---
title: 'SEMAP-curvature: Bayesian Dose-Finding with Sigmoid Emax'
url: https://www.emergentmind.com/topics/semap-curvature
type: topic
---

# SEMAP-curvature: Bayesian Dose-Finding with Sigmoid Emax

SEMAP-curvature is a particular instantiation of the MAP-curvature framework in which the default dose-response shape is the sigmoid Emax model. It is model-free in the sense that it directly estimates the mean responses at the prespecified doses, while smoothness is enforced by penalising curvature after mapping the curve into the Emax scale. Within Phase II dose-finding, the method was introduced as a Bayesian model-free approach for continuous outcomes with known common variance, and was further extended by integrating MAP-curvature with a Bayesian hierarchical model to enable flexible borrowing of historical data [2509.23777].

## 1. Definition and position within MAP-curvature

SEMAP-curvature is defined within a parallel-group Phase II trial with placebo and \(M\) active doses, using standardised doses
\[
x_0, x_1, \dots, x_M \in [0,1],
\]
with \(x_0\) the placebo. At dose \(x_i\), patient \(j=1,\dots,N_i\) has continuous outcome
\[
Y_{ij}\mid \mu_i \sim N(\mu_i,\sigma^2), \qquad i=0,\dots,M,
\]
where \(\mu_i=f(x_i)\) is the mean response at dose \(x_i\) for an unknown dose-response curve \(f\) on \([0,1]\). The variance \(\sigma^2\) is assumed known and common to all arms [2509.23777].

The general MAP-curvature construction chooses a default dose-response function
\[
\phi(x;\boldsymbol{\theta}),\quad x\in[0,1],
\]
with parameter vector \(\boldsymbol{\theta}\), and measures deviation of the true curve from that default shape through the second derivative of the transformed function \(\phi^{-1}(f(x);\boldsymbol{\theta})\). The associated \(L^2\) total curvature is
\[
S\phi^{-1}(f)=\left(\int_{x_0}^{x_M}\left(\frac{d^2}{dx^2}\phi^{-1}\bigl(f(x);\boldsymbol{\theta}\bigr)\right)^2dx\right)^{1/2}.
\]
If \(f(x)=\phi(x;\boldsymbol{\theta})\) exactly, then \(\phi^{-1}(f(x);\boldsymbol{\theta})=x\), so the integrand vanishes up to numerical approximation and the curvature measure is small. Large values indicate strong deviation from the default shape.

SEMAP-curvature specializes this framework by taking the sigmoid Emax model as the default. LiMAP-curvature is the linear special case in which \(\phi(x;\boldsymbol{\theta})=x\), whereas SEMAP-curvature employs the sigmoid Emax model and is intended to provide greater flexibility for nonlinear pharmacological patterns. A plausible implication is that the method occupies an intermediate position between fully parametric Emax fitting and nonparametric smoothing: it preserves dose-level mean estimation while regularising toward a pharmacologically plausible family.

## 2. Sigmoid Emax transformation and curvature penalty

In SEMAP-curvature, the default dose-response model is the sigmoid Emax function
\[
\phi(x;\boldsymbol{\theta})=E_0+E_{\max}\frac{x^\lambda}{x^\lambda+ED_{50}^\lambda},
\]
with inverse
\[
\phi^{-1}(y;\boldsymbol{\theta})=ED_{50}\left(\frac{E_{\max}}{E_{\max}+E_0-y}-1\right)^{1/\lambda},
\]
where
\[
\boldsymbol{\theta}=\{E_0,E_{\max},ED_{50},\lambda\}.
\]
Here \(E_0\) is the baseline effect, \(E_{\max}\) the maximum achievable treatment effect, \(ED_{50}\) the dose achieving \(50\%\) of \(E_{\max}\), and \(\lambda\) the Hill coefficient controlling steepness [2509.23777].

The curvature measure in SEMAP-curvature remains
\[
S\phi^{-1}(f)=\left(\int_{x_0}^{x_M}\left(\frac{d^2}{dx^2}\phi^{-1}\bigl(f(x);\boldsymbol{\theta}\bigr)\right)^2dx\right)^{1/2},
\]
but in practice \(f\) is only observed at the discrete doses \(\{x_i\}\). Using a second-order central difference approximation, for \(i=1,\dots,M-1\),
\[
\frac{d^2}{dx^2}\phi^{-1}\bigl(f(x_i);\boldsymbol{\theta}\bigr)\approx 2\left(\frac{\phi^{-1}(\mu_{i+1};\boldsymbol{\theta})-\phi^{-1}(\mu_i;\boldsymbol{\theta})}{(x_{i+1}-x_i)(x_{i+1}-x_{i-1})}-\frac{\phi^{-1}(\mu_i;\boldsymbol{\theta})-\phi^{-1}(\mu_{i-1};\boldsymbol{\theta})}{(x_i-x_{i-1})(x_{i+1}-x_{i-1})}\right).
\]
This yields the discrete approximation
\[
S_{\boldsymbol{\mu}}=
2\left(\sum_{i=1}^{M-1}\left(\frac{\phi^{-1}(\mu_{i+1};\boldsymbol{\theta})-\phi^{-1}(\mu_i;\boldsymbol{\theta})}{(x_{i+1}-x_i)(x_{i+1}-x_{i-1})}-\frac{\phi^{-1}(\mu_i;\boldsymbol{\theta})-\phi^{-1}(\mu_{i-1};\boldsymbol{\theta})}{(x_i-x_{i-1})(x_{i+1}-x_{i-1})}\right)^2\Delta x_i\right)^{1/2},
\]
where
\[
\Delta x_i=
\begin{cases}
(x_2+x_1)/2-x_0,& i=1,\\[2pt]
(x_{i+1}-x_{i-1})/2,& i=2,\dots,M-2,\\[2pt]
x_M-(x_{M-1}+x_{M-2})/2,& i=M-1.
\end{cases}
\]

