I have a unconventional question arising from my current master thesis on regression modeling.
Suppose we have fitted a (linear) relationship between a dependent variable $y$ and an independent variable $x$. Now we choose two points on the $x$-axis, i.e. according to percentiles $x_{10}$ and $x_{90}$. These two points are chosen to select the data points for two groups in order to perform pairwise comparisons. In the first run, we choose 20 data pairs around the $x$-values of $x_{10}$ and $x_{90}$ and run a statistical test in order to infere if their $y$-values differ significantly. In a subsequent step, we choose, say 30 data points around $x_{90}$ and test against $x_{10}$; in the next step 40 data points around $x_{90}$ and test against $x_{10}$ and so on. The intention for this is to determine the group size and consequently the corresponding lowest $x$-value where the comparison turns out to be significant.
How would you approach this problem. I thought of a many-to-one procedure like the Dunnett test where multiple groups are tested against the same control (which could be $x_{10}$ in our example). However, these multiple groups would contain partly the same subjects. Or is some sort of adaptive design the right choice. Or something completely different?
I'd be very happy for any kind of help since I'm completely stuck with this question and have no idea how to solve it.
Edit: The idea behind this procedure would be to determine the lowest x-value where the pairwise comparison would turns out to be significant. Suppose the first and smallest group around $x_{90}$ is tested against the control $x_{10}$ and is not significant. So you take a look at the next group comprising the data points from group1 plus additional 10 and so on. If the pairwise comparison is significant, one determines the smallest x-value of the corresponding group as some sort of "change point". How could such a problem be approached? Thanks a lot! Andres
