Because the varieties are randomly selected, the regression model for each variety is a random model selected from the population of variety models. The intercept and slope are random for each variety and might be correlated. The random coefficients are centered at the fixed effects. The fixed effects are the population intercept and the slope, which are the expected values of the population of the intercepts and slopes of the varieties. This example is taken from Littell et al. (2006, p. 320).
Fitting the model using REML in the Standard Least Squares personality lets you view the variation in intercepts and slopes (Figure 7.2). Note that the slopes do not have much variability, but the intercepts have quite a bit. The intercept and slope might be negatively correlated; varieties with lower intercepts seem to have higher slopes.
Figure 7.2 Standard Least Squares Regression
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Select Analyze > Fit Model.
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Select Mixed Model from the Personality list. Alternatively, you can select the Mixed Model personality first, and then click Y to add Yield.
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Select the Random Effects tab.
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Select Variety from the Select Columns list, select Moisture from the Random Effects tab, and then click Nest Random Coefficients.
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Click Run.
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The Fit Mixed report is shown in Figure 7.5. Note that some of the constituent reports are closed because of space considerations. The Actual by Predicted plot shows no discrepancy in terms of model fit and underlying assumptions.
Yield = 33.43 + 0.66 * Moisture
Figure 7.5 Fit Mixed Report
Figure 7.6 Random Coefficients Report
In the Model Specification window, the Center Polynomials option is selected by default. Because of this, the Moisture effect is centered at its mean of 35.583, as stated in the note at the top of the Variety report. From the Fixed Effects Parameter Estimates and Random Coefficients reports, you obtain the following prediction equation for Variety 2:
Variety 2 starts with a lower yield than the population average and increases with Moisture at a slower rate than the population average.