How to do ANOVA repeated measures?

I have 12 Female and 11 Male rats which participate in a sucrose tracking experiment for 4 different levels of sucrose concentration. I have approach rate data (between 0 and 1) for each animal for 4 different concentration. I have attached the figure. I want to do ANOVA repeated measures for the experiment to check the effect of concentration as well as gender. I wrote this following code. But I'm not sure if this is correct or how to interpret the result.

dataForFemale = loadFile.featureForEachSubjectId{1};
dataForMale = loadFile.featureForEachSubjectId{2};
for i = 1:4
    dataMatrix(:,i) = [dataForFemale{i} dataForMale{i}]';
end
Gender = [repmat({'F'},1,length(dataForFemale{1})),repmat({'M'},1,length(dataForMale{1}))]';
t = [Gender, array2table(dataMatrix)];
t.Properties.VariableNames = {'Gender','c1','c2','c3','c4'};
rm = fitrm(t, 'c1-c4 ~ Gender', 'WithinDesign', table([1, 2, 3, 4]','VariableNames', {'Concentration'}));
result = ranova(rm);
multcompare(rm, 'Concentration', 'By', 'Gender');

[Scott MacKenzie]

Your ANOVA code is correct.

Bear in mind that you have a mixed design with one within-subjects factor (concentration with four levels) and one between-subjects factor (gender with two levels). The "repeated-measures" aspect of the ANOVA only pertains to concentration.

[Atanu]

Thanks for your response!

[Scott MacKenzie]

You're welcome. I just had another look and notice that you omitted the within-subjects IV (sucrose concentration) in the ranova function. I've added it, rearranged the code slightly, and posted an answer. Hope this helps.

Answer 1

[Scott MacKenzie]

Here's a script for the ANOVA that includes both the within-subjects IV (sucrose concentration and the between-subjects IV (gender).

As you can see, the effect of gender on approach_avoid was not statistically significant (F1,21 = 3.83, p > .05), but the effect of sucrose_concentration on approach_avoid was (F3,63 = 118.5, p < .001). If you examine m (the output from multcompare), you'll see which pairwise differences by gender were statistically significant. There were lots!

Note: I don't like the ANOVA table generated by ranova. The script below below includes a function to create a more conventional ANOVA table from the ranova output. Of course, for your research, "Participants" is "Rats".

load(websave('tmp', 'https://www.mathworks.com/matlabcentral/answers/uploaded_files/1370074/BLsexDiffApproachavoid.mat'));

dataForFemale = featureForEachSubjectId{1};
dataForMale = featureForEachSubjectId{2};
for i = 1:4
    dataMatrix(:,i) = [dataForFemale{i} dataForMale{i}]';
end

Gender = [repmat({'F'},1,length(dataForFemale{1})),repmat({'M'},1,length(dataForMale{1}))]';
t = [Gender, array2table(dataMatrix)];
t.Properties.VariableNames = {'Gender','c1','c2','c3','c4'};

% setup and do the ANOVA
withinDesign = table([1 2 3 4]', 'VariableNames', {'Sucrose_Concentration'});
withinDesign.Sucrose_Concentration = categorical(withinDesign.Sucrose_Concentration);
rm = fitrm(t,'c1-c4 ~ Gender', 'WithinDesign', withinDesign);
result = ranova(rm, 'WithinModel', 'Sucrose_Concentration');

% do post hoc multiple comparisons (check m for significant pairwise differences)
m = multcompare(rm, 'Sucrose_Concentration', 'By', 'Gender');

% output a conventional ANOVA table
disp(anovaTable(result, 'Approach_Avoid'));

ANOVA table for Approach_Avoid
===========================================================================================
Effect                               df          SS             MS         F         p
-------------------------------------------------------------------------------------------
Gender                                1        0.07668        0.07668      3.827     0.0639
Participant                          21        0.42081        0.02004                      
Sucrose_Concentration                 3        7.05085        2.35028    118.527     0.0000
Gender:Sucrose_Concentration          3        0.13201        0.04400      2.219     0.0946
Participant(Sucrose_Concentration)   63        1.24924        0.01983                      
===========================================================================================

% -------------------------------------------------------------------------
% function to create a conventional ANOVA table from the overly-complicated 
% and confusing anova table created by the ranova function.

function [s] = anovaTable(AT, dvName)

c = table2cell(AT);

% remove erroneous entries in F and p columns
for i=1:size(c,1)
    if c{i,4} == 1
        c(i,4) = {''};
    end
    if c{i,5} == .5
        c(i,5) = {''};
    end
end

% use conventional labels in Effect column
effect = AT.Properties.RowNames;
for i=1:length(effect)
    tmp = effect{i};
    tmp = erase(tmp, '(Intercept):');
    tmp = strrep(tmp, 'Error', 'Participant');
    effect(i) = {tmp};
end

% determine the required width of the table
fieldWidth1 = max(cellfun('length', effect)); % width of Effect column
fieldWidth2 = 57; % width for df, SS, MS, F, and p columns
barDouble = repmat('=', 1, fieldWidth1 + fieldWidth2);
barSingle = repmat('-', 1, fieldWidth1 + fieldWidth2);

% re-organize the data
c = c(2:end,[2 1 3 4 5]);
c = [num2cell(repmat(fieldWidth1, size(c,1), 1)), effect(2:end), c]';

% create the ANOVA table
s = sprintf('ANOVA table for %s\n', dvName);
s = [s sprintf('%s\n', barDouble)];
s = [s sprintf('%-*s %4s %11s %14s %9s %9s\n', fieldWidth1, 'Effect', 'df', 'SS', 'MS', 'F', 'p')];
s = [s sprintf('%s\n', barSingle)];
s = [s sprintf('%-*s %4d %14.5f %14.5f %10.3f %10.4f\n', c{:})];
s = [s sprintf('%s\n', barDouble)];

end

[Atanu]

Thank you for the detailed explanation. However, for me output of ranova table is fine, cause I am not very familiar with either of them. I was wondering why it is necessary to use "WithinModel",'Concentration' in ranova like the following?

result = ranova(rm,"WithinModel",'Concentration');

It produces the following table.

Whereas,

result = ranova(rm);

produces the following result:

What does the extra first 3 rows stand for in the first table? And which p-values should I report for effect of concentration and gender? Is it the p-values on row 4 and 5 in the first table?

[Scott MacKenzie]

The WithinModel option with ranova is needed to include the within-subjects independent varariable in the analysis. Your design is a 2 x 4 mixed design with one between-subjects factor (gender with 2 levels) and one within-subjects factor (concentration with 4 levels). So, there are three results of interest: (1) the main effect of gender on approach_avoid, (2) the main effect of concentration on approach_avoid, and (3) the Gender x Concentration interaction effect on approach_avoid. That's why there are three F-statistics in the ANOVA table in my answer. If the WithinModel option is omitted, the analysis is only for the between-subjects factor and only one F-statistic appears.

Note that in both the tables in your comment, the ranova output table includes an extra F-statistic for the intercept. This is rarely of interest and is typically not included in the ANOVA table.

[Atanu]

Okay, I understand. Thank you very much for the explanation and code again!

[Scott MacKenzie]

You're welcome. Good luck with your research.

[Atanu]

Thank you!