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Biometrika 1985 72(2):429-439; doi:10.1093/biomet/72.2.429
© 1985 by Biometrika Trust
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Approximating the level probabilities in order restricted inference: The simple tree ordering

F. T. WRIGHT and TUAN TRAN

Department of Mathematics and Statistics, University of Missouri-Rolla Rolla, Missouri 65401, U.S.A.

Distribution theory for the likelihood ratio statistics for the comparison of several treatments with a control is discussed. These test statistics account for prior information that the treatments are at least as effective as the control. Their null distributions are mixtures of chi-squared or beta distributions and the mixing coefficients, which are ‘level probabilities’, are intractable for even a moderate number of treatments and unequal weights, which are typically the sample sizes. The distribution corresponding to equal weights is considered as an approximation and an approximation based on the pattern of large and small weights is developed. Both are adequate for moderate variations in the weights, but the approximation based on patterns in the weight set is considerably more accurate for larger variations.

Key Words: Chi-bar-squared distribution • Comparison with control • Level probability • Order restricted inference • Simple tree ordering


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