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Bayesian inference for a principal stratum estimand on recurrent events truncated by death

Tianmeng Lyu, Björn Bornkamp, Guenther Mueller-Velten and Heinz Schmidli

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March 31, 2023

Statistical Methodology

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Agenda

  • Introduction
    • The estimands framework
    • Principal stratum estimand
  • Recurrent event project
    • Motivating example
    • Proposed analysis method
    • Real data application: PARAGON-HF
  • Concluding remarks

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Introduction

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Motivating example

  • The following example is a simplified version of the example in the ICH E9(R1) training materials available on the ICH website
  • Imagine that there is a clinical trial to compare treatment vs. control in patients with diabetes and the endpoint of interest is change from baseline in HbA1c at week 24
  • A certain proportion of patients in both arms took rescue medication before week 24

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In my analysis, I excluded the HbA1c values collected after rescue medication

In my analysis, I included HbA1c values regardless of rescue medication

  • They got different results
  • Which one will you choose?

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Motivating example

  • The difference between their analysis approaches may seem to be a pure statistical question, however, they are actually answering different questions!

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  • For the approach where data collected after use of rescue are excluded (and some imputed values are used instead), this is trying to answer the question “what would be the treatment effect if rescue medications were not taken/not allowed”
  • For the approach where data collected after use of rescue are included, this is trying to answer the question “what is the treatment effect regardless of rescue medication”, i.e. it is comparing “treatment with rescue (taken as required) vs. control with rescue (taken as required)”
  • The estimands framework helps to clarify such ambiguity

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The estimands framework

  • It was introduced in ICH E9 (R1) “Addendum on estimands and sensitivity analysis in clinical trials”, in 2019
  • Estimand is defined as “a precise description of the treatment effect reflecting the clinical question posed by the trial objective. It summarises at a population-level what the outcomes would be in the same patients under different treatment conditions being compared”

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Trial objective

Estimand

Estimator

  • Trial planning:

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The estimands framework

  • The description of an estimand involves specification of five attributes:
    • Treatment
    • Population
    • Variable
    • Intercurrent events
    • Population-level summary

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  • Intercurrent event: events occurring after treatment initiation that affect either the interpretation or the existence of the measurements associated with the clinical question of interest
    • See next slide for examples

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Intercurrent events

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Patient 2

Patient 1

Patient 3

Patient 4

Death

Patient 5

TIMELINE

Randomisation

Data collection for the variable

Change in background therapy

Treatment switch

Treatment discontinuation due to lack of efficacy

Use of rescue medication

Treatment discontinuation due to adverse events

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Strategies to handle intercurrent events

  • ICH E9 (R1) introduced five strategies to handle intercurrent events:

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    • Treatment policy: the occurrence of the intercurrent event is considered irrelevant in defining the treatment effect of interest
    • Hypothetical: a scenario is envisaged in which the intercurrent event would not occur
    • Composite: an intercurrent event is considered in itself to be informative about the patient’s outcome and is therefore incorporated into the definition of the variable
    • While on treatment: response to treatment prior to the occurrence of the intercurrent event is of interest
    • Principal stratum: the target population might be taken to be the “principal stratum” in which an intercurrent event would (or would not) occur. The clinical question of interest relates to the treatment effect only within the principal stratum

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Potential outcomes framework

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Patient ID

Y(0)

Y(1)

1

2.5

3.5

2

1.8

2.3

3

3.0

4.2

*Note: the numbers in the table above are just some random fake numbers for illustration purpose only

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Principal stratum – motivation

  • Motivation: interested in the treatment effect in a subpopulation defined by a post-treatment variable (S)
    • For example, “S” could be intake of rescue medication, death, etc.

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Principal stratum – introduction�Frangakis & Rubin (2002)

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0 (treatment non-tolerance)

1 (tolerance)

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0 (treatment non-tolerance)

Tolerate neither

Tolerate test treatment only

1 (tolerance)

Tolerate control only

Tolerate both

Principal strata in terms of treatment tolerance

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Principal stratum – introduction

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0 (treatment non-tolerance)

1 (tolerance)

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0 (treatment non-tolerance)

Tolerate neither

Tolerate test treatment only

1 (tolerance)

Tolerate control only

Tolerate both

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Recurrent event project

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Recurrent events with death as an intercurrent event

  • Recurrent events: An event that can be observed repeatedly for a single subject
  • Death complicates defining the estimand for recurrent events
  • Examples of patient journey:

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  • Different follow up times
  • Data after death doesn’t exist
  • E.g. patient 1 lives longer than patient 3, but also has more recurrent events

Start of study

End of study

Recurrent event

Death

Drop out

1

2

3

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Introduction to PARAGON-HF study

  • A multicenter, randomized, double-blind, parallel group, active-controlled phase III study completed in 2019
    • Population: heart failure patients (NYHA Class II-IV) with preserved ejection fraction
    • Treatment: sacubitril/valsartan vs. valsartan
    • Primary endpoint: composite endpoint of CV death and total heart failure hospitalizations
  • Results show that there is no obvious difference between the two arms on death
  • Treatment effect on recurrent hospitalizations is clinically relevant
    • Analysis of total hospitalizations in CSR (joint frailty model): RR=0.85, 95% CI: 0.72 - 1.00 

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Principal stratum strategy

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Proposed analysis method – full Bayesian approach

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Proposed analysis method – full Bayesian approach

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Models

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Likelihood

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Estimation of the causal estimand

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For each bth posterior sample:

...

