I'm trying to figure out the most appropriate way of modelling repeated observations. In my case I have 5 annual observations of 20 brands in 10 product categories, and some sources suggest a mixed effects model, say in SPSS, in which you specify the type of correlation between each observation .... and then other sources suggest clustering the errors ... despite lots of reading, I can't quite figure which approach is most appropriate. Then an added complexity is some brands (like retailer's brands) are present in more than one product category which means I would ideally accomodate the dependence across observations at two levels (?). Any advice gratefully appreciated.
II will have a go at different forms of mixed models ( which I think is much better than robust / clustered standard errors).
The issue can be seen as goodness of fit and complexity, that is parsimony.
The simplest two level model( occasions within brands) is a compound symmetry random intercepts (CS) model with just one term defining the degree of implied dependence ( the ratio of the level 2 variance to the sum of the level 2 variance plus the level 1 variance). Say you got an estimated value of 0.20 then 20 percent of the variation lies at the higher level. The similarity in the response between pairs of observations on different occasions is 0.2 .
The most complex model that you can fit is called the multivariate or unstructured model. This allows a different implied correlation between any pair of of observations when pairs are defined by how far apart in time the observations are . So you could get a correlation of 0.2 for occasions 1 and 2, 0.15 for 1 and 3, and 0.13 for occasions 2 and 3. The number of terms required grows rapidly with this specification.
There are a number of in-between models and I like the Toeplitz model - not as complex as the unstructured model , not a simple as the compound symmetry. Now the implies correlation is constrained by the lag between variables, and those with the same lag get the same estimated correlation. So the pair lagging by 1 unit (occasion 1 and 2, 2 and 3 , 3 and 4) will have the same estimate, but those 2 lags apart ( 1 and 3 , 2 and 4 etc) can potentially have a different value.
So what you can do is start with the simplest CS model and then build up greater complexity, performing a likelihood ratio test to see if there justification for this complexity. The idea being that you will get better standard errors but also a better idea about the underlying processes such as brand loyalty from examining the correlation structure.
I have a long example in this
Book Developing multilevel models for analysing contextuality, he...
and this site is very helpful across lots of software, and the pace of the material is quite measured (pun!)
http://www.pilesofvariance.com/
the majority of published applications are in reality CS models.
Hi John, a multi-level model would suit well. We did something similar in our latest ML paper.
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