My question concerns a typical design in my area – a researcher takes a group of subjects (say 10) and then applies three different conditions to them to measure the change in a response variable, e.g. vertical jump height performed after drinking a glucose drink, coloured plain water, and fruit juice (say). Every subject has every treatment, but in a random order with enough time between for effects to ‘wash out’.
Kuehl (2000) (Kuehl, R. O. (2009) Design of Experiments: Statistical principles of research design and analysis, Duxbury Press, CA, p497 2nd Ed.) states:
When each of the treatments is administered in a random order to each subject... then subjects are random blocks in a randomised complete block design”
and then shows the corresponding analysis.
In this case, the subject is a random effect, but a nuisance or blocking factor, and although our statistical model will test the significance of the block factor, we aren’t really interested in its significance. However, many researchers (and reviewers!) think that such a design should be analysed as a repeated measures design with a Mauchly test for the Huynh-Feldt condition (with the treatment as the repeated measure). However, this seems more appropriate when a time factor is being analysed – for example when observations are taken at 0 minutes, 10 minutes, 30 minutes and 60 minutes, for example. In this case the covariance between pairs of time points might reasonably be expected to change, particularly when unequal time intervals are used. [In fact, I use SAS to model different covariance structures in this case (e.g. autoregressive) and use the AIC to choose the best structure, though this is not an approach that is well received by many reviewers.]
I understood that when the subject is a block factor, and the different treatments are administered in a random order that is different for different subjects, this means that the correlation between observations is different for each subject so compound symmetry can be assumed.
- How should repeated measures ANOVAs with 3 or more conditions presented in random order be analysed?
- Is it reasonable to assume compound symmetry?