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A distribution-free test of parallelism for two-sample repeated measurements. (English) Zbl 1487.62040

Summary: In this paper, we propose a new two-sample distribution-free procedure for testing group-by-time interaction effect in repeated measurements from a linear mixed model setting. The test statistic is based on the maximum difference of partial sums (MDPS) over time points between the two groups. Although the test has a biomedical focus, it can be applied in fields that the study is designed and monitored to be balanced and complete with equal sample sizes as would be generally done in a controlled experiment. The asymptotic null distribution of the test statistic was also derived based on the maxima of Brownian bridge under two different conditions. The simulations revealed that MDPS performed markedly better than the commonly used unstructured multivariate approach (UMA) to profile analysis. However, the empirical powers of MDPS test were convincingly close to those of the best-fitting linear mixed model (LMM).

MSC:

62G10 Nonparametric hypothesis testing
62E20 Asymptotic distribution theory in statistics

Software:

MIXED; S-PLUS; MEMSS
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References:

[1] Billingsley, P., Convergence of Probability Measures (1968), Wiley: Wiley New York · Zbl 0172.21201
[2] Boik, R., Scheffés mixed model for multivariate repeated measures: a relative efficiency evaluation, Comm. Statist. Theory Methods, 20, 1233-1255 (1991) · Zbl 0751.62027
[3] Christensen, W. F.; Rencher, A. C., A comparison of Type I error rates and power levels for seven solutions to the multivariate Behrens-Fisher problem, Comm. Statist. Simulation Comput., 26, 1251-1273 (1997) · Zbl 1100.62575
[4] Davis, C., Statistical Methods for the Analysis of Repeated Measurements (2002), Springer Verlag · Zbl 0985.62002
[5] Dudley, R. M., Real Analysis and Probability (2002), Cambridge Univ. Pr. · Zbl 1023.60001
[6] Everitt, B. S., A Monte Carlo investigation of the robustness of Hotelling’s one-and two-sample T 2 tests, J. Amer. Statist. Assoc., 74, 48-51 (1979) · Zbl 0398.62029
[7] Everitt, B., The analysis of repeated measures: a practical review with examples, Statistician, 44, 113-135 (1995)
[8] Fai, A.; Cornelius, P., Approximate F-tests of multiple degree of freedom hypotheses in generalized least squares analyses of unbalanced split-plot experiments, J. Stat. Comput. Simul., 54, 363-378 (1996) · Zbl 0925.62081
[9] Feller, W., On the Kolmogorov-Smirnov limit theorems for empirical distributions, Ann. Math. Statist., 19, 177-189 (1948) · Zbl 0032.03801
[10] Frison, L.; Pocock, S., Repeated measures in clinical trials: analysis using mean summary statistics and its implications for design, Stat. Med., 11, 1685-1704 (1992)
[11] Heagerty, P. J.; Kurland, B. F., Misspecified maximum likelihood estimates and generalised linear mixed models, Biometrika, 88, 973 (2001) · Zbl 0986.62060
[12] Jacqmin-Gadda, H.; Sibillot, S.; Proust, C.; Molina, J. M.; Thiebaut, R., Robustness of the linear mixed model to misspecified error distribution, Comput. Statist. Data Anal., 51, 5142-5154 (2007) · Zbl 1162.62319
[13] Kenward, M.; Roger, J., Small sample inference for fixed effects from restricted maximum likelihood, Biometrics, 53, 983-997 (1997) · Zbl 0890.62042
[14] Kolmogorov, A., Sulla determinazione empirica delle leggi di probabilita, G. Ist. Ital. Attuari, 4, 1-11 (1933)
[15] Laird, N. M.; Ware, J. H., Random-effects models for longitudinal data, Biometrics, 39, 963-974 (1982) · Zbl 0512.62107
