# Covariance matrix in multilevel repeated measures model

I'm trying to fit a multilevel model for a repeated measures design with three levels: Subjects - conditions - trials. Each subject passes the test in two conditions, and there are 21 trials in each condition. Each trial is expected to be correlated in some manner with the previous and following trial.

My problem is that I don't know which covariance type for repeated measures should I select to the model... Can anyone tell me how can I find out which is correct? (By the way, I'm using SPSS and not familiar with other software ...)

• I'm unclear what "covariance type" means here - what are the options? Commented May 23, 2012 at 12:55
• When fitting the mixed model in SPSS, at the first steps it ask for the "Repeated Covariance Type", and offers options as "Diagonal", "Unestructured", "Scaled Identity", "Compound symmetry", and many more. Later, when defining Random effects, it also asks for "Covariance Type", and offers "variance components", "diagonal", etc. Commented May 25, 2012 at 8:21

## 1 Answer

Disclaimer: I'm no expert is the mixed procedure and I simply happened to have similar questions analysing my own data.

According to SPSS 25 manual p. 22 the repeated covariance type is the covariance structure for the residuals. Among others, SPSS provides following structures:

• AR(1)
• Compound Symmetry
• Diagonal
• Unstructured

The only reason I picked those 4 (out of 22) is that I found those more relevant to my own research

On p.80 of the same manual we can find a brief explanations:

1. AR(1). This is a first-order autoregressive structure with homogenous variances. The correlation between any two elements is equal to rho for adjacent elements, rho2 for elements that are separated by a third, and so on. is constrained so that –1<<1.
2. Compound Symmetry. This structure has constant variance and constant covariance.
3. Diagonal. This covariance structure has heterogeneous variances and zero correlation between elements.
4. Unstructured. This is a completely general covariance matrix.

I would not go as far as recommending any structure for your data (this question is 6 years old, so you probably sorted it out) but: for a repeated measure design the default in SPSS is Diagonal.

A. Field (in Discovering Statistics Using SPSS) p. 738 suggests testing different structures on a final model (estimated with ML, not the default REML) and comparing their goodness-of-fit indices (AIC, AICC)

Also this answer might be helpful as well: https://stats.stackexchange.com/a/49786/133561

If anyone can provide a plain English explanation I'd love to hear it and understand some more.