In short#
- 4PL: Y = Bottom + (Top − Bottom) / (1 + (X / EC50)^h). Parameters: two plateaux, EC50 and the Hill slope h. The sign of h sets the direction of the curve.
- EC50 or IC50 is the dose at which the fitted response lies halfway between the plateaux. It is not necessarily the dose where Y = 50.
- The model is nonlinear. Doses must cover the transition and each plateau being estimated. Linza requires at least six distinct positive doses spanning at least a tenfold range.
- A fixed plateau is treated as exact, so its uncertainty is not included. Linza estimates an approximate 95% log-Wald interval for EC50.
- Linza first tests whether series can share an EC50 or IC50 and Hill slope while keeping plateaux separate. If this test is not significant, it also tests a single common curve. Holm adjustment is applied to the planned test family. A non-significant result does not prove equality.
In plain words#
A dose-response relationship can be S-shaped, with a plateau at each end and a transition between them. A logarithmic X axis conveniently displays a wide range of positive doses. It changes how doses are spaced on the plot; it does not validate the model.
EC50 is the concentration at which the fitted response lies halfway between the lower and upper plateaux. For an inhibitory effect, the midpoint is usually called IC50. This point does not have to correspond to Y = 50.
If measured doses miss the transition or the plateaux being estimated, EC50 and the slope may be poorly identified. Linza rejects fits that fail its numerical checks, but passing those checks does not by itself guarantee a suitable experimental design.
The full model gives each series its own plateaux, EC50 or IC50, and Hill slope. The first reduced model shares the midpoint and slope but keeps plateaux separate. The second also shares the plateaux, so one curve describes all series.
Each comparison uses an F test of nested models on the original response scale. It assesses how much worse the simpler model fits than the full model. The p-value describes evidence against the simpler model under its assumptions; it does not measure effect size or scientific importance.
In Linza#
- Create an XY table and enter positive concentrations as they are, not logarithms.
- In Analyses, select Dose-response, choose the curve direction, then choose which plateaux to estimate and which to fix.
- Open the decreasing-curve example. It includes data and displays the X axis on a logarithmic scale.
- Choose all pairwise comparisons or each series versus a control. Series must use the same direction and plateau settings, and their dose ranges must overlap around the transition. Review raw and Holm-adjusted p-values, then the EC50 or IC50 ratio and simultaneous interval for each planned pair.
Common mistakes#
- Entering the logarithm of dose instead of the dose.
- Fixing a plateau to get a nicer curve without external justification.
- Treating a non-significant curve comparison as proof that the curves are identical.
Check yourself#
Sources#
- Hill A.V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. Journal of Physiology, 40. Search for publication Opens in a new tab
- Sebaugh J.L. (2011) Guidelines for accurate EC50/IC50 estimation. Pharmaceutical Statistics, 10(2). Search for publication Opens in a new tab
- Ritz C., Baty F., Streibig J.C., Gerhard D. (2015) Dose-response analysis using R. PLoS ONE, 10(12). Open publication Opens in a new tab
- GraphPad Prism 11 Curve Fitting Guide, Comparing nonlinear model fits. Open publication Opens in a new tab
References are for further reading. A method’s suitability depends on the study design and its assumptions.