Determining which covariance structures are nested within each other for constructing likelihood-ratio tests
I'm running a linear mixed model in the SPSS MIXED procedure. I would like to use likelihood-ratio tests to help me determine the appropriate covariance structure. I know how to construct there using differences in -2 restricted log-likelihoods for nested models, but I'm not sure how to tell which models are nested within each other. Can you provide some guidance here?
Resolving the problem
The general answer is that structure A is nested within structure B if you can put restrictions on some covariance parameter values in B and the structure then simplifies to A.
Note as preliminary that UNR is just a rescaling of UN, and CSR is a rescaling of CS, so we won't discuss the two correlation metric structures specifically. Anything that applies to UN applies to UNR, and similarly for CS and CSR.
VC is the same as DIAG when considering the R matrix for repeated measures. For a RANDOM subcommand with only one effect listed, VC is the same as ID. With multiple effects listed on a RANDOM subcommand, it's the same as DIAG if each effect has only one degree of freedom, and is nested within DIAG otherwise, with ID nested within it.
One thing that should be obvious is that for one structure to contain another, it must have more free parameters to estimate. A side note to remember about this is that some nesting relationships that hold in the general case will turn into equality relationships when the number of levels of the effect is only two. For example, CSH is nested within UN with more than two levels, but with only two levels, there's only one off-diagonal parameter to estimate, so they become the same. This won't generally cause problems, because you'll get the same -2 restricted log-likelihood and number of covariance parameters estimated in both cases.
UN is the most general structure, containing all other structures as special cases. ID is the simplest, contained in all other structures. DIAG is contained in anything that fits heterogeneous variances, including AD1, ARH1, CSH, FAH1, and TPH.