Estimation of Random Effect Parameters Example

Random effects have a dual character. In one characterization, they represent residual error, such as the error associated with a whole-plot experimental unit. In another characterization, they are like fixed effects, associating a parameter to each level of the random effect. As parameters, you have extra information about them—they are derived from a normal distribution with mean zero and the variance estimated by the variance component. The effect of this extra information is that the estimates of the parameters are shrunken toward zero.

The parameter estimates associated with random effects are called BLUPs (Best Linear Unbiased Predictors). Some researchers consider these BLUPs as parameters of interest, and others consider them uninteresting by-products of the methodology.

BLUP parameter estimates are used to estimate random-effect least squares means, which are therefore also shrunken toward the grand mean. The degree of shrinkage depends on the variance of the effect and the number of observations per level in the effect. With large variance estimates, there is little shrinkage. If the variance component is small, then more shrinkage takes place. If the variance component is zero, the effect levels are shrunk to exactly zero. It is even possible to obtain highly negative variance components where the shrinkage is reversed. You can consider fixed effects as a special case of random effects where the variance component is very large.

The REML method balances the information about each individual level with the information about the variances across levels. If the number of observations per level is large, the estimates shrink less. If there are very few observations per level, the estimates shrink more. If there are infinitely many observations, there is no shrinkage and the estimates are identical to fixed effects.

Suppose that you have batting averages for different baseball players. The variance component for the batting performance across players describes how much variation is typical between players in their batting averages. Suppose that the player only plays a few times and that the batting average is unusually small or large. Then you tend not to trust that estimate, because it is based on only a few at-bats. But if you mix that estimate with the grand mean, that is, shrink the estimate toward the grand mean, you would trust the estimate more. For players who have a long batting record, you would shrink much less than those with a short record.

You can explore this behavior yourself.

Select Help > Sample Data Library and open Baseball.jmp.

Select Analyze > Fit Model.

Select Batting and click Y, Response.

Select Player and click Add.

Select Player in the Construct Model Effects box, and select Random Effect from the Attributes list.

Click Run.

Table 3.3 shows the Least Squares Means from the Player report for a REML (Recommended) fit. Also shown are the Method of Moment estimates, obtained using the EMS Method. The Method of Moment estimates are the ordinary Player means. Note that the REML estimate for Suarez, who has only three at-bats, is shrunken more towards the grand mean than estimates for other players with more at-bats.

Table 3.3 Comparison of Estimates between Method of Moments and REML
	Method of Moments	REML	N
Variance Component	0.01765	0.019648
Anderson Jones Mitchell Rodriguez Smith Suarez	0.29500000 0.20227273 0.32333333 0.55000000 0.35681818 0.55000000	0.29640407 0.20389793 0.32426295 0.54713393 0.35702094 0.54436227	6 11 6 6 11 3
Least Squares Means	same as ordinary means	shrunken from means