Dose-response curves. Which is more Gaussian, EC50 or log(EC50)?

Scientists in many fields fit data to sigmoid dose-response curves. Is it better to find the best-fit value of EC50 or log(EC50)?  We used simulations to find out.

The simulated curves each had ten data points, equally spaced on a log scale from 1 nM to 10 mM. The true curve had a bottom plateau at 0.0, a top plateau at 100, and an EC50 of 1 mM.  Random noise was added to each point, using a Gaussian distribution with an SD of 15, which simulates data in a system with lots of scatter. Three typical data sets are superimposed in the graph below.

Five thousand data sets were simulated and fit to the two alternative expressions of the dose-response equation. The distribution of the EC50 and log(EC50) values are shown below.

Clearly, the distribution of log(EC50) values is much closer to Gaussian. The normality test confirms this impression. The distribution of log(EC50) passes the normality test. The distribution of EC50 values fails the normality test with P<0.0001.

How accurate are the confidence intervals of EC50 and logEC50?

If you express the equation in terms of EC50, which is less Gaussian, only 91.20% of the 5000 "95%" confidence intervals contained the true value. In contrast, 94.20% of the "95%" confidence intervals of log(EC50) contained the true value. This is one reason to prefer an equation written in terms of logEC50.

The confidence intervals reported by Prism (and most other nonlinear regression programs) are always symmetrical around the best-fit value. When computing the confidence interval, Prism just applies the equation (see How certain are the best-fit values? ) and doesn't have any common sense about whether the results make sense. If you express the results of these simulations as an EC50 value, in fact most of the results do not make sense. In 78% of the simulations, the lower confidence limit was negative! Of course negative concentrations are impossible, so you would call the lower limit zero in these cases. This means that in most of our simulations, the 95% confidence interval gives you no information at all about the lower limit of the EC50 and only gives an estimate of the upper limit. In contrast, if you express the dose-response equation in terms of the log(EC50), it is impossible for the confidence interval to include negative values of the EC50. The 95% CI of logEC50 gives you information about both the lower and upper limit of the value.

These simulations show a clear advantage to expressing the dose-response equation in terms of log(EC50).  This makes sense. Since the concentrations of drug are equally spaced on a log scale, it makes sense that the uncertainty of the log(EC50) will be symmetrical, but the uncertainty of the EC50 will not be. Prism always reports symmetrical confidence intervals of best-fit values. If the true uncertainty is not symmetrical, then the 95% confidence interval reported by Prism will not be very useful.