Use of copula to model within‐study association in bivariate meta‐analysis of binomial data at the aggregate level: A Bayesian approach and application to surrogate endpoint evaluation

Abstract

Bivariate meta-analysis provides a useful framework for combining information across related studies and has been utilized to combine evidence from clinical studies to evaluate treatment efficacy on two outcomes. It has also been used to investigate surrogacy patterns between treatment effects on the surrogate endpoint and the final outcome. Surrogate endpoints play an important role in drug development when they can be used to measure treatment effect early compared to the final outcome and to predict clinical benefit or harm. The standard bivariate meta-analytic approach models the observed treatment effects on the surrogate and the final outcome outcomes jointly, at both the within-study and between-studies levels, using a bivariate normal distribution. For binomial data, a normal approximation on log odds ratio scale can be used. However, this method may lead to biased results when the proportions of events are close to one or zero, affecting the validation of surrogate endpoints. In this article, we explore modeling the two outcomes on the original binomial scale. First, we present a method that uses independent binomial likelihoods to model the within-study variability avoiding to approximate the observed treatment effects. However, the method ignores the within-study association. To overcome this issue, we propose a method using a bivariate copula with binomial marginals, which allows the model to account for the within-study association. We applied the methods to an illustrative example in chronic myeloid leukemia to investigate the surrogate relationship between complete cytogenetic response and event-free-survival. K E Y W O R D S binary outcomes, bivariate meta-analysis, copula modeling, surrogate endpointsThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Four alternative methodologies for simulated treatment comparison: How could the use of simulation be re‐invigorated?

“…This method is very similar to a double bootstrap approach which has been previously used to, for example, estimate the correlation with an interval used as an input in the multivariate meta-analysis of mixed outcomes. 22,23 3.4 | The infinite population method (STC-IP)…”

“…This can have implications for the stability of the standard error, as we will illustrate in our simulation study in Section 5. This method is very similar to a double bootstrap approach which has been previously used to, for example, estimate the correlation with an interval used as an input in the multivariate meta‐analysis of mixed outcomes 22,23 …”

Bridging disconnected networks of first and second lines of biologic therapies in rheumatoid arthritis with registry data: bayesian evidence synthesis with target trial emulation

“…However, modeling jointly non-normal outcomes, such as binomial responses, would require transforming data, which can lead to biased results [12]. Papanikos et al carried out a simulation study showing that when the within-study correlation is weak, a multivariate meta-analysis model with independent binomial likelihoods is preferable [13]. An exploratory analysis of the BSRBR-RA dataset, estimating the within-study correlation using the bootstrapping approach [14,15], showed that the within-study correlation between the treatment effects for the two lines of therapy transformed onto the log odds ratio (OR) scale was close to zero.…”

“…An exploratory analysis of the BSRBR-RA dataset, estimating the within-study correlation using the bootstrapping approach [14,15], showed that the within-study correlation between the treatment effects for the two lines of therapy transformed onto the log odds ratio (OR) scale was close to zero. We, therefore, adapted the approaches to multivariate/bivariate NMA by Achana et al [16] and Bujkiewicz et al [17] by assuming independent binomial likelihoods at the within-study level, as in Papanikos et al [13], to model the proportions of responders to treatment in each line of therapy. To predict treatment effects in the second line when data are only available for the therapy in first line, additional assumptions of exchangeability needed to be made, where instead of placing prior distributions on basic parameters, we added another level of hierarchy to the model as in Bujkiewicz et al [17].…”