LMM: what effect does the repeated variable have on the model?

#1
When making a linear mixed model what does the repeated variable do? I want to be able to relate what I'm clicking on and the statistics that happen. Conceptually, I understand what a repeated measure is, but what is actually happening when I specify a repeated variable in the "Specify Subjects and Repeated" dialogue? I've seen examples in textbooks and online where there are repeated measures, but they don't specify a repeated variable (i.e. there's no '/REPEATED' statement). It's certainly not a trivial field because in my own analyses, specifying a variable in the "Repeated:" box affects the model. Can someone offer an explanation of how the repeated variable affects the general matrix specification, Yi=Xib+Ziui+e? Does it affect the variance-covariance matrix? Does it change how Yi is indexed?
 

Jake

Cookie Scientist
#2
You can specify effects as either "REPEATED" or "RANDOM" and what this basically means is the following. The mixed model is just as you wrote it, where b is the vector of fixed effects, u is the vector of random effects, and e is the vector of errors. The conventional notation is to let G be the covariance matrix of u, and let R be the covariance matrix of e. You may have heard the phrases "G-side covariance structure" or "R-side covariance structure" and this is what these refer to. When you specify an effect as RANDOM, you are modifying the matrix G, either by letting some of the diagonal elements (i.e., the random effects) have some non-zero variance to be estimated, or letting some of the off-diagonal elements (i.e., covariance between random effects) have some non-zero value to be estimated, or whatever. When you specify an effect as REPEATED, you are instead modifying the matrix R, that is, you are allowing the errors to be correlated in some way, for example by estimating a block covariance structure such that errors from the same subject are potentially correlated while errors from different subjects are uncorrelated. In certain simple cases you can get the same substantive answer either way (e.g., by adding subject-level random effects or adding non-zero correlations between errors for a given subject), but they are not the same in general. Hopefully this is not too confusing.
 
#3
Thanks! This is very helpful. To be clear though, even when you don't specify a repeated effect, you still have the error vector, right? If I don't put anything in the "repeated" box, then does SPSS just assume R is a diagonal matrix?
P.S. I also live in Boulder! Are you a student at CU?