6.12 Repeated Measures Analysis 87are interested in the time evolution of the measurements for many
patients over a short number of visits, say 3 to 6, we are doing longi-
tudinal analysis, and the measurements over time for a particular patient
are called repeated measures.
The correlation structure within a patient must be modeled and
estimated parametrically from the data. Common parametric structures
for the correlation matrix are AR(1), Toeplitz, and compound symme-
try, among others. These patterns correspond to statistical dependency
models. For example, AR(1) is a fi rst - order autoregressive time series
model, where Y ( t ) = ρ Y ( t − 1 ) + ε ( t ), − 1 < ρ < 1, and ε ( t ) is an inde-
pendent random variable with mean 0 and constant variance for all
times t. Sometimes, if there is suffi cient data, the covariance can be
estimated without modeling a particular correlation structure. In soft-
ware packages, such as SAS, Proc Mixed is to declare the covariance
to be unspecifi ed.
In SAS, repeated measures analysis of variance can be done using
the GLM procedure or the procedure “ Mixed, ” but the two procedures
handle various similar statements differently, and some cases can only
be done with Proc Mixed. The Mixed Procedure is intended to do mixed
effects analysis of variance, where mixed effects means that some of
the effects can be treated as fi xed, but other may be best modeled as
random effects.
Why might we be interested in random effects? In many clinical
trials, many different centers from different parts of the country or in
different countries enroll patients for the trial. However, sometimes
there is signifi cant variation between the sites. We may want to see if
these differences do exist, and so to do that, we model the site as a
factor in the ANOVA model. Usually, it makes sense to consider the
sites chosen as though they represented a random sample from the
population of all potential sites. In such cases, the site becomes a
random effect. Other factors may also in a similar way need to be
modeled as random effects.
This topic is fairly advanced and beyond the scope of the course.
But it is such an important part of clinical trials analysis. Also, missing
data is a common practical problem, and mixed models provide a way
to handle missing data that is sometimes appropriate. The following
references provide detailed treatment of longitudinal data analysis and
missing data modeling and analysis. Hardin and Hilbe ( 2003 ), Verbeke
and Molenberghs ( 1997 ), Hand and Crowder ( 1996 ), and Little and