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.

中文翻译：

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使用 copula 在聚合级别的二项式数据的双变量元分析中对研究内关联进行建模：贝叶斯方法及其在替代终点评估中的应用

双变量荟萃分析为合并相关研究的信息提供了一个有用的框架，并已被用于合并临床研究的证据以评估两种结果的治疗效果。它还被用于研究替代终点的治疗效果与最终结果之间的替代模式。替代终点在药物开发中起着重要作用，因为它们可用于与最终结果相比早期衡量治疗效果并预测临床益处或危害。标准的双变量元分析方法使用双变量正态分布，在研究内和研究间水平上共同模拟观察到的对替代物的治疗效果和最终结果结果。对于二项式数据，可以使用对数优势比标度的正态近似。然而，当事件的比例接近于 1 或 0 时，这种方法可能会导致有偏差的结果，从而影响替代终点的验证。在本文中，我们探讨了在原始二项式尺度上对两个结果进行建模。首先，我们提出了一种方法，该方法使用独立的二项式似然来模拟研究内的变异性，避免近似观察到的治疗效果。但是，该方法忽略了研究内关联。为了克服这个问题，我们提出了一种使用具有二项式边缘的双变量 copula 的方法，该方法允许模型考虑研究内关联。我们将这些方法应用于慢性粒细胞白血病的一个说明性示例，以研究完全细胞遗传学反应和无事件生存之间的替代关系。