Skip to main content

Advertisement

SpringerLink
Log in

Testing marginal homogeneity of a continuous bivariate distribution with possibly incomplete paired data

  • Published:
Metrika Aims and scope Submit manuscript

Abstract

We discuss the testing problem of homogeneity of the marginal distributions of a continuous bivariate distribution based on a paired sample with possibly missing components (missing completely at random). 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Akritas MG, Antoniou ES, Osgood DW (2002) A nonparametric approach to matched pairs with missing data. Sociol Methods Res 30:425–454

    MathSciNet  Google Scholar 

  • Akritas MG, Antoniou ES, Kuha J (2006) Nonparametric analysis of factorial designs with random missingness: bivariate data. J Am Stat Assoc 101:1513–1526

    MathSciNet  MATH  Google Scholar 

  • Amro L, Pauly M (2016) Permuting incomplete paired data: a novel exact and asymptotic correct randomization test. J Stat Comput Simul 87:1148–1159

    MathSciNet  Google Scholar 

  • Amro L, Konietschke F, Pauly M (2018) Multiplication-combination tests for incomplete paired data. arxiv:1801.08821

  • Anderson TW (1962) On the distribution of the two-sample Cramer–von Mises criterion. Ann Math Stat 33:1148–1159

    MathSciNet  MATH  Google Scholar 

  • Bhoj DS (1978) Testing equality of means of correlated variates with missing observations on both responses. Biometrika 65:225–228

    MATH  Google Scholar 

  • Bhoj DS (1984) On difference of means of correlated variates with incomplete data on both responses. J Stat Comput Simul 19:275–289

    Google Scholar 

  • Bhoj DS (1987) On testing equality of means of correlated variates with incomplete data. Biometrical J 29:589–594

    MathSciNet  MATH  Google Scholar 

  • Bhoj DS (1989) On comparing correlated means in the presence of incomplete data. Biometrical J 31:279–288

    MathSciNet  Google Scholar 

  • Bhoj DS (1991) Testing equality of means in the presence of correlation and missing data. Biometrical J 33:63–72

    MathSciNet  Google Scholar 

  • Derrick B, Russ B, Toher D, White P (2017) 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

    Google Scholar 

  • Dubnicka SR, Blair RC, Hettmansperger TP (2002) Rank-based procedures for mixed paired and two-sample designs. J Mod Appl Stat Methods 1:32–41

    Google Scholar 

  • Dudley RM (1984) A course on empirical processes. Lecture Notes in Mathematics 1097. Springer, New York, pp 1–142

  • 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

  • Einsporn RL, Habtzghi D (2013) Combining paired and two-sample data using a permutation test. J Data Sci 11:767–779

    MathSciNet  Google Scholar 

  • Ekbohm G (1976) On comparing means in the paired case with incomplete data on both responses. Biometrika 63:299–304

    MATH  Google Scholar 

  • Ekbohm G (1981) On testing equality of means in the paired case with incomplete data on both responses. Biometrical J 23:251–259

    MATH  Google Scholar 

  • Fong Y, Huang Y, Lemos MP, Mcelrath MJ (2018) Rank-based two-sample tests for paired data with missing values. Biostatistics 19:281–294

    MathSciNet  Google Scholar 

  • Gänßler P, Ziegler K (1994) A uniform law of large numbers for set-indexed processes with applications to empirical and partial-sum processes. Probab Banach Spaces 9:385–400

    MathSciNet  MATH  Google Scholar 

  • Gao X (2007) A nonparametric procedure for the two-factor mixed model with missing data. Biometrical J 49:774–788

    MathSciNet  Google Scholar 

  • Gibbons JD, Chakraborti S (2011) Nonparametric statistical inference. CRC Press, Boca Raton

    MATH  Google Scholar 

  • Guo B, Yuan Y (2017) A comparative review of methods for comparing means using partially paired data. Stat Methods Med Res 26:1323–1340

    MathSciNet  Google Scholar 

  • Hamdan MA, Khuri AI, Crews SL (1978) A test for equality of means of two correlated normal variates with missing data on both responses. Biometrical J 20:667–674

    MATH  Google Scholar 

  • Howard AG (2012) Missing data in non-parametric tests of correlated data. Ph.D. thesis, The University of North Carolina at Chapel Hill

  • Kiefer J (1959) K-sample analogues of the Kolmogorov–Smirnov and Cramer–V. Mises tests. Ann Math Stat 30:420–447

    MathSciNet  MATH  Google Scholar 

  • Konietschke F, Harrar SW, Lange K, Brunner E (2012) Ranking procedures for matched pairs with missing data—asymptotic theory and a small sample approximation. Comput Stat Data Anal 56:1090–1102

    MathSciNet  MATH  Google Scholar 

  • Koul HK, Müller UU, Schick A (2013) The transfer principle: a tool for complete case analysis. Ann Stat 40:3031–3049

