Design and Analysis of Experiments by Douglas Montgomery. Heath Rushing
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Rate and select Regression Reports > Show All Confidence Intervals.
7. In addition to viewing the Expanded Estimates, you can also click the red triangle next to Response Etch Rate and select Effect Screening > Scaled Estimates.
8. The Scaled Estimates report produces the same output as the Expanded Estimates report, in addition to a graphical representation of the magnitude of the treatment effects.
9. By clicking the red triangle next to Response Etch Rate and selecting Factor Profiling > Profiler, you obtain an interval plot of the mean responses and their confidence intervals.
The Prediction Profiler provides the estimate mean response together with a confidence interval for each power setting. We will not explore the full functionality of the Prediction Profiler here, but it may be used for optimizing parameter settings to achieve a desired response.
10. Leave the Fit Model platform open for the next exercise.
Section 3.4 Model Adequacy Checking
1. In the Fit Model platform from the previous exercise, scroll down to the Residuals by Predicted plot. This plot is discussed in Section 3.4.3 of the textbook.
The variance appears to be constant across the range of predicted etch rates, and no patterns emerge from the plot. Because only a single categorical factor, Power, is included in the model, the validity of the assumption of constant residual variance may also be checked with formal tests, as described in Example 3.4.
2. To check the normality assumption, a quantile plot is commonly used. In JMP, the first step is to generate residuals for each observation. Click the red triangle next to Response Etch Rate and select Save Columns > Residuals.
3. Return to the Etch-Rate data table (you can use a shortcut by clicking the table icon
4. Click Analyze > Distribution.
5. Select Residual Etch Rate for Y, Columns.
6. Click OK.
7. Click the red triangle next to Residual Etch Rate and select Continuous Fit > Normal.
8. Scroll down and click the red triangle next to Fitted Normal and select Diagnostic Plot.
The error distribution appears to be approximately normal as the points fall relatively close to a straight line. We may also perform a Shapiro-Wilk test for the hypothesis that the residuals are from a normal distribution.
9. Click the red triangle next to Fitted Normal and select Goodness of Fit.
With a p-value of 0.2152, the residuals do not display a significant number of departures from normality.
10. Section 3.4.2 of the text discusses plotting the residuals in a time sequence to look for correlations between subsequent runs, which would represent a violation of the (important) independence assumption. To generate this plot, select Analyze > Modeling > Time Series.
11. Select Residual Etch Rate for Y, Time Series.
12. Click OK.
13. Click OK for the next dialog setting the number of autocorrelation lags to 19.
The first four residuals are all greater than zero while the next seven are all less than zero. There could be a systematic cause for this behavior, such as an omitted covariate (e.g. operator or ambient temperature). Though it is beyond the scope of our discussion, the Time Series platform may be used to detect correlations between subsequent runs. Furthermore, detecting patterns in a residual by time plot is analogous to detecting out-of-control conditions on a control chart (e.g. using the Western Electric Rules). If the residual by time plot signals as out of control according to these rules, it could indicate a shift in the behavior of the process during the course of the experiment.
14. Leave Etch-Rate open for the next exercise.
Example 3.4 Test for Equal Variances
1. Select Analyze > Fit Y by X.
2. Select Etch Rate and click Y, Response.
3. Select Power and click X, Factor.
4. Click OK.
5. Click the red arrow next to One-way Analysis of Etch Rate By Power and select Unequal Variances.
As discussed in the textbook, the Levene test is robust to the assumption of normality, whereas the Bartlett test is extremely sensitive to this assumption. We saw in the previous example that the data appear to have been generated from a process that can be modeled with the normal distribution, so we may use Bartlett’s test, which has a p-value of .9332. There is no evidence that the variance of etch rate differs across the levels of the power setting. Further discussion of the tests for equal variances produced by JMP is available from the JMP help documentation.
6. Select Window > Close All.
Example 3.5 Analysis of Variance
1. Open Peak-Discharge.jmp.
2. Select Analyze > Fit Y by X.
3. Select Discharge and click Y, Response.
4. Select Method and click X, Factor.
5. Click OK.
6. Click the red triangle next to One-way Analysis of Discharge By Method and select Unequal Variances.
The Levene test rejects the hypothesis of equal variances with a p-value of 0.0032. By default, JMP produces the result of Welch’s test, which is a generalization of ANOVA with unequal population variances to factors with more than two levels. Instead, we will apply a variance-stabilizing transformation to the Discharge variable.
7. Select Analyze > Fit Model.
8. Check Keep dialog open. This will enable us to return to the model dialog to make changes to the model.
9. Click Run.
The p-value of the F test is <.0001 indicating that the treatment means are not all equal. However, the Residual by Predicted Plot shows that the assumption of constant variance (homoscedasticity) is violated: the variance of the residuals seems to grow in proportion with the level of discharge (heteroskedasticity). To remedy this, we will take an appropriate transformation, the square root transformation, of Discharge and perform an ANOVA on the transformed variable.
10. Return to the Fit Model dialog.
11. Select Discharge under