balancing for nuisance effects in cluster-randomization stats

[Michael Wibral]

Dear Eric, dear Fieldtrippers,

I have a questions regading the control of certain effects that are usually
controlled by balancing over subjects in old fashioned analyses:
Imagine subjects see stimuli A and B and have to respond with the buttons C
(for seeing A) and D (for seeing B). Of course, one could then not
distinguish between perceptual effects (response to A,B) and motor effcts
(pressing C,D).
Usually one would now balance the button presses over subjects such that one
(random) half of the subjects gets the inverted instruction (press D when
seeing A and press C when seeing B). For an experiment with , say, six
subjects one would get then the following set of correct 'trials':
set1:
{1-AC, 2-AC, 3-AC, 4-AD, 5-AD, 6-AD}
set2:
{1-BD, 2-BD, 3-BD, 4-BC, 5-BC, 6-BC}

If one now tries to check parametrically whether there is an A versus B
effect this should work given the balancing has worked.
In permutation testing using dependend samples the following happens: One of
the permutations will be (last three subjects with exchanged conditions):
permutation set1:
{1-AC, 2-AC, 3-AC, 4-BC, 5-BC, 6-BC}
permutation set2:
{1-BD, 2-BD, 3-BD, 4-AD, 5-AD, 6-AD}

Hence one will get the full C versus D effect in this permutation sample and
similar ones in all permutations that are not too far away from it. If the C
versus D effect is a large one (as e.g. button presses tend to be) this will
definitely dominate the extreme ends of the cluster-t distribution, killing
any chance of detecting an A versus B effect. (I assume that
clusterrandomisation also shouldn't work in this case because the
prerequsite of exchangeability is violated even when the A/B null hypothesis
was true.)

Hence, my question how to design an experiment to control for the
omnipresent button presses (or motor readiness potentials if one chooses a
delayed response paradigm)?

Any ideas appreciated,
Michael Wibral

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