Design and Analysis of Experiments by Douglas Montgomery. Heath Rushing

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      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 Pick Role Variables, and then click the red triangle to the right of Transform. Select Sqrt.

      It would also be possible to create an additional column in the data table that contains the transformed values. An advantage of using the transform option, however, is that the predicted values from the Fit Model report are automatically transformed back to the original scale for prediction.

      12. Click Run.

      The Residual by Predicted Plot for the ANOVA of the transformed response does not show the same increasing “funnel” of variance that appears in the plot for the analysis of the original response.

      13. Select Window > Close All.

      Example 3.7 Tukey Multiple Comparisons

      NOTE: It is possible to perform the multiple comparisons procedures of the next three examples either using the Fit Y by X platform or the Fit Model platform. The next three examples use Fit Y by X, and the example labeled Section 3.8.2 uses Fit Model.

      1. Open Etch-Rate.jmp.

      2. Select Analyze > Fit Y by X.

      3. Select Etch Rate and select Y, Response.

      4. Click Power and select X, Factor.

      5. Click OK.

      6. Click the red arrow next to One-way Analysis of Etch Rate By Power and select Compare Means > All Pairs, Tukey HSD.

      The LSD (least significant difference) Threshold Matrix reports positive values in components that correspond to significantly different pairs of treatment effects. This information is summarized in the Connecting Letters Report. The Ordered Differences Report lists the pairwise comparisons according to magnitude of the difference between the treatment means. The Tukey procedure indicates that all pairs of means are significantly different.

      7. Leave the Fit Y by X report open for the next two examples.

      Example 3.8 Fisher Multiple Comparison

      1. Return to the Fit Y by X platform from the previous example.

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