hypothetical question

Okay, so let's say you run the following experiment:

You want to compare different states of adaptation. The yardstick you're going to use to compare them is is a matching function. You have two stimuli, x and y, and you're going to assume that the associated matching function - your matching function model - is simple, like y = mx + b. You want to know how those function parameters, m and b, vary when the adaptation state changes.

To do the experiment, you keep one adaptation state constant in all conditions. You can do this because you have two stimuli, and you can adapt them separately. So, you have two adaptors, X and Y. You keep adaptor X the same in all conditions, but you vary adaptor Y. Since X doesn't change, you can then compare the effects of Y across conditions. Adaptor X is your baseline.

Within a subject, this design is fine. You can take your xy data from different X conditions and plot them on the same axes. You look at how the data for X1 differs from X2, for example. You fit your model to the X1 and X2 data, and find that mX1 is higher than mX2. You repeat the experiment with another subject and find the same pattern - the m values are different across subjects, but you see the same relative difference between mX1 and mX2 for every subject you test. You average the results together to show that mX1 is higher than mX2. This constitutes a result of your study.

But then...

You start to look at the individual data, at how the m values vary so much across individual subjects, but that within-subject difference is always there. You think, something is covarying between these two things, what could it be? Why is it that whatever value mX2 takes for a particular subject, mX1 is always higher?

Then you realize: Y. mX1 and mX2 might not vary at all, at least not to the extent that they appear to. Maybe its mY that's varying.

Look at that model from the point of view of Y. Then you have x = (mY)y + (bY). Turn it around, and you get y = (1/(mY))x + (bY)/(mY). This means that mX is inversely proportional to mY, so that measured values of mX1 and mX2 will be similarly affected by differences, across individual subjects, in the value of mY.

Well, this led somewhere, anyways.