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A Levene-type test of homogeneity of variances against ordered alternatives

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Abstract

Investigations focusing on differences in scale parameters across multiple samples appear in various scientific fields, e.g., when assessing measurement precision. In such cases a natural order of groups often suggests itself, which can be exploited by tests involving ordered alternatives. We propose multiple contrast tests with order-restriction based on robust Levene residuals as a powerful approach to detect scale differences between several groups. The proposed method is applied to a real data set and also extensive simulation is conducted to assess power and robustness against non-normality.

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References

  • Agresti A, Coull BA (1998) Order-restricted inference for monotone trend alternatives in contingency tables. Comput Stat Data Anal 28:139–155

    Article  MATH  Google Scholar 

  • Blackman MJ, Stein GJ, Vandiver PB (1993) The standardization hypothesis and ceramic mass production: technological, compositional and metric indexes of craft specialization at Tell Leilan, Syria. Am Antiq 58:60–80

    Article  Google Scholar 

  • Bretz F (2006) An extension of the Williams trend test to general unbalanced linear models. Comp Stat Data Anal 50:1735–1748

    Article  MATH  MathSciNet  Google Scholar 

  • Bretz F, Hothorn LA (2003) Statistical analysis of monotone or non-monotone dose-response data from in vitro toxicological assays. Altern Lab Anim: ATLA 31:81–96

    Google Scholar 

  • Brown MB, Forsythe AB (1974) Robust tests for equality of variances. J Am Stat Assoc 69:364–367

    Article  MATH  Google Scholar 

  • Cahoy DO (2010) A bootstrap test for equality of variances. Comput Stat Data Anal 54:2306–2316

    Article  MATH  MathSciNet  Google Scholar 

  • Dilba G, Schaarschmidt F, Hothorn LA (2007) Inferences for ratios of normal means. R News 7:20–23

    Google Scholar 

  • Djira GD, Hothorn LA (2009) Detecting relative changes in multiple comparisons with an overall mean. J Qual Technol 41:60–65

    Google Scholar 

  • Djira GD, Hasler M, Gerhard D, Schaarschmidt F (2012) mratios: Inferences for ratios of coefficients in the general linear model. R package version 1.3.17, available at http://cran.r-project.org/package=mratios

  • Dunnett C (1955) A multiple comparison procedure for comparing several treatments with a control. J Am Stat Assoc 50:1096–1121

    Article  MATH  Google Scholar 

  • Fujino Y (1979) Tests for the homogeneity of a set of variances against ordered alternatives. Biometrika 66:133–139

    Article  MATH  MathSciNet  Google Scholar 

  • Gastwirth JL, Gel YR, Miao W (2009) The impact of Levene’s test of equality of variances on statistical theory and practice. Stat Sci 24:343–360

    Article  MathSciNet  Google Scholar 

  • Gastwirth JL, Gel YR, Hui WLW, Lyubchich V, Miao W, Noguchi K (2013) lawstat: An R package for biostatistics, public policy, and law. R package version 2.4.1, available at http://cran.r-project.org/package=lawstat

  • Genz A, Bretz F (2009) Computation of multivariate normal and \(t\) probabilities. Springer, Heidelberg

    Book  MATH  Google Scholar 

  • Genz A, Bretz F, Miwa T, Mi X, Leisch F, Scheipl F, Hothorn T (2012) mvtnorm: Multivariate normal and \(t\) distributions. R package version 0.9-9992, available at http://cran.r-project.org/package=mvtnorm

  • Gould SJ (1988) Trends as changes in variance: a new slant on progress and directionality in evolution. J Paleontol 62:319–329

    Google Scholar 

  • Hayter AJ (1990) A one-sided studentized range test for testing against a simple ordered alternative. J Am Stat Assoc 85:778–785

    MATH  MathSciNet  Google Scholar 

  • Hirotsu C, Yamamoto S, Hothorn LA (2011) Estimating the dose-response pattern by the maximal contrast type test approach. Stat Biopharm Res 3:40–53

    Article  Google Scholar 

  • Hothorn LA, Libiger O, Gerhard D (2012) Model-specific tests on variance heterogeneity for detection of potentially interacting genetic loci. BMC Genet 13:article 59

  • Hothorn T, Bretz F, Westfall P (2008) Simultaneous inference in general parametric models. Biom J 50:346–363

    Article  MathSciNet  Google Scholar 

  • Keyes TK, Levy MS (1997) Analysis of Levene’s test under design imbalance. J Educ Behav Stat 22:227–236

    Article  Google Scholar 

  • Konietschke F, Hothorn LA (2012) Evaluation of toxicological studies using a nonparametric Shirley-type trend test for comparing several dose levels with a control group. Stat Biopharm Res 4:14–27

    Article  Google Scholar 

  • Konietschke F, Hothorn LA, Brunner E (2012) Rank-based multiple test procedures and simultaneous confidence intervals. Electron J Stat 6:738–759

    Article  MATH  MathSciNet  Google Scholar 

  • Konietschke F, Bösiger S, Brunner E, Hothorn LA (2013) Are multiple contrast tests superior to the ANOVA? Int J Biostat 9:1–11

