How to describe characteristics of the curve fit by a four parameter non-linear regression? I've fit a non-linear mixed effects model with a four parameter logistic function. My specific interest is in characterizing the point on the curve at which the horizontal component of the curve meets the oblique component. For instance, in the following fixed effects plot I've indicated with blue arrows the region on the curve that I would be interested in deriving a value for. 

One option I have come across would be to use a cumulative Gaussian and then use something like $\mu - 1.65\sigma $ to describe this point. However, if possible I would like a way to describe the part of the curve with the parameters of the four-parameter logistic that I am using: 
$$
f(x) = \left(\frac{A-D}{1+\left(\frac{x}{C}\right)^B}\right) + D
$$
 A: If we approximate the log-scale logistic function $1/(1 + \exp(-B(t - \log C))$ with the linear function $1/2 + (t - \log C)B/4$, the approximation intersects the asymptotes at $t = \log C \pm 2/B$. Translating that to the $x$-scale puts the "corners" of the function at $x = C \exp(\pm2/B)$.
A: I think your question is based on a flaw. You ask where is "the point on the curve at which the horizontal component of the curve meets the oblique component"? But there are no "components". It is one curve. Asking where two components intersect is a meaningless question when there are no such components. 
Here are some similar questions that might be worth asking. 
What the X value where...


*

*...the second derivative is largest? (@Whuber suggested this in a comment)

*... Y is the smallest value that is large enough to matter scientifically? This threshold would need to be defined scientifically, not statistically, based on your knowledge of the system being studied.

*... Y is 10% of the way from the bottom plateau (A) to the top plateau (D)? Or 5% or 20%. Again, you'd need to set this threshold based on the context of the work.

A: Are you trying to compare both curves?
Well, if I am correct A and D represent the upper and lower asymptotes, C is a scaling coefficient and B is a rate parameter. A simple, "quick and dirty" way to compare the curves for the points you highlight is to solve your function for x, and plug in your solved parameter values, but instead of using your solved A, use 5% of A or so (use something relevant to your research questions). That will give you the X at which 5% of the Max (i.e. A) is reached. 
Now, to do this more thoroughly, take a look at:
http://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/viewer.htm#statug_nlin_sect037.htm
This document explains how to compare non-linear trends among groups. With these procedures you can test if the curves are different, and then identify which parameters are different between groups. Once you have this information, you can improve the analysis in the "quick and dirty" approach by adding confidence intervals for your solved values of X at a given fraction of A. 
A word of caution: Check your assumptions and other measures (e.g. Skewness) before utilizing the confidence intervals for parameter estimates, since violations can render your inferences using standard errors and confidence intervals invalid.
