Statistics and Its Interface

Volume 9 (2016)

Number 1

Modeling multiple responses via bootstrapping margins with an application to genetic association testing

Pages: 47 – 56



Jiwei Zhao (Department of Biostatistics, University at Buffalo, State University of New York, Buffalo, N.Y., U.S.A.)

Heping Zhang (Department of Biostatistics, Yale School of Public Health, Yale University, New Haven, Connecticut, U.S.A.)


The need for analysis of multiple responses arises from many applications. In behavioral science, for example, comorbidity is a common phenomenon where multiple disorders occur in the same person. The advantage of jointly analyzing multiple correlated responses has been examined and documented. Due to the difficulties of modeling multiple responses, nonparametric tests such as generalized Kendall’s Tau have been developed to assess the association between multiple responses and risk factors. These procedures have been applied to genomewide association studies of multiple complex traits. Unfortunately, those nonparametric tests only provide the significance of the association but not the magnitude. We propose a Gaussian copula model with discrete margins for modeling multivariate binary responses. This model separates marginal effects from between-trait correlations. We use a bootstrapping margins approach to constructing Wald’s statistic for the association test. Although our derivation is based on the fully parametric Gaussian copula framework for simplicity, the underlying assumptions to apply our method can be weakened. The bootstrapping margins approach only requires the correct specification of the model margins. Our simulation and real data analysis demonstrate that our proposed method not only increases power over some existing association tests, but also provides further insight into genetic association studies of multivariate traits.


multiple traits, marginal approach, bootstrap, Gaussian copula

2010 Mathematics Subject Classification

Primary 62P10. Secondary 62G09.

Published 22 October 2015