Propensity score-integrated power prior approach for incorporating real-world evidence in single-arm clinical studies

ABSTRACT

We are now at an amazing time for medical product development in drugs, biological products and medical devices. As a result of dramatic recent advances in biomedical science, information technology and engineering, ``big data’’ from health care in the real-world have become available. Although big data may not necessarily be attuned to provide the preponderance of evidence to a clinical study, high-quality real-world data can be transformed into scientific evidence for regulatory and healthcare decision-making using proven analytical methods and techniques, such as propensity score methodology and Bayesian inference. In this paper, we extend the Bayesian power prior approach for a single-arm study (the current study) to leverage external real-world data. We use propensity score methodology to pre-select a subset of real-world data containing patients that are similar to those in the current study in terms of covariates, and to stratify the selected patients together with those in the current study into more homogeneous strata. The power prior approach is then applied in each stratum to obtain stratum-specific posterior distributions, which are combined to complete the Bayesian inference for the parameters of interest. We evaluate the performance of the proposed method as compared to that of the ordinary power prior approach by simulation and illustrate its implementation using a hypothetical example, based on our regulatory review experience.

KEYWORDS:

• Covariate balance
• overlapping coefficient
• power prior
• propensity score
• real-world data
• real-world evidence

Acknowledgments

The first author (CW) was a consultant for the Food and Drug Administration on this project and was compensated for his consultation services. The authors are grateful to the referees for their great suggestions and comments which helped to improve the article.

A.1. Derivation of the PS-power prior for normal and binary cases

Let $Y_i$ denote the outcome of patient $i$ ($i=1,\dots ,N_0$) in an existing dataset.

First, consider $Y_i\sim N({\theta }_s,{\sigma }_s^2)$ in stratum $s$ with ${\sigma }_s^2$ known. In practice, we set ${\sigma }_s$ to be the standard deviation of $Y_i$ in stratum $s$.

The likelihood in stratum $s$ is given by

$L({\theta }_s|D_{s,0})={\left(\frac{1}{\sqrt{2\pi {\sigma }_s^2}}\right)}^{n_{s,0}}exp\left[-\frac{{\sum }_{i=1}^{n_{s,0}}{(Y_i-{\theta }_s)}^2}{2{\sigma }_s^2}\right].$

Note that

$\int [L({\theta }_s|D_{s,0}){]}^{{\alpha }_s}d{\theta }_s={\left(\frac{1}{\sqrt{2\pi {\sigma }_s^2}}\right)}^{{\alpha }_s{n}_{s,0}}exp\left[-\frac{{\sum }_{i=1}^{n_{s,0}}{(Y_i-{\bar{Y}}_s)}^2}{2\frac{{\sigma }_s^2}{{\alpha }_s}}\right]\int exp\left[-\frac{{({\theta }_s-{\bar{Y}}_s)}^2}{2\frac{{\sigma }_s^2}{{\alpha }_s{n}_{s,0}}}\right]d{\theta }_s$

$={\left(\frac{1}{\sqrt{2\pi {\sigma }_s^2}}\right)}^{{\alpha }_s{n}_{s,0}}\sqrt{\frac{2\pi {\sigma }_s^2}{{\alpha }_s{n}_{s,0}}}exp\left[-\frac{{\sum }_{i=1}^{n_{s,0}}{(Y_i-{\bar{Y}}_s)}^2}{2\frac{{\sigma }_s^2}{{\alpha }_s}}\right].$

Thus,

$\frac{{[L({\theta }_s|D_{s,0})]}^{{\alpha }_s}}{\int {[L({\theta }_s|D_{s,0})]}^{{\alpha }_s}d{\theta }_s}=\frac{1}{\sqrt{2\pi \frac{{\sigma }_s^2}{{\alpha }_s{n}_{s,0}}}}exp\left[-\frac{{({\theta }_s-{\bar{Y}}_s)}^2}{2\frac{{\sigma }_s^2}{{\alpha }_s{n}_{s,0}}}\right]=\varphi \left(\frac{{\theta }_s-{\bar{Y}}_s}{\frac{{\sigma }_s}{\sqrt{{\alpha }_s{n}_{s,0}}}}\right).$

Assign $\pi ({\theta }_s)\propto 1$. Then,

$\pi ({\theta }_1,\dots ,{\theta }_S,v_1,\dots ,v_S)\propto \prod _s\varphi \left(\frac{{\theta }_s-{\bar{Y}}_s}{\frac{{\sigma }_s}{\sqrt{{\alpha }_s{n}_{s,0}}}}\right)v_s^{\frac{r_s}{R}}.$

Next, consider $Y_i\sim \mathrm{B}\mathrm{e}\mathrm{r}\mathrm{n}({\theta }_s)$ in stratum $s$. The likelihood in stratum $s$ is given by

$L({\theta }_s|D_{s,0})={\theta }_s^{n_{s,0}{\bar{Y}}_s}(1-{\theta }_s{)}^{n_{s,0}-n_{s,0}{\bar{Y}}_s}.$

Assign $\pi ({\theta }_s)=\mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(1,1)$. Then,

$\int L({\theta }_s|D_{s,0}{)}^{{\alpha }_s}d{\theta }_s=\int {\theta }_s^{{\alpha }_s{n}_{s,0}{\bar{Y}}_s}(1-{\theta }_s{)}^{{\alpha }_s{n}_{s,0}-{\alpha }_s{n}_{s,0}{\bar{Y}}_s}d{\theta }_s=B({\alpha }_s{n}_{s,0}{\bar{Y}}_s+1,{\alpha }_s{n}_{s,0}-{\alpha }_s{n}_{s,0}{\bar{Y}}_s+1)$

where $B(\alpha ,\beta )=\frac{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(\beta )}{\mathrm{\Gamma }(\alpha +\beta )}$. Thus,

$\pi ({\theta }_1,\dots ,{\theta }_S,v_1,\dots ,v_S)\propto \prod _s\mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}({\alpha }_s{n}_{s,0}{\bar{Y}}_s+1,{\alpha }_s{n}_{s,0}-{\alpha }_s{n}_{s,0}{\bar{Y}}_s+1)v_s^{\frac{r_s}{R}}.$