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How can i understand the within and between model factors of a mixed model ANOVA

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Hi everyone,
I am trying to run a mixed model ANOVA and struggle to define the within and between subject terms of the fitrm function.
Data description:
Lets say I have temerature data from 3 timepoints (10min, 30min and 60min) and three locations (left ear, right ear, forehead), additionally I have two groups (1,0) where group 1 gets a treatment and group 0 not. I want to run a mixed model ANOVA to see whether or not there are effects over time, i.e. does the temperature decrease, increase or stay the same. Simultaneously I want to see whether this is dependant on the group, i.e. does it matter if I am getting the treatment as well as the location that the measurement comes from, i.e. does it matter where am I measuring. In my heaad a mixed model ANOVA was teh right way to do so. I have created a dataset that should replicate the type of data I have and attached the file.
I have troubles wrapping my head around the within and between factors and how to specify them. At the moment my model runs with this specification:
withinFactors = 1:3; % 1,2,3 for Meas1, Meas2 and Meas3
% Define the model formula
modelFormula = 'Meas1-Meas3 ~ Group:Location';
% Fit mixed-design ANOVA model
rmModel = fitrm(data_tbl, modelFormula, 'WithinDesign', withinFactors);
And I get a decent output with it. However my struggel is that I would like to have the Location Variable as a withinFactor and I do not know how to handle this, at least it does not appear very intuitively. I have tried to understand the matlab examples where two withinfactors are supplied, but I simply cannot transfer it to my code since I do not understand what going on. Any help would be much appreciated.
I hope the data supplied is sufficient, if you need anything else, pleas elet me know.
  2 Comments
Jeff Miller
Jeff Miller on 19 Apr 2024
I don't understand how the data_tbl created by your attached script matches up with the design you describe. For example, the different locations appear in different rows of data_tbl, which makes it look like location is a between factor rather than a within factor.
From your data description I was expecting 9 data points per row of data_tbl (3 time points x 3 locations, both within), with one row per subject (different rows labelled 1 or 0 to indicate group).
Julian Ostertag
Julian Ostertag on 19 Apr 2024
Edited: Julian Ostertag on 19 Apr 2024
Hi Jeff,
thank you for your answer. I know what you mean, and it may be the better way of organizing the data. I just have one concern regarding the naming of the columns. It would be something like:
"Group", "Left1", "Left2","Left3", "Right1", "Right2","Right3", "Front1", "Front2","Front3"
But then how would I be able to have time and location as separate factors, since in that data setup I cannot specify the timepoint specifically? Or is this then something that can be accounted for in the withinDesign?
Thats why in my setup I basically defined a new row to accoount for the different locations and a new column to account for the different timepoints. Another option I was thinking about would be to have a third column named "Timepoint" which is then a 1,2 or 3 and then subsequently only have one column for the data. But certainly, I would also like to understand your idea better, since it can for sure be also organized in that way. I just want to tell you my concern before doing so.
Best
Julian

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Accepted Answer

Jeff Miller
Jeff Miller on 19 Apr 2024
Yes, this is something that can be accounted for using withinDesign. See https://au.mathworks.com/matlabcentral/answers/467222-ranova-with-two-within-factors for an example. After you rearrange your data to have 9 data columns per subject, I think you will want something like this:
WithinStructure = table([1 1 1 2 2 2 3 3 3]',[1 2 3 1 2 3 1 2 3]','VariableNames',{'Location','TimePoint'});
WithinStructure.TimePoint = categorical(WithinStructure.TimePoint); % probably you want this
WithinStructure.Location = categorical(WithinStructure.Location); % probably you want this
rm = fitrm(data_tbl,'Left1,Left2,Left3,Right1,Right2,Right3,Front1,Front2,Front3~Group','WithinDesign',WithinStructure,'WithinModel','Location*TimePoint');
ranovatable = ranova(rm,'WithinModel','Group*Location*TimePoint');

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