restricted maximum likelihood (REML)in place of the least squares approaches that
we have focused on up to now and will focus on again in the next two chapters.^7
In this section I will discuss a small part of the broader topic of hierarchical or multi-
level models. For these models the repeated measure (e.g., Time or Trials) is a fixed factor
while Subjects is a random factor. The between-subjects factor is also usually a fixed fac-
tor. By approaching the problem using restricted maximum likelihood (REML) as the
method of parameter estimation, the solution can take cognizance from the very beginning
of the analysis that one or more factors are fixed and one or more factors are random. Least
squares solutions of standard analysis of variance treats all factors as fixed until it comes to
determining error terms for Fstatistics.
No one would seriously attempt to do a mixed model analysis by hand. You must use
computer software to perform the analysis. However, there are many software programs
available, some of them even free. The ones that you will have most access to are probably
SPSS Mixedand SAS Proc Mixed.I will use SPSS for our example, though SAS proc
mixed is probably more flexible. A more complete discussion of the analysis of alternative
designs can be found at http://www.uvm.edu/~dhowell/StatPages/More_Stuff/Missing_
Data / Mixed Models for Repeated Measures.pdf. For an example I have chosen a design with
one between subject variable and one within subject variable. The example has missing data
because that will illustrate an analysis that you can not do with standard analysis of variance.
The Data
I created data to have a number of characteristics. There are two groups—a Control group
and a Treatment group, measured at 4 times. These times are labeled as 0 (pretest), 1 (one
month posttest), 3 (three months follow-up), and 6 (six months follow-up). I had a study of
treatment of depression in mind, so I created the treatment group to show a sharp drop in
depression at post-test and then sustain that drop (with slight regression) at 3 and 6 months.
The Control group declines slowly over the 4 intervals but does not reach the low level of
the Treatment group.
The data are shown in Table 14.15. A period is used to indicate missing values.
500 Chapter 14 Repeated-Measures Designs
Table 14.15 Data for mixed model analysis
Group Subj Time0 Time1 Time3 Time6
1 1 296 175 187 242
1 2 376 329 236 126
1 3 309 238 150 173
1 4 222 60 82 135
1 5 150. 250 266
1 6 316 291 238 194
1 7 321 364 270 358
1 8 447 402. 266
1 9 220 70 95 137
1 10 375 335 334 129
1 11 310 300 253.
1 12 310 245 200 170
(^7) In previous editions I used the MANOVA approach under SPSS/Univariate/Repeated measures as a way of
avoiding assumptions of compound symmetry. This approach does not require compound symmetry, but it does
require balanced designs. I have dropped it in favor of the mixed model precisely because the mixed model will
handle missing data much better.
(continues)
SPSS Mixed
SAS Proc Mixed
restricted
maximum
likelihood (REML)