Confidence intervals for values estimated from the nonlinear regression model

Question

I have a question about nonlinear regression and confidence intervals for values estimated from the model. Here is my problem. I have sets of data where $X$ is the logarithm of the dose of a chemical substance and $Y$ is the response readout (normalized to be between 0 and 100%).

I am fitting logistic function to this data using nonlinear regression with 4 parameters:

$$y = A + \frac{B-A}{1+e^{-ax+b}}$$

Once I have estimated the $A, B, a,$ and $b$ parameters, I can estimate the value of $X$ at which $Y$ should take 20%, which is what I need. Everything up to this point is clear.

However, something is unclear to me:

How do I calculate 95% confidence interval for that estimated $X$ value?

I know there is software out there, which can probably do this for me, but I have to understand the principle and algorithm myself, because I have to implement it in my own library.

Any help will be appreciated.

EDIT Dear Eupraxis and Maarten, thank you for your response:

@Eupraxis - Here is the picture, the way I understood your explanation about using function prediction intervals to calculate 95% CI bounds of estimated X value. Is it what you proposed (sorry for hand drawings)?

What if my function cannot be re-expressed in linear way? I am not even sure how to do it for logistic curve anyway. Please also see my comment for Maarten.

@Maarten - Thank you for the article! I tried to read it, but frankly it was quite hard to comprehend. I found what it seems to be a much simpler description of delta method for calculation of prediction interval in nonlinear regression:

How to compute prediction bands for non-linear regression?

Then I was planning to use these 95% PI to calculate 95 CI for x' as Eupraxis suggested above. However, there is one thing I do not quite get in this delta method. Specifically, as described:

First, let's define G|x, which is the gradient of the parameters at a particular value of X and using all the best-fit values of the parameters. The result is a vector, with one element per parameter. For each parameter, it is defined as dY/dP, where Y is the Y value of the curve given the particular value of X and all the best-fit parameter values, and P is one of the parameters.)

Could you maybe explain it, perhaps with some numeric example I can follow? The rest of the algorithm seems to be clear to me.

Answer 1

You are probably looking for the delta method, see for example:

Oehlert, G. W. 1992. A note on the delta method. American Statistician 46: 27–29.

Answer 2

If you're fitting a logistic curve, then first re-express (via a transform) so that the predictor and predicted varialbes are linearly related. Then, form a predicion interval using standard linear regression techniques. In particular, you will want to find the smallest transformed X value whose upper 95% PI cutoff is equal to the transformed Y=20% value, as well as the largest transformed X value whose 95% PI is equal to the transformed Y=20% value. Then, you can back-tranform these values to the original scale to get you X interval.

Answer 3

Consider trying to express your model so one of the parameters you fit is the concentration that brings you up to 20%. If you can fit that as one of the parameters, then the CI comes out of the nonlinear regression easily. Here are some notes I wrote for a related equation that might help.

Two notes:

• If you really know your normalized data go from 0 to 100, and the normalization is based on solid controls, then you don't need to fit A and B, but can set those to constant values of 0 and 100. If you don't have good controls to normalize to, then there really is no need for normalization at all.
• I'd suggest renaming your parameters, so you don't have both b and B in your model. Depending only on upper/lower case to distinguish entirely separate values seems likely to cause a bug someday.