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I'm planning a study in which I want to test the effect of a treatment on a dependent psychometric variable. I expect subjects who score lower at baseline to benefit more from the treatment (larger score difference pre-post), and I expect the treatment to have less of an effect on subjects who score high at baseline (smaller score difference pre-post). This is not due to a shortcoming of the psychometric measure but due to the underlying psychometric construct. I have empirical data of a small (n = 10) pre-study with identical design that I could use to quantify this effect.

Is there a more elegant solution to test this than to perform a median split?

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    $\begingroup$ I would be wary of issues like regression to the mean. See also subtleties in causal inference and bias in causal inference: Glymour, M., Weuve, J., Berkman, L., Kawachi, I., and Robins, J. (2005). When is baseline adjustment useful in analyses of change? an example with education and cognitive change. American Journal of Epidemiology, 162(3):267–278. $\endgroup$
    – Alexis
    Commented May 13, 2014 at 20:11
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    $\begingroup$ Recommended reading: Best practice when analysing pre-post treatment-control designs $\endgroup$ Commented May 13, 2014 at 20:30

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What you're describing is a negative correlation between pre- and post-treatment values, so that would be the most straight-forward test. Instead of treating the factor "baseline" as a categorical predictor, you could employ it as being continuous. That way, you don't act as if the point to the left of the median would be expected to behave completely differently from the point one step t the right of it.

Moreover, you should visualise the data once it's in. While this would be post-hoc exploration and should be marked so, it is possible that visualisation shows you a more subtle pattern than an inverse linear dependence of post- on pre-scores. In that case, a suitable regression model could be evaluated.

I agree with @Alexis that regression to the mean is the first thing coming to mind here. This you could maybe deal with by checking for an interaction with group (treatment vs. control) in this negative correlation. Regression to the mean should be observable in both groups, but in the treatment group, you should have an "overshoot"-like effect on top of the regression to the mean, which you should observe in isolation in the control group. I have no practical experience with such an analysis though.

Though you will likely need a very large sample to see anything with such a derived outcome.

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