Discussiones Mathematicae Probability and Statistics 22(1,2) (2002) 73-104


Viktor Witkovský

Institute of Measurement Science
Slovak Academy of Sciences
Dúbravská cesta 9
842 19 Bratislava, Slovak Republic


Weerahandi (1995b) suggested a generalization of the Fisher's solution of the Behrens-Fisher problem to the problem of multiple comparisons with unequal variances by the method of generalized p-values. In this paper, we present a brief outline of the Fisher's solution and its generalization as well as the methods to calculate the p-values required for deriving the conservative joint confidence interval estimates for the pairwise mean differences, refered to as the generalized Scheffé intervals. Further, we present the corresponding tables with critical values for simultaneous comparisons of the mean differences of up to k = 6 normal populations with unequal variances based on independent random samples with very small sample sizes.

Keywords: Behrens-Fisher distribution, pairwise comparisons, unequal variances, generalized p-values.

2000 Mathematics Subject Classification: 62F04; 62E15.


D.E. Amos, A portable package for bessel functions of a complex argument and nonnegative order, ACM Transactions on Mathematical Software 12 (1986), 265-273.
M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York 1965.
G. Bacová, A. Gábelová, O. Babusíková and D. Slamenová, The single cell gel electrophoresis: A potential tool for DNA analysis of the patients with hematological maligancies, Neoplasma 45 (1998), 349-359.
G.A. Barnard, Comparing the means of two independent samples, Applied Statistics 33 (1984), 266-271.
R.A. Fisher, The fiducial argument in statistical inference, Annals of Eugenics 6 (1935), 391-398.
R.A. Fisher and F. Yates, Statistical Tables, Longman, London 1975.
C.W. Dunnett, Pairwise multiple comparisons in the unequal variance case, Journal of the American Statistical Association 75 (1980), 796-800.
J. Gil-Pelaez, Note on the inversion theorem, Biometrika 38 (1951), 481-482.
A.F.S. Lee and J. Gurland, Size and power of tests of equality of two normal populations with unequal variances, Journal of the American Statistical Association 70 (1975), 933-941.
X.-L. Meng, Posterior predictive p-values, Annals of Statistics 22 (1994), 1142-1160.
G.K. Robinson, Properties of Student t and of the Behrens-Fisher solution to the two means problem, Annals of Statistics 4 (1976), 963-971.
G.K. Robinson, Behrens-Fisher problem, Encyclopedia of Statistical Sciences, 9 volumes plus Supplement, Wiley, New York, (1982), 205-209.
H. Scheffé, Practical solutions of the Behrens-Fisher problem, Journal of the American Statistical Association 65 (1970), 1501-1508.
K.W. Tsui, and S. Weerahandi, Generalized p values in significance testing of hypotheses in the presence of nuisance parameters, Journal of the American Statistical Association 84 (1989), 602-607.
G.A. Walker and J.G. Saw, The distribution of linear combinations of t variables, Journal of the American Statistical Association 73 (1978), 876-878.
S. Weerahandi, ANOVA under unequal error variances, Biometrics 51 (1995a), 589-599.
S. Weerahandi, Exact Statistical Methods for Data Analysis, Springer-Verlag, New York 1995b.
B.L. Welch, The generalization of `Student's' problem when several different population variances are involved, Biometrika 34 (1947), 28-35.
V. Witkovský, On the exact computation of the density and of the quantiles of linear combinations of t and F random variables, Journal of Statistical Planning and Inference 94 (2001a), 1-13.
V. Witkovský, Computing the distribution of a linear combination of inverted gamma variables, Kybernetika 37 (2001b), 79-90.
V. Witkovský, Exact distribution of positive linear combinations of inverted chi-square random variables with odd degrees of freedom, Submitted to Statistics & Probability Letters 2001c.

Received 5 December 2002