Fitting dose-response curve when X is dose, rather than log(dose).
There are two sets of dose-response equations built-in to Prism: one set designed to accept the X values as log(dose), and a second set designed to accept the X values as just dose. When using the first set of equations (with X as log(dose)), you can either enter the data with X values as logarithms of doses, or use the Transform analysis to create a results table with the data arranged that way which can then be graphed and fit. You can label the X axis using antilog or powers-of-ten numbering, so the axis is labeled as concentrations rather than log(concentration).
Here is the equation built-in to Prism for fitting a variable slope (four-parameter) log(dose) response curve:
Y=Bottom + (Top-Bottom)/(1+10^((LogEC50-X)*HillSlope))
Using the second set of equations, it is possible to fit data where X values are concentrations, rather than log(concentrations). Here is the corresponding equation for fitting a variable slope (four-parameter) dose-response curve. Because this equation expects the X values to be concentrations, not logarithms, the concentration is not raised to the tenth power:
Y=Bottom + (X^Hillslope)*(Top-Bottom)/(X^HillSlope + EC50^HillSlope)
In this case, the equation fits the EC50, rather than the LogEC50. However, for both sets of equations, Prism will report both the LogEC50 and the transformed EC50 in the analysis results.
Ultimately, though, these two sets of equations are equivalent (so long as you use the right form of input for X). In fact, if you have X values as dose and fit them with the equation expecting X as dose, you'll get the exact same results as you would if you first log-transformed the X values, and fit the transformed data with the corresponding equation expecting X as Log(dose). Download this Prism file to see how it works. The same data are fit and graphed twice.
- In one version the X values are transformed to logarithms, and then fit to the log(dose) equation built-in to Prism. Here the graph has a linear X axis, but the numbering is converted to powers-of-ten to show that the X values represent logarithms.
- In the other version, the data are fit with the X values remaining as concentrations and fit to the dose version of the equation built-in to Prism. Here the X axis is stretched to a logarithmic scale (top right of Format Graph dialog).
The two graphs look identical. The results of the two fits are identical.
Notes:
- The second graph will look ragged in Prism 4 (and earlier), which was not smart about plotting curves on axes stretched to a logarithmic scale.
- This page assumes that the log(dose) vs. response relationship is symmetrical, so the dose vs. response relationship does not form a symmetrical sigmoidal curve. In the rare situation where the dose vs. response relationship forms a sigmoidal curve (when the X axis is linear), see this page.
Why is the HillSlope applied to the EC50 as well as the X values?