    Article  Google Scholar 

  • Kvamme KL, Stark MT, Longacre WA (1996) Alternative procedures for assessing standardization in ceramic assemblages. Am Antiq 61:116–126

    Article  Google Scholar 

  • Levene H (1960) Robust tests for equality of variances. In: Olkin I (ed) Contributions to probability and statistics. Stanford University Press, Palo Alto, pp 278–292

    Google Scholar 

  • Loh WY (1987) Some modifications of Levene’s test of variance homogeneity. J Stat Comput Simul 28:213–226

    Article  Google Scholar 

  • Marcus R (1982) Some results on simultaneous confidence-intervals for monotone contrasts in one-way ANOVA model. Commun Stat A-Theory Methods 11:615–622

    Article  Google Scholar 

  • Marozzi M (2011) Levene type tests for the ratio of two scales. J Stat Comput Simul 81:815–826

    Article  MATH  MathSciNet  Google Scholar 

  • Marozzi M (2012) A distribution free test for the equality of scales. Commun Stat-Simul Comput 41:878–889

    Article  MATH  MathSciNet  Google Scholar 

  • Marozzi M (2013) Nonparametric simultaneous tests for location and scale testing: a comparison of several methods. Commun Stat-Simul Comput 42:1298–1317

    Article  MATH  MathSciNet  Google Scholar 

  • McDermott MP (1999) Generalized orthogonal contrast tests for homogeneity of ordered means. Can J Stat-Rev Can Stat 27:457–470

    Article  MATH  Google Scholar 

  • Neuhäuser M, Hothorn LA (2000) Parametric location-scale and scale trend tests based on Levene’s transformation. Comput Stat Data Anal 33:189–200

    Article  MATH  Google Scholar 

  • Neuhäuser M, Leuchs AK, Ball D (2011) A new location-scale test based on a combination of the ideas of Levene and Lepage. Biom J 53:525–534

    Article  MATH  MathSciNet  Google Scholar 

  • Noguchi K, Gel YR (2010) Combination of Levene-type test and a finite-intersection method for testing equality of variances against ordered alternatives. J Nonparametr Stat 22:897–913

    Article  MATH  MathSciNet  Google Scholar 

  • O’Neill ME, Mathews K (2000) A weighted least squares approach to Levene’s test of homogeneity of variance. Aust N Z J Stat 42:81–100

    Article  MATH  MathSciNet  Google Scholar 

  • Parra-Frutos I (2009) The behaviour of the modified Levene’s test when data are not normally distributed. Comput Stat 24:671–693

    Article  MATH  MathSciNet  Google Scholar 

  • Parra-Frutos I (2013) Testing homogeneity of variances with unequal samples sizes. Comput Stat 28:1269–1297

    Article  MathSciNet  Google Scholar 

  • Peng J, Lee CIC, Liu L, Jiang J (2006) Orthogonal multiple contrast test for testing monotone on the average. Stat Probab Lett 76:161–168

    Article  MATH  MathSciNet  Google Scholar 

  • Development Core Team R (2012) R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna

  • Rublík F (2010) A note on simultaneous confidence intervals for ratio of variances. Commun Stat-Theory Methods 39:1038–1045

    Article  MATH  Google Scholar 

  • Singh P (2006) Simultaneous confidence intervals for the successive ratios of scale parameters. J Stat Plan Inf 136:1007–1019

    Article  MATH  Google Scholar 

  • Singh P, Gill AN, Kumar N (2009) A test of homogeneity of several variances against tree ordered alternative. Stat Probab Lett 79:2315–2320

    Article  MATH  MathSciNet  Google Scholar 

  • Singh P, Goyal A, Gill AN (2010) A note on comparing several variances with a control variance. Stat Probab Lett 80:1995–2002

    Article  MATH  MathSciNet  Google Scholar 

  • Williams DA (1971) A test for differences between treatment means when several dose levels are compared with a zero dose control. Biometrics 27:103–117

    Article  Google Scholar 

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Authors and Affiliations

  1. Institute of Biostatistics, Leibniz University Hannover, Herrenhaeuser Strasse 2, 30419 , Hannover, Germany

    Philip Pallmann & Ludwig A. Hothorn

  2. Department of Mathematics and Statistics, South Dakota State University, Brookings, SD , 57007, USA

    Gemechis D. Djira

Authors
  1. Philip Pallmann
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  2. Ludwig A. Hothorn
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  3. Gemechis D. Djira
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Corresponding author

Correspondence to Philip Pallmann.

Additional information

The work of the second author (LAH) was partly supported by the German Science Foundation grant DFG-HO1687. The authors wish to thank the editor and two anonymous referees for hints and suggestions that helped improve the manuscript.

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Pallmann, P., Hothorn, L.A. & Djira, G.D. A Levene-type test of homogeneity of variances against ordered alternatives. Comput Stat 29, 1593–1608 (2014). https://doi.org/10.1007/s00180-014-0508-z

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  • DOI: https://doi.org/10.1007/s00180-014-0508-z

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