...

 

 

 

...

...

...

...

Summarize across the B posterior samples: point estimate (median) and 95% credible interval

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Revisit PARAGON-HF example

  • Dataset overview:
    • 4796 patients included
    • Baseline covariates were determined based on data availability and analyses from previous heart failure studies (Simpson et al., 2020*), e.g. gender, NYHA III/IV, NT-proBNP, etc.

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*Simpson, J., Jhund, P.S., Lund, L.H., Padmanabhan, S., Claggett, B.L., Shen, L., Petrie, M.C., Abraham, W.T., Desai, A.S., Dickstein, K., Køber, L., Packer, M., Rouleau J.L., Mueller-Velten G., Solomon, S.D., Swedberg, K., Zile, M.R., and McMurray, J.J.V., 2020. Prognostic models derived in PARADIGM-HF and validated in ATMOSPHERE and the Swedish Heart Failure Registry to predict mortality and morbidity in chronic heart failure. JAMA Cardiology, 5(4), pp.432-441.

 

 

 

 

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Estimation results

  • Estimated causal effect:
    • Rate ratio among patients who would be alive under both arms by the end of t year(s):

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𝒕=𝟏

𝒕=𝟐

𝒕=𝟑

P(principal stratum)

93.3%

85.6% 

78.5% 

Rate ratio 

(95% CI)

0.773 

(0.639, 0.933)

0.858 

(0.725, 1.009)

0.891 

(0.753, 1.044)

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Model checking

  • Assess the model fit by posterior predictive checks
  • Main idea: generate replicate datasets from the fitted models with random frailty and compare non-parametric measures based on the replicate data vs. observed data
  • We check the model fit with the following:
    • Kaplan-Meier estimator for time to all-cause death (KM)
    • Ghosh-Lin estimator for the mean number of recurrent events (HF hospitalizations) (GL)
  • Next we show how to do the checking with KM as an example

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Model checking – detailed steps

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Observed data

 

For each bth posterior sample of parameters

Compare based on posterior predictive p-value

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Model checking – p value

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Model checking – visual comparisons

  • KM plot for observed data (sold line) vs. KM plots for 100 replicate data (dashed lines)

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  • Mean number of recurrences plot for observed data (sold line) vs. mean number of recurrences plots for 100 replicate data (dashed lines)

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Concluding remarks

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Concluding remarks

  • The proposed estimator for principal stratum estimand:
    • Straightforward and easy interpretation by joint modeling of survival and recurrent event processes
    • Flexible by including covariates and allowing time dependent intensity/hazard functions (Weibull)
    • Computationally efficient by working with the marginalized likelihood
  • The proposed model fits the PARAGON-HF data reasonably well
  • The discussed estimand is also called survivor average causal effects (SACE)
  • Limitations:
    • Relies on the assumption that all processes share the same frailty term – unverifiable
      • Hopefully can be mitigated by allowing different parameter values in recurrent event/survival process and also in treatment/control group
    • Parametric models – model checking is recommended

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References

  • ICH (2019) ICH E9 (R1): addendum on estimands and sensitivity analysis in clinical trials to the guideline on statistical principles for clinical trials.
  • Lyu, T., Bornkamp, B., Mueller‐Velten, G. and Schmidli, H., Bayesian inference for a principal stratum estimand on recurrent events truncated by death. Biometrics.
  • Frangakis, C.E. & Rubin, D.B. (2002) Principal stratification in causal inference. Biometrics, 58(1), 21–29.
  • Comment, L., Mealli, F., Haneuse, S. & Zigler, C. (2019) Survivor average causal effects for continuous time: a principal stratification approach to causal inference with semicompeting risks. arXiv:1902.09304.
  • Simpson, J., Jhund, P.S., Lund, L.H., Padmanabhan, S., Claggett, B.L., Shen, L., et al. (2020) Prognostic models derived in PARADIGM-HF and validated in ATMOSPHERE and the Swedish Heart Failure Registry to predict mortality and morbidity in chronic heart failure. JAMA Cardiology, 5(4), 432–441.
  • Solomon, S.D., McMurray, J.J.V., Anand, I.S., Ge, J., Lam, C.S.P., Maggioni, A.P., et al. & PARAGON-HF Investigators and Committees (2019) Angiotensin–neprilysin inhibition in heart failure with preserved ejection fraction. New England Journal of Medicine, 381(17), 1609–1620.

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Thank you