Tetsuhisa Miwa (National Institute of Agro-Environmental Sciences, Japan)

A. J. Hayter (Georgia Institute of Technology, USA),

Wei Liu (University of Southampton, UK)


Exact calculations of the level probabilities in the unbalancedone-way models with applications to Bartholomew's test

An easy and quick procedure is presented to calculate the level probabilities under simple order of k independent normal random variables with unequal variances. A crucial step in calculating the level probabilities is the calculation of orthant probabilities of the form , where a recursive method (Hayter and Liu, 1996) and a cubic polynomial approximation method are employed.

These level probabilities have an application in the unbalanced one-way models for comparing k treatment effects, where Bartholomew (1959, 1961) proposed the likelihood ratio test for testing the homogeneity of the treatment effects against the simply ordered alternative hypothesis. Although there is some literature showing that Bartholomew's test has good power properties, its null distribution for the unbalanced models has been difficult to calculate except for small k. The problem in the evaluation of this null distribution has been the difficulty in calculating these level probabilities. Our procedure to calculate the level probabilities allows the computation of the p-values and the critical points of Bartholomew's test for the unbalanced models.


  1. Bartholomew, D. J. (1959).A test of homogeneity for ordered alternatives, Biometrika, 41, 36-48.

  1. Bartholomew, D. J. (1961).Ordered tests in the analysis of variance, Biometrika, 48, 325-332.
  2. Hayter, A. J. and Liu, W. (1996).A note on the calculation of
, Amer. Statist., 50, 365