Sample Size Determination and Power by Ryan Thomas P.;

Author:Ryan, Thomas P.;
Language: eng
Format: epub
ISBN: 1207569
Publisher: Wiley
Published: 2013-05-23T16:00:00+00:00

6.2 ONE FACTOR—MORE THAN TWO FIXED LEVELS

When a factor has more than two levels, analysis of variance (ANOVA) is used to analyze the data. We will assume that the design is a completely randomized design (CRD). That is, the levels are assigned at random to the experimental units and the design is run with random order. For example, if there were six experimental units for each of three levels, the 18 experimental runs would be made in a random order.

The sample size hand computation is naturally more involved when there are more than two levels, but it can be done without too much difficulty, if hand computation is desired to gain understanding.

With more than two levels, one obvious question is: What “effect” is used in determining sample size? That is, should it be the largest pairwise effect, or should it be the F-statistic for testing the equality of the means, or something else? The user of Lenth's applet, for example, can determine sample size based on either the F-test or on multiple comparisons such as Tukey's Honestly Significant Difference [see, e.g., http://en.wikipedia.org/wiki/Tukey's_test or Gravetter and Wallnau (2009)] or the set of t-tests. It might also be of interest to solve for n based on the smallest difference of the effects of two levels of interest. This is discussed, for example, in Sahai and Ageel (2000, p. 63).

When an F-statistic is the criterion, power is computed using the value of the noncentrality parameter. Unfortunately, it is not possible to give a simple definition of a noncentrality parameter, and this is due in large part to the fact that noncentrality parameters for various tests in experimental design are defined differently by different writers. For example, in one-way ANOVA with k means, the noncentrality parameter is often given as , where ni denotes the number of observations in level i, k is the number of means, , with μi the mean for the ith level and μ the overall mean (i.e., the expected value of each individual observation under the null hypothesis that the treatment means are all equal, and ) Of course, σ2 is the variance of the individual observations under the null hypothesis. Note that the expression simplifies to when there are an equal number of observations for each level. Some authors (e.g., Sahai and Ageel, 2000, p. 57) have given as the form of the noncentrality parameter, and other authors have used the square root of this expression. [See., for example, the discussion of this in Giesbrecht and Gumpertz (2004, p. 61).] These different, conflicting expressions do impede understanding somewhat.

One very general and simple explanation of a noncentrality parameter given by Minitab, Inc. at http://www.minitab.com/en-US/support/answers/answer.aspx?log=0&id=733&langType=1033 is that noncentrality parameters “reflect the extent to which the null hypothesis is false.” More specifically, it is a standardized measure of the difference between the hypothesized value of a parameter and the actual (assumed) value. We can see this if we think about the noncentrality parameter for a t-test, although it isn't necessary to

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