Modeling and Prediction of Multiple Correlated Functional Outcomes
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We propose a copula-based approach for analyzing functional data with correlated multiple functional outcomes exhibiting heterogeneous shape characteristics. To accommodate the possibly large number of parameters due to having several functional outcomes, parameter estimation is performed in two steps: first, the parameters for the marginal distributions are estimated using the skew t family, and then the dependence structure both within and across outcomes is estimated using a Gaussian copula. We develop an estimation algorithm for the dependence parameters based on the Karhunen–Loève expansion and an EM algorithm that significantly reduces the dimension of the problem and is computationally efficient. We also demonstrate prediction of an unknown outcome when the other outcomes are known. We apply our methodology to diffusion tensor imaging data for multiple sclerosis (MS) patients with three outcomes and identify differences in both the marginal distributions and the dependence structure between the MS and control groups. Our proposed methodology is quite general and can be applied to other functional data with multiple outcomes in biology and other fields. Supplementary materials accompanying this paper appear online.
KeywordsDiffusion tensor imaging Gaussian copulas Multiple sclerosis Skewed functional data Tractography data
The authors are very grateful for the constructive comments of the Editor, the Associate Editor and two reviewers, which are extremely helpful for us to improve our work. The authors also thank Daniel Reich and Peter Calabresi and their research teams, who were instrumental in collecting the data for this study. Scans were funded by grants from the National Multiple Sclerosis Society and EMD Serono. We are grateful to Ciprian Crainiceanu for providing access to the data and for meaningful discussions and personal communications. Carroll was supported by a Grant U01-CA057030 from the National Cancer Institute (NCI). Ruppert was supported by the NCI Grant U01-CA057030 and a NSF Grant AST-1312903.
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