I am currently working on a project where we are administering different treatments to subjects, but with some potential side effects. I am trying to measure the effect each method has on the well being of the subjects based on some welfare parameters. One problem is that, due to reasons, I am not able to test the same subjects before and after treatment. In reality, an already "not well" subject cannot get better during treatment, (as we are measuring some form of physical damage), but as we are not measuring the same subjects before and after it can appear so from the data (measure one "not well" before and then measure a subject that was very well off before the treatment, even though it might have gotten worse during treatment it would appear to have "healed").
Now comes the main problem, as it is a strictly positive dataset, we can measure a subject to be perfectly fine before treatment, and then it can only get worse. However, if I measure a subject to already be "not well" and then I measure it to have gotten better after, I will get a skewed effect. It will appear that the effect on well being is stronger for subjects that are of good health to begin with.
I have tried to think about how to ameliorate this but I can't think of, or find a good solution. Do any of you know of some methods that deal with this? In essence, I want to push down the effect around zero, but then push it less and less down when I get further away from zero. Please look at the picture. I believe the lines would be lower close to zero, but that they are correct further from zero. image of the data and regression lines
Thank you very much for any help.