ANPP: the Adapted Normalized Power Prior for Borrowing Information from Multiple Historical Datasets in Clinical Trials

TL;DR

This paper addresses how to borrow information from multiple historical clinical datasets using Bayesian priors. It introduces the Adapted Normalized Power Prior (ANPP), which imposes dependent discounting across histories via a global parameter $a_0$ and transformations to align with a Bayesian hierarchical model (BHM); this yields posterior inferences for the main parameter $ heta$ that are equivalent to those from the BHM under suitable priors. Theoretical results establish explicit mappings between the NPP/ANPP priors and the BHM, enabling semi-automatic prior elicitation for dynamic borrowing (e.g., $a_0=f(v)$ and $h_k(a_0)$). Simulations show the ANPP’s borrowing behavior tracks the overall heterogeneity across histories and is more sensitive to conflicts than independent discounting; an application to pediatric lupus demonstrates practical equivalence to BHM with informed borrowing from adult trials. Overall, the ANPP provides a principled, interpretable framework for borrowing from multiple historical datasets with easier prior calibration than direct BHM specification, with avenues for extending to non-normal data and GLMs.

Abstract

The power prior is a popular class of informative priors for incorporating information from historical data. It involves raising the likelihood for the historical data to a power, which acts as a discounting parameter. When the discounting parameter is modeled as random, the normalized power prior (NPP) is recommended. When there are multiple historical datasets, there has been limited research on how to choose priors for the multiple discounting parameters of the NPP to induce desirable information borrowing behavior. In this work, we address this question by investigating the analytical relationship between the NPP and the Bayesian hierarchical model (BHM), which is a widely used method for synthesizing information from different sources. We develop the adapted normalized power prior (ANPP), which establishes dependence between the dataset-specific discounting parameters of the NPP, leading to inferences that are identical to the BHM. We establish a direct relationship between the prior for the dataset-specific discounting parameters of the ANPP and the prior for the variance parameter of the BHM. Establishing this relationship not only justifies the NPP from the perspective of hierarchical modeling, but also achieves easy prior elicitation for the NPP for the purpose of dynamic borrowing. We examine the borrowing properties of the ANPP through simulations, and apply it to a case study for a pediatric lupus trial.

Figures (10)

• Figure 1: Marginal posteriors for $a_{01}$, $a_{02}$ and $\theta$ using the ANPP and the iNPP for simulated i.i.d. normal data. In the top row, the three datasets are fully compatible. In the bottom row, the two historical datasets are compatible but they are both incompatible with the current dataset. The first column includes plots of the densities of the current data (black line) and two historical datasets (yellow and blue lines). The second column includes plots of the posterior densities of $a_{01}$ and $a_{02}$ using the ANPP (yellow and blue solid lines) and the iNPP (yellow and blue dashed lines). The third column includes plots of the posterior densities of $\theta$ using four different priors, the ANPP (purple), the iNPP (grey), the power prior with $a_0=0$ (black) and the power prior with $a_0=1$ (pink).
• Figure 2: Marginal posteriors for $a_{01}$, $a_{02}$ and $\theta$ using the ANPP and the iNPP for simulated i.i.d. normal data. In these scenarios, one of the historical datasets is compatible with the current data while the other is not. The first column includes plots of the densities of the current data (black line) and two historical datasets (yellow and blue lines). The second column includes plots of the posterior densities of $a_{01}$ and $a_{02}$ using the ANPP (yellow and blue solid lines) and the iNPP (yellow and blue dashed lines). The third column includes plots of the posterior densities of $\theta$ using four different priors, the ANPP (purple), the iNPP (grey), the power prior with $a_0=0$ (black) and the power prior with $a_0=1$ (pink).
• Figure 3: We choose four beta distributions as the prior for $a_0$ in the NPP, and find the best approximating inverse gamma distribution prior for $v$ in a BHM. The black line represents the induced prior on $v$ using Theorem \ref{['single']}, and the blue line represents the inverse gamma distribution that best approximates the induced prior.
• Figure 4: We choose four inverse gamma distributions as the prior for $v$ in the BHM, and find the best approximating beta prior for $a_0$ in a NPP. We draw samples of $v$ from the inverse gamma distributions and transform them to samples of $a_0$ using the formula in Theorem \ref{['single']}. We use beta regression to solve for the beta distribution that best fits the $a_0$ samples. The black line represents the induced prior on $a_0$ using Theorem \ref{['single']}, and the blue line represents the beta distribution that best approximates the induced prior.
• Figure 5: Pediatric lupus trial: the histogram represents the posterior of $\theta$ using a ANPP with a uniform prior on $a_0$ and $h_k(a_0)$ chosen according to Theorem \ref{['multiple']}. The black density curve represents the posterior of $\theta$ using the BHM where the prior on $v$ is induced using Theorem \ref{['multiple']}. We observe that the two posteriors are equivalent. They are also equivalent to the posterior of $\theta$ using a Bernoulli model (blue curve) for the induced BHM. We ran four independent chains of 10,000 iterations with 5,000 burn-ins using RStan.

Theorems & Definitions (6)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• proof
• proof