Modeling and Multivariate Methods - SAS

Chapter 3 Fitting Standard Least Squares Models 115
Restricted Maximum Likelihood (REML) Method
Table 3.14 Types of Effects in a Split Plot Model
Split Plot Model Type of Effect Repeated Measures Model
whole plot treatment fixed effect across subjects treatment
whole plot ID random effect subject ID
subplot treatment fixed effect within subject treatment
subplot ID random effect repeated measures ID
Each of these cases can be treated as a layered model, and there are several traditional ways to fit them in a
fair way. The situation is treated as two different experiments:
1. The whole plot experiment has whole plot or subjects as the experimental unit to form its error term.
2. Subplot treatment has individual measurements for the experimental units to form its error term (left as
residual error).
The older, traditional way to test whole plots is to do any one of the following:
• Take means across the measurements and fit these means to the whole plot effects.
• Form an F-ratio by dividing the whole plot mean squares by the whole plot ID mean squares.
• Organize the data so that the split or repeated measures form different columns and do a MANOVA
model, and use the univariate statistics.
These approaches work if the structure is simple and the data are complete and balanced. However, there is
a more general model that works for any structure of random effects. This more generalized model is called
the mixed model, because it has both fixed and random effects.
Another common situation that involves multiple random effects is in measurement systems where there are
multiple measurements with different parts, different operators, different gauges, and different repetitions.
In this situation, all the effects are regarded as random.
Generalizability
Random effects are randomly selected from a larger population, where the distribution of their effect on the
response is assumed to be a realization of a normal distribution with a mean of zero and a variance that can
be estimated.
Often, the exact effect sizes are not of direct interest. It is the fact that they represent the larger population
that is of interest. What you learn about the mean and variance of the effect tells you something about the
general population from which the effect levels were drawn. That is different from fixed effects, where you
only know about the levels you actually encounter in the data.