--- QUESTION SOLVED (thanks for your help!)---
I am interested in the effect of different exposures on participants' performance (% correct) in two tasks. Specifically, I would like to compare the effects of different exposures as a function of task. To do so, I build the following model (simplified):
- performance ~ exposure (4 levels) X task (2 levels)
However, results are difficult to interpret as performance distributions of task 1 and 2 differ from one another. As the attached exemplatory graphs show, participants generally performed better in task 1 than in task 2.
Does anyone have a suggestion how I could reasonably investigate whether there is an interaction between exposure and task? As an example, I am hoping to get to a conclusion like: "compared to the control (exposure A), exposure B had a significantly stronger effect on performance in task 2 than task 1." Z-transformation could be the key, but I don't quite know how to proceed.
Thanks in advance for your suggestions.
There are many things that need to be considered and discussed to come up with a sensible analysis that would give you interpretable and reliable answers. Some of these questions are about what are between- and within-subject factors in your design, if there was a cross-over or if time-and/training effects need to be considered (and how). Another question is if "exposure" is really categorical or rather metric.
It seems to me that would need some kind of a mixed (multilevel) binomial model adressing the interaction between task and exposure.
A z-transformation is not ok. Using the proportions rather than the (known?) number of trials and successes per subject is already not recommended. If you don't have that information and the response is not binomial (like correct vs. not correct) but measured as a proportion (%), you would use a beta model rather than a binomial model. Although the relationship between predictors and response can be linearized using a logit-transfromation, I would not recommend this. It is better to link the mean via the logit rather then to transform the response values.
These comments should encourage you to follow my main suggestion: seek help from a local statistician.
Suppose you have a model:
Y = b0 + b1x1 + b2x2 + error
What is the interaction between x1 and x2? if there is interaction, the model above will become:
Y = b0 + b1x1 + b2x2 + b3x1x2 + error
How to determine b3?
b3 = b^ - b1 - b2 - b0
b^ = Y^ - (b2 - b0) - b1
LINK:
Working Paper Customer Loyalty Modeling under Logistic Function
Dear Jochen Wilhelm,
Thank you for taking the time to respond to my question and for your valuable feedback.
I agree without regarding the transformation of data. In fact, I ran all my other analyses using generalized linear mixed effects models with a binomial family (responses were indeed coded into correct vs. incorrect) and a logit link function.
I just decided to not specify this here, to simplify things. Because irrespective of whether I build a model with my binary response variable (using the glmer function with the fixed interaction term Task X Condition and several random effects including a random intercept for participant) or whether I average my data across items and calculate percentages, the problem remains the same.
I wish to account for the effect of task difficulty (as reflected in the respective Scores for C1 - my control condition) to evaluate whether the magnitude of the effects C2, C3, and C4 (which are indeed categorical) varies as a function of task.
Regarding your advice to consult a local statistician, I respond in a private message.
Again, thank you for your help.
Best,
Isabel
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