= Graded dose response
<Graded dose response> describes how the magnitude of a defined biological effect changes with dose or concentration under a specified protocol. The observable can be an activity, force, concentration or other quantitative endpoint. It differs from <quantal dose response>, which records whether a specified criterion is met. The EPA’s primary terminology explicitly separates degree of response from incidence or probability.<IRIS Glossary by the United States Environmental Protection Agency>[Dose-Response Assessment, Dose-Response Relationship and Endpoint entries]{ref}
For an original saturating concentration-response model, write
$$
E(c)=E_0+(E_{\mathrm{top}}-E_0)\frac{c}{K+c},\qquad c\ge0.
$$
The plateau $E_{\mathrm{top}}$ is an absolute response level, while $E_{\mathrm{top}}-E_0$ is the maximum change above baseline. At $c=K$, the response is halfway between baseline and plateau. If they are $20$ and $100$ response units, that midpoint is $60$, not $50$. Stating the baseline convention is essential when comparing midpoint parameters.
The graded mean also does not determine a binary responder fraction. Two original four-observation examples, $(20,20,20,20)$ and $(0,0,40,40)$, both have mean $20$. With a response criterion of at least $30$, their responder fractions are $0$ and $1/2$. Converting a continuous measurement into a binary outcome discards information about response magnitude.
A controlled assay measures matched concentrations at stated observation times with suitable baseline and reference conditions. <Pharmacokinetics> can make an administered dose produce different concentration histories, and biological time dependence can change the effect even at the same nominal concentration. ICH-E4 discusses concentration-response information and the interaction of response with time.<Dose-Response Information to Support Drug Registration by ICH>[Sections I.C and I.E, concentration-response data and time]{ref}
A fitted curve is a model for the observed endpoint, not proof of a particular receptor mechanism. An apparent plateau may instead reflect an instrument limit or the tested range. <Graded dose response> is most useful when the measured quantity, normalization, exposure definition and uncertainties are all explicit.
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