To test the equality of means of non-normal data, both parametric test and non-parametric test may be applied. In this paper, the analysis of variance (ANOVA) test, the likelihood ratio test and the Kruskal-Wallis test are investigated in order to test the means of several Pareto distributions. Since Pareto data are non-normal, they must be transformed to normal with homogeneous variances so that ANOVA test can be applied. The alternative transformation for any sets of Pareto data to normal with equal variances is used. The power of the ANOVA test, the likelihood ratio test and the Kruskal-Wallis test are compared. It is found that the power depends on the location and the shape parameters. The results are divided into three cases. First, if the location parameters are the same but the shape parameters are different, for samples of sizes 10 to 30, the power of the likelihood ratio test is the highest. The power of all the tests is almost the same when the population means are largely different. Second, if both the location parameters and the shape parameters are different, for samples of sizes 10, the power of the likelihood ratio test is still the highest. For samples of sizes 20 to 30, the power of the ANOVA test is the highest. Moreover, the power of all tests are almost the same when the population means are much different. Last, with the different location parameters but the same shape parameters, for samples of sizes 10 to 30, the power of the Kruskal-Wallis test is the highest. The power of the Kruskal-Wallis test and the likelihood ratio test are almost equal when the differences of the population means are large, but the power of ANOVA test is still the lowest. For unequal sample sizes, the conclusion is the same as the case of samples of sizes 10.
Keywords: Pareto data, Power, The alternative transformation
[...] The Three Tests under Consideration There are many methods for testing the equality of parameters in several populations, most of which can be applied to test for equality of just one parameter, while the others are held fixed. If the data are non-normal, we usually apply the likelihood ratio test to scale, location, or shape parameter. Nagarsenker derived the exact distribution of the likelihood ratio statistic for testing the equality of k one parameter exponential populations. Chaudhuri and Chandra presented an alternative procedure for testing the equality of scale parameters of Weibull populations with a common shape based on sample quantiles. [...]
[...] The power of all three tests are almost the same if the differences among the population means are large If the populations have different values of both location and shape parameters, the likelihood ratio test has the highest power for the simulations with equal samples of sizes 10 and unequal sample sizes but ANOVA test is higher than the likelihood ratio test a little for the simulations with equal samples of sizes 20 and 30. However, all the three tests have almost the same power when the differences among the population means are large. [...]
[...] In this case, if the differences among the population means are large, the likelihood ratio test and the Kruskal-Wallis test have almost the same power and higher power than ANOVA test Conclusions and Recommendations To test the equality of means of Pareto data, ANOVA test, likelihood ratio test and Kruskal-Wallis test are investigated. The use of ANOVA test for testing equality of means for several Pareto distributions, transformed data to normal with constant variances is needed. The data sets transformed by an alternative transformation meet the assumptions required for the application of ANOVA test. [...]
[...] For the set of Pareto data that the null hypothesis, the alternative hypothesis, the level of significance, α , and the sample size are the same, the power of the three tests are compared. The values of parameters, θi , γ i and the significant value, α , are set as follows: k = number of the populations = 3 n i = sample sizes from the ith Pareto population are between 10 and 30 θi , the location parameter of the ith Pareto population, is between 1,000 and 6,400 γ i , shape parameter of the ith Pareto population, is between 1.1 and 60 fixed by location parameter and coefficient of variation of population means α = Finding Critical Values To find the critical value for reject on of the null hypothesis, Pareto populations of size Ni = 4,000 are generated for θ1 = θ2 = . [...]
[...] n i The power function of this test is given by 1 χ γ θ ( 2 γ k 2 β(ξ ξ k , σξ ) = 2 χα e e s i ( ξi ξ ) 2 i (χ ) ( k + t 2 t k 2 12 γ s i (ξ i ξ ) θ ( 2γ + i t Γ + t 22t .t! 3. Comparison of the Power Simulation Results Comparison of the power of the ANOVA test with those of the likelihood ratio and the Kruskal-Wallis tests for Pareto data is carried out by simulation Simulation Method Karlis and Xekalaki suggested a power comparison of several tests for Poisson data in two steps. First, critical values for three tests and all distributions are obtained for different sample sizes. [...]
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