Power Curves for the Analysis of Means for Variances - ASQ

Power Curves for the Analysis of Means for Variancesy

The analysis of means for variances (ANOMV) is used for testing the k-sample homogeneity of variances hypothesis for samples from normal populations. The test can be done in a graphical form and has been shown to have power comparable to the best competing tests. In order to be able to determine the power for various combinations of k, degrees of freedom (v), and level of significance (), the least favorable configuration for the k variances is discussed; power curves are presented for = 0.1,0.05,0.01, k = 3(1)8,10,12, and = 3(1)15,20,30,60. These power curves are useful for determining appropriate sample sizes and for interpreting experimental results.

Keywords: Homogeneity of Variances, Power, Sample Size.

by PETER S. WLUDYKA, University of North Florida, Jacksonville, 32224-2645
PETER R. NELSON, Clemson University, Clemson, SC 29634-0975 and
PETER R. SILVA, University of North Florida, Jacksonville, FL 32224-2645

INTRODUCTION

Often a test for homogeneity of variances among several populations is desired. For example, suppose six alloys can be used to manufacture a component. If one wishes to compare the lifetimes of the components (alloys) with the goal of process improvement, as one might in a total quality management environment, both measures of location and variability of the lifetimes are likely to be of interest. The claim that all of the populations under study (the six alloys) have the same variance will be called the homogeneity of variance (HOV) hypothesis. The HOV hypothesis for a single factor experiment with k > 2 factor levels can be represented as

where is the variance of the ith population. The alternative hypothesis is Ha : not H0.

A key step in testing the HOV hypothesis is determining an appropriate sample size. Choosing the level of significance and the desired power for detecting a specified difference among the variances determines the necessary sample size. Alternatively, for a fixed level of significance and fixed sample size the resulting power for detecting a specific difference among the variances can be useful for interpreting experimental results.

Currently, five well-known HOV tests—proposed by Bartlett (1937), Hartley (1940, 1950), Cochran (1941), Doornbos (1956), and Foster (1964)—are frequently employed by practitioners when normality can safely be assumed. Only the first two of these tests are general purpose tests that exhibit good power for all variance configurations. Wludyka and Nelson (1997) proposed the Analysis of Means for Variances (ANOMV) test and showed via a Monte Carlo study that this test exhibits good power for all variance configurations , has power roughly comparable to the tests proposed by Bartlett and Hartley, and that no test outperforms the others for all variance configurations. Wludyka and Nelson (1997) showed that ANOMV was, comparatively, a good HOV test. In this paper, however, the power associated with the ANOMV is systematically investigated, and guidance is offered regarding the choice of sample size and the interpretation of results based on power considerations.

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