Means Grouping based on Studentized Midrange

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Ben Dêivide de Oliveira Batista
Daniel Furtado Ferreira
Matheus Fernando Rodrigues Santos
Henrique José de Paula Alves


The purpose of this paper was to develop two procedures of multiple comparisons based on methods of clustering means, that is, MGM and MGR tests. The first is based on the studentized midrange distribution, and the second based on the studentized range distribution. The tests showed performance equivalent to the performance of the compared tests. Like the tests presented that were based on methods of grouping averages, the MGM and MGR tests did not control the experimentwise error rate for almost all evaluated scenarios. However, under the complete $H_1$ hypothesis, these tests showed high power, with emphasis on the MGM test. Thus, what we propose is yet another test alternative without ambiguity in its results, and not a substitution of the traditional tests already present in the literature.

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Batista, B. D. de O., Furtado Ferreira, D., Fernando Rodrigues Santos, M. ., & José de Paula Alves, H. (2023). Means Grouping based on Studentized Midrange. Brazilian Journal of Biometrics, 41(4), 361–397.


Batista, B. D. O. & Ferreira, D. F. Alternative to Tukey test. 44, 1–10 (2020).

Batista, B. D. O. & Ferreira, D. F. SMR: An R Package for Computing the Externally Studentized Normal Midrange Distribution. The R Journal 6, 123–136 (2014).

Batista, B. D. O. & Ferreira, D. F. SMR: Externally Studentized Midrange Distribution R package version 2.0.1 (Vienna, Austria, 2014). http://cran.r-

Batista, B. D. O. & Ferreira, D. F. The Externally StudentizedNormal Midrange Distribution. Ciência e Agrotecnologia 41, 378–389 (2017).

Bernhardson, C. S. 375: Type I Error Rates When Multiple Comparison Procedures Follow a Significant F Test of ANOVA. English. Biometrics 31, 229–232 (1975).

Bhering, L. L., Cruz, C. D. a., Vasconcelos, E. S. d., Ferreira, A. & Resende Jr, M. F. R. d. Alternative methodology for Scott Knott test. Crop Breeding and Applied Biotechnology 8, 9–16 (2008).

Boardman, T. J. & Moffitt, D. R. Graphical Monte Carlo Type I Error Rates for Multiple Comparison Procedures. Biometrics 27, 738–744 (1971).

Borges, L. C. & Ferreira, D. F. Poder e taxas de erro tipo I dos tetes Scott-Knott, Tukey e Student-Newman-Keuls sob Distribuições normal e não normais dos resíduos. Revista Matemática e Estatística 21. (Portuguese), 67–83 (2003).

Calinski, T. & Corsten, L. C. A. Clustering Means in Anova by Simultaneous Testing. Biometrics 41, 39–48 (1985).

Carmer, S. G. & Swanson, M. R. An Evaluation of Ten Pairwise Multiple Comparison Procedures by Monte Carlo Methods. Journal of the American Statistical Association 68, 66–74 (1973).

Conrado, T. V., Ferreira, D. F., Scapim, C. A. & Maluf, W. R. Adjusting the Scott-Knott cluster analyses for unbalanced designs. Crop Breeding and Applied Biotechnology 17, 1–9 (2017).

Cui, X., Dickhaus, T., Ding, Y. & Hsu, J. Handbook of Multiple Comparisons 418 (CRC Press, Boca Raton, 2021).

David, H. A., Hartley, H. O. & Pearson, E. S. The Distribution of the Ratio, in a SingleNormal Sample, of Range to Standard Deviation. English. Biometrika 41, 482–493 (1954).

David, H. A. & Nagaraja, H. N. Order Statistics 458 (JohnWiley & Sons, Canada, 2003).

Duncan, D. B. Multiple range and multiple F tests. Biometrics 11, 1–42 (1955).

Einot, I. & Gabriel, K. R. A Study of the Powers of Several Methods of Multiple Comparisons. Journal of the American Statistical Association 70, 574–583 (1975).

