# Nonlinear regression: Confidence intervals on transformed or untransformed parameters?

Suppose I am using a standard inhibition model to find biochemical parameters that fit my data. The equation is:

$y = \frac{A}{{1 + \exp \left( {\ln \left[ S \right] - \ln IC_{50}} \right)}}$

where $\left[ S \right]$ is the concentration of my inhibitor and $IC_{50}$ is the concentration of my inhibitor at which the measurement (with a maximum $A$) is reduced by half. Which of these approaches should I take?

1. enter the equation into the NLR procedure as-is (FYI: I am using SPSS) and let it fit the values of $A$ and $IC_{50}$ in the same manner, i.e. with all of the assumptions of OLS regression parameters. Or...

2. enter the equation as

$y = \frac{A}{{1 + \exp \left( {\ln \left[ S \right]- \ IC_{50}^* } \right)}}$

where $IC_{50}^* = \ln IC_{50}$. This of course would require a transform on the output parameter and confidence limits of $IC_{50}^*$ giving me asymmetric error bars.

Which of these strategies is most rigorous? My instincts suggest the 2nd option, as $IC_{50}$ is actually bound by 0 and is thus more likely to be log-normal rather than normal. Any help (direct answers, references, etc) is appreciated.

• assuming that it is truely log-normal then this would be the back transformation you are looking for, also with respect to error. amstat.tandfonline.com/doi/full/10.1080/… Commented Sep 19, 2019 at 18:03

• By symmetrical, you mean the variance of $y$ is symmetrical around the mean of $y$, correct? In the generalized case where this is true, is the uncertainty of each parameter also symmetrical around the prediction of that parameter? If that's true, then what you say makes perfect sense. Commented May 14, 2013 at 17:27