## Electronic Journal of Statistics

### Accurate inference for repeated measures in high dimensions

#### Abstract

This paper proposes inferential methods for high-dimensional repeated measures in factorial designs. High-dimensional refers to the situation where the dimension is growing with sample size such that either one could be larger than the other. The most important contribution relates to high-accuracy of the methods in the sense that p-values, for example, are accurate up to the second-order. Second-order accuracy in sample size as well as dimension is achieved by obtaining asymptotic expansion of the distribution of the test statistics, and estimation of the parameters of the approximate distribution with second-order consistency. The methods are presented in a unified and succinct manner that it covers general factorial designs as well as any comparisons among the cell means. Expression for asymptotic powers are derived under two reasonable local alternatives. A simulation study provides evidence for a gain in accuracy and power compared to limiting distribution approximations and other competing methods for high-dimensional repeated measures analysis. The application of the methods are illustrated with a real-data from Electroencephalogram (EEG) study of alcoholic and control subjects.

#### Article information

Source
Electron. J. Statist., Volume 13, Number 2 (2019), 4916-4944.

Dates
First available in Project Euclid: 10 December 2019

https://projecteuclid.org/euclid.ejs/1575946866

Digital Object Identifier
doi:10.1214/19-EJS1644

Mathematical Reviews number (MathSciNet)
MR4040758

Zentralblatt MATH identifier
07147368

Subjects
Primary: 62H10: Distribution of statistics
Secondary: 62H15: Hypothesis testing

#### Citation

Kong, Xiaoli; Harrar, Solomon W. Accurate inference for repeated measures in high dimensions. Electron. J. Statist. 13 (2019), no. 2, 4916--4944. doi:10.1214/19-EJS1644. https://projecteuclid.org/euclid.ejs/1575946866

