Dear all experts,
I'm currently doing a fMRI study and compare two groups in two conditions (switch and non-switch), and wondering if they have differences in one ROI (region of interest).
Therefore, I ran a Mixed ANOVA (2 groups by 2 conditions) to see if they have interaction. The result is ''yes' they do have interaction between groups and conditions, then I found that there's no groups main effect but has conditions main effect. So I further compare the simple effect (Paired t test) between two conditions in two groups.
The result is that both groups have significant differences between two conditions. According to the graph, however, one group has larger difference (bigger gap) compare to the other group, and the values of Cohen'd does indicate that one group is larger than the other.
So could I use Cohen's d say although both groups reach statically significant differences between two condition, but based on the effect size, the difference in B group is much powerful than A group. In the other word, B group had larger difference than A group.
If I can explain it on this way, how to say the effect size of group B is 'significant bigger' than A?
thanks in advance for your patient in my not clever question.
Peter
You should have an estimate of the coefficient of the interaction. This is what you should interpret. The (statistical) significance of the interaction tells you that you think your data is sufficient to interpret the sign of this coefficient.
I haven't heard about using Cohen's d as an effect size measure of the interaction (and I am not a big fan of "standardized effect sizes"). But if you want something like Cohen's d for the interaction, then this should be the ratio of the coefficient estimate and the residual standard deviation.
Note that Cohen's "recommendation" of what d stands for rather strong or weak effect relates to phychological experiments! It is completely unclear if and how this might be transferred to other subjects (like fMRI data)! It may not even be very sensible in all cases of phychological data.
Hi Dr Wilhelm
Thank you for your kind words.
Sorry that I'm not really sure if the estimate of the coefficient of the interaction you said is coefficient r?
As above, there's an interaction between groups and conditions (F(1, 54) = 7.72, p = .008) in this ROI, then I used paired t to compare if each group had difference between 2 conditions (switch and non-switch).
Group A (t(27) = 4.48, p < .001, r = .65)
Group B (t(27) = 7.38, p < .001, r = .82)
and the plot also shown that group B had a bigger gap (increase switch and decrease non-switch) compared to group A.
HOWEVER both groups had significant differences between 2 conditions.
so if I want to emphasise this result, could I use r values to say the trend of BOLD had larger difference in group B?
thank you very much,
Peter
You have a 2x2 factorial model (your response [Y] is to be explained by 2 factors [A,B], each of which can take two different values/levels [a1/a2 and b1,b2]). To write this down as a formula, one has to code these levels numerically. One frequently used way is the dummy- or treatment-coding:
say,
XA = 0 if A=a1 and 1 id A=a2,
XB = 0 if B=b1 and 1 id B=b2,
A simple model can then be written as
Y = β0 + βA*XA + βB*XB
This model has three coefficients: β0, βA, and βB
As you can easily see, β0, will be the mean response for A=a1/B=b1, βA is the mean difference between A=a1 to A=A2 (independent of B), and βB is the mean difference for B=b1 to B=b2 (independent of A).
The "problem" with this model is that it assumes that the effects of A and B simply add, that the effects of A and B are independent of each other. This is resolved by adding an interaction term in the model:
Y = β0 + βA*XA + βB*XB + βAB*XA*XB
The coefficient βAB is the coefficient of the interaction. It quantifies how much the effect of A depends on the level of B (or vice versa). The value of βAB can be estimated from the data, and you can test the hypothesis βAB = 0 to see if the data allows you to at least interpret the sign of this estimate. This is called the "test for interaction" in 2x2-ANOVA.
For more background and details please read some books/resources on linear models and ANOVA, e.g. http://www.unc.edu/courses/2010fall/ecol/563/001/docs/lectures.html
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