Coefficients of variations are unit-free measures that can, for example, be used to compare the variability of different samples. To this end, we study inference methods for them as well as their reciprocal given by standardised means in general heterogeneous one-way ANOVA designs. As no specific model assumptions are made, a permutation method is proposed to guarantee good finite sample performance. Building on recent limit theorems for randomisation techniques, we prove that the permutation procedure is asymptotically correct in general and finitely exact when data is exchangeable. These results are fostered in extensive simulation studies and two illustrative data analyses.
FeltzCJMillerGE.An asymptotic test for the equality of coefficients of variation from k population. Stat Med1996;
15: 647–658.
2.
FerriMGJonesWH.Determinants of financial structure: a new methodological approach. J Finance1979;
34: 631–644.
3.
WeberEUShafirSBlaisAR.Predicting risk sensitivity in humans and lower animals: risk as variance or coefficient of variation.Psychol Rev2004;
111: 430–445.
4.
Wright S. The genetics of quantitative variability. In: Reeve ECR and Waddington C (eds) Quantitative inheritance. London: Her Majesty's Stationary Office, 1952, pp.5–41.
5.
CastagliolaPAchouriATalebH, et al.
Monitoring the coefficient of variation using control charts with run rules. Quality Technol Quantitative Manage2013;
10: 75–94.
6.
WankePFZinnW.Strategic logistics decision making. Int J Physical Distribution Logistics Management2004;
34: 466–478.
7.
PangR.A practical guide to internal quality control (iqc) for quantitative tests in medical laboratories (proposed guidelines). Version 2.0. Hong Kong: Hong Kong Association ofMedical Laboratories Limited2015, pp.1–65.
8.
FungWKTsangTS.A simulation study comparing tests for the equality of coefficients of variation.Stat Med1998;
17: 2003–2014.
9.
ForkmanJ.Estimator and tests for common coefficients of variation in normal distributions. Comm Statist Theory Meth2009;
38: 233–251.
10.
KrishnamoorthyKLeeM.Improved tests for the equality of normal coefficients of variation. Comput Statist2014;
29: 215–232.
11.
AertsSHaesbroeckG.Robust asymptotic tests for the equality of multivariate coefficients of variation. Test2017;
26: 163–187.
12.
ShiXWongA.Accurate tests for the equality of coefficients of variation. J Stat Computat Simulat2018;
88: 3529–3543.
13.
NeuhausG.Conditional rank tests for the two-sample problem under random censorship. Ann Statist1993;
21: 1760–1779.
14.
JanssenA.Studentized permutation tests for non-i.i.d. hypotheses and the generalized Behrens-Fisher problem. Statist Probab Lett1997;
36: 9–21.
15.
JanssenAPaulyM. Asymptotics and effectiveness of conditional tests with applications to randomization tests. Technical report, University of Duesseldorf, 2009.
16.
PaulyM.Discussion about the quality of F-ratio resampling tests for comparing variances. Test2011;
20: 163–179.
17.
ChungEYRomanoJP.Exact and asymptotically robust permutation tests. Ann Statist2013;
41: 484–507.
18.
PaulyMBrunnerEKonietschkeF.Asymptotic permutation tests in general factorial designs. J R Stat Soc Ser B Stat Methodol2015;
77: 461–473.
19.
SmagaŁ.Wald-type statistics using {2}-inverses for hypothesis testing in general factorial designs. Statist Probab Lett2015;
107: 215–220.
20.
ChungEYRomanoJP.Asymptotically valid and exact permutation tests based on two-sample U-statistics. J Statist Plann Inference2016;
168: 97–105.
21.
SmagaŁ.Diagonal and unscaled Wald-type tests in general factorial designs. Electron J Stat2017;
11: 2613–2646.
22.
DoblerDPaulyM.Bootstrap- and permutation-based inference for the Mann-Whitney effect for right-censored and tied data. Test2018;
27: 639–658.
23.
JamesGS.The comparison of several groups of observations when the ratios of the population variances are unknown. Biometrika1951;
38: 324–329.
24.
BrownMBForsytheAB.The small sample behavior of some statistics which test the equality of several means. Technometrics1974;
16: 129–132.
25.
LehmannELRomanoJP.Testing statistical hypotheses.
New York, NY:
Springer, 2005.
26.
HothornTHornikKVan De WielMA, et al.
Implementing a class of permutation pests: the coin package. J Stat Software2008;
28: 1–23.
27.
ArborettiRFontanaRPesarinF, et al.
Nonparametric combination tests for comparing two survival curves with informative and non-informative censoring.Stat Meth Med Res2018;
27: 3739–3769.
28.
BoosDJanssenPVeraverbekeN.Resampling from centered data in the two-sample problem. J Statist Plann Inference1989;
21: 327–345.
29.
AndersonMBraakCT.Permutation tests for multi-factorial analysis of variance. J Stat Comput Simul2003;
73: 85–113.
30.
OmelkaMPaulyM.Testing equality of correlation coefficients in two populations via permutation methods. J Statist Plann Inference2012;
142: 1396–1406.
31.
MarwickBKrishnamoorthyK. cvequality: Tests for the equality of coefficients of variation from multiple groups, https://CRAN.R-project.org/package=cvequality (2019). R package version 0.2.0.
32.
BaderRSLehmanWH.Phenotypic and genotypic variation in odontometric traits of the house mouse. Am Midlands Naturalist1965;
74: 28–38.
33.
R Core Team. R: A language and environment for statistical computing.
Vienna, Austria:
R Foundation for Statistical Computing, 2018, https://www.R-project.org/
DuchesnePFrancqC.Multivariate hypothesis testing using generalized and {2}-inverses - with applications. Statistics2015;
49: 475–496.
36.
BradleyJV.Robustness?Br J Math Stat Psychol1978;
31: 144–152.
37.
SmithHGnanadesikanRHughesJB.Multivariate analysis of variance (MANOVA). Biometrics1962;
18: 22–41.
38.
MorrisonD.Multivariate statistical methods.
New York, NY:
McGraw-Hill Series in Probability and Statistics, McGraw-Hill, 1990.
39.
GóreckiTŁuczakM.Linear discriminant analysis with a generalization of the Moore-Penrose pseudoinverse. Int J Appl Math Comput Sci2013;
23: 463–471.
40.
AlbertAZhangL.A novel definition of the multivariate coefficient of variation.Biom J2010;
52: 667–675.
41.
AertsSHaesbroeckGRuwetC.Multivariate coefficients of variation: comparison and influence functions. J Multivariate Anal2015;
142: 183–198.
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