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In the context of multilevel modelling, Field (2013) p. 827 provides the following representation of a variance-covariance matrix to illustrate Variance Components

Field p827

and writes

This covariance structure assumes that all random effects are independent (hence, the covariances in the matrix are 0). Variances of random effects are assumed to be the same (hence, they are 1 in the matrix) and sum to the variance of the outcome variable. In SPSS this is the default covariance structure for random effects and is sometimes called the independence model. In SPSS this is the default covariance structure for random effects and is sometimes called the independence model.

Heck, Thomas, and Tabata (2013) p. 91 write

The default covariance structure is Variance Components (VC). VC is the default covariance structure for random effects. This specifies a diagonal covariance matrix for the random effects; that is, it provides a separate variance estimate for each random effect, but not covariances between random effects.

These two definitions seem to be inconsistent with each other, inasmuch as the former suggests the variances are assumed the same, whereas the latter does not.

The IBM documentation for SPSS says here of Variance Components that

This structure assigns a scaled identity (ID) structure to each of the specified random effects.

The IBM documentation also says here that Variance Components

is the default covariance structure for random effects. When the variance components structure is specified on a RANDOM subcommand, a scaled identity (ID) structure is assigned to each of the effects specified on the subcommand. If the variance components structure is specified on the REPEATED subcommand, it is replaced by the diagonal (DIAG) structure.

Heck et al. define the scaled identity structure in a couple of different ways, stating on p. 136 that

The Scaled Identity covariance structure has constant variance and assumes no correlation between any elements.

and on p. 210 that

The Scaled Identity covariance structure has heterogenous variances and zero correlation between elements

These seemingly contradictory definitions are making it difficult for me to understand Variance Components, and I have the following questions.

Which of the Field, Heck, and IBM descriptions of Variance Components are consistent with one another? Which, if any, are correct?

What would it mean for a scaled identity structure to be assigned to each of the effects specified? Which of the two Heck et al. definitions of a scaled identity structure are correct?

I’m also interested to know if this issue is some SPSS-specific thing, or whether Variance Components has a canonical definition that should hold across all programs.

Field, A. (2013). Discovering statistics using IBM SPSS statistics. London, UK: Sage.

Heck, R. H., Thomas, S. L., & Tabata, L. N. (2013). Multilevel and longitudinal modeling with IBM SPSS. Routledge.

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    $\begingroup$ Reviewing a paper, I just ran across this "scaled identity" covariance structure. It was helpful to see this question. It does seem to be the default in SPSS. I know that Stata also has a default variance-covariance matrix for mixed (multilevel) models that assumes the random effects are uncorrelated and you have to specify cov(un) if you want to override this. I always do, personally. R's lme4 has a default unstructured variance-covariance matrix for the random effects. Appropriately so, IMO. $\endgroup$ – Erik Ruzek Jul 12 '20 at 21:36
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The difference between those two variance-covariances matrixes is that SPSS assigned PER random effect one Scaled Identity Matrix when you indicate Variance Component. When you indicate Scaled Identity, the covariance structure aplies for each specification of the random effects.

Below there is an example; two random effects (Intercept and Time with 2 time points).

Scaled Identity Covariance Matrix SPSS

Variance Components Covariance Matrix SPSS

Variance Components Covariance Matrix SPSS

For the Scaled Identity Matrix, you will see that only one parameter (=Variance of the residuals) is estimated for both random effects. For the Variance Component Matrix, there are two parameters for two random effects. Put bluntly, you can see one value estimated in the Identity Matrix, and two values estimated in the Variance Component Matrix.

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