1´
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PROC GLM;
CLASS A B;
MODEL A B A*B;
Using the JMP Fit Model command and requesting a factorial model for columns A and B produces the following design matrix. Note that A13 in this matrix is A1–A3 in the previous matrix. However, A13B13 is A13*B13 in the current matrix.
Least squares means are the predicted values corresponding to some combination of levels, after setting all the other factors to some neutral value. The neutral value for direct continuous regressors is defined as the sample mean. The neutral value for an effect with uninvolved nominal factors is defined as the average effect taken over the levels (which happens to result in all zeros in our coding). Ordinal factors use a different neutral value in Ordinal Least Squares Means. The least squares means might not be estimable, and if not, they are marked nonestimable. The least squares means in JMP agree with those in SAS PROC GLM (Goodnight and Harvey 1978) in all cases except when a weight is used. When a weight variable is used, JMP uses a weighted mean and SAS PROC GLM uses an unweighted mean for its neutral values.
In SAS PROC GLM, the hypothesis tests for Types III and IV are constructed using the general form of estimable functions and finding functions that involve only the effects of interest and effects contained by the effects of interest (Goodnight 1978).
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JMP implements the effective hypothesis tests described by Hocking (1985, pp. 80–89, 163–166), although JMP uses structural rather than cell-means parameterization. Effective hypothesis tests start with the hypothesis desired for the effect and include “as much as possible” of that test. Of course, if there are containing effects with missing cells, then this test has to drop part of the hypothesis because the complete hypothesis would not be estimable. The effective hypothesis drops as little of the complete hypothesis as possible.
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With linear dependencies, a least squares solution for the parameters might not be unique and some tests of hypotheses cannot be tested. The strategy chosen for JMP is to set parameter estimates to zero in sequence as their design columns are found to be linearly dependent on previous effects in the model. A special column in the report shows what parameter estimates are zeroed and which parameter estimates are estimable. A separate singularities report shows what the linear dependencies are.
Note that this shows that a test on the β1 parameter is equivalent to testing that the least squares means are the same. But because β1 is not estimable, the test is not testable, meaning there are no degrees of freedom for it.
a1
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a2
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b1
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ab11
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ab21
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4.
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The tests are whole marginal tests, meaning they always go completely across other effects in interactions.
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Figure 12.1 Expected Values of the Cell Means
Table 12.10, shows the test of the main effect A in terms of the GLM parameters. The first set of columns is the test done by JMP. The second set of columns is the test done by GLM Type IV. The third set of columns is the test equivalent to that by JMP; it is the first two columns that have been multiplied by a matrix: