  Bivariate meta-analysis provides a useful framework for combining information
across related studies and has been utilised 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 paper, we explore modelling the two outcomes on the original binomial
scale. Firstly, 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
(CCyR) and event-free-survival (EFS).
