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This paper examines the robustness of the Welch test, the James test as well as Tan's ANOVA test (to be referred as Fβ test) for testing parallelism in k straight lines under heteroscedasticity and nonnormality. Results of Monte Carlo studies demonstrate the robustness of all tests with respect to departure from normality. Further, there is hardly any difference between these methods with respect to both power and size of the test.

By using Monte Carlo studies, this paper compares the Welch test, the James test and the Brown-Forsythe test for comparing several means under heteroscedasticity. It appears that all the tests are quite robust with respect to departure from normality. The Brown-Forsythe test is at least as good as the James test and the Welch test; but the differences are so small that the choice is immaterial for practical purposes.

Numerous studies have documented the robustness of t and F to heterogeneous variances under the restricted condition of equal n’s. Likewise, the distortion of α in the presence of unequal n’s and variances has been demonstrated in both mathematical and empirical studies. Several investigations, however, have shown the Welch technique to be robust to this disturbing situation in the two group case. The present study was addressed to the k group AOV situation. Monte Carlo methods were employed to contrast several procedures with respect to a.) control over Type I errors and b.) power. Results indicate that the generalized Welch technique may be substituted for the AOV when variances are heterogeneous and n’s are unequal.

A comparative study is made of three tests, developed by James (1951), Welch (1951) and Brown & Forsythe (1974). James presented two methods of which only one is considered in this paper. It is shown that this method gives better control over the size than the other two tests. None of these methods is uniformly more powerful than the other two. In some cases the tests of James and Welch reject a false null hypothesis more often than the test of Brown & Forsythe, but there are also situations in which it is the other way around. We conclude that for implementation in a statistical software package the very complicated test of James is the most attractive. A practical disadvantage of this method can be overcome by a minor modification.

This paper examines the robustness of the Welch test , the James test as well as Tan's Anova test (to be referred as Fβ test) for testing parallelism in k straight lines under heteroscedasticity and nonnormality. Results of Monte Carlo studies demonstrate the robustness of all tests with respect to departure from normality. Further, there is hardly any difference between these methods with respect to both power and size of the test.

The method in Krutchkoff (1988b) for one-way ANOVA is extended to the two-way situation. The K- ANOVA application program is discussed.

SUMMARY A brief survey of the recent literature leads to the conclusion that approximately similar tests for the Behrens-Fisher problem retain their practical relevance. The asymptotic expansion of the power function of Welch's test is compared with the asymptotic expansion of the power functions of other asymptotically similar tests. Finally, it is shown that in the class of all translation invariant tests the Welch test is asymptotically uniformly most powerful against one-sided alternatives.

The probabilities of the Type I errors of the Welch approximate-t test and the Aspin-Welch test for the Behrens-Fisher problem have been calculated for selected sets of degrees of freedom and nominal significance levels. The results show satisfactory agreement to the desired level. Thus, in practice, one can just use the usual t-table to carry out the Welch approximate t-test without much loss of accuracy.

To verify the accuracy and reliability of the computational fluid dynamics simulation of sludge anaerobic digester, fluid flow visualization experiment is a reliable approach. However, it cannot be used for anaerobic digester due to the opaque nature of sludge. Xanthan gum (XG) aqueous solution was selected as the analog fluid of the digested sludge (DS) due to its good optical clarity and good stability. The rheological properties of DS and the XG aqueous solution were measured by applying a rotational viscometer and the measuring results qualitatively and quantitatively compared by graphics and mathematical statistics analysis. The experimental results revealed the rheological behaviour of XG aqueous solution was similar to DS. They were both typical pseudo-plastic non-Newtonian fluid and showed the same trend in the rheological curves with similar curvatures. Carreau model (R 2 =0.9578) was proposed to describe the rheological behaviour and calculate the rheological parameters of the analog fluid. The limiting viscosity of the XG aqueous solution linearly increased with the increases of the concentration within the concentration range of 0.20g/L∼1.60g/L. According to the linear relationship, 0.50g/L, 1.00g/L and 1.50g/L XG aqueous solution were selected to compare with DS. The result of Welch's t-test demonstrated that there is no significant difference in the rheological parameters of DS and the 1.00g/L XG aqueous solution at the significant level of α=0.05, which indicated the feasibility of choosing the 1.00g/L XG aqueous solution as transparent analog fluid of DS, and build a foundation for visualization experiment in anaerobic digester.