### Context:

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’.

### Analysis:

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.

### Question:

- How should repeated measures ANOVAs with 3 or more conditions presented in random order be analysed?
- Is it reasonable to assume compound symmetry?

#### Best Answer

Repeated measures is kind of an overloaded term. To some people it refers to a particular statistical analysis method; to others it refers to the structure of the design.

This is a variant on a three period, three treatment crossover design.

It is a variant because usually in a crossover design you randomize subjects to sequences. In this case the sequence is determined randomly for each subject. Since there are six possible sequences, it might be that some sequences are not observed, especially with 10 subjects. Maybe this is formally the same as randomizing subjects to sequences, but I haven't looked at that yet.

The considerations for crossover designs are:

*Carryover effects*: Also known as residual effects, where prior treatment may affect response to current treatment. The goal of the washout periods is to remove this from consideration. You could also have (in theory) second-order residual effects, where the treatment given in the first period potentially affects the response to treatment given in the third period.*Period effects*: Response to treatment(s) may change as the study goes on for a given subject.*Autocorrelation:*Serial correlation in errors is usually an issue with more closely measured data. In simple balanced designs, having a random effect for subject is going to imply equal correlaation of errors from each subject.*Subject effects*: Subjects may differ in mean response from each other regardless of treatments. You could conceive of a situation where measurement error was serially correlated separate from a random subject effect.*Sequence effect*: In cases where you randomize subjects to sequences, subjects are considered nested in sequence.

A minimal analysis for this would be the suggested randomized complete block design. That is, a fixed effect for treatment and a random effect for subject. With a skimpy sample size that might be *all* you can really do.

I would argue for a bit more structure to the analysis, if possible. Assuming no carryover effects on scientific grounds, it seems like a good idea to have at fixed effects for treatment, period, and treatment $times$ period interaction, and a random effect for subjects. For small data sets, if this model can't be fit, I would drop the treatment $times$ period interaction first.

Period should be included because it represents a restriction on the randomization. You cannot "randomize" periods — they always happen in the same order. Treatment $times$ period interaction might be indicative of some sort of carryover effect.

With tons of data one could work up terms that would allow estimation of various specific carryover effects. My notes on this are gone, though I know I've seen it handled in some texts.

The strategy of additionally modelling the correlation structure on the R-side seems reasonable to me. That allows one to claim that one is handling the possible dependence structure induced by repeated measures on the same subject, which I would also probably claim about the random effect for subject if the analysis devolved to that level… It is also nice if various analysis strategies provide broadly or very similar results.

For implementation, I'd use `PROC MIXED`

in `SAS`

and likely `nlme`

or `lme4`

in `R`

.

I'll punt on the compound symmetry question, since that seems more like a holdover from the days where MANOVA was the only "correct" analysis for repeated measures.

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