# Question about permutation testing of an interaction in a two by two design

Eric Maris e.maris at DONDERS.RU.NL
Tue Nov 24 23:39:18 CET 2009

```Dear Lin,

Thanks for your clear answer. I also have a question about the permutation
testing of the interactions in the mixed factorial design.

Suppose I have three factors, with two levels in each:
2Group(boys,girls)*2A(A1,A2)*2B(B1,B2). Factor Group is a between-subject
factor, while factor A and B are within-subject factors. What I want to test
is whether factors B influences factor A differently on boys and girls (a
three-way interaction).

Can I test the interaction between factor B and Group separately in B1 and
B2 conditions? That is, first split the data into two parts (B1, B2), and
then for each part test the interaction between factor A and Group by
subtracting the observations of A1 from A2 and taking it as a new dependent
variable.

Furthermore, if I have another factor C(C1, C2), and I'm interested in
whether the combination of B and C influences factor A differently on boys
and girls. Can I first split the data into four parts according to the
combination of factor B and C (B1C1, B1C2, B2C1, B2C2) and do the
interaction between factor A and Group separately for each part?

I do not think it is necessary to split the data and do separate analyses on
the different parts.

Reformulating an interacting effect null hypothesis as a main effect null
hypothesis for a different dependent variable is possible if you have at
most a single between-subjects variable in your design. Thus, it also works
in the case you describe, with two within-subjects and one between-subjects
variable. The dependent variable that must be calculated is the two-way
interaction effect contrast for the 2-factorial within-subjects design with
the variables A and B: [A1,B1] + [A2,B2] - [A1,B2] - [A2,B1]. You then
perform a independent samples T-test on the interaction contrast variable,
comparing the two levels of your between-subjects variable.

This reformulation is also possible when there are more than two
within-subjects variables in the design. The only thing that changes is the
interaction effec contrast.

Best,

Eric

dr. Eric Maris
Donders Institute for Brain, Cognition and Behavior

Center for Cognition and F.C. Donders Center for Cognitive Neuroimaging

P.O. Box 9104
6500 HE Nijmegen
The Netherlands
T:+31 24 3612651
Mobile: 06 39584581

F:+31 24 3616066
E: e. <mailto:e.maris at donders.ru.nl> maris at donders.ru.nl

MSc Cognitive Neuroscience:  <http://www.ru.nl/master/cns/>
www.ru.nl/master/cns/

Thank you in advance.

Lin

2009/11/24 Eric Maris <e.maris at donders.ru.nl>

Dear Stephan,

It is sometimes possible to reformulate an interaction effect null
hypothesis such that it becomes a main effect null hypothesis (however, for
a different dependent variable, obtained be calculating the difference
between conditions). Such a reformulation is possible for a two-factorial
design in which one independent variable is manipulated within subjects (in
your case, this is condition), and the other between subjects (in your case,
this is group). You do this by calculating subject-specific difference
scores, [condition1 - condition2], and using these as the dependent variable
in a between-groups comparison. This is possible by means of the standard
independent-samples T-test, but also by means of a permutation test
(involving permutation of the difference scores), which allows you to deal
with the multiple comparison problem.

For a factorial design that only involves between-subject independent
variables, such a reformulation is not possible (at least, I am not aware of
it).

Best,

Eric

> -----Oorspronkelijk bericht-----

> Van: FieldTrip discussion list [mailto:FIELDTRIP at NIC.SURFNET.NL] Namens

> Stephan Moratti

> Verzonden: dinsdag 24 november 2009 10:20

> Aan: FIELDTRIP at NIC.SURFNET.NL

> Onderwerp: Re: [FIELDTRIP] Question about permutation testing of an
interaction

> in a two by two design

>

> Hi all,

>

> Thanks for the interesing discussion about the interaction with respect to

> permutation. I have been also struggling with this question. As we want to

> test the interaction (and not the main effect condition), I am not sure if

> resampling condition would produce the distribution of our null
hypothesis.

> Regarding the interaction, we want to check if the mean values across

> conditions follow the same pattern in two groups (if we consider a group x

> condition interaction). If we make a line plot an interaction would be
indicated

> by a line crossing if each line represents a group (ideally). No
interaction

> would be represented by parallel lines for each group. So I wonder if by

> resampling the subject values between the groups keeping the condition

> structure intact, would create our null distribution for the interaction.
If we

> resample condition, we would destroy the condition structure.

>

> What do you think? I would be happy for any input.

>

> Best,

>

> Stephan

>

> ----------------------------------

> The aim of this list is to facilitate the discussion between users of the
FieldTrip

> toolbox, to share experiences and to discuss new ideas for MEG and EEG
analysis.

> http://www.ru.nl/neuroimaging/fieldtrip.

----------------------------------

The aim of this list is to facilitate the discussion between users of the
FieldTrip toolbox, to share experiences and to discuss new ideas for MEG and
EEG analysis.

http://listserv.surfnet.nl/archives/fieldtrip.html

http://www.ru.nl/fcdonders/fieldtrip/

----------------------------------

The aim of this list is to facilitate the discussion between users of the
FieldTrip toolbox, to share experiences and to discuss new ideas for MEG and
EEG analysis.

http://listserv.surfnet.nl/archives/fieldtrip.html

http://www.ru.nl/fcdonders/fieldtrip/

----------------------------------
The aim of this list is to facilitate the discussion between users of the FieldTrip  toolbox, to share experiences and to discuss new ideas for MEG and EEG analysis. See also http://listserv.surfnet.nl/archives/fieldtrip.html and http://www.ru.nl/neuroimaging/fieldtrip.
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