Pairwise post-hoc testing using coefTest (MATLAB) - matlab

Question

I found this post from back in 2015 but it doesn't seem to answer the question (and if it does, I am sorry that I am missing it!). I am trying to understand how I can test for differences within a level of an interaction. For this purpose I have created a random dataset with known properties. In my mock dataset, I have a group of subjects who participate in something-or-other. They can belong to one of three Groups (ABC) at the time of test. Then, we have three factors (ABC) under which we have three levels for each (A-DEF, B-GHI, C-JKL). I arranged the data such that of the groupings, only Group C has any effect (i.e., 2) and then there is a significant interaction factor C, in that if you have level L, you get a big boost.
Thus, any analysis should hopefully detect a signficiant main effect of Group and a significant Group x Factor C interaction effect, but no other obvious effects. Within the effect of Group, it should be level C that stands out. Within the interaction, it should be level L that stands out.
My mock dataset can be found here.
To test my understanding, I generated the below script. In it, I am (very sure) I am correctly using coefTest to detect the significant main and interaction effects. Within the main effect, I am (somewhat sure) I am correctly using coefTest to perform pairwise comparisons among Groups ABC; as predicted, I find that C is different from both A and B, while A and B do not differ.
However, within the interaction effect, I would like to test for differences among levels JKL. Can this be done? If so, can someone please help me to do so?
Thank you!!!
load table.mat
mdl = fitlme(d, ...
'Value ~ Group*FactorA + Group*FactorB + Group*FactorC + (1|Subject)', ...
'DummyVarCoding', 'effects', 'FitMethod', 'REML');
disp(mdl)
%% Generate predictions for ALL data
g = unique(d.Group);
a = unique(d.FactorA);
b = unique(d.FactorB);
c = unique(d.FactorC);
cv = sortrows(combvec(1:numel(g), 1:numel(a), 1:numel(b), 1:numel(c))');
predTable = table();
predTable.Subject(1:size(cv, 1),1) = categorical({'Generic'});
predTable.Group = g(cv(:,1),1);
predTable.FactorA = a(cv(:,2),1);
predTable.FactorB = b(cv(:,3),1);
predTable.FactorC = c(cv(:,4),1);
predTable.Value = predict(mdl, predTable, 'Conditional', false);
% Trim to only Groups and then only groupxfactor C interaction
[~, ia, ~] = unique(predTable.Group);
groupTable = removevars(predTable(ia,:), ...
{'Subject', 'FactorA', 'FactorB', 'FactorC'});
groupTable.Value = splitapply(#mean, predTable.Value, ...
findgroups(predTable.Group));
[~, ia, ~] = unique(predTable.FactorC);
AxFactorCTable = removevars(predTable(ia,:), ...
{'Subject', 'Group', 'FactorA', 'FactorB'});
AxFactorCTable.Value = splitapply(#mean, ...
predTable.Value(predTable.Group == 'A'), ...
findgroups(predTable.FactorC(predTable.Group == 'A')));
BxFactorCTable = removevars(predTable(ia,:), ...
{'Subject', 'Group', 'FactorA', 'FactorB'});
BxFactorCTable.Value = splitapply(#mean, ...
predTable.Value(predTable.Group == 'B'), ...
findgroups(predTable.FactorC(predTable.Group == 'B')));
CxFactorCTable = removevars(predTable(ia,:), ...
{'Subject', 'Group', 'FactorA', 'FactorB'});
CxFactorCTable.Value = splitapply(#mean, ...
predTable.Value(predTable.Group == 'C'), ...
findgroups(predTable.FactorC(predTable.Group == 'C')));
figure('units', 'normalized', 'outerposition', [0; 0; 1; 1])
subplot(2, 3, 2)
bar(groupTable.Group, groupTable. Value)
title('Group Means')
ylim([-3, 6])
axis square
subplot(2, 3, 4)
bar(AxFactorCTable.FactorC, AxFactorCTable. Value)
title('Factor C Means for Group A')
ylim([-3, 6])
axis square
subplot(2, 3, 5)
bar(BxFactorCTable.FactorC, BxFactorCTable. Value)
title('Factor C Means for Group B')
ylim([-3, 6])
axis square
subplot(2, 3, 6)
bar(CxFactorCTable.FactorC, CxFactorCTable. Value)
title('Factor C Means for Group C')
ylim([-3, 6])
axis square
% Perform pairwise testing among the main effects for each group.
% Because Group C is not assigned to an effect, it can be tricky to do
% pairwise comparisons with it. However, following the comparison
% approach found at
% https://www.mathworks.com/help/stats/generalizedlinearmixedmodel.coeftest.html,
% we can determine what the post hoc arrangements should be:
% syms A B C
% f1 = A + B + C == 0;
% C = solve(f1, C)
% fAB = A - B
% fAC = A - C
% fBC = B - C
% The results are:
% C = -A - B
% fAB = A - B
% fAC = 2*A + B
% fBC = A + 2*B
% [Int, G_A, G_B, FA_D, FA_E, FB_G, FB_H, FC_J, FC_K, G_A:FA_D, G_B:FA_D, G_A:FA_E, G_B:FA_E, G_A:FB_G, G_B:FB_G, G_A:FB_H, G_B:FB_H, G_A:FC_J, G_B:FC_J, G_A:FC_K, G_B:FC_K]
HAB = [ 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0];
HAC = [ 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0];
HBC = [ 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0];
pG = coefTest(mdl, [HAB; HAC; HBC]);
pAB = coefTest(mdl, HAB);
pAC = coefTest(mdl, HAC);
pBC = coefTest(mdl, HBC);
disp(['Significance of Group effect: ', num2str(pG, 3)])
disp([' * Significance of Group A vs. Group B difference: ', ...
num2str(pAB, 3)])
disp([' * Significance of Group A vs. Group C difference: ', ...
num2str(pAC, 3)])
disp([' * Significance of Group B vs. Group C difference: ', ...
num2str(pBC, 3)])
% [Int, G_A, G_B, FA_D, FA_E, FB_G, FB_H, FC_J, FC_K, G_A:FA_D, G_B:FA_D, G_A:FA_E, G_B:FA_E, G_A:FB_G, G_B:FB_G, G_A:FB_H, G_B:FB_H, G_A:FC_J, G_B:FC_J, G_A:FC_K, G_B:FC_K]
HG_FA = [[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]; ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]; ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]; ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]];
HG_FB = [[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0]; ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]; ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0]; ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0]];
HG_FC = [[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0]; ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0]; ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]; ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]];
pG_FA = coefTest(mdl, HG_FA);
pG_FB = coefTest(mdl, HG_FB);
pG_FC = coefTest(mdl, HG_FC);
disp(' ')
disp(['Significance of Group vs. Factor A interaction: ', ...
num2str(pG_FA, 3)]);
disp(['Significance of Group vs. Factor B interaction: ', ...
num2str(pG_FB, 3)]);
disp(['Significance of Group vs. Factor C interaction: ', ...
num2str(pG_FC, 3)]);
disp(' * Here''s where I would like to test for differences among JKL within Group C')
% Uncomment to check above against MATLAB anova
% disp(' ')
% disp(anova(mdl))
Some helpful output to visualize expected outcomes