The repeated effect is used to model a correlation structure on the residuals. Specifying the repeated effect is optional. If no repeated effect is specified, then R=s2 IN is used, where N denotes the number of observations, and IN is the N ´ N identity matrix.
All variables used on this tab of the Linear Mixed Effects object must be classification variables.
To specify a particular repeated effect, one must have a classification model term that uniquely identifies each individual observation. Put the model term in the Repeated Specification field. Note that a model term can be a variable name, an interaction term (e.g., Time*Subject), or a nested term (e.g., Time(Subject)).
The variance blocking model term creates a block diagonal R matrix. Suppose the variance blocking model term has b levels, and further suppose that the data are sorted according to the variance blocking variable. Then R would have the form:
where S is the variance structure specified.
The variance S is specified using the Type pull-down menu. Several variance structures are possible, including unstructured, autoregressive, heterogeneous compound symmetry, and no-diagonal factor analytic. See the Help file for the details of the variance structures. The autoregressive is a first-order autoregressive model.
To model heterogeneity of variances, use the group variable. Group will accept any model term. The effect is to create additional parameters of the same variance structure. If a group variable has levels g=1, 2,…, ng, then the variance for observations within group g will be Sg.
An example will make this easier to understand. Suppose five subjects are randomly assigned to each of three treatment groups; call the treatment groups T1, T2, and T3. Suppose further that each subject is measured in periods t=0, 1, 2, 3, and that serial correlations among the measurements are likely. Suppose further that this correlation is affected by the treatment itself, so the correlation structure will differ among T1, T2, and T3. Without loss of generality, one may assume that the data are sorted by treatment group, then subject within treatment group.
First consider the effect of the group. It produces a variance structure that looks like:
where each element of R is a 15´15 block. Each Rg has the same form. Because the variance blocking variable is specified, the form of each Rg is:
I5 is used because there are five subjects within each treatment group. Within each subject, the variance structured specified is:
This structure is the autoregressive variance type. Other variance types are also possible. Often compound symmetry will effectively mimic autoregressive in cases where autoregressive models fail to converge.
The output will consist of six parameters: s2 and r for each of the three treatment groups.