A colleague of mine asked me to analyse some data from a movement experiment.
Participants were asked to turn their head three times to the left (toL), three times to the right (toR), with both the eyes open (EO) and the eyes closed (EC). Thus, each participant completed 12 movements. Their movement was recorded with optoelectronics and different movement's measures have been obtained (e.g. amplitude, maximum speed...).
Things are actually more complex, given that the same experiment has been run on healthy controls and neurological patients and, in the latter, in multiple sessions several days apart. Moreover, patients received different treatments in between some sessions.
For the moment, just consider a single session with patients only.
I have a dataset like this:
patient trial side eyes measure
1 1 toR EO 45
1 2 toR EO 38
1 3 toR EO 20
1 4 toR EC 16
1 5 toR EC 30
1 6 toR EC 40
1 7 toL EO 26
1 8 toL EO 30
1 9 toL EO 16
1 10 toL EC 10
1 11 toL EC 29
1 12 toL EC 9
2 1 toR EO 41
2 2 toR EO 26
2 3 toR EO 42
2 4 toR EC 27
2 5 toR EC 39
2 6 toR EC 28
2 7 toL EO 3
...
n 12 toL EC 5
Due to the overall complexity of the experimental design, I'd like to use linear mixed effects (LME) models to analyse these data.
However, I have a major perplexity due to repeated trials.
As often happens when human movement is studied, participants are asked to repeat the very same movement several times (three times in the current work). This is done to let participants to get used to the experimental setting, to reduce the measurement error and to take into consideration some learning effect which is (more or less) always present in movement.
If ANOVA were used to analyse these data, the three "identical" movements would be averaged (i.e. aggregated) and the effects of "eyes" and "side" on measures would be studied.
It seems to me that one of the strength of the LME models is that they are able to deal with full data (i.e. not aggregated).
Therefore, I would try in R (lme4 library, lmer function) the following maximal model on the full dataset reported above:
full_model <- measure ~ eyes * side + (1 + eyes * side|patient)
With "full dataset" I mean the dataset with 12n rows. Note that "trial" does not appear on the right side of the formula.
I know that full_model could be too complex for the current dataset. Therefore, I would use the buildmer() function to find the most complex model that is supported by data.
I know that the repeated measure's problem has been already discussed in Cross Validated single trial vs aggregation. However, I think that our data are peculiar.
We introduce a source of repetition (i.e. trial), which is not taken into account in the model. Is this approach correct?
Note that, if this is ok in LME models, it should be acceptable to drop another source of repetition (e.g. eyes or side) and still analyse the full dataset (i.e. with 12n rows).
In other words, I should be able to use the full database (i.e. with 12n rows) to analyse the following model: full_model <- measure ~ eyes + (1 + eyes|patient)
I'm asking you for your help because it seems to me that not taking into account "trial" in the formula of the LME model could lead to increase the type I error probability.
Could you please give me an advice? References with analyses similar to the present one would be highly appreciated.
ADDENDUM (16 03 2021) I explain better the model selection, prompted by Robert's comment.
I had a lot of thinking about the best model for these data. I agree that the choice of the model should be on some theoretical ground. And I see as well that an automated choice could lead to a model poorly generalisable. The best model, but for the current dataset only. However, I really think that the right model should have random intercepts and slopes for any predictor and interaction.
Consider patient 1 and 2 moving to the left with the eyes open.
I cannot find a theoretical reason to assume that, for example, movement speed (i.e. the "measure" in the toy dataset) is the same in the two patients. For this reason, random intercepts are used in the model.
Now consider these two patients moving to the left with the eyes open and then with the eyes closed.
Again, I cannot assume that movement speed decreases the same in the two patients when moving with the eyes closed. Thus, random slopes should be included in the model.
Note that an interaction term is needed to describe this condition (movement to the left, eyes open, eyes closed).
The right model seems to be the most complex one. I simple ask to buildermer to check if data support this model. If not, buildmer finds for me the closest model to the most complex one.
