Confidence intervals of the EC50

Two ways to compute the 95% confidence interval of the EC50

The sample data above were fit to a dose-response curve with a Hill slope of 1. The best-fit value for logEC50 is -6.059. Converting to the EC50 is no problem – simply take the antilog, which is 0.87 mM.

The standard error of the logEC50 is 0.2717. It is used to calculate a 95% confidence interval, which ranges from -6.657 to -5.461. Take the antilog of both of those limits to express that confidence interval on a concentration scale -- from 0.22 to 3.46 mM. This is the interval that Prism presents. Note that it is not centered on the best-fit value (0.87 mM). Switching from linear to log scale converted the symmetrical confidence interval into an asymmetrical interval.

If you fit the same data to an equation describing a dose-response curve in terms of the EC50 rather than the logEC50, the EC50 remains 0.87 mM. But now Prism computes the SE of the EC50 (0.5459 mM), and uses this to compute the 95% confidence interval of the EC50, which ranges from -0.3290 to +2.074 mM. Note that the lower limit of the confidence interval is negative! Since the EC50 is a concentration, negative values are nonsense. Even setting aside the negative portion of the confidence interval, it includes all values from zero on up, which isn't terribly useful.

The problem is that the uncertainty of the EC50 really isn't symmetrical, especially when you space your doses equally on a log scale. It only makes sense to compute the 95% CI of the logEC50, and then transform both confidence limits to a concentration scale, knowing that the confidence interval will not be symmetrical on the concentration scale.

Do not transform the standard error of the logEC50

When some people see the SE of the logEC50, they are tempted to convert this to the standard error of the EC50 by taking the antilog. This is invalid.

In the example, the SE of the logEC50 is 0.2717. The antilog of 0.2717 equals 1.869. What does this mean? It certainly is not the SE of the EC50. The SE does not represent a point on the axis; rather it represents a distance along the axis. A distance along a log axis does not represent a consistent distance along a linear (standard) axis. For example, increasing the logEC50 1 unit from -9 to -8 increases the EC50 9nM, while increasing the logEC50 1 unit from -3 to -2 increases the EC50 by 9 mM, which equals 9,000,000 nM.

Averaging the EC50 from several experiments

The uncertainty is symmetrical when you express the midpoint of a dose-response curve as a logEC50, but is far from symmetrical (often) when you express it as the EC50. When pooling several experiments, therefore, it is best to average the logEC50 values, which will give a different result than averaging the EC50 values.