    MathSciNet  MATH  Google Scholar 

  • Kuan PF, huang B (2013) A simple and robust method for partially matched samples using the p-values pooling approach. Stat Med 32:3247–3259

    MathSciNet  Google Scholar 

  • Lin P-E, Stivers LE (1975) On difference of means with incomplete data. Biometrika 61:325–334

    MathSciNet  MATH  Google Scholar 

  • Little RJA, Rubin DB (2014) Statistical analysis with missing data. Wiley, Hoboken

    MATH  Google Scholar 

  • Looney S, Jones P (2003) A method for comparing two normal means using combined samples of correlated and uncorrelated data. Stat Med 22:1601–1610

    Google Scholar 

  • Martinez-Camblor P, Corral N, de la Hera J (2012) Hypothesis test for paired samples in the presence of missing data. J Appl Stat 40:76–87

    MathSciNet  Google Scholar 

  • Maritz JM (1995) A permutation paired test allowing for missing values. Aust N Z J Stat 37:153–159

    MATH  Google Scholar 

  • Modarres R (2008) Tests of bivariate exchangeability. Int Stat Rev 76:203–213

    Google Scholar 

  • Morrison DF (1973) A test for equality of means of correlated variates with missing data on one response. Biometrika 60:101–105

    MathSciNet  MATH  Google Scholar 

  • Rempala GA, Looney SW (2006) Asymptotic properties of a two sample randomized test for partially dependent data. J Stat Plan Inference 68–89

  • Samawi HM, Vogel R (2014) Notes on two sample tests for partially correlated (paired) data. J Appl Stat 41:109–117

    MathSciNet  Google Scholar 

  • Samawi HM, Vogel R (2015) On some nonparametric tests for partially observed correlated data: proposing new tests. J Stat Theory Appl 14:131–155

    MathSciNet  Google Scholar 

  • Student (1908) The probable error of a mean. Biometrika 6:1–25

    MATH  Google Scholar 

  • Tang X (2007) New test statistic for comparing medians with incomplete paired data. Ph.D. thesis, University of Pittsburgh

  • The Comprehensive R Archive Network (2018). https://cran.r-project.org/web/packages/copula/copula.pdf

  • Uddin N, Hasan MS (2017) Testing equality of two normal means using combined samples of paired and unpaired data. Commun Stat Comput Simul 46:2430–2446

    MathSciNet  MATH  Google Scholar 

  • van der Vaart A, Wellner JA (1996) Weak convergence and empirical processes. Springer, New York

    MATH  Google Scholar 

  • Wilcoxon F (1945) Individual comparisons by ranking methods. Biometrics Bull 1:80–83

    Google Scholar 

  • Woolson R, Leeper J, Cole J, Clarke W (1976) A Monte Carlo investigation of a statistic for a bivariate missing data problem. Commun Stat Theory Methods 5:681–688

    MATH  Google Scholar 

  • Xu J, Harrar SW (2012) Accurate mean comparisons for paired samples with missing data: an application to a smokingcessation trial. Biometrical J 54:281–295

    MathSciNet  MATH  Google Scholar 

  • Yu D, Lim Y, Liang F, Kim K, Kim BS, Jang W (2012) Permutation test for incomplete paired data with application to cDNA microarray data. Comput Stat Data Anal 56:510–521

    MathSciNet  MATH  Google Scholar 

  • Ziegler K (1997) Functional central limit theorems for triangular arrays of function-indexed processes under uniformly integrable entropy conditions. J Multivar Anal 62:233–272

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author wishes to thank the Editor, an Associate Editor, and two Referees for very helpful comments and suggestions. Special thanks goes to Y. Fong from the Vaccine and Infectious Disease Division of the Fred Hutchinson Cancer Research Center in Seattle, WA, USA, for the support.

Author information

Authors and Affiliations

  1. Institut für Mathematische Stochastik, Leibniz Universität Hannover, Postfach 60 09, 30060, Hannover, Germany

    Daniel Gaigall

Authors
  1. Daniel Gaigall
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Gaigall.

Ethics declarations

Conflict of interest

The author states that there is no conflict of interest.

Additional information

Publisher's Note

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

Electronic supplementary material

Below is the link to the electronic supplementary material.

Online Resource 1

A simple example is given which certifies the fact that tests for verifying symmetry about zero or tests for verifying exchangeability are not applicable for the treatment of the testing problem of marginal homogeneity. (PDF 106KB).

Online Resource 2

R code for the implementation of simulations is given. (PDF 114KB).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gaigall, D. Testing marginal homogeneity of a continuous bivariate distribution with possibly incomplete paired data. Metrika 83, 437–465 (2020). https://doi.org/10.1007/s00184-019-00742-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00184-019-00742-5

Keywords

Mathematics Subject Classification

Search

Navigation