Signature
k = D / ED_50; S = k^h / (1 + k^h)
| Inputs | Definition | Unit |
|---|---|---|
D | Dose of the drug | mg |
ED_50 | Dose giving half the maximum added effect | mg |
h | Steepness of the curve around ED_50 | none |
k | Dose divided by ED_50 | ratio |
|---|---|---|
S | Added effect at the dose divided by E_max | proportion |
Function
Dose-response curves and the dose-dependent drug cost behind incremental cost per extra responder
Links the expected response to the dose through the sigmoid Emax curve and the dose to the drug cost actually paid, through whole single-use vials and relative dose intensity, so that doses can be compared by the incremental cost per extra unit of effect. The incremental ratio between adjacent doses is HE-FM-ICER-001 with the sequential rules of HE-FM-CEA-001 and HE-FM-DOM-002, and the average ratio against placebo HE-FM-ACER-002. Notation follows the Dose Response article.
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Implementations
Excel
Share of the maximum effect from named cells
With DoseMg, ED50Dose and HillCoef named as in HE-FM-DOSE-001, the formulas return the dose multiple and the share of the maximum effect, held in DoseMultiple and MaxShare.
=DoseMg/ED50Dose; =DoseMultiple^HillCoef/(1+DoseMultiple^HillCoef)
Assumptions
Same sigmoid Emax curve as the response formula
The share follows from HE-FM-DOSE-001 by subtracting E_0 and dividing by E_max, so it carries the same assumptions about the curve.
Worked examples
Doubling the dose from ED50 with a hyperbolic curve
With h = 1, a dose of twice ED_50 reaches 2 / 3 of the maximum effect, 0.667, against 0.5 at ED_50, as in the article.
D = 200; ED_50 = 100; h = 1; k = 2; S = 0.667
Doubling the dose from ED50 with a Hill coefficient of 2
With h = 2 the same doubling reaches 4 / 5 of the maximum, 0.80, as in the article.
D = 200; ED_50 = 100; h = 2; k = 2; S = 0.8
Half the maximum effect at ED50
At D = ED_50 the share is one half whatever the Hill coefficient, as the definition of ED_50 requires.
D = 100; ED_50 = 100; h = 2; k = 1; S = 0.5
Common errors
Assuming that doubling the dose doubles the effect
From ED_50, doubling the dose raises the share of the maximum effect from 0.5 to 0.667 with h = 1 and to 0.80 with h = 2, never to 1, so a linear dose-response assumption overstates the benefit of higher doses.
Sources
DoseFinding parameters of the sigmoid Emax model
Bornkamp B, Pinheiro J, Bretz F, et al. DoseFinding: Planning and Analyzing Dose Finding Experiments. R package version 1.4-1. 2025. Reference manual, help page drmodels (full text read). drmodels, Arguments and Details: ed50 is the dose giving half of the asymptotic maximum effect and h the Hill parameter determining the steepness of the model at the ED50; the sigmoid Emax model describes monotonic, sigmoid dose-response relationships.
Canonical Identity
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