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Note on the utilization of the generalized student ratio in the analysis of variance or dispersion

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Summary

In this note, we shall consider the usage of the generalized Student ratioT orT 2, which is the statistic introduced by H. Hotelling [3, 4], in the analysis of variance or dispersion. F. A. Graybill and J. L. Folks [1, 2] have shown that in a randomized block experiment and in the situation of heterogeneneity of error variances, the equality of main effect constants can be tested by using the T2P-statistiç. It will be shown that their approach is easily extended for testing the significance of the interaction in a two-factor experiment with different error variances. For the multivariate analysis of dispersion, the use of the generalized T2 measure is illustrated by a simple example and the asymptotic power of the test based on this statistic is also considered. The tables of 5% and1% significance points for this test are prepared for three and four dimensional cases.

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References

  1. Folks. J. L. and Graybill, F. A., “Heterogeneity of error variances in a randomized block design.”Biometrika, Vol. 44, (1957), pp, 275–277.

    Article  Google Scholar 

  2. Graybill, F. A., “Variance heterogeneity in a randomized block design”Biometrics, Vol. 10, (1954), pp. 516–520.

    Article  Google Scholar 

  3. Hotelling, H., “The generalization of Student’s ratio”Ann. Math. Stat. Vol. 2, (1931), pp. 360–378.

    Article  Google Scholar 

  4. Hotelling, H. “A generalizedT test and measure of multivariate dispersion”Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, edited by J. Neyman, University of California Press, (1951), pp. 23–41.

  5. Ito, K., “Asymptotic formulae for the distribution of Hotelling’s generalized To 2 statistic”Ann. Math. Stat. Vol. 27, (1956), pp. 1091–1105.

    Article  MathSciNet  Google Scholar 

  6. James, G. S., “Tests of linear hypotheses in univariate and multivariate analysis when the ratios of the population variances are unknown,”Biometrika, Vol. 41 (1954), pp. 19–43.

    MathSciNet  MATH  Google Scholar 

  7. Siotani, M., Un the distribution of the Hotelling’s T2-statistics”The Proceedings of the Institute of Statistical Mathematics, (in Japanese), Vol. 4, No. 1, (1956), pp. 33–42.

    Google Scholar 

  8. Siotani, M., “On the distributions of the Hotelling’s T2-statistics”Annals of the Institute of Statistical Mathematics, Vol. 8, (1956), pp. 1–14.

    Article  MathSciNet  Google Scholar 

  9. Welch, B. L., “The generalization of Student’s problem when several different population variances are involved,”Biometrika, Vol. 34, (1947), pp. 28–35.

    MathSciNet  MATH  Google Scholar 

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  1. The Institute of Statistical Mathematics, USA

    Minoru Siotani

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  1. Minoru Siotani
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Siotani, M. Note on the utilization of the generalized student ratio in the analysis of variance or dispersion. Ann Inst Stat Math 9, 157–171 (1957). https://doi.org/10.1007/BF02892503

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  • DOI: https://doi.org/10.1007/BF02892503

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