Efficient Bayesian Power Prior Survival Modeling with BayesPPDSurv | HackerNoon

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Abstract and 1 Introduction: BayesPPDSurv

2 Theoretical Framework

2.1 The Power Prior and the Normalized Power Prior

2.2 The Piecewise Constant Hazard Proportional Hazards (PWCH-PH) Model

2.3 Power Prior for the PWCH-PH Model

2.4 Implementing the Normalized Power Prior for the PWCH-PH Model

2.5 Bayesian Sample Size Determination

2.6 Data Simulation for the PWCH-PH Model

3 Using BayesPPDSurv

3.1 Sampling Priors

4 Case Study: Melanoma Clinical Trial Design

5 Discussion and References

Abstract

The BayesPPDSurv (Bayesian Power Prior Design for Survival Data) R package supports Bayesian power and type I error calculations and model fitting using the power and normalized power priors incorporating historical data with for the analysis of time-to-event outcomes. The package implements the stratified proportional hazards regression model with piecewise constant hazard within each stratum. The package allows the historical data to inform the treatment effect parameter, parameter effects for other covariates in the regression model, as well as the baseline hazard parameters. The use of multiple historical datasets is supported. A novel algorithm is developed for computationally efficient use of the normalized power prior. In addition, the package supports the use of arbitrary sampling priors for computing Bayesian power and type I error rates, and has built-in features that semi-automatically generate sampling priors from the historical data. We demonstrate the use of BayesPPDSurv in a comprehensive case study for a melanoma clinical trial design.

1 Introduction: BayesPPDSurv

The incorporation of historical information in clinical trial design has become increasingly popular due to its potential of making trials more efficient. If the historical trial is sufficiently similar to the current trial, one can achieve more accurate point estimates and increased power (Viele et al., 2014). One natural way of integrating historical information is through informative priors in a Bayesian framework. The power prior (Ibrahim and Chen, 2000) is a popular class of informative priors that allow the incorporation of historical data through a discounted likelihood. It is constructed by raising the historical data likelihood to a power a0, where 0 ≤ a0 ≤ 1. The discounting parameter a0 can be fixed or modeled as random. When it is modeled as random and estimated jointly with other parameters of interest (denoted by θ), the normalized power prior (Duan et al., 2006) is recommended, as normalization is critical to enabling the prior to factor into a conditional distribution of θ given a0 and a marginal distribution of a0.

The power prior is widely used due to its easy construction and intuitive interpretation (Ibrahim et al., 2015). The theoretical properties of the power prior and the normalized power prior have been extensively studied. For example, Ibrahim et al. (2003) show that the power prior is an optimal class of informative priors in the sense that it minimizes a convex sum of the Kullback–Leibler (KL) divergences between two posterior densities, in which one density is based on no incorporation of historical data, and the other density is based on pooling the historical and current data. Shen et al. (2024) show that the marginal posterior for the discounting parameter converges to a point mass at zero if there is any discrepancy between the historical and current data. They also show that the marginal posterior for a0 does not converge to a point mass at one when the datasets are fully compatible, and yet, for an i.i.d normal model and finite sample size, the marginal posterior for a0 always has most mass around one when the datasets are fully compatible. Chen and Ibrahim (2006) and Shen et al. (2024) establish the analytic connection between the the power and the normalized power priors and Bayesian hierarchical models. The normalizing constant in the normalized power prior is analytically intractable when there are covariates, except in the case of the normal linear model. Carvalho and Ibrahim (2021) propose a bisection-type algorithm for computing the normalizing constant. In our package, BayesPPDSurv (Bayesian Power Prior Design for Survival Data), we propose a novel algorithm that provides an arbitrarily accurate approximation to the normalized power prior for θ itself, avoiding the need to compute the normalizing constant and thus providing significant computational efficiency.

The focus of the BayesPPDSurv package is on applying the power prior and normalized power prior to time-to-event outcomes for sample size determination and/or data analysis. In particular, we implement the proportional hazards model (Cox, 1972) with piecewise constant baseline hazard (Friedman, 1982). Modeling the baseline hazard with a piecewise constant function is a widely used approach in Bayesian analysis (Ibrahim et al., 2001). Several R packages on CRAN implement the piecewise constant hazard model. The pch package (Frumento, 2024) implements the piecewise constant hazard model for censored and truncated data. The gsDesign package (Anderson, 2024) supports the piecewise constant hazard model for group sequential clinical trial design. These packages do not allow the inclusion of historical data. There are several R packages that implement the power prior or variations of the power prior for the purpose of model estimation or design. The BayesCTDesign package (Eggleston et al., 2021) supports two-arm randomized Bayesian trial design using historical control data with the power prior for a variety of outcome models, including the Weibull and piesewise constant hazard models for time-to-event outcomes. However, it does not allow using the historical data to inform the treatment effect parameter or parameter effects for other covariates. The bayesDP package (Balcome et al., 2021) implements the discounted power prior for single arm and two-arm clinical trials where the discounting parameter is determined by a discounting function estimated based on a measure of prior-data conflict. While it accommodates borrowing historical information for the treatment effect with the piecewise constant hazard model, it does not allow additional covariates, and it must be used in conjunction with the package bayesCT (Chandereng et al., 2020) for trial design. There are three R packages that implement the normalized power prior where a0 is modeled as random. The NPP package (Han et al., 2021) supports posterior sampling using the normalized power prior for Bernoulli, normal, multinomial and Poisson models, as well as for the normal linear model. The hdbayes (Alt, 2022) package implements several methods that leverage historical data for generalized linear models, including the power prior and the normalized power prior. These two packages do not accommodate models for time-to-event outcomes, nor do they perform sample size determination. The R package BayesPPD (Shen et al., 2023b) supports trial analysis and design using the power prior and the normalized power prior for generalized linear models, but it does not support models for time-to-event outcomes.

Our package BayesPPDSurv (Shen et al., 2024) addresses an important gap by providing a suite of functions for Bayesian power and type I error rate calculations and model fitting after incorporating historical data with the power prior and the normalized power prior for time-to-event data. It implements the stratified proportional hazards regression model with piecewise constant hazard within each stratum. The package allows the historical data to inform the treatment effect parameter, parameter effects for other covariates in the hazard ratio regression model, as well as the baseline hazard parameters. The time interval partition for the piecewise baseline hazards can be stratified. The discounting parameter a0 can be fixed or modeled as random. The use of multiple historical datasets is supported. For sample size determination, we consider the simulation-based method developed in Chen et al. (2011) utilizing the sampling and fitting priors (Wang and Gelfand, 2002) as applied in Psioda and Ibrahim (2019). The package supports the use of arbitrary sampling priors for computing Bayesian power and type I error rates, and has built-in features that semi-automatically generate sampling priors from the historical data. BayesPPDSurv is computationally efficient. It implements the slice sampler (Neal, 2003) with Rcpp (Eddelbuettel and Francois, 2011), and functions for analysis take less than a minute to execute in most instances.

The rest of the article proceeds as follows. In section 2, we describe the theoretical details of the methods implemented in the package. In section 3, we provide details on using the package and its various features. In section 4, we present a comprehensive case study for a melanoma clinical trial design with example code. The article is concluded with a brief discussion.