#### References

• Bai, Z. and Saranadasa, H. (1996). Effect of high dimension: By an example of a two sample problem., Statistica Sinica, 6(2):311–329.
• Bock, R. (1963)., Multivariate analysis of variance of repeated measurements. (Problems in measuring change. Editor C. W. Harris). University of Wisconsin Press, Madison, Wisconsin.
• Box, G. E. P. (1954). Some theorems on quadratic forms applied in the study of analysis of variance problems, I. Effect of inequality of variance in the one-way classification., The Annals of Mathematical Statistics, 25(2):290–302.
• Brunner, E., Bathke, A. C., and Placzek, M. (2012). Estimation of box’s $\varepsilon$ for low- and high-dimensional repeated measures designs with unequal covariance matrices., Biometrical Journal, 54(3):301–316.
• Brunner, E., Dette, H., and Munk, A. (1997). Box-type approximations in nonparametric factorial designs., Journal of the American Statistical Association, 92(440):1494–1502.
• Brunner, E., Konietschke, F., Pauly, M., and Puri, M. L. (2017). Rank-based procedures in factorial designs: hypotheses about non-parametric treatment effects., Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79(5):1463–1485.
• Brunner, E., Munzel, U., and Puri, M. L. (1999). Rank-score tests in factorial designs with repeated measures., Journal of Multivariate Analysis, 70(2):286–317.
• Cai, T. T., Liu, W., and Xia, Y. (2014). Two-sample test of high dimensional means under dependence., Journal of the Royal Statistical Society: Series B (Statistical Methodology), 76(2):349–372.
• Chen, S. X. and Qin, Y. L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing., The Annals of Statistics, 38(2):808–835.
• Chi, Y. Y., Gribbin, M., Lamers, Y., Gregory, J. F., and Muller, K. E. (2012). Global hypothesis testing for high-dimensional repeated measures outcomes., Statistics in Medicine, 31(8):724–742.
• Collier, R. O., Baker, F. B., Mandeville, G. K., and Hayes, T. F. (1967). Estimates of test size for several test procedures based on conventional variance ratios in the repeated measures design., Psychometrika, 32(3):339–353.
• Fujikoshi, Y. (1973). Asymptotic formulas for the distributions of three statistics for multivariate linear hypothesis., Annals of the Institute of Statistical Mathematics, 25(1):423–437.
• Geisser, S. and Greenhouse, S. W. (1958). An extension of box’s results on the use of the $F$ distribution in multivariate analysis., The Annals of Mathematical Statistics, 29(3):885–891.
• Happ, M., Harrar, S. W., and Bathke, A. C. (2015). Inference for low- and high-dimensional multi-group repeated measures designs with unequal covariance matrices., Biometrical Journal, 58(4):810–830.
• Harrar, S. W. and Kong, X. (2016). High-dimensional multivariate repeated measures analysis with unequal covariance matrices., Journal of Multivariate Analysis, 145:1–21.
• Hill, G. W. and Davis, A. W. (1968). Generalized asymptotic expansions of cornish-fisher type., The Annals of Mathematical Statistics, 39(4):1264–1273.
• Huynh, H. (1978). Some approximate tests for repeated measurement designs., Psychometrika, 43(2):161–175.
• Huynh, H. and Feldt, L. S. (1970). Conditions under which mean square ratios in repeated measurements designs have exact $F$-distributions., Journal of the American Statistical Association, 65(332):1582–1589.
• Huynh, H. and Feldt, L. S. (1976). Estimation of the box correction for degrees of freedom from sample data in randomized block and split-plot designs., Journal of Educational Statistics, 1(1):69–82.
• Hyodo, M., Takahashi, S., and Nishiyama, T. (2014). Multiple comparisons among mean vectors when the dimension is larger than the total sample size., Communications in Statistics – Simulation and Computation, 43(10):2283–2306.
• Katayama, S., Kano, Y., and Srivastava, M. S. (2013). Asymptotic distributions of some test criteria for the mean vector with fewer observations than the dimension., Journal of Multivariate Analysis, 116:410–421.
• Konietschke, F., Bathke, A. C., Harrar, S. W., and Pauly, M. (2015). Parametric and nonparametric bootstrap methods for general manova., Journal of Multivariate Analysis, 140:291–301.
• Lecoutre, B. (1991). A correction for the formula approximate test in repeated measures designs with two or more independent groups., Journal of Educational and Behavioral Statistics, 16(4):371–372.
• Li, J. and Chen, S. X. (2012). Two sample tests for high-dimensional covariance matrices., The Annals of Statistics, 40(2):908–940.
• Maxwell, S. E. and Arvey, R. D. (1982). Small sample profile analysis with many variables., Psychological Bulletin, 92(3):778–785.
• Pauly, M., Ellenberger, D., and Brunner, E. (2015). Analysis of high-dimensional one group repeated measures designs., Statistics, 49(6):1243–1261.
• Rencher, A. C. and Christensen, W. F. (2012)., Methods of multivariate analysis. John Wiley & Sons, United States, 3rd edition.
• Shiryaev, A. and Chibisov, D. (2016)., Probability-1. Graduate Texts in Mathematics. Springer New York.
• Srivastava, M. S. (2005). Some tests concerning the covariance matrix in high dimensional data., Journal of the Japan Statistical Society, 35(2):251–272.
• Stoloff, P. H. (1970). Correcting for heterogeneity of covariance for repeated measures designs of the analysis of variance., Educational and Psychological Measurement, 30(4):909–924.
• Takahashi, S. and Shutoh, N. (2016). Tests for parallelism and flatness hypotheses of two mean vectors in high-dimensional settings., Journal of Statistical Computation and Simulation, 86(6):1150–1165.
• Wang, H. and Akritas, M. G. (2010a). Inference from heteroscedastic functional data., Journal of Nonparametric Statistics, 22(2):149–168.
• Wang, H. and Akritas, M. G. (2010b). Rank test for heteroscedastic functional data., Journal of Multivariate Analysis, 101(8):1791–1805.
• Wang, H., Higgins, J., and Blasi, D. (2010). Distribution-free tests for no effect of treatment in heteroscedastic functional data under both weak and long range dependence., Statistics & Probability Letters, 80(5-6):390–402.
• Watamori, Y. (1990). On the moments of traces of wishart and inverted wishart matrices., South African Statistical Journal, 24(2):153–176.
• Yang, X. M., Yang, X. Q., and Teo, K. L. (2001). A matrix trace inequality., Journal of Mathematical Analysis and Applications, 263(1):327–331.
• Zhang, C., Bai, Z., Hu, J., and Wang, C. (2018). Multi-sample test for high-dimensional covariance matrices., Communications in Statistics – Theory and Methods, 48(13):3161–3177.