I’m analyzing some data per GLMM with a probit link function and I'm getting some weird inconsistencies between two GLMM specifications that, in my understanding, shouldn't be all that different.
Let me give you some context about the experiment. This is the stimulus (YouTube). The big ball can have one out of two speeds (and two directions), and the speed of the ball cloud is chosen dynamically based on participant responses such that I have a range of values with a larger concentration of values close to the speed where the cloud is perceived as as fast as the big ball. While the big ball is moving across the screen, participants experience visual self-motion in the same or opposite direction as the target stimulus or no self-motion at all. Participants judge on each trial whether the big target was faster or the ball cloud.
With my analysis, I want to figure out if the self-motion shifts the graph to the left or right and if self-motion makes the slope more shallow.
I was originally using the following specification for the GLMM (in lme4 for R/Wilkinson & Rogers syntax):
Probability ~ SelfmotionProfile * Difference + (Difference | Participant) + (Difference | Speed_BigBall)
Probability is the probability to judge that the big target was faster; SelfmotionProfile is a categorical variable with the values “Static”, “Same Direction”, “Opposite Directions”; Difference is the difference between the velocity of the big target and the velocity of the ball cloud. The regression coefficient for the fixed effect “SelfmotionProfile” should correspond to how much the self-motion condition shifts the function to the left/right, and the interaction between “SelfmotionProfile” and “Difference” should correspond to how much the self-motion condition changes the slope of the function.
The problem is that the results I get with this specification don’t make sense on a conceptual level. They are also inconsistent with the results I am obtaining with a more traditional method in my field, in which you fit the data for each condition and participant to a Cumulative Gaussian function and then use the means and standard deviations of the fitted Cumulative Gaussians to address the question I am asking my data.
Now, when I use the speed of the ball cloud as effect in the model instead of the difference between the speed of the main target (which is constant) and the ball cloud (which is variable) ...
Probability ~ SelfmotionProfile * Speed_BallCloud + (Speed_BallCloud| Participant) + (Speed_BallCloud | Speed_BigBall)
... the results I get do make sense conceptually and are much more consistent with the traditional analysis I performed as sanity check.
So the big question: Why the hell does this change in the model make for such huge differences? By subtracting the speed of the big target, shouldn’t I just be shifting everything to the left and center it around zero (-ish, because Speed_BigBall is typically > Speed_BallCloud)? E.g., the impact of “SelfmotionProfile” on position and slope of the fitted function shouldn’t be affected at all?
