# Discussion and Future Research

Copula models have wide applicability in clinical trials where correctly accounting for dependence between multiple outcomes is important to answer the underlying scientific questions. Exploration of the joint distribution can provide deeper insight into the effect of the intervention across multiple dimensions. Even if the dependence structure is not of interest, accounting for dependence with a copula model can improve marginal inferences compared to performing separate analyses or other less flexible joint modeling approaches if the data contains enough information to estimate the additional copula parameters well [5,63,69].

Copula models have been fairly well explored in the context of early phase dose-finding trials, survival analysis, and longitudinal data, but their application – especially using a Bayesian framework – has been limited in other clinical trial settings where they may be advantageous. Potential avenues of future research include Bayesian copula models which incorporate information from historical controls, heterogeneous populations (e.g., pediatric vs. adult, multiple countries or regions), or large observational data sources. We are currently researching the conditions under which Bayesian power for a marginal efficacy assessment can be improved by borrowing information from other correlated efficacy endpoints. A further area of interest is a structured comparison to other methods for quantitative benefit-risk analysis (e.g., net benefit:risk index) or as an alternative to composite or ordinal outcome approaches which transform a multidimensional outcome into a unidimensional outcome.

While many software resources exist for creation, manipulation, and estimation of basic unconditional copulas (R packages copula and VineCopula, Python libraries copulas and pycopula, MATLAB Statistics and Machine Learning Toolbox and Multivariate Copula Analysis Toolbox, SAS PROC COPULA) there is a lack of software to perform general purpose conditional copula regression modeling which necessitates writing ad hoc code for each application. This requirement creates a barrier to wider adoption and exploration of these models in clinical trials but also presents an opportunity for useful contributions. The need for more accessible software for Bayesian copula modeling is especially acute as there are fewer off-the-shelf options and the analyst must handle model specification in addition to priors and potentially more complex methods to sample the posterior.

Copulas are a valuable tool for the study of multivariate data collected for clinical trials. They are unique among joint modeling approaches for several reasons, including: the ability to separately specify the marginal models and the dependence model while maintaining the marginal parameter interpretations; flexible, transformation-invariant and interpretable dependence structures; and symmetric treatment of the outcomes of interest. Despite these advantages, their potential in clinical trials has not been fully realized due to gaps in knowledge about when they are most beneficial and a dearth of user-friendly software which creates a challenge for implementation. Future research addressing these areas will provide a valuable contribution to the literature on design and analysis of clinical trials data.

### References

5. Leon AR de, Wu B. Copula-based regression models for a bivariate mixed discrete and continuous outcome. Statistics in Medicine [Internet] 2011 [cited 2018 Oct 5];30:175–85. Available from: http://doi.wiley.com/10.1002/sim.4087

63. Cunanan K, Koopmeiners JS. Evaluating the performance of copula models in phase i-II clinical trials under model misspecification. BMC Medical Research Methodology [Internet] 2014 [cited 2018 Jul 26];14. Available from: http://bmcmedresmethodol.biomedcentral.com/articles/10.1186/1471-2288-14-51

69. Song PX-K, Li M, Yuan Y. Joint regression analysis of correlated data using gaussian copulas. Biometrics [Internet] 2009 [cited 2018 Oct 12];65:60–8. Available from: http://doi.wiley.com/10.1111/j.1541-0420.2008.01058.x