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.
This is potentially not the most robust solution. But you may be able to seriously improve the quality of the envelope using something along these lines:

1. break down your data into n intervals (where the number of intervals depends on the density of your data)
2. find the max of you data within each interval
3. Pass linear regression model though the selected maximum data points.