Question: I am looking for a way to test whether a boundary threshold exists in a physiological response – a sample of the data is plotted below. My hypothesis is that the X-variable imposes a physiological constraint on Y-values, thus producing a boundary 'ceiling' for maximum Y-values that decreases at higher X-values (indicated by the red line on figure). I assume any Y-values below the boundary are limited by some other factor not included in this model.

Essentially, my goal is to determine if the boundary exists and if so to derive a confidence interval for the boundary line model – similar to a linear regression model, but describing the upper bound of the Y-values, rather than the center of mass.

I'm sure something like this exists, but I haven't come across it before. Also, I would appreciate any suggestions on a better title or tags for this post – I assume there are more accurate terms for what I'm describing that would help folks find this post.

![threshold](https://i.stack.imgur.com/1BQBk.jpg)

Answer: 
Such a pattern would often occur when no "boundary" actually exists.

Here I generate X and Y as *independent* right-skew random variates, yet such a pattern occurs:

![enter image description here](https://i.stack.imgur.com/VKEcy.png)

The impression of any sense of a boundary in my plot is *completely bogus*, yet it looks very similar to yours. (There's an actual *vertical* boundary in this bivariate distribution at $x=80$, but I could generate very similar looking plots without any boundaries at all.)

Here's the code I used to generate the plot (in R):

```
x = rbeta(1000,1,10)*80
y = rbeta(1000,1,3)/1.5+.3
plot(x,y,ylim=c(0,1))

```

Trying it a few more times it looks like about a third of the time it gives a plot that seems to have such a slanting boundary.

No doubt a little fiddling with distributions could improve the proportion of times it occurs and at the same time make it look even more like your picture (this shifted/scaled beta(1,10)$\times$beta(1,3) was the very first counterexample I tried).

Given my picture doesn't actually have any boundary there, one should be careful of over-interpreting such a pattern. You'd need a characterization of what makes it a boundary that *wouldn't* generate lots of false positives on examples like the one I give.
I would start by finding the "upper envelope" of your data and then representing the "envelope" as a straight line or piece-wise linear function. 

For starters, you could estimate the "envelope" as a piece-wise constant function f(x) =max(yk, given abs(x-xk) is below delta), where delta is a parameter, say 3 and (xk, yk) are your data points. Drawing a straight line through points (xk, f(xk)) should be straightforward :)