Methods, boundaries, and numerical meaning

Model and integrated response

Let the fixed dose axis be $[0,D]$, where $D>0$ has dose units. The baseline $a$ is unrestricted, the endpoint excess response $\delta$ is nonnegative, and the dimensionless shape $q$ lies in $[0,1]$. For normalized dose $u=x/D$, the mean response is defined by

$$\mu(x)=a+\delta h_q(x/D),\qquad h_q(u)=\frac{u}{1-q+qu}\quad(q<1).$$

Here $h_q$ is dimensionless and emax_shape() evaluates it. The closure $h_1(0)=0$, $h_1(u)=1$ for $u>0$ represents immediate saturation. At $q=0$ the shape is linear. These endpoint definitions are part of the model, not finite-dose approximations.

The average integrated excess response $T$ is defined on this linear axis by

$$T=D^{-1}\int_0^D\{\mu(x)-a\}\,dx=\delta G(q),\qquad G(q)=\int_0^1 h_q(u)\,du.$$

The change of variables $u=x/D$ explains why normalization removes the factor $D$. Thus $T$ and $\delta$ have response units. emax_shape_area() computes $G$ with its endpoint values $G(0)=1/2$ and $G(1)=1$, using the stable series and logarithmic formula from the research implementation.

Gaussian confidence region

Suppose $m\geq3$ distinct fixed doses satisfy $0=x_1<\cdots<x_m=D$, with positive integer counts $n_j$ and total $n=\sum_{j=1}^m n_j>m$. Let $\bar Y$ be the vector of dose means, $W=\operatorname{diag}(n_j)$, and $S^2$ the pooled within-dose variance with $\nu=n-m$ degrees of freedom. With $v=W^{1/2}\mathbf 1/\sqrt n$ and $P=I-vv^\mathsf{T}$, define centered coordinates $y=PW^{1/2}\bar Y$ and $H(q)=PW^{1/2}h_q$. Here $h_q$ denotes the vector of values $h_q(x_j/D)$ at dose levels indexed by $j=1,\ldots,m$. The identity matrix $I$ has size $m$, and $\mathbf 1$ is the length-$m$ vector of ones.

Under independent Gaussian errors of common variance, the exact centered F region accepts parameters satisfying

$$\|y-\delta H(q)\|^2\leq(m-1)S^2 F_{m-1,\nu}(1-\alpha).$$

The norm is Euclidean, $\alpha\in(0,1)$ is the chosen error probability, and $F_{r,\nu}(1-\alpha)$ denotes an F quantile with numerator degrees of freedom $r$. The rank $m-1$ follows from removing the unrestricted baseline. emax_region() constructs this region; emax_project() encloses the hull of its accepted values of $\delta G(q)$. The region pivot is standard Gaussian inference. The accompanying manuscript develops the integrated-target boundary theory and computational construction; no new Emax model is claimed.

calibration = "mean" instead uses the full mean-region radius $m S^2F_{m,\nu}(1-\alpha)$. The larger radius provides the manuscript’s full mean-region benchmark after minimizing over the intercept. Both are finite-sample constructions under the same Gaussian assumptions.

Boundary behavior

At a flat curve, $\delta=0$ and shape need not be identifiable. At positive immediate saturation, the integrated target has logarithmic sensitivity near $q=1$, even though fixed positive-dose shapes have finite one-sided derivatives. The package includes both closures. It does not replace their inference with an interior-only normal approximation.

For unknown physical doses, equally spaced ordinal codes describe an illustrative code-axis integral. They do not establish physical-dose area, clinical benefit, or a treatment recommendation.

Numerical enclosure and status

The adaptive search uses fixed-shape quadratic slices, analytic outer bounds, necessary shape-box inequalities, and feasible witnesses. The refined search also intersects the original finite-budget enclosure. Defaults are endpoint tolerance 0.001 and a budget of 4096 splits; no cap on the statistical effect parameter is introduced.

endpoint_gaps reports the outer-to-witness differences. The result includes the numerical trace and termination reason. The exact-arithmetic reasoning does not make ordinary floating-point results formal directed-rounding certificates. outward_rounding_certified is always false.

An empty model intersection has zero reported width and no endpoints. An unresolved search is not an empty set. Coverage studies must retain empty sets as noncoverage and report unresolved computations separately.

Citation and scope

The scientific reference is Bahadır Yüzbaşı’s unpublished manuscript, Confidence-region projection for integrated Emax responses: boundary behavior and finite-sample inference (2026). No accepted-paper status or DOI is claimed. The package ports its validated R construction to RcppArmadillo; it does not establish robustness to heteroscedasticity, dependence, or non-Gaussian errors.