= Quantal dose response
<Quantal dose response> describes the probability or proportion of subjects meeting a specified binary response criterion at each exposure level. The response is yes/no for that criterion, rather than the magnitude of the underlying change. A curve therefore requires both the exposure definition and the endpoint used to classify a responder.<IRIS Glossary by the United States Environmental Protection Agency>[Dose-Response Assessment and Endpoint entries]{ref}
Let $B_i(d)$ be $1$ when subject $i$ meets the criterion at dose $d$ and $0$ otherwise. The population quantity is
$$
p(d)=\Pr(B_i(d)=1),\qquad \widehat p(d)=\frac{k(d)}{n(d)}.
$$
Under independent subjects with the same probability, $k(d)$ has a <binomial distribution>. With $12$ responders among $20$ subjects, the estimate is $0.60$ and its plug-in standard error is
$$
\sqrt{\frac{0.60(1-0.60)}{20}}\approx0.110.
$$
That is a sampling calculation, not a claim that each person has a $60\%$ effect magnitude or that the estimate is exact.
An original threshold model illustrates a possible origin of the curve. Suppose subjects have response thresholds $T$, with response occurring when $d\ge T$. Then $p(d)=\Pr(T\le d)$, a <cumulative distribution function>. Equal proportions with thresholds $1$, $2$, $4$ and $8$ arbitrary dose units give a responder fraction $1/2$ at dose $2$. Those subjects’ effect magnitudes are still unspecified. Real response need not follow a deterministic threshold for each person.
A dose associated with $50\%$ responders is consequently not the same parameter as a dose producing half a maximum <graded dose response>. Changing the responder criterion changes the quantal curve even if the underlying observations do not change.
The study design must handle observation time, population composition and repeated measurements. Several observations from one person are not automatically independent extra subjects. ICH-E4’s distinction between individual and population-average dose-response information is relevant to interpreting the resulting curve.<Dose-Response Information to Support Drug Registration by ICH>[Section I.B, individual and population-average relationships]{ref} The curve summarizes a defined population and endpoint rather than supplying an individual dosing rule.
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