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CHAPTER 13 DESIGN AND ANALYSIS OF SINGLE-FACTOR EXPERIMENTS: THE ANALYSIS OF VARIANCE
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where LSD, the least signi cant difference, is
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If the sample sizes are different in each treatment, the LSD is de ned as LSD EXAMPLE 13-2 t
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We will apply the Fisher LSD method to the hardwood concentration experiment. There are a 4 means, n 6, MSE 6.51, and t0.025,20 2.086. The treatment means are y1. y2. y3. y4. 10.00 psi 15.67 psi 17.00 psi 21.17 psi
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The value of LSD is LSD t 0.025,20 12MSE n 2.0861216.512 6 3.07. Therefore, any pair of treatment averages that differs by more than 3.07 implies that the corresponding pair of treatment means are different. The comparisons among the observed treatment averages are as follows: 4 vs. 1 4 vs. 2 4 vs. 3 3 vs. 1 3 vs. 2 2 vs. 1 21.17 21.17 21.17 17.00 17.00 15.67 10.00 15.67 17.00 10.00 15.67 10.00 11.17 5.50 4.17 7.00 1.33 5.67 3.07 3.07 3.07 3.07 3.07 3.07
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From this analysis, we see that there are signi cant differences between all pairs of means except 2 and 3. This implies that 10 and 15% hardwood concentration produce approximately the same tensile strength and that all other concentration levels tested produce different tensile strengths. It is often helpful to draw a graph of the treatment means, such as in Fig. 13-2, with the means that are not different underlined. This graph clearly reveals the results of the experiment and shows that 20% hardwood produces the maximum tensile strength. The Minitab output in Table 13-5 shows the Fisher LSD method under the heading Fisher s pairwise comparisons. The critical value reported is actually the value of t0.025,20
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Figure 13-2 Results of Fisher s LSD method in Example 13-2.
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13-2 THE COMPLETELY RANDOMIZED SINGLE-FACTOR EXPERIMENT
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2.086. Minitab implements Fisher s LSD method by computing con dence intervals on all pairs of treatment means using Equation 13-12. The lower and upper 95% con dence limits are shown at the bottom of the table. Notice that the only pair of means for which the con dence interval includes zero is for 10 and 15. This implies that 10 and 15 are not signi cantly different, the same result found in Example 13-2. Table 13-5 also provides a family error rate, equal to 0.192 in this example. When all possible pairs of means are tested, the probability of at least one type I error can be much greater than for a single test. We can interpret the family error rate as follows. The probability is 1 0.192 0.808 that there are no type I errors in the six comparisons. The family error rate in Table 13-5 is based on the distribution of the range of the sample means. See Montgomery (2001) for details. Alternatively, Minitab permits you to specify a family error rate and will then calculate an individual error rate for each comparison.
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More About Multiple Comparisons (CD Only) Residual Analysis and Model Checking
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The analysis of variance assumes that the observations are normally and independently distributed with the same variance for each treatment or factor level. These assumptions should be checked by examining the residuals. A residual is the difference between an observation yij and its estimated (or tted) value from the statistical model being studied, denoted as yij. For the completely randomized design yij yi. and each residual is eij yij yi., that is, the difference between an observation and the corresponding observed treatment mean. The residuals for the paper tensile strength experiment are shown in Table 13-6. Using yi. to calculate each residual essentially removes the effect of hardwood concentration from the data; consequently, the residuals contain information about unexplained variability. The normality assumption can be checked by constructing a normal probability plot of the residuals. To check the assumption of equal variances at each factor level, plot the residuals against the factor levels and compare the spread in the residuals. It is also useful to plot the residuals against yi. (sometimes called the tted value); the variability in the residuals should not depend in any way on the value of yi. Most statistics software packages will construct these plots on request. When a pattern appears in these plots, it usually suggests the need for a transformation, that is, analyzing the data in a different metric. For example, if the variability in the residuals increases with yi., a transformation such as log y or 1y should be considered. In some problems, the dependency of residual scatter on the observed mean yi. is very important information. It may be desirable to select the factor level that results in maximum response; however, this level may also cause more variation in response from run to run. The independence assumption can be checked by plotting the residuals against the time or run order in which the experiment was performed. A pattern in this plot, such as sequences of positive and negative residuals, may indicate that the observations are not independent.
Table 13-6 Residuals for the Tensile Strength Experiment Hardwood Concentration (%) 5 10 15 20 3.00 3.67 3.00 2.17 2.00 1.33 1.00 3.83 Residuals 5.00 2.67 2.00 0.83 1.00 2.33 0.00 1.83 1.00 3.33 1.00 3.17 0.00 0.67 1.00 1.17