Example 5.25: (page 228) Screenshots from JMP. These examples assume post-hoc analysis using a Protected LSD or Protected Welch strategy.

Transcription

1 JMP Output from Chapter 5 Factorial Analysis through JMP Example 5.25: (page 228) Screenshots from JMP. These examples assume post-hoc analysis using a Protected LSD or Protected Welch strategy. Fitting the Model and checking conditions After fitting the model, use the [red arrow] to save the residuals like we have done before. Then you can get a QQ plot of the residuals (at right). The Brown-Forsythe test in a Factorial model is a little bit tricky. The null hypothesis is that all of the cell variances are equal. So, we need a factor that represents all of the cell variances. Create a factor called Trt (or something), and make a formula for it. The formula will need to paste together the factor levels using the Concat function. The formula to do this is on the next page.

2 In this particular example, the values of the factor levels are numeric (even though we have set them to be categories), so we need to tell JMP to convert them to character strings. This uses the Char function just above Concat. The final formula is below (I used Concat twice, because I like to put an underscore in the middle, but you don t have to). Then we run Fit Y by X using our raw data as the response, and this new Trt factor as the factor. After running this model, we can get the Brown-Forsythe test as we would have done for 1 Factor ANOVA. In this case, the Brown-Forsythe test says that the treatment variances are NOT equal. So, our post-hoc test should use the Protected Welch analysis (this document will show both LSD and Welch). For the conditions: Assuming Normality of the residuals is OK, but there is evidence that the variances are unequal. The global tests should still be fairly reliable, but we will need to be careful on the posthoc analysis.

3 Global tests: Interaction and Main Effects In a factorial model, we first test the highest order interactions, and them move down the chain. A term only gets tested if there are no higher-order interactions which contain it. The Effects Tests table contains the global tests. In this case, there is evidence of interaction (testing interaction at the 0.25 level of significance, if you test everything at 0.05, then there is no evidence of interaction). The global test results are: Protected LSD for the Interaction (Simple Effects) When interaction is significant, we default to the Simple Effects. This means looking for differences between the levels of Factor A within the levels of Factor B. If we think of the factorial design as a table of means as below, the Simple Effects imply that we change ONE subscript at a time. So we might test: H! : μ!! = μ!" or H! : μ!! = μ!" or H! : μ!" = μ!" But we would NOT test: H! : μ!! = μ!! μ!! μ!" μ!" μ!" μ!! μ!" This last hypothesis is changing BOTH subscripts at a time. The two means are in neither the same row, nor in the same column. Sometimes this might be a comparison of interest, but by default we do not look at that comparison (just like in 1-Factor ANOVA, by default we don t look at arbitrary contrasts). The idea of the Simple Effects is to look at all of the pairwise comparisons within each row, and all of the pairwise comparisons within each column. In this example, that means there are 9 comparisons of interest (3 for each row, and 1 for each column). For the Protected LSD Simple Effects, we get the results from the Fit Model output. Using the interaction term, we request the output from LSMeans Student s t. One problem with the Protected LSD results from Fit Model is that they provide ALL of the pairwise comparisons, not just the Simple Effects. So, before they are appropriate to submit with a report, etc., we need to filter out only the Simple Effects (or other comparisons of interest).

4 The Crosstab from Simple Effects: The Connecting Letters Report The Ordered Comparisons table is also a useful way to get the results, but again, it contains ALL of the pairwise comparisons, not just the Simple Effects, so some filtering is required. Also, right-clicking on the table lets you remove some columns (like the Std Err, and upper/lower CL). From these p-values or confidence intervals, we can create an appropriate underline report for the simple effects. If there had been no interaction, we would simply have looked at each (significant) factor individually, in the same way that we would when using Fit Model to do a 1-Factor ANOVA.

5 Protected Welch for the Interaction (Simple Effects) Since the HoV test (Brown- Forsythe) was significant, we aught to have used the Protected Welch approach for the post-hoc analysis. This is a little tricky in JMP, but follows basically the same procedure as the Protected Welch for a 1-Factor ANOVA. Using Fit Y by X, put both (all) of the factors into the model as if it was a factorial model (Fit Y by X will run a 1-factor model for each). Then use Script > Local Data Filter. Select both (all) of the factors, and click Add. Using CTRL or CMD (PC vs Mac), select the levels of interest. For example, to compare Anneal Temp 900 to 950, for the doping level 1e20, the selection looks like: The results should show up as t Test Assuming unequal variances. If they don t, use the [red arrow] to make sure t test is selected (note: this option will not appear if JMP sees more than 2 categories for that factor). To run the Protected Welch Main Effects, just select ALL of the categories of the other Factor.