[16] Lim, J.; Wang, X., Response to letter to the Editor by Dr. Vossoughi, Stat. Med., 32, 717 (2013)
[17] Lim, J.; Wang, X.; Lee, S.; Jung, S.-H., A distribution-free test of constant mean in linear mixed effects models, Stat. Med., 27, 3833-3846 (2008)
[18] Litière, S.; Alonso, A.; Molenberghs, G., Type I and Type II error under random-effects misspecification in generalized linear mixed models, Biometrics, 63, 1038-1044 (2007) · Zbl 1274.62822
[19] Littell, R. C.; Pendergast, J.; Natarajan, R., Tutorial in biostatistics: Modelling covariance structure in the analysis of repeated measures data, Stat. Med., 19, 1793-1819 (2000)
[20] Little, R.; Freund, R.; Spector, P., SAS System for Linear Models (1991)
[21] Matthews, J. N.S.; Altman, D. G.; Campbell, M. J.; Royston, P., Analysis of serial measurements in medical research, Br. Med. J., 300, 230-235 (1990)
[22] Nachtsheim, C. J.; Johnson, M. E., A new family of multivariate distributions with applications to Monte Carlo studies, J. Amer. Statist. Assoc., 83, 984-989 (1988) · Zbl 0669.62030
[23] Newrnan, R. M.; Smitha, W. B.; Speed, F. M., Properties of profile parallelism tests in repeated measures designs, Comm. Statist. Simulation Comput., 28, 1073-1098 (1999) · Zbl 0968.62531
[25] Park, T., A comparison of the generalized estimating equation approach with the maximum likelihood approach for repeated measurements, Stat. Med., 12, 1723-1732 (1993)
[26] Park, T.; Park, J.; Davis, C., Effects of covariance model assumptions on hypothesis tests for repeated measurements: analysis of ovarian hormone data and pituitary pteryomaxillary distance data, Stat. Med., 20, 2441-2453 (2001)
[27] Patterson, H.; Thompson, R., Recovery of inter-block information when block sizes are unequal, Biometrika, 58, 545-554 (1971) · Zbl 0228.62046
[29] Pinheiro, J.; Bates, D., Mixed-Effects Models in S and S-PLUS (2000), Springer: Springer New York, NY · Zbl 0953.62065
[30] Rencher, A. C., Methods of Multivariate Analysis (1995), Wiley: Wiley New York · Zbl 0836.62039
[31] Rencher, A. C., Multivariate Statistical Inference and Applications (1998), Wiley: Wiley New York · Zbl 0932.62065
[33] Smirnov, N., Table for estimating the goodness of fit of empirical distributions, Ann. Math. Statist., 19, 279-281 (1948) · Zbl 0031.37001
[34] Tandona, P. K.; Moeschberger, M. L., Comparison of nonparametric and parametric methods in repeated measures designs—A simulation study, Comm. Statist. Simulation Comput., 18, 777-792 (1989) · Zbl 0695.62050
[35] Verbeke, G.; Lesaffre, E., The effect of misspecifying the random-effects distribution in linear mixed models for longitudinal data, Comput. Statist. Data Anal., 23, 541-556 (1997) · Zbl 0900.62374
[36] Vossoughi, M., Comments on ‘A distribution-free test of constant mean in linear mixed effects models’, Stat. Med., 32, 714-716 (2013)
[37] Vossoughi, M.; Ayatollahi, S. M.T.; Towhidi, M.; Ketabchi, F., On summary measure analysis of linear trend repeated measures data: performance comparison with two competing methods, BMC Med. Res. Methodol., 12, 33 (2012)
[38] Ware, J. H., Linear models for the analysis of longitudinal studies, Amer. Statist., 95-101 (1985)
[39] Wishart, J., Growth-rate determination in nutrition studies with the bacon pig, and their analysis, Biometrika, 30, 16-28 (1938)
[40] Zhang, D.; Davidian, M., Linear mixed models with flexible distributions of random effects for longitudinal data, Biometrics, 57, 795-802 (2001) · Zbl 1209.62087
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