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Figure 13-4 Plot of residuals versus factor levels (hardwood concentration).
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Figure 13-3 Normal probability plot of residuals from the hardwood concentration experiment.
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Figure 13-5
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This suggests that time or run order is important or that variables that change over time are important and have not been included in the experimental design. A normal probability plot of the residuals from the paper tensile strength experiment is shown in Fig. 13-3. Figures 13-4 and 13-5 present the residuals plotted against the factor levels and the tted value yi. respectively. These plots do not reveal any model inadequacy or unusual problem with the assumptions.
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In any experimental design problem, the choice of the sample size or number of replicates to use is important. Operating characteristic curves can be used to provide guidance in making this selection. Recall that the operating characteristic curve is a plot of the probability of a
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type II error ( ) for various sample sizes against a measure of the difference in means that it is important to detect. Thus, if the experimenter knows the magnitude of the difference in means that is of potential importance, the operating characteristic curves can be used to determine how many replicates are required to achieve adequate sensitivity. The power of the ANOVA test is 1 P5Reject H0 0 H0 is false6 P5F0 f ,a 1, a 1n 12 0 H0 is false6
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To evaluate this probability statement, we need to know the distribution of the test statistic F0 if the null hypothesis is false. It can be shown that, if H0 is false, the statistic F0 MSTreatments MSE is distributed as a noncentral F random variable, with a 1 and a(n 1) degrees of freedom and a noncentrality parameter . If 0, the noncentral F-distribution becomes the usual or central F-distribution. Operating characteristic curves are used to evaluate de ned in Equation 13-17. These curves plot against a parameter , where
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2 i 1
2 i 2
The parameter 2 is (apart from n) the noncentrality parameter . Curves are available for 0.05 and 0.01 and for several values of the number of degrees of freedom for numerator (denoted v1) and denominator (denoted v2). Figure 13-6 gives representative O.C. curves, one for a 4 (v1 3) and one for a 5 (v1 4) treatments. Notice that for each value of a there are curves for 0.05 and 0.01. O.C. curves for other values of a are in Section 13-2.7 on the CD. In using the operating curves, we must de ne the difference in means that we wish to a detect in terms of g i 1 2 . Also, the error variance 2 is usually unknown. In such cases, we i a must choose ratios of g i 1 2 2 that we wish to detect. Alternatively, if an estimate of 2 i is available, one may replace 2 with this estimate. For example, if we were interested in the sensitivity of an experiment that has already been performed, we might use MSE as the estimate of 2. EXAMPLE 13-3 Suppose that ve means are being compared in a completely randomized experiment with 0.01. The experimenter would like to know how many replicates to run if it is impor5 tant to reject H0 with probability at least 0.90 if g i 1 2 2 5.0 . The parameter 2 is, in i this case,
i 1 2
n 152 5
and for the operating characteristic curve with v1 a 1 5 1 4, and v2 a(n 1) 5(n 1) error degrees of freedom refer to the lower curve in Figure 13-6. As a rst guess, try n 4 replicates. This yields 2 4, 2, and v2 5(3) 15 error degrees of freedom. Consequently, from Figure 13-6, we nd that 0.38. Therefore, the power of the test is approximately 1 1 0.38 0.62, which is less than the required 0.90, and so we