# On the simultaneous anova and manova tests

- 135 Downloads
- 38 Citations

## Summary

In the present paper, the simultaneous confidence intervals associated with the Simultaneous Multivariate Analysis of Variance (SMANOVA) tests are derived when the hypotheses can be tested in a “quasi-independentrd manner. With respect to the Simultaneous Analysis of Variance (SANOVA) test, the quasi-independence is weakened and the distribution problems are investigated when the joint distribution of the sums of squares associated with various hypotheses is a “multivariate chi-square ” distribution. It is shown that the lengths of the confidence intervals associated with the SANOVA test (dependent case) are shorter than the lengths of the corresponding confidence intervals obtained by using Scheffé’s method when the hypotheses on several contrasts are tested simultaneously. It is also shown that the SMANOVA test yields narrower confidence intervals than the MANOVA test when several quasi-independent hypotheses are tested simultaneously.

## Keywords

Joint Distribution Large Root Frequency Function Confidence Bound Wishart Distribution## References

- [1]T. W. Anderson,
*An Introduction to Multivariate Statistical Analysis*, John Wiley and Sons, New York, 1958.MATHGoogle Scholar - [2]S. Bose, “On the distribution of the ratio of variances of two samples drawn from a given normal bivariate correlated population,”
*Sankhyā*, 2 (1935), 65–72.Google Scholar - [3]R. C. Bose,
*Least Square Aspects of Analysis of Variance*, (mimeographed notes), University of North Carolina, 1958.Google Scholar - [4]H. Cramér,
*Mathematical Methods of Statistics*, Princeton university Press, 1946.Google Scholar - [5]Olive Jean Dunn, “Estimation of the means of dependent variables,”
*Ann. Math. Statist*., 29 (1958), 1095–1111.MathSciNetCrossRefMATHGoogle Scholar - [6]—, “Multiple comparisons among means,”
*J. Amer. Statist. Ass.*, 56 (1961), 52–64.MathSciNetCrossRefMATHGoogle Scholar - [7]C. W. Dunnett, “multiple comparison procedure for comparing several treatments with a control,”
*J. Amer. Statist. Ass.*, 50 (1955), 1091–1121.CrossRefMATHGoogle Scholar - [8]C. W. Dunnett and M. Sobel, “A bivariate generalization of Student’s t-distribution, with tables for certain special cases,”
*Biometrika*, 41 (1954), 153–169.MathSciNetCrossRefMATHGoogle Scholar - [9]—, “Approximations to the probability integral and certain percentage points of a multivariate analogue of Student’s t-distribution,”
*Biometrika*, 42 (1955), 258–260.MathSciNetCrossRefMATHGoogle Scholar - [10]W. Feller,
*An Introduction to Probability Theory and Its Applications*, Vol. I, John Wiley and Sons, New York, 1957.MATHGoogle Scholar - [11]D. J. Finney, “The joint distribution of variance ratios based on a common error mean square,”
*Annals of Eugenics*, 11 (1941), 136–140.MathSciNetCrossRefMATHGoogle Scholar - [12]M. N. Ghosh, “Simultaneous tests of linear hypotheses,”
*Biometrika*, 42 (1955), 441–449.MathSciNetCrossRefMATHGoogle Scholar - [13]R. Gnanadesikan,
*Contributions to Multivariate Analysis Including Univariate and Multivariate Components Analysis and Factor Analysis*, Institute of Statistics, University of North Carolina, Mimeo. Series No. 158, 1956.Google Scholar - [14]—, “Equality of more than two variances and of more than two dispersion matrices against certain alternatives,”
*Ann. Math. Statist.*, 30 (1958), 177–184. Correction,*ibid.*, 31 (1959), 227-228.MathSciNetCrossRefMATHGoogle Scholar - [15]S. S. Gupta and M. Sobel, “On a statistic which arises in selection and ranking problems,”
*Ann. Math. Statist.*, 28 (1957), 957–967.MathSciNetCrossRefMATHGoogle Scholar - [16]S. S. Gupta, “On a selection and ranking procedure for Gamma populations,”
*Ann. Inst. Stat. Math.*, 14 (1963), 199–216.MathSciNetCrossRefMATHGoogle Scholar - [17]H. O. Hartley, “Studentization and large-sample theory,” J. Roy. Stat. Soc., Suppl., 5 (1938), 80–88.Google Scholar
- [18]—, “Some recent developments in analysis of variance,”
