Modeling and Multivariate Methods - SAS

114 Fitting Standard Least Squares Models Chapter 3
Restricted Maximum Likelihood (REML) Method
For historical interest only, the platform also offers the Method of Moments (EMS), but this is no longer a
recommended method, except in special cases where it is equivalent to REML.
If you have a model where all of the effects are random, you can also fit it in the Variability Chart platform.
The REML method for fitting mixed models is now the mainstream, state-of-the-art method, supplanting
older methods.
In the days before availability of powerful computers, researchers needed to restrict their interest to
situations in which there were computational short cuts to obtain estimates of variance components and
tests on fixed effects in a mixed model. Most books today introduce mixed models using these short cuts
that work on balanced data. See McCulloch, Searle, and Neuhaus (2008), Poduri (1997), and Searle,
Casella, and McCulloch(1992). The Method of Moments provided a way to calculate what the expected
value of Mean Squares (EMS) were in terms of the variance components, and then back-solve to obtain the
variance components. It was also possible using these techniques to obtain expressions for test statistics that
had the right expected value under the null hypotheses that were synthesized from mean squares.
If your model satisfies certain conditions (that is, it has random effects that contain the terms of the fixed
effects that they provide random structure for) then you can use the EMS choice to produce these
traditional analyses. However, since the newer REML method produces identical results to these models,
but is considerably more general, the EMS method is never recommended.
The REML approach was pioneered by Patterson and Thompson in 1974. See also Wolfinger, Tobias, and
Sall (1994) and Searle, Casella, and McCulloch(1992). The reason to prefer REML is that it works without
depending on balanced data, or shortcut approximations, and it gets all the tests right, even contrasts that
work across interactions. Most packages that use the traditional EMS method are either not able to test
some of these contrasts, or compute incorrect variances for them.
Introduction to Random Effects
Levels in random effects are randomly selected from a larger population of levels. For the purpose of testing
hypotheses, the distribution of the effect on the response over the levels is assumed to be normal, with mean
zero and some variance (called a variance component).
In one sense, every model has at least one random effect, which is the effect that makes up the residual error.
The units making up individual observations are assumed to be randomly selected from a much larger
population, and the effect sizes are assumed to have a mean of zero and some variance, σ 2 .
The most common model that has random effects other than residual error is the repeated measures or split
plot model. Table 3.14 on page 115, lists the types of effects in a split plot model. In these models the
experiment has two layers. Some effects are applied on the whole plots or subjects of the experiment. Then
these plots are divided or the subjects are measured at different times and other effects are applied within
those subunits. The effects describing the whole plots or subjects are one random effect, and the subplots or
repeated measures are another random effect. Usually the subunit effect is omitted from the model and
absorbed as residual error.