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A Comparison of Five Bootstrap and Non-Bootstrap Levene-Type Tests of Homogeneity of Variances

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Abstract

The test of homogeneity of variances arises in various statistical problems. In this paper, we compared five Levene-type tests including Levene’s test, Welch’s test, James’s test, Brown and Forsythe’s test and Alexander–Govern’s test in terms of power and size. The simulation results show that the bootstrapping after applying Noguchi and Gel (J Nonparametr Stat 22:897–913, 2010) correction and removing structural zeros significantly improve the performance of Type I error rates. We identify the best in terms of controlling the Type I error rate and power for each number of groups.

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References

  • Alexander RA, Govern DM (1994) A new and simpler approximation and ANOVA under variance heterogeneity. J Educ Stat 19:91–101

    Article  Google Scholar 

  • Bartlett MS (1937) Properties of sufficiency and statistical tests. Proc R Soc Ser A 160:268–282

    MATH  Google Scholar 

  • Boos DD, Brownie C (1989) Bootstrap methods for testing homogeneity of variance. Technometrics 31:69–82

    Article  MathSciNet  MATH  Google Scholar 

  • Boos DD, Brownie C (2004) Comparing variances and other measures of dispersion. Stat Sci 19:571–578

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Brown MB, Forsythe AB (1974b) The small sample behavior of some statistics which test the equality of several means. Technometrics 16:129–132

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Conover WJ, Johnson ME, Johnson MM (1981) A comparative study of tests for homogeneity of variances, with applications to the outer continental shelf bidding data. Technometrics 23:351–361

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Hines WGS, O’Hara Hines RJ (2000) Increased power with modified forms of Levene (Med) test for heterogeneity of variance. Biometrics 56:451–454

    Article  MATH  Google Scholar 

  • Jaffe PR, Parker FL, Wilson DJ (1982) Distribution of toxic substances in rivers. J Environ Eng Div 108:639649

    Google Scholar 

  • James GS (1951) The comparison of several groups of observations when the ratios of the population variances are unknown. Biometrika 38:324–329

    Article  MathSciNet  MATH  Google Scholar 

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

    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, Ghurye SG, Hoeffding W, Madow WG, Mann HB (eds) Contributions to probability and statistics. Stanford University Press, Stanford, pp 278–292

    Google Scholar 

  • Lim TS, Loh WY (1996) A comparison of tests of equality of variances. Comput Stat Data Anal 22:287–301

    Article  MathSciNet  MATH  Google Scholar 

  • Lix LM, Keselman JC, Keselman HJ (1996) Consequences of assumption violations revisited, a quantitative review of alternatives to the one-way analysis of variance \(F\) test. Rev Educ Res 66:579–619

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Marozzi M (2012b) A modified Hall–Padmanabhan test for the homogeneity of scales. Commun Stat Simul Comput 41:16–17

    Article  MathSciNet  MATH  Google Scholar 

  • Marozzi M (2016) Multivariate tests based on interpoint distances with application to magnetic resonance imaging. Stat Methods Med Res 25:2593–2610

    Article  MathSciNet  Google Scholar 

  • Neyman J, Pearson ES (1931) On the problem of \(k\) samples. Bull Int Acad Crac Ser A:460–481

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  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  MathSciNet  MATH  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  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Pesarin F, Salmaso L (2010) Permutation tests for complex data: theory, applications and software. Wiley, Chichester

    Book  MATH  Google Scholar 

  • Plourde A, Watkins GC (1998) Crude oil prices between 1985 and 1994: how volatile in relation to other commodities? Resour Energy Econ 20:245–262

    Article  Google Scholar 

  • Sharma D, Kibria G (2013) On some test statistics for testing homogeneity of variances: a comparative study. J Statist Comput Simul 83:1944–1963

    Article  MathSciNet  Google Scholar 

  • Shoemaker LH (2003) Fixing the \(F\) test for equal variances. Am Stat 57:105–114

    Article  MathSciNet  MATH  Google Scholar 

  • Welch BL (1951) On the comparison of several mean values, an alternative approach. Biometrika 38:330–336

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Statistics, University of Kurdistan, Sanandaj, Iran

    Nabaz Esmailzadeh

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  1. Nabaz Esmailzadeh
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Correspondence to Nabaz Esmailzadeh.

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Esmailzadeh, N. A Comparison of Five Bootstrap and Non-Bootstrap Levene-Type Tests of Homogeneity of Variances. Iran J Sci Technol Trans Sci 43, 979–989 (2019). https://doi.org/10.1007/s40995-018-0485-0

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  • DOI: https://doi.org/10.1007/s40995-018-0485-0

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