Repeated measures designs are widely used in practice to increase power, reduce sample size, and increase efficiency in data collection. Correlation between repeated measurements is one of the first research questions that needs to be addressed in a repeated-measure study. In addition to an estimate for correlation, confidence interval should be computed and reported for statistical inference. The asymptotic interval based on the delta method is traditionally calculated due to its simplicity. However, this interval is often criticized for its unsatisfactory performance with regards to coverage and interval width. Bootstrap could be utilized to reduce the interval width, and the widely used bootstrap intervals include the percentile interval, the bias-corrected interval, and the bias-corrected with acceleration interval. Wilcox (Comput Stat Data Anal 22:89–98,1996) suggested a modified percentile interval with the interval levels adjusted by sample size to have the coverage probability close to the nominal level. For a study with repeated measures, more parameters in addition to sample size would affect the coverage probability. For these reasons, we propose modifying the percentiles in the percentile interval to guarantee the coverage probability based on simulation studies. We analyze the correlation between imaging volumes and memory scores from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) study to illustrate the application of the considered intervals. The proposed interval is exact with the coverage probability guaranteed, and is recommended for use in practice.
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The authors are very grateful to Editor, Associate Editor and two referees for their insightful comments that help improve the manuscript. Data collection and sharing for this project was funded by the Alzheimer’s Disease Neuroimaging Initiative (ADNI) (National Institutes of Health Grant U01 AG024904) and DOD ADNI (Department of Defense award number W81XWH-12-2-0012). ADNI is funded by the National Institute on Aging, the National Institute of Biomedical Imaging and Bioengineering, and through generous contributions from the following: AbbVie, Alzheimer’s Association; Alzheimer’s Drug Discovery Foundation; Araclon Biotech; BioClinica, Inc.; Biogen; Bristol-Myers Squibb Company; CereSpir, Inc.; Cogstate; Eisai Inc.; Elan Pharmaceuticals, Inc.; Eli Lilly and Company; EuroImmun; F. Hoffmann-La Roche Ltd and its affiliated company Genentech, Inc.; Fujirebio; GE Healthcare; IXICO Ltd.; Janssen Alzheimer Immunotherapy Research & Development, LLC.; Johnson & Johnson Pharmaceutical Research & Development LLC.; Lumosity; Lundbeck; Merck & Co., Inc.; Meso Scale Diagnostics, LLC.; NeuroRx Research; Neurotrack Technologies; Novartis Pharmaceuticals Corporation; Pfizer Inc.; Piramal Imaging; Servier; Takeda Pharmaceutical Company; and Transition Therapeutics. The Canadian Institutes of Health Research is providing funds to support ADNI clinical sites in Canada. Private sector contributions are facilitated by the Foundation for the National Institutes of Health (www.fnih.org). The grantee organization is the Northern California Institute for Research and Education, and the study is coordinated by the Alzheimer’s Therapeutic Research Institute at the University of Southern California. ADNI data are disseminated by the Laboratory for Neuro Imaging at the University of Southern California.
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Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at: http://adni.loni.usc.edu/wp-content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf
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Shan, G., Zhang, H., Barbour, J. et al. Bootstrap confidence intervals for correlation between continuous repeated measures. Stat Methods Appl (2021). https://doi.org/10.1007/s10260-020-00555-1
- Bootstrap confidence interval
- Correction for repeated measures
- Coverage probability
- Longitudinal data
- Proc mixed