Modeling and Multivariate Methods - SAS

Chapter 3 Fitting Standard Least Squares Models 117
Restricted Maximum Likelihood (REML) Method
Scripting Note
The JSL option for Unbounded Variance Components is No Bounds( Boolean ). Setting this option
to true (1) is equivalent to checking the Unbounded Variance Components option.
Random Effects BLUP Parameters
Random effects have a dual character. In one perspective, they appear like residual error, often the error
associated with a whole-plot experimental unit. In another respect, they are like fixed effects with a
parameter to associate with each effect category level. 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 shrunk
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 by-products of the method that are not interesting in themselves. In JMP, these estimates are available,
but in an initially closed report.
BLUP parameter estimates are used to estimate random-effect least squares means, which are therefore also
shrunken toward the grand mean, at least compared to what they would be if the effect were treated as a
fixed effect. 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.
If the number of observations per level is large, the estimate shrinks less. If there are very few observations
per level, the estimates shrink more. If there are infinite observations, there is no shrinkage and the answers
are the same as fixed effects.
The REML method balances the information about each individual level with the information about the
variances across levels.
For example, suppose that you have batting averages for different baseball players. The variance component
for the batting performance across player describes how much variation is usual between players in their
batting averages. If the player only plays a few times and if 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; the estimate has a high
standard error. But if you mixed it with the grand mean, that is, shrunk the estimate toward the grand mean,
you would trust the estimate more. For players that have a long batting record, you would shrink much less
than those with a short record.
You can run this example and see the results for yourself. The example batting average data are in the
Baseball.jmp sample data file. To compare the Method of Moments (EMS) and REML, run the model
twice. Assign Batting as Y and Player as an effect. Select Player in the Construct Model Effects box, and
select Random Effect from the Attributes pop-up menu.
Run the model and select REML (Recommended) from the Method popup menu. The results show the
best linear unbiased prediction (BLUP) for each level of random effects.