Figueiredo, U. J., Nunes, J. A. R., Parrella, R. A. C., Souza, E. D., Silva, A. R., Emygdio, B. M., Machado, J. R. A. & Tardin, F. D. Adaptability and stability of genotypes of sweet sorghum by GGEBiplot and Toler methods. Genetics and Molecular Research 14, 11211–11221 (2015).

Graybill, F. An introduction to linear statistical models 1, 463 (McGraw-Hill, New York, 1961).

Gumbel, E. J. Statistics of Extremes 375 (Columbia University Press, New York, 1958).

Hartley, H. O. The Range in Random Samples. English. Biometrika 32, 334–348 (1942).

Keuls, M. The use of the “studentized range” in connection with an analysis of variance. Euphytica 1, 112 122 (1952).

Leemis, L. M. & Trivedi, K. S. A Comparison of Approximate Interval Estimators for the Bernoulli Parameter. The American Statistician 50, 63–68 (1996).

Newman, D. The Distribution of Range in Samples from a Normal Population, Expressed in Terms of an Independent Estimate of Standard Deviation. Biometrika 31, 20–30 (1939).

Oliveira, I. R. C. & Ferreira, D. F. Multivariate extension of chi-squared univariate normality test. Journal of Statistical Computation and Simulation 80, 513–526 (2010).

Pearson, E. S. & Haines, J. The Use of Range in Place of Standard Deviation in Small Samples. Supplement to the Journal of the Royal Statistical Society 2, 83–98 (1935).

Pearson, E. S. & Hartley, H. O. Tables of the Probability Integral of the Studentized Range. English. Biometrika 33, 89–99 (1943).

Pearson, E. S. & Hartley, H. O. The Probability Integral of the Range in Samples of n Observations From a Normal Population. Biometrika 32, 301–310 (1942).

Pearson, E. S. A Further Note on the Distribution of Range in Samples Taken from a Normal Population. Biometrika 18, 173–194 (1926).

Pearson, E. S. The Percentage Limits for the Distribution of Range in Samples from a Normal Population. Biometrika 24, 404–417 (1932).

Perecin, D. & Malheiros, E. B. Uma avaliação de seis procedimentos para comparações múltiplas in 3° Simpósio de Estatística aplicada à Experimentação Agonômica (Portuguese) (Lavras, MG, 1989), 66.

R CORE TEAM. R. A Language and Environment for Statistical Computing R Foundation for Statistical Computing (Vienna, Austria, 2022).

Ramos, P. S. & Ferreira, D. F. Agrupamento de médias via bootstrap de populações normais e não-normais. 56. (Portuguese), 140–149 (2009).

Ramos, P. S. & Vieira, M. T. Bootstrap multiple comparison procedure based on the F distribution. 31, 529 546 (2014).

Rider, P. R. The midrange of a sample as an estimator of the population midrange. Journal of the American Statistical Association 52, 537–542 (1957).

Sauder, D. C. & DeMars, C. E. An Updated Recommendation for Multiple Comparisons. Advances in Methods and Practices in Psychological Science 2, 26–44 (2019).

Scott, A. J. & Knott, M. A Cluster Analysis Method for Grouping Means in the Analysis of Variance. English. Biometrics 30, 507–512 (1974).

Searle, S. R. Linear models for unbalanced data 536 (Wiley, New York, 1987).

Shimokawa, T. & Goto, M. Hierarchical cluster analyis for multi-sample comparisons based on the power-normal distribution. Behaviometrika 38, 125–138 (2011).

Silva, E. C. d., Ferreira, D. F. & Bearzoti, E. Avaliação do poder e taxas de erro tipo I do teste de Scott-Knott por meio do método de Monte Carlo. Ciência e Agrotecnologia 23. (Portuguese), 687–696 (1999).

Student. Errors in routine analysis. Biometrika 19, 151–164 (1927).

Tukey, J. W. The problem of multiple comparisons. Mimeographed monograph. Unpublished memorandum in private circulation (1953).

Waller, R. A. & Duncan, D. B. A Bayes Rule for the Symmetric Multiple Comparisons Problems. Journal of the American Statistical Association 64, 1484–1503 (Dec. 1969).

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