Modeling and Multivariate Methods - SAS

118 Fitting Standard Least Squares Models Chapter 3
Restricted Maximum Likelihood (REML) Method
Run the model again with EMS (Traditional) as Method. The least square mean for fixed affects and each
level of random effects and is the same.
Table 3.15 on page 118, summarizes the estimates between Method of Moments and REML across a set of
baseball players in this simulated example. Note that Suarez, with only three at-bats, is shrunk more than
the others with more at-bats.
Table 3.15 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
REML and Traditional Methods Agree on the Standard Cases
It turns out that in balanced designs, the REML F-test values for fixed effects is the same as with the
Method of Moments (Expected Means Squares) approach. The degrees of freedom could differ in some
cases. There are a number of methods of obtaining the degrees of freedom for REML F-tests; the one that
JMP uses is the smallest degrees of freedom associated with a containing effect.
F-Tests in Mixed Models
Note: This section details the tests produced with REML.
The REML method obtains the variance components and parameter estimates, but there are a few
additional steps to obtain tests on fixed effects in the model. The objective is to construct the F statistic and
associated degrees of freedom to obtain a p-value for the significance test.
Historically, in simple models using the Method of Moments (EMS), standard tests were derived by
construction of quadratic forms that had the right expectation under the null hypothesis. Where a mean
square had to be synthesized from a linear combination of mean squares to have the right expectation,
Satterthwaite's method could be used to obtain the degrees of freedom to get the p-value. Sometimes these
were fractional degrees of freedom, just as you might find in a modern (Aspin-Welch) Student's t-test.
With modern computing power and recent methods, we have much improved techniques to obtain the
tests. First, Kackar, and Harville (1984) found a way to estimate a bias-correction term for small samples.