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Testing for ordered alternatives in heteroscedastic ANOVA under normality

Author

Listed:
  • Anjana Mondal

    (Indian Institute of Technology Kharagpur)

  • Markus Pauly

    (Technical University Dortmund
    Research Center Trustworthy Data Science and Security, UA Ruhr)

  • Somesh Kumar

    (Indian Institute of Technology Kharagpur)

Abstract

In a one-way design with unknown and unequal error variances, we are interested in testing the null hypothesis $$H_0:\mu _1=\mu _2=\cdots =\mu _k$$ H 0 : μ 1 = μ 2 = ⋯ = μ k against the ordered alternative $$H_1:\mu _1\le \mu _2\le \cdots \le \mu _k$$ H 1 : μ 1 ≤ μ 2 ≤ ⋯ ≤ μ k (with at least one strict inequality). We propose the likelihood ratio test (LRT) as well as two Min-T tests (Min-T and AMin-T) and study their asymptotic behaviour. For better finite sample properties, a parametric bootstrap procedure is used. Asymptotic accuracy of the parametric bootstrap is established. Simulation results show that all the tests control the nominal level in case of small, moderate, and highly unbalanced sample sizes. This even holds for combinations with large variances. However, the asymptotic likelihood ratio test (ALRT) achieves the nominal size only for large samples and $$k=3$$ k = 3 . Power comparisons between LRT and Min-T test show that the LRT has more power than Min-T. However, the Min-T test is easier to use as it does not require evaluation of MLEs. Similar observations are made while comparing ALRT and AMin-T test. The robustness of these tests under departure from normality is also investigated by considering five non-normal distributions (t-distribution, Laplace, exponential, Weibull, and lognormal). When high variances are combined with small or highly unbalanced sample sizes, the tests are observed to be robust only for symmetric distributions. If variances of groups are equal and sample sizes of each group are at least 10, the proposed tests perform satisfactorily for symmetric and skewed distributions. R packages are developed for all tests and a practical example is provided to illustrate their application.

Suggested Citation

  • Anjana Mondal & Markus Pauly & Somesh Kumar, 2023. "Testing for ordered alternatives in heteroscedastic ANOVA under normality," Statistical Papers, Springer, vol. 64(6), pages 1913-1941, December.
    • Handle: RePEc:spr:stpapr:v:64:y:2023:i:6:d:10.1007_s00362-022-01366-0
    DOI: 10.1007/s00362-022-01366-0
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    References listed on IDEAS

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    1. Markus Pauly & Edgar Brunner & Frank Konietschke, 2015. "Asymptotic permutation tests in general factorial designs," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 77(2), pages 461-473, March.
    2. Shi, Ning-Zhong & Jiang, Hua, 1998. "Maximum Likelihood Estimation of Isotonic Normal Means with Unknown Variances," Journal of Multivariate Analysis, Elsevier, vol. 64(2), pages 183-195, February.
    3. Bretz, Frank, 2006. "An extension of the Williams trend test to general unbalanced linear models," Computational Statistics & Data Analysis, Elsevier, vol. 50(7), pages 1735-1748, April.
    4. Esther Herberich & Johannes Sikorski & Torsten Hothorn, 2010. "A Robust Procedure for Comparing Multiple Means under Heteroscedasticity in Unbalanced Designs," PLOS ONE, Public Library of Science, vol. 5(3), pages 1-8, March.
    5. Konietschke, Frank & Bathke, Arne C. & Harrar, Solomon W. & Pauly, Markus, 2015. "Parametric and nonparametric bootstrap methods for general MANOVA," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 291-301.
    6. Guogen Shan & Daniel Young & Le Kang, 2014. "A New Powerful Nonparametric Rank Test for Ordered Alternative Problem," PLOS ONE, Public Library of Science, vol. 9(11), pages 1-10, November.
    7. Chang, Ching-Hui & Pal, Nabendu, 2008. "Testing on the common mean of several normal distributions," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 321-333, December.
    8. Cohen, Arthur & Kemperman, J. H. B. & Sackrowitz, Harold B., 2000. "Properties of Likelihood Inference for Order Restricted Models," Journal of Multivariate Analysis, Elsevier, vol. 72(1), pages 50-77, January.
    9. Cabilio, P. & Peng, J., 2008. "Multiple rank-based testing for ordered alternatives with incomplete data," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2609-2613, November.
    10. Ahad Malekzadeh & Mahmood Kharrati-Kopaei, 2018. "Inferences on the common mean of several normal populations under heteroscedasticity," Computational Statistics, Springer, vol. 33(3), pages 1367-1384, September.
    11. Pal, Nabendu & Lin, Jyh-Jiuan & Chang, Ching-Hui & Kumar, Somesh, 2007. "A revisit to the common mean problem: Comparing the maximum likelihood estimator with the Graybill-Deal estimator," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 5673-5681, August.
    12. Hayter, A. J. & Liu, W., 1996. "Exact calculations for the one-sided studentized range test for testing against a simple ordered alternative," Computational Statistics & Data Analysis, Elsevier, vol. 22(1), pages 17-25, June.
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