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Testing marginal homogeneity of a continuous bivariate distribution with possibly incomplete paired data. (English) Zbl 1437.62162

Let \[ \left[\frac{X_{1,1}}{X_{2,1}}\right],\ldots,\left[\frac{X_{1,n}}{X_{2,n}} \right] \] be a sample of size \(n \in \mathbb{N}\) of independent and identically distributed bivariate random vectors with unknown bivariate distribution function \(F\). Denote by \(F_{1}\) and \(F_{2}\) the first and second marginal distribution of \(F\), respectively.
The author considers the full nonparametric testing problem of marginal homogeneity \[ \mathcal{H}: F_{1} = F_{2}, \quad \mathcal{K}: F_{1} \ne F_{2} \] with possibly missing data. He assumes that the missing components are missing completely at random (MCAR).
From the author’s abstract: “Applying the well-known two-sample Crámer-von-Mises distance to the remaining data, we determine the limiting null distribution of our test statistic in this situation. It is seen that a new resampling approach is appropriate for the approximation of the unknown null distribution.We prove that the resulting test asymptotically reaches the significance level and is consistent. Properties of the test under local alternatives are pointed out as well. Simulations investigate the quality of the approximation and the power of the new approach in the finite sample case. As an illustration, we apply the test to real data sets.”

MSC:

62G10 Nonparametric hypothesis testing
62G09 Nonparametric statistical resampling methods
62H15 Hypothesis testing in multivariate analysis
62D10 Missing data

Software:

copula; CRAN
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Full Text: DOI

References:

