×

Permuting incomplete paired data: a novel exact and asymptotic correct randomization test. (English) Zbl 07191993

Summary: Various statistical tests have been developed for testing the equality of means in matched pairs with missing values. However, most existing methods are commonly based on certain distributional assumptions such as normality, 0-symmetry or homoscedasticity of the data. The aim of this paper is to develop a statistical test that is robust against deviations from such assumptions and also leads to valid inference in case of heteroscedasticity or skewed distributions. This is achieved by applying a clever randomization approach to handle missing data. The resulting test procedure is not only shown to be asymptotically correct but is also finitely exact if the distribution of the data is invariant with respect to the considered randomization group. Its small sample performance is further studied in an extensive simulation study and compared to existing methods. Finally, an illustrative data example is analysed.

MSC:

62G09 Nonparametric statistical resampling methods
62G10 Nonparametric hypothesis testing

Software:

R
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Little RJ, Rubin DB. Statistical analysis with missing data. New York: John Wiley and Sons; 2014. [Google Scholar] · Zbl 0665.62004
[2] Verbeke G, Molenberghs G. Linear mixed models for longitudinal data. New York: Springer Science & Business Media; 2009. [Google Scholar] · Zbl 1162.62070
[3] Akritas MG, Kuha J, Osgood DW. A nonparametric approach to matched pairs with missing data. Sociol Methods Res. 2002;30(3):425-454. [Crossref], [Web of Science ®], [Google Scholar]
[4] Lin P-E, Stivers LE. On difference of means with incomplete data. Biometrika. 1974;61:325-334. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 0283.62026
[5] Ekbohm G. On comparing means in the paired case with incomplete data on both responses. Biometrika. 1976;63(2):299-304. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 0342.62010
[6] Bhoj DS. Testing equality of means of correlated variates with missing observations on both responses. Biometrika. 1978;65(1):225-228. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 0371.62084
[7] Looney SW, Jones PW. A method for comparing two normal means using combined samples of correlated and uncorrelated data. Stat Med. 2003;22(9):1601-1610. [Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[8] Kim BS, Kim I, Lee S, et al. Statistical methods of translating microarray data into clinically relevant diagnostic information in colorectal cancer. Bioinformatics. 2005;21(4):517-528. [Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[9] Samawi HM, Vogel R. Notes on two sample tests for partially correlated (paired) data. J Appl Stat. 2014;41(1):109-117. [Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1514.62841
[10] Pesarin F. Multivariate permutation tests: with applications in biostatistics. Chichester: Wiley; 2001. [Google Scholar] · Zbl 0972.62037
[11] Good PI. Extensions of the concept of exchangeability and their applications. J Mod Appl Stat Methods. 2002;1(2):34. [Crossref], [Google Scholar]
[12] Good P. Permutation, parametric and bootstrap tests of hypotheses. New York: Springer; 2005. [Google Scholar] · Zbl 1076.62043
[13] Lehmann EL, Romano JP. Testing statistical hypotheses. New York: Springer Science & Business Media; 2006. [Google Scholar] · Zbl 1076.62018
[14] Pesarin F, Salmaso L. Permutation tests for complex data: theory, applications and software. New York: John Wiley & Sons; 2010. [Crossref], [Google Scholar] · Zbl 1359.62158
[15] Winkler AM, Webster MA, Vidaurre D , et al. Multi-level block permutation. NeuroImage. 2015;123:253-268. [Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[16] Janssen A. Testing nonparametric statistical functionals with applications to rank tests. J Statist Plann Inference. 1999;81(1):71-93. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 0951.62037
[17] Konietschke F, Pauly M. Bootstrapping and permuting paired t-test type statistics. Stat Comput. 2014;24(3):283-296. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1325.62097
[18] Janssen A. Studentized permutation tests for non-iid hypotheses and the generalized Behrens-Fisher problem. Statist Probab Lett. 1997;36(1):9-21. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1064.62526
[19] Janssen A. Nonparametric symmetry tests for statistical functionals. Math Methods Stat. 1999;8(3):320-343. [Google Scholar] · Zbl 1103.62343
[20] Neubert K, Brunner E. A studentized permutation test for the nonparametric Behrens-Fisher problem. Comput Statist Data Anal. 2007;51(10):5192-5204. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1162.62351
[21] Pauly M. Discussion about the quality of F-ratio resampling tests for comparing variances. TEST. 2011;20(1):163-179. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1331.62245
[22] Omelka M, Pauly M. Testing equality of correlation coefficients in two populations via permutation methods. J Statist Plann Inference. 2012;142(6):1396-1406. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1242.62036
[23] Chung E, Romano JP. Exact and asymptotically robust permutation tests. Ann. Statist. 2013;41(2):484-507. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1267.62064
