Bootstrap confidence intervals for correlation between continuous repeated measures


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.

This is a preview of subscription content, access via your institution.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8


  1. Aarts E, Verhage M, Veenvliet JV, Dolan CV, Van Der Sluis S (2014) A solution to dependency: using multilevel analysis to accommodate nested data. Nat Neurosci 17(4):491–496

    Article  Google Scholar 

  2. Bakdash JZ, Marusich LR (2017) Repeated measures correlation. Front Psychol 8(MAR):456

    Article  Google Scholar 

  3. Bernick C, Zetterberg H, Shan G, Banks S, Blennow K (2018) Longitudinal performance of plasma neurofilament light and tau in professional fighters: the professional fighters brain health study. J Neurotrauma 35(20):2351–2356

    Article  Google Scholar 

  4. Bland JM, Altman DG (1995) Calculating correlation coefficients with repeated observations: part 1—correlation within subjects. BMJ (Clin Res Ed) 310(6977):446

    Article  Google Scholar 

  5. Bland JM, Altman DG (1995) Calculating correlation coefficients with repeated observations: part 2—correlation between subjects. BMJ 310(6980):446

    Article  Google Scholar 

  6. Casella G, Berger RL (2002) Statistical inference, 2nd edn. Thomson Learning, Belmont, CA

    Google Scholar 

  7. Crowder M (1995) On the use of a working correlation matrix in using generalised linear models for repeated measures. Biometrika 82(2):407–410

    Article  Google Scholar 

  8. Cummings J (2018) Lessons learned from Alzheimer disease: clinical trials with negative outcomes. Clin Transl Sci 11(2):147–152

    Article  Google Scholar 

  9. Efron B (1985) Bootstrap confidence intervals for a class of parametric problems. Biometrika 72(1):45–58

    MathSciNet  Article  Google Scholar 

  10. Efron B, Tibshirani RJ (1994) An introduction to the bootstrap, softcover edition edn. Monographs on statistics and applied probability. Chapman and Hall/CRC, London

    Google Scholar 

  11. Hall P (1986) On the number of bootstrap simulations required to construct a confidence interval. Ann Stat 14(4):1453–1462

    MathSciNet  Article  Google Scholar 

  12. Hamlett A, Ryan L, Wolfinger R (2004) On the use of PROC MIXED to estimate correlation in the presence of repeated measures. In: Proceedings of statistics and data analysis, pp 129–198

  13. Irimata K, Wakim P, Li X (2018) Estimation of correlation coefficient in data with repeated measures. In: SAS paper, p 2424

  14. Lam M, Webb KA, Donnell DE(1999) Correlation between two variables in repeated measures. In: Proceedings-American statistical association biometrics section, pp 213–218

  15. Pearson K (1900) On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philos Mag Ser 5 50(302):157–175

    Article  Google Scholar 

  16. Roy A (2006) Estimating correlation coefficient between two variables with repeated observations using mixed effects model. Biom J Biom Z 48(2):286–301

    MathSciNet  Article  Google Scholar 

  17. Shan G (2013) A note on exact conditional and unconditional tests for Hardy–Weinberg equilibrium. Hum Hered 76(1):10–17

    Article  Google Scholar 

  18. Shan G, Banks S, Miller JB, Ritter A, Bernick C, Lombardo J, Cummings JL (2018) Statistical advances in clinical trials and clinical research. Alzheimer’s Dement Transl Res Clin Interv 4:366–371

    Article  Google Scholar 

  19. Shan G, Bayram E, Caldwell JZK, Miller JB, Shen JJ, Gerstenberger S (2020) Partial correlation coefficient for a study with repeated measurements. Stat Biopharm Res.

  20. Shan G, Ma C (2014) Exact methods for testing the equality of proportions for binary clustered data from otolaryngologic studies. Stat Biopharm Res 6(1):115–122

    Article  Google Scholar 

  21. Shan G, Vexler A, Wilding GE, Hutson AD (2011) Simple and exact empirical likelihood ratio tests for normality based on moment relations. Commun Stat Simul Comput 40(1):129–146

    MathSciNet  Article  Google Scholar 

  22. Wilcox Rand (2011) Modern statistics for the social and behavioral sciences: a practical introduction. CRC Press, Boca Raton

    Google Scholar 

  23. Wilcox RR (1996) Confidence intervals for the slope of a regression line when the error term has nonconstant variance. Comput Stat Data Anal 22(1):89–98

    Article  Google Scholar 

Download references


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 ( 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.

Author information




Corresponding author

Correspondence to Guogen Shan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database ( 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:

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shan, G., Zhang, H., Barbour, J. et al. Bootstrap confidence intervals for correlation between continuous repeated measures. Stat Methods Appl (2021).

Download citation


  • Bootstrap confidence interval
  • Correction for repeated measures
  • Coverage probability
  • Longitudinal data
  • Proc mixed