[1] Akritas, MG; Antoniou, ES; Osgood, DW, A nonparametric approach to matched pairs with missing data, Sociol Methods Res, 30, 425-454 (2002)
[2] Akritas, MG; Antoniou, ES; Kuha, J., Nonparametric analysis of factorial designs with random missingness: bivariate data, J Am Stat Assoc, 101, 1513-1526 (2006) · Zbl 1171.62339
[3] Amro, L.; Pauly, M., Permuting incomplete paired data: a novel exact and asymptotic correct randomization test, J Stat Comput Simul, 87, 1148-1159 (2016) · Zbl 07191993
[4] Amro L, Konietschke F, Pauly M (2018) Multiplication-combination tests for incomplete paired data. arxiv:1801.08821
[5] Anderson, TW, On the distribution of the two-sample Cramer-von Mises criterion, Ann Math Stat, 33, 1148-1159 (1962) · Zbl 0116.37601
[6] Bhoj, DS, Testing equality of means of correlated variates with missing observations on both responses, Biometrika, 65, 225-228 (1978) · Zbl 0371.62084
[7] Bhoj, DS, On difference of means of correlated variates with incomplete data on both responses, J Stat Comput Simul, 19, 275-289 (1984)
[8] Bhoj, DS, On testing equality of means of correlated variates with incomplete data, Biometrical J, 29, 589-594 (1987) · Zbl 0618.62052
[9] Bhoj, DS, On comparing correlated means in the presence of incomplete data, Biometrical J, 31, 279-288 (1989)
[10] Bhoj, DS, Testing equality of means in the presence of correlation and missing data, Biometrical J, 33, 63-72 (1991)
[11] Derrick, B.; Russ, B.; Toher, D.; White, P., Test statistics for the comparison of means for two samples which include both paired observations and independent observations, J Mod Appl Stat Methods, 16, 137-157 (2017)
[12] Dubnicka, SR; Blair, RC; Hettmansperger, TP, Rank-based procedures for mixed paired and two-sample designs, J Mod Appl Stat Methods, 1, 32-41 (2002)
[13] Dudley RM (1984) A course on empirical processes. Lecture Notes in Mathematics 1097. Springer, New York, pp 1-142 · Zbl 0554.60029
[14] Dunu ES (1994) Comparing the powers of several proposed tests for testing the equality of the means of two populations when some data are missing. Ph.D. thesis, University of North Texas
[15] Einsporn, RL; Habtzghi, D., Combining paired and two-sample data using a permutation test, J Data Sci, 11, 767-779 (2013)
[16] Ekbohm, G., On comparing means in the paired case with incomplete data on both responses, Biometrika, 63, 299-304 (1976) · Zbl 0342.62010
[17] Ekbohm, G., On testing equality of means in the paired case with incomplete data on both responses, Biometrical J, 23, 251-259 (1981) · Zbl 0484.62037
[18] Fong, Y.; Huang, Y.; Lemos, MP; Mcelrath, MJ, Rank-based two-sample tests for paired data with missing values, Biostatistics, 19, 281-294 (2018)
[19] Gänßler, P.; Ziegler, K., A uniform law of large numbers for set-indexed processes with applications to empirical and partial-sum processes, Probab Banach Spaces, 9, 385-400 (1994) · Zbl 0812.60029
[20] Gao, X., A nonparametric procedure for the two-factor mixed model with missing data, Biometrical J, 49, 774-788 (2007) · Zbl 1442.62372
[21] Gibbons, JD; Chakraborti, S., Nonparametric statistical inference (2011), Boca Raton: CRC Press, Boca Raton · Zbl 1278.62004
[22] Guo, B.; Yuan, Y., A comparative review of methods for comparing means using partially paired data, Stat Methods Med Res, 26, 1323-1340 (2017)
[23] Hamdan, MA; Khuri, AI; Crews, SL, A test for equality of means of two correlated normal variates with missing data on both responses, Biometrical J, 20, 667-674 (1978) · Zbl 0402.62030
[24] Howard AG (2012) Missing data in non-parametric tests of correlated data. Ph.D. thesis, The University of North Carolina at Chapel Hill
[25] Kiefer, J., K-sample analogues of the Kolmogorov-Smirnov and Cramer-V. Mises tests, Ann Math Stat, 30, 420-447 (1959) · Zbl 0134.36707
[26] Konietschke, F.; Harrar, SW; Lange, K.; Brunner, E., Ranking procedures for matched pairs with missing data—asymptotic theory and a small sample approximation, Comput Stat Data Anal, 56, 1090-1102 (2012) · Zbl 1241.62066
[27] Koul, HK; Müller, UU; Schick, A., The transfer principle: a tool for complete case analysis, Ann Stat, 40, 3031-3049 (2013) · Zbl 1296.62040
[28] Kuan, PF; huang, B., A simple and robust method for partially matched samples using the p-values pooling approach, Stat Med, 32, 3247-3259 (2013)
[29] Lin, P-E; Stivers, LE, On difference of means with incomplete data, Biometrika, 61, 325-334 (1975) · Zbl 0283.62026
[30] Little, RJA; Rubin, DB, Statistical analysis with missing data (2014), Hoboken: Wiley, Hoboken
[31] Looney, S.; Jones, P., A method for comparing two normal means using combined samples of correlated and uncorrelated data, Stat Med, 22, 1601-1610 (2003)
[32] Martinez-Camblor, P.; Corral, N.; de la Hera, J., Hypothesis test for paired samples in the presence of missing data, J Appl Stat, 40, 76-87 (2012) · Zbl 1514.62745
[33] Maritz, JM, A permutation paired test allowing for missing values, Aust N Z J Stat, 37, 153-159 (1995) · Zbl 0850.62369
[34] Modarres, R., Tests of bivariate exchangeability, Int Stat Rev, 76, 203-213 (2008)
[35] Morrison, DF, A test for equality of means of correlated variates with missing data on one response, Biometrika, 60, 101-105 (1973) · Zbl 0256.62026
[36] Rempala GA, Looney SW (2006) Asymptotic properties of a two sample randomized test for partially dependent data. J Stat Plan Inference 68-89 · Zbl 1078.62042
[37] Samawi, HM; Vogel, R., Notes on two sample tests for partially correlated (paired) data, J Appl Stat, 41, 109-117 (2014) · Zbl 1514.62841
[38] Samawi, HM; Vogel, R., On some nonparametric tests for partially observed correlated data: proposing new tests, J Stat Theory Appl, 14, 131-155 (2015)
[39] Student, The probable error of a mean, Biometrika, 6, 1-25 (1908) · Zbl 1469.62201
[40] Tang X (2007) New test statistic for comparing medians with incomplete paired data. Ph.D. thesis, University of Pittsburgh
[41] The Comprehensive R Archive Network (2018). https://cran.r-project.org/web/packages/copula/copula.pdf
[42] Uddin, N.; Hasan, MS, Testing equality of two normal means using combined samples of paired and unpaired data, Commun Stat Comput Simul, 46, 2430-2446 (2017) · Zbl 1364.62131
[43] van der Vaart, A.; Wellner, JA, Weak convergence and empirical processes (1996), New York: Springer, New York · Zbl 0862.60002
[44] Wilcoxon, F., Individual comparisons by ranking methods, Biometrics Bull, 1, 80-83 (1945)
[45] Woolson, R.; Leeper, J.; Cole, J.; Clarke, W., A Monte Carlo investigation of a statistic for a bivariate missing data problem, Commun Stat Theory Methods, 5, 681-688 (1976) · Zbl 0333.62021
[46] Xu, J.; Harrar, SW, Accurate mean comparisons for paired samples with missing data: an application to a smokingcessation trial, Biometrical J, 54, 281-295 (2012) · Zbl 1242.62125
[47] Yu, D.; Lim, Y.; Liang, F.; Kim, K.; Kim, BS; Jang, W., Permutation test for incomplete paired data with application to cDNA microarray data, Comput Stat Data Anal, 56, 510-521 (2012) · Zbl 1239.62053
[48] Ziegler, K., Functional central limit theorems for triangular arrays of function-indexed processes under uniformly integrable entropy conditions, J Multivar Anal, 62, 233-272 (1997) · Zbl 0895.60035
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