[24] DiCiccio CJ, Romano JP. Robust permutation tests for correlation and regression coefficients; J Am Stat Assoc., to appear (http://dx.doi.org/10.1080/01621459.2016.1202117). [Google Scholar] · Zbl 1524.62200 · doi:10.1080/01621459.2016.1202117
[25] Pauly M, Brunner E, Konietschke F. Asymptotic permutation tests in general factorial designs. J R Stat Soc Ser B Statist Methodol. 2015;77(2):461-473. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1414.62339
[26] Chung E, Romano JP. Asymptotically valid and exact permutation tests based on two-sample U-statistics. J Statist Plann Inference. 2016;168:97-105. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1333.62124
[27] Maritz JS. A permutation paired test allowing for missing values. Aust J Stat. 1995;37(2):153-159. [Crossref], [Google Scholar] · Zbl 0850.62369
[28] Yu D, Lim J, Liang F, et al. Permutation test for incomplete paired data with application to cDNA microarray data. Comput Statist Anal. 2012;56(3):510-521. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1239.62053
[29] Janssen A. Resampling Student’s t-type statistics. Ann Inst Statist Math. 2005;57(3):507-529. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1095.62050
[30] Janssen A, Pauls T. How do bootstrap and permutation tests work? Ann Statist. 2003;31(3):768-806. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1028.62027
[31] Placzek M, Konietschke F, Pauly M. Studentisierte Permutationstests für verbundene und unverbundene 2-Stichprobenprobleme. KSFE 2014; 2014. [Google Scholar]
[32] Perlman MD, Wu L, et al. The emperor’s new tests. Statist Sci. 1999;14(4):355-369. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1059.62515
[33] Ekbohm G. On testing equality of means in the paired case with incomplete data on both responses. Biom J. 1981;23(3):251-259. [Crossref], [Google Scholar] · Zbl 0484.62037
[34] Woolson RF, Leeper JD, Cole JWL, et al. A monte carlo investigation of a statistic for a bivariate missing data problem. Comm Statist Theory Methods. 1976;5(7):681-688. [Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 0333.62021
[35] Hamdan MA, Khuri AI, Crews SL. A test for equality of means of two correlated normal variates with missing data on both responses. Biom J. 1978;20:667-674. [Crossref], [Google Scholar] · Zbl 0402.62030
[36] Bhoj DS. On comparing correlated means in the presence of incomplete data. Biom J. 1989;31(3):279-288. [Crossref], [Web of Science ®], [Google Scholar]
[37] Uddin N, Hasan M. Testing equality of two normal means using combined samples of paired and unpaired data. Communications in Statistics-Simulation and Computation. 2015; (just-accepted). [Taylor & Francis Online], [Google Scholar] · Zbl 1364.62131
[38] Bhoj DS. On difference of means of correlated variates with incomplete data on both responses. J Stat Comput Simul. 1984;19(4):275-289. [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
[39] Bhoj DS. Testing equality of means in the presence of correlation and missing data. Biom J. 1991;33(1):63-72. [Crossref], [Web of Science ®], [Google Scholar]
[40] Dunu ES. Comparing the powers of several proposed tests for testing the equality of the means of two populations when some data are missing [dissertation]. University of North Texas; 1994. [Google Scholar]
[41] Guo B, Yuan Y. A comparative review of methods for comparing means using partially paired data. Stat Methods Med Res, to appear (http://dx.doi.org/10.1177/0962280215577111). [Google Scholar] · doi:10.1177/0962280215577111
[42] R Core Team. R: a language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing. 2015. Available from: https://www.R-project.org/[Google Scholar]
[43] Karnofsky DA, Burchenal JH. The clinical evaluation of chemotherapeutic agents in cancer. In: MacLeod CM, editor. Evaluation of Chemotherapeutic Agents. New York: Columbia University Press; 1949. p. 191-205. [Google Scholar]
[44] Hermann C, Looney S. The effectiveness of symptom management in hospice patients during the last seven days of life. J Hosp Palliat Nurs. 2001;3(3):88-96. [Crossref], [Google Scholar]
[45] Rempala GA, Looney SW. Asymptotic properties of a two sample randomized test for partially dependent data. J Statist Plann Inference. 2006;136(1):68-89. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1078.62042
[46] Konietschke F, Pauly M. A Studentized permutation test for the nonparametric Behrens-Fisher problem in paired data. Electron J Statist. 2012;6:1358-1372. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1334.62084
[47] Harrar SW, Gupta AK. Asymptotic expansion for the null distribution of the f-statistic in one-way anova under non-normality. Ann Inst Statist Math. 2007;59(3):531-556. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1122.62011
[48] Xu J, Harrar SW. Accurate mean comparisons for paired samples with missing data: an application to a smoking-cessation trial. Biom J. 2012;54(2):281-295. [Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1242.62125
[49] Pauly M. Eine Analyse bedingter Tests mit bedingten Zentralen Grenzwertsätzen für Resampling-Statistiken [inaugural Dissertation]. University of Duesseldorf; 2009. [Google Scholar]
[50] Janssen A, Völker D. Most powerful conditional tests. Statist Decisions. 2007;25(1):41-62. [Google Scholar] · Zbl 1130.62042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.