Open Access Research article

Evaluating the performance of copula models in phase I-II clinical trials under model misspecification

Kristen Cunanan and Joseph S Koopmeiners*

Author Affiliations

Division of Biostatistics, School of Public Health, University of Minnesota, A460 Mayo Building, MMC 303, 420 Delaware St. SE, 55455 Minneapolis, MN, USA

For all author emails, please log on.

BMC Medical Research Methodology 2014, 14:51  doi:10.1186/1471-2288-14-51

Published: 14 April 2014



Traditionally, phase I oncology trials are designed to determine the maximum tolerated dose (MTD), defined as the highest dose with an acceptable probability of dose limiting toxicities(DLT), of a new treatment via a dose escalation study. An alternate approach is to jointly model toxicity and efficacy and allow dose escalation to depend on a pre-specified efficacy/toxicity tradeoff in a phase I-II design. Several phase I-II trial designs have been discussed in the literature; while these model-based designs are attractive in their performance, they are potentially vulnerable to model misspecification.


Phase I-II designs often rely on copula models to specify the joint distribution of toxicity and efficacy, which include an additional correlation parameter that can be difficult to estimate. We compare and contrast three models for the joint probability of toxicity and efficacy, including two copula models that have been proposed for use in phase I-II clinical trials and a simple model that assumes the two outcomes are independent. We evaluate the performance of the various models through simulation both when the models are correct and under model misspecification.


Both models exhibited similar performance, as measured by the probability of correctly identifying the optimal dose and the number of subjects treated at the optimal dose, regardless of whether the data were generated from the correct or incorrect copula, even when there is substantial correlation between the two outcomes. Similar results were observed for a simple model that assumes independence, even in the presence of strong correlation. Further simulation results indicate that estimating the correlation parameter in copula models is difficult with the sample sizes used in Phase I-II clinical trials.


Our simulation results indicate that the operating characteristics of phase I-II clinical trials are robust to misspecification of the copula model but that a simple model that assumes independence performs just as well due to difficulty in estimating the copula model correlation parameters from binary data.

Bayesian Adaptive Design; Phase I-II; Copula models