Example of a Mixed Model ANOVA
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A company ran an experiment to see how several conditions affect the thickness of a coating substance that it manufactures. The experiment was run at two different times, in the morning and in the afternoon. Three operators were chosen from a large pool of operators employed by the company. The manufacturing process was run at three settings, 35, 44, and 52. Two determinations of thickness were made by each operator at each time and setting. Thus, the three factors are crossed. One factor, operator, is random; the other two, time and setting, are fixed.

The statistical model is:

Yijkl = m + Ti + Oj + Sk + TOij + TSik + OSjk + TOSijk + eijkl,

where Ti is the time effect, Oj is the operator effect, and Sk is the setting effect, and TOij, TSik, OSjk, and TOSijk are the interaction effects.

Operator, all interactions with operator, and error are random. The random terms are:

Oj TOij OSjk TOSijk eijkl

These terms are all assumed to be normally distributed random variables with mean zero and variances given by

These variances are called variance components. The output from expected means squares contains estimates of these variances.

In the unrestricted model, all these random variables are independent. The remaining terms in this model are fixed.

In the restricted model, any term which contains one or more subscripts corresponding to fixed factors is required to sum to zero over each fixed subscript. In the example, this means:

Your choice of model does not affect the sums of squares, degrees of freedom, mean squares, or marginal and cell means. It does affect the expected mean squares, error term for the F-tests, and the estimated variance components.

Step 1: Fit the restricted form of the model

1    Open the worksheet EXH_AOV.MTW.

2    Choose Stat > ANOVA > Balanced ANOVA.

3    In Responses, enter Thickness.

4    In Model, enter Time | Operator | Setting.

5    In Random Factors, enter Operator.

6    Click Options. Check Use the restricted form of the mixed model. Click OK.

7    Click Results. Check Display expected mean squares and variance components.

8    Click OK in each dialog box.

Step 2: Fit the unrestricted form of the model

1    Repeat steps 1-8 above except that, in 6, uncheck Use the restricted form of the mixed model.

Session window output for restricted case

Session window output for unrestricted case

Interpreting the results

The organization of the output is the same for restricted and unrestricted models: a table of factor levels, the analysis of variance table, and as requested, the expected mean squares. The differences in the output are in the expected means squares and the F-tests for some model terms. In this example, the F-test for Operator is synthesized for the unrestricted model because it could not be calculated exactly.

Examine the 3 factor interaction, Time*Operator*Setting. The F-test is the same for both forms of the mixed model, giving a p-value of 0.001. This implies that the coating thickness depends upon the combination of time, operator, and setting. Many analysts would go no further than this test. If an interaction is significant, any lower order interactions and main effects involving terms of the significant interaction are not considered meaningful.

Let's examine where these models give different output. The Operator*Setting F-test is different, because the error terms are Error in the restricted case and Time*Operator*Setting in the unrestricted case, giving p-values of < 0.0005 and 0.088, respectively. Likewise, the Time*Operator differs for the same reason, giving p-values of 0.002 and 0.369, respectively, for the restricted and unrestricted cases, respectively. The estimated variance components for Operator, Time*Operator, and Operator*Setting also differ.