Bayesian framework to correct for bias. We present simulation results from applying various novel Bayesian hierarchical models for subgroup analysis to a phase II basket trial. For different scenarios considered, we compare the average total sample size, and frequentist-like operating characteristics of power and familywise type I error rate. We compare the precision of the model estimates of the treatment effect by assessing average relative bias and the width of the 95% credible interval for the bias. We also demonstrate flexible Bayesian hierarchical models in a case study of a phase III oncology trial for subgroup treatment effect estimation to help with regulatory decision making. Finally, we conclude our findings in the discussion section and give recommendations on how these methods could be implemented in confirmatory and early phase clinical trials.

A semiparametric Bayesian approach to population finding with time-to-event and toxicity data in a randomized clinical trial

A utility-based Bayesian population finding (BaPoFi) method was proposed by Morita and Müller to analyze data from a randomized clinical trial with the aim of identifying good predictive baseline covariates for optimizing the target population for a future study. The approach casts the population finding process as a formal decision problem together with a flexible probability model using a random forest to define a regression mean function. BaPoFi is constructed to handle a single continuous or binary outcome variable. In this paper, we develop BaPoFi-TTE as an extension of the earlier approach for clinically important cases of time-to-event (TTE) data with censoring, and also accounting for a toxicity outcome. We model the association of TTE data with baseline covariates using a semiparametric failure time model with a Pólya tree prior for an unknown error term and a random forest for a flexible regression mean function. We define a utility function that addresses a trade-off between efficacy and toxicity as one of the important clinical considerations for population finding. We examine the operating characteristics of the proposed method in extensive simulation studies. For illustration, we apply the proposed method to data from a randomized oncology clinical trial. Concerns in a preliminary analysis of the same data based on a parametric model motivated the proposed more general approach.

Noninferiority Trials to Evaluate Drug Effects in Rheumatoid Arthritis

OBJECTIVE

The increased availability of highly effective treatments in rheumatoid arthritis (RA) necessitates a reexamination of study designs evaluating new treatments. We undertook this study to discuss possible specifications and considerations of noninferiority (NI) trials assessing drug effects in RA.

METHODS

We focused on the use of approved tumor necrosis factor inhibitors (TNFi) as potential active controls and reviewed previous placebo-controlled studies. We summarized the similarities in baseline characteristics and study design of the historical placebo-controlled studies used. After performing meta-analyses to estimate the effects of TNFi on symptoms, physical function, and radiographic progression in RA, we proposed NI margins and evaluated the feasibility of NI trials in this therapeutic setting.

RESULTS

We determined that an NI trial comparing an experimental treatment to a TNFi using the symptomatic end point of the American College of Rheumatology 20% improvement criteria response can feasibly provide evidence of a treatment effect, with a 12% absolute difference as one possible appropriate NI margin. For change from baseline in the Health Assessment Questionnaire disability index score, reasonable margins range from 0.10 to 0.12. In evaluating radiographic progression, an appropriate margin and the corresponding feasibility of the trial are dependent on the selected active control and the expected variability in progression.

CONCLUSION

Active-controlled studies in RA with justified NI margins can provide persuasive evidence of treatment effects on symptomatic, functional, and radiographic end points. Such studies can also provide reliable, controlled safety data and relevant information for treatment decisions in clinical practice. Thus, we recommend considering NI designs in future clinical trials in RA.

Individualized treatment effects with censored data via fully nonparametric Bayesian accelerated failure time models

Individuals often respond differently to identical treatments, and characterizing such variability in treatment response is an important aim in the practice of personalized medicine. In this article, we describe a nonparametric accelerated failure time model that can be used to analyze heterogeneous treatment effects (HTE) when patient outcomes are time-to-event. By utilizing Bayesian additive regression trees and a mean-constrained Dirichlet process mixture model, our approach offers a flexible model for the regression function while placing few restrictions on the baseline hazard. Our nonparametric method leads to natural estimates of individual treatment effect and has the flexibility to address many major goals of HTE assessment. Moreover, our method requires little user input in terms of model specification for treatment covariate interactions or for tuning parameter selection. Our procedure shows strong predictive performance while also exhibiting good frequentist properties in terms of parameter coverage and mitigation of spurious findings of HTE. We illustrate the merits of our proposed approach with a detailed analysis of two large clinical trials (N = 6769) for the prevention and treatment of congestive heart failure using an angiotensin-converting enzyme inhibitor. The analysis revealed considerable evidence for the presence of HTE in both trials as demonstrated by substantial estimated variation in treatment effect and by high proportions of patients exhibiting strong evidence of having treatment effects which differ from the overall treatment effect.

Bayesian Approaches to Subgroup Analysis and Related Adaptive Clinical Trial Designs

We review Bayesian and Bayesian decision theoretic approaches to subgroup analysis and applications to subgroup-based adaptive clinical trial designs. Subgroup analysis refers to inference about subpopulations with significantly distinct treatment effects. The discussion mainly focuses on inference for a benefiting subpopulation, that is, a characterization of a group of patients who benefit from the treatment under consideration more than the overall population. We introduce alternative approaches and demonstrate them with a small simulation study. Then, we turn to clinical trial designs. When the selection of the interesting subpopulation is carried out as the trial proceeds, the design becomes an adaptive clinical trial design, using subgroup analysis to inform the randomization and assignment of treatments to patients. We briefly review some related designs. There are a variety of approaches to Bayesian subgroup analysis. Practitioners should consider the type of subpopulations in which they are interested and choose their methods accordingly. We demonstrate how subgroup analysis can be carried out by different Bayesian methods and discuss how they identify slightly different subpopulations.

Heterogeneity of treatment effect of stress ulcer prophylaxis in ICU patients: A secondary analysis protocol

BACKGROUND

In the Stress Ulcer Prophylaxis in the Intensive Care Unit (SUP-ICU) trial, 3291 adult ICU patients at risk for gastrointestinal (GI) bleeding were randomly allocated to intravenous pantoprazole 40 mg or placebo once daily in the ICU. No difference was observed between the groups in the primary outcome 90-day mortality or the secondary outcomes, except for clinically important gastrointestinal bleeding. However, heterogeneity of treatment effect (HTE) not detected by conventional subgroup analyses could be present.

METHODS

This is a protocol and statistical analysis plan for a secondary, post hoc, exploratory analysis of the SUP-ICU trial. We will explore HTE in one set of subgroups based on severity of illness (using the Simplified Acute Physiology Score [SAPS] II) and another set of subgroups based on the total number of risk factors for GI bleeding in each patient using Bayesian hierarchical models. We will summarise posterior probability distributions using medians and 95% credible intervals and present probabilities for different levels of benefit and harm of the intervention in each subgroup. Finally, we will assess if the treatment effect interacts with SAPS II and the number of risk factors separately on the continuous scale using marginal effects plots.

CONCLUSIONS

The outlined post hoc analysis will explore whether HTE was present in the SUP-ICU trial and may help answer some of the remaining questions regarding the balance between benefits and harms of pantoprazole in ICU patients at risk of GI bleeding. CLINICALTRIALS.

GOV REGISTRATION

NCT02467621.

Application of structured statistical analyses to identify a biomarker predictive of enhanced tralokinumab efficacy in phase III clinical trials for severe, uncontrolled asthma

Background

Tralokinumab is an anti–interleukin (IL)-13 monoclonal antibody investigated for the treatment of severe, uncontrolled asthma in two Phase III clinical trials, STRATOS 1 and 2. The STRATOS 1 biomarker analysis plan was developed to identify biomarker(s) indicative of IL-13 activation likely to predict tralokinumab efficacy and define a population in which there was an enhanced treatment effect; this defined population was then tested in STRATOS 2.

Methods

The biomarkers considered were blood eosinophil counts, fractional exhaled nitric oxide (FeNO), serum dipeptidyl peptidase-4, serum periostin and total serum immunoglobulin E. Tralokinumab efficacy was measured as the reduction in annualised asthma exacerbation rate (AAER) compared with placebo (primary endpoint measure of STRATOS 1 and 2). The biomarker analysis plan included negative binomial and generalised additive models, and the Subgroup Identification based on Differential Effect Search (SIDES) algorithm, supported by robustness and sensitivity checks. Effects on the key secondary endpoints of STRATOS 1 and 2, which included changes from baseline in standard measures of asthma outcomes, were also investigated. Prior to the STRATOS 1 read-out, numerous simulations of the methodology were performed with hypothetical data.

Results

FeNO and periostin were identified as the only biomarkers potentially predictive of treatment effect, with cut-offs chosen by the SIDES algorithm of > 32.3 ppb and > 27.4 ng/ml, respectively. The FeNO > 32.3 ppb subgroup was associated with greater AAER reductions and improvements in key secondary endpoints compared with the periostin > 27.4 ng/ml subgroup. Upon further evaluation of AAER reductions at different FeNO cut-offs, ≥37 ppb was chosen as the best cut-off for predicting tralokinumab efficacy.

Discussion

A rigorous statistical approach incorporating multiple methods was used to investigate the predictive properties of five potential biomarkers and to identify a participant subgroup that demonstrated an enhanced tralokinumab treatment effect. Using STRATOS 1 data, our analyses identified FeNO at a cut-off of ≥37 ppb as the best assessed biomarker for predicting enhanced treatment effect to be tested in STRATOS 2. Our findings were inconclusive, which reflects the complexity of subgroup identification in the severe asthma population.

Trial registration

STRATOS 1 and 2 are registered on ClinicalTrials.gov (NCT02161757 registered on June 12, 2014, and NCT02194699 registered on July 18, 2014).

Electronic supplementary material

The online version of this article (10.1186/s12890-019-0889-4) contains supplementary material, which is available to authorized users.