*Communications in Pure and Applied Mathematics*, 8 (1955), 47–72.MathSciNetCrossRefMATHGoogle Scholar - [19]S. John, “On the evaluation of the probability integral of the multivariate t-distribution, ”
*Biometrika*, 48 (1961), 409–417.MathSciNetMATHGoogle Scholar - [20]W. F. Kibble, “A two variable Gamma type distribution,”
*Sankhyā*, 5 (1941), 137–150.MathSciNetMATHGoogle Scholar - [21]A. W. Kimball, “On dependent tests of significance in the analysis of variance,”
*Ann. Math. Statist.*, 22 (1951), 600–602.MathSciNetCrossRefMATHGoogle Scholar - [22]P. R. Krishnaiah, and M. M. Rao, “Remarks on a multivariate Gamma distribution,”
*The American Mathematical Monthly*, 68 (1961), 342–346.MathSciNetCrossRefMATHGoogle Scholar - [23]P. R. Krishnaiah, Peter Hagis, Jr., and L. Steinberg,
*The Bivariate Chi Distribution*, Applied Mathematics Department, Remington Rand Univac, Philadelphia, Technical Report No. 3, 1961.MATHGoogle Scholar - [24]A. S. Krishnamoorthy and M. Parthasarathy, “A multivariate Gamma-type distribution,”
*Ann. Math, Statist.*, 22 (1951), 549–557. Correction,*ibid.*, 31 (1960), 229.MathSciNetCrossRefMATHGoogle Scholar - [25]K. R. Nair, “The studentized form of the extreme mean square test in the analysis of variance,”
*Biometrika*, 35 (1948), 16–31.MathSciNetCrossRefMATHGoogle Scholar - [26]K. C. S. Pillai and K. V. Ramachandran, “On the distribution of the ratio of the ith observation in an ordered sample from a normal population to an independent estimate of the standard deviation,”
*Ann. Math, Statist.*, 25 (1954), 565–572.MathSciNetCrossRefMATHGoogle Scholar - [27]K. V. Ramachandran, “On the simultaneous analysis of variance test,”
*Ann. Math. Statist.*, 27 (1956), 521–528.MathSciNetCrossRefMATHGoogle Scholar - [28]S. N. Roy and J. Roy,
*Analysis of Variance with Univariate or Multivariate, Fixed or Mixed Classical Models*, Institute of Statistics, University of North Carolina, Mimeo. Series No. 208, 1958.Google Scholar - [29]S. N. Roy, “The individual sampling distribution of the maximum, the minimum and any intermediate one of the p-statistics on the null hypothesis,”
*Sankhyā*, 7 (1945), 133–158.MATHGoogle Scholar - [30]—, “On a heuristic method of test construction and its use in multivariate analysis,”
*Ann. Math. Statist.*, 24 (1953), 220–238.MathSciNetCrossRefMATHGoogle Scholar - [31]—, “A note on some further results in simultaneous confidence interval estimation,”
*Ann. Math. Statist.*, 27 (1956), 856–858.MathSciNetCrossRefMATHGoogle Scholar - [32]—,
*Some Aspects of Multivariate Analysis*, John Wiley and Sons, Inc., New York, 1957.Google Scholar - [33]S. N. Roy and R. C. Bose, “Simultaneous confidence interval estimation,”
*Ann. Math. Statist.*, 24 (1953), 513–536.MathSciNetCrossRefMATHGoogle Scholar - [34]S. N. Roy and R. Gnanadesikan, “Further contributions to multivariate confidence bounds”
*Biometrika*, 44 (1957), 399–410.MathSciNetCrossRefMATHGoogle Scholar - [35]—, “A note on ‘further contributions to multivariate confidence bounds’”
*Biometrika*, 45 (1958), 581.Google Scholar - [36]— “Some contributions to ANOVA in one or more dimensions—I”
*Ann. Math. Statist.*, 30 (1959), 304–317.MathSciNetCrossRefMATHGoogle Scholar - [37]—, “Some contributions to ANOVA in one or more dimensions—II”
*Ann. Math. Statist.*, 30 (1959), 318–340.MathSciNetCrossRefMATHGoogle Scholar - [38]—, “A method for judging all contrasts in the analysis of variance”
*Biometrika*, 20 (1953), 87–104.MathSciNetMATHGoogle Scholar - [39]—,
*The Analysis of Variance*, John Wiley and Sons, New York, 1960.MATHGoogle Scholar - [40]M. Siotani, “The extreme value of the generalized distances of the individual points in the multivariate normal sample”
*Ann. Inst. Stat. Math.*, 10 (1959), 183–203.MathSciNetCrossRefMATHGoogle Scholar - [41]—, “Notes on multivariate confidence bounds”
*Ann. Inst. Stat. Math.*, 11 (1960), 167–182.MathSciNetCrossRefMATHGoogle Scholar - [42]J. W. Tukey,
*The Problems of Multiple Comparisons*, unpublished dittoed notes, Princeton University.Google Scholar