Semi-parametric Bayesian regression for subgroup analysis in clinical trials

Determining whether there are differential treatment effects in subgroups of trial participants remains an important topic in clinical trials as precision medicine becomes ever more relevant. Any assessment of differential treatment effect is predicated on being able to estimate the treatment response accurately while satisfying constraints of balancing the risk of overlooking an important subgroup with the potential to make a decision based on a false discovery. While regression models, such as marginal interaction model, have been widely used to improve accuracy of subgroup parameter estimates by leveraging the relationship between treatment and covariate, there is still a possibility that it can lead to excessively conservative or anti-conservative results. Conceivably, this can be due to the use of the normal distribution as a default prior, which forces outlying subjects to have their means over-shrunk towards the population mean, and the data from such subjects may be excessively influential in estimation of both the overall mean response and the mean response for each subgroup, or a model mis-specification. To address this issue, we investigate the use of nonparametric Bayes, particularly Dirichlet process priors, to create semi-parametric models. These models represent uncertainty in the prior distribution for the overall response while accommodating heterogeneity among individual subgroups. They also account for the effect and variation due to the unaccounted terms. As a result, the models do not force estimates to excessively shrink but still retain the attractiveness of improved precision given by the narrower credible intervals. This is illustrated in extensive simulations investigating bias, mean squared error, coverage probability and credible interval widths. We applied the method on a simulated data based closely on the results of a cystic fibrosis Phase 2 trial.

Bayesian hierarchical classification and information sharing for clinical trials with subgroups and binary outcomes

Bayesian hierarchical models have been applied in clinical trials to allow for information sharing across subgroups. Traditional Bayesian hierarchical models do not have subgroup classifications; thus, information is shared across all subgroups. When the difference between subgroups is large, it suggests that the subgroups belong to different clusters. In that case, placing all subgroups in one pool and borrowing information across all subgroups can result in substantial bias for the subgroups with strong borrowing, or a lack of efficiency gain with weak borrowing. To resolve this difficulty, we propose a hierarchical Bayesian classification and information sharing (BaCIS) model for the design of multigroup phase II clinical trials with binary outcomes. We introduce subgroup classification into the hierarchical model. Subgroups are classified into two clusters on the basis of their outcomes mimicking the hypothesis testing framework. Subsequently, information sharing takes place within subgroups in the same cluster, rather than across all subgroups. This method can be applied to the design and analysis of multigroup clinical trials with binary outcomes. Compared to the traditional hierarchical models, better operating characteristics are obtained with the BaCIS model under various scenarios.

Considerations when assessing heterogeneity of treatment effect in patient-centered outcomes research

When baseline risk of an outcome varies within a population, the effect of a treatment on that outcome will vary on at least one scale (e.g., additive, multiplicative). This treatment effect heterogeneity is of interest in patient-centered outcomes research. Based on a literature review and solicited expert opinion, we assert the following: (1) Treatment effect heterogeneity on the additive scale is most interpretable to health-care providers and patients using effect estimates to guide treatment decision-making; heterogeneity reported on the multiplicative scale may be misleading as to the magnitude or direction of a substantively important interaction. (2) The additive scale may give clues about sufficient-cause interaction, although such interaction is typically not relevant to patients' treatment choices. (3) Statistical modeling need not be conducted on the same scale as results are communicated. (4) Statistical testing is one tool for investigations, provided important subgroups are identified a priori, but test results should be interpreted cautiously given nonequivalence of statistical and clinical significance. (5) Qualitative interactions should be evaluated in a prespecified manner for important subgroups. Principled analytic plans that take into account the purpose of investigation of treatment effect heterogeneity are likely to yield more useful results for guiding treatment decisions.

Subgroup analysis with semiparametric models toward precision medicine

In analyzing clinical trials, one important objective is to classify the patients into treatment-favorable and nonfavorable subgroups. Existing parametric methods are not robust, and the commonly used classification rules ignore the fact that the implications of treatment-favorable and nonfavorable subgroups can be different. To address these issues, we propose a semiparametric model, incorporating both our knowledge and uncertainty about the true model. The Wald statistics is used to test the existence of subgroups, while the Neyman-Pearson rule to classify each subject. Asymptotic properties are derived, simulation studies are conducted to evaluate the performance of the method, and then method is used to analyze a real-world trial data.

Estimating individual benefits of medical or behavioral treatments in severely ill patients

There is a need for statistical methods appropriate for the analysis of clinical trials from a personalized-medicine viewpoint as opposed to the common statistical practice that simply examines average treatment effects. This article proposes an approach to quantifying, reporting and analyzing individual benefits of medical or behavioral treatments to severely ill patients with chronic conditions, using data from clinical trials. The approach is a new development of a published framework for measuring the severity of a chronic disease and the benefits treatments provide to individuals, which utilizes regression models with random coefficients. Here, a patient is considered to be severely ill if the patient’s basal severity is close to one. This allows the derivation of a very flexible family of probability distributions of individual benefits that depend on treatment duration and the covariates included in the regression model. Our approach may enrich the statistical analysis of clinical trials of severely ill patients because it allows investigating the probability distribution of individual benefits in the patient population and the variables that influence it, and we can also measure the benefits achieved in specific patients including new patients. We illustrate our approach using data from a clinical trial of the anti-depressant imipramine.