Regularisation is imposed through a half-normal prior on total curvature. Operationally, the prior density on \(S_{\boldsymbol{\mu}}\) is
\[
p(S_{\boldsymbol{\mu}}\mid \gamma,\boldsymbol{\theta})\propto \exp\left(-\frac{S_{\boldsymbol{\mu}}^2}{2\gamma^2}\right),\quad S_{\boldsymbol{\mu}}\ge 0,
\]
with hyperprior
\[
\gamma\sim HN(\tau^2).
\]
Smaller \(\tau\) favours smaller \(\gamma\) and hence stronger penalisation of curvature; larger \(\tau\) allows more flexible curves. The paper explicitly interprets this as penalising departure from the Emax shape rather than departure from a straight line. This suggests that SEMAP-curvature should be most effective when the true dose-response is monotone, saturating, and concave downward over clinically relevant regions.

## 3. Bayesian specification, posterior objective, and computation

The model assigns independent diffuse uniform priors
\[
\mu_i\sim U(0,1),\quad i=0,\dots,M,
\]
and a prior \(p(\boldsymbol{\theta})\) determined by the sigmoid Emax specification. The generic guidance for \(p(\boldsymbol{\theta})\) is as follows: \(E_0\) and \(E_{\max}\) receive Normal priors, with means informed by historical or expert knowledge when available; \(\lambda\) may receive gamma, beta, or log-normal priors chosen so that values near \(2\) are most probable and values below \(1\) or above \(5\) have low probability; \(ED_{50}\) receives a truncated normal prior on \([0,1]\). In the main simulation study, the authors fix the placebo response at \(0\), omit \(E_0\), and use
\[
E_{\max}\sim N(0.5,0.2^2),\qquad ED_{50}\sim N_{[0,1]}(0.5,0.15^2),\qquad \lambda\sim \Gamma(2.5,1.18),
\]
together with
\[
\gamma\sim HN(\tau^2),\qquad \tau=0.5
\]
for the main results, plus sensitivity analysis over \(\tau\in\{0.05,0.1,0.5,1,2,3,4,5,6\}\) [2509.23777].

The joint prior is
\[
p(\boldsymbol{\mu},\gamma,\boldsymbol{\theta})=p(\gamma)p(\boldsymbol{\theta})p(S_{\boldsymbol{\mu}}\mid \gamma,\boldsymbol{\theta})\prod_{i=0}^{M}p(\mu_i),
\]
and the likelihood factorises as
\[
p(\boldsymbol{Y}\mid \boldsymbol{\mu},\gamma,\boldsymbol{\theta})
=
\prod_{i=0}^M\prod_{j=1}^{N_i}\frac{1}{\sqrt{2\pi}\sigma}\exp\Bigl\{-\frac{(Y_{ij}-\mu_i)^2}{2\sigma^2}\Bigr\}.
\]
Hence the posterior is
\[
p(\boldsymbol{\mu},\gamma,\boldsymbol{\theta}\mid \boldsymbol{Y})
\propto
p(\boldsymbol{\mu},\gamma,\boldsymbol{\theta})\,p(\boldsymbol{Y}\mid \boldsymbol{\mu},\gamma,\boldsymbol{\theta}).
\]

Taking logs and discarding constants gives the MAP objective
\[
\begin{aligned}
\log p(\boldsymbol{\mu},\gamma,\boldsymbol{\theta}\mid \boldsymbol{Y})
&=
-\frac{\gamma^2}{2\tau^2}
+\log p(\boldsymbol{\theta})
+\log \gamma
-\frac{S_{\boldsymbol{\mu}}^2}{2\gamma^2}
-\sum_{i=0}^{M}\sum_{j=1}^{N_i}\frac{(Y_{ij}-\mu_i)^2}{2\sigma^2}
+\text{const}.
\end{aligned}
\]
The MAP estimates are
\[
\hat{\boldsymbol{\mu}},\hat{\gamma},\hat{\boldsymbol{\theta}}
=
\arg\max_{\boldsymbol{\mu},\gamma,\boldsymbol{\theta}}
\log p(\boldsymbol{\mu},\gamma,\boldsymbol{\theta}\mid \boldsymbol{Y}),
\]
obtained numerically via a quasi-Newton method such as BFGS. No full MCMC algorithm is specified. Once \(\hat{\boldsymbol{\mu}}\) is obtained, the dose-response curve is constructed by interpolation between the points \((x_i,\hat{\mu}_i)\), and target doses such as the minimum effective dose are derived from this curve.

The implementation steps stated in the paper are operationally simple: specify doses, sample sizes, and observed data; initialise \(\boldsymbol{\mu}\), \(\boldsymbol{\theta}\), and \(\gamma\); at each objective evaluation compute \(\phi^{-1}(\mu_i;\boldsymbol{\theta})\), finite differences, the likelihood, and prior terms; optimise the posterior; and use the fitted \(\hat{\boldsymbol{\mu}}\) to construct the estimated curve and MED. No public software is described.

## 4. Relation to LiMAP-curvature and MCP-Mod

LiMAP-curvature is recovered by choosing
\[
\phi(x;\boldsymbol{\theta})=x,\qquad \phi^{-1}(y;\boldsymbol{\theta})=y.
\]
Then \(S_{\boldsymbol{\mu}}\) reduces to the discrete \(L^2\) norm of the second differences of \(\boldsymbol{\mu}\), so the method penalises deviations from linearity. The stated interpretation is that LiMAP-curvature works well when the true dose-response is approximately linear or only mildly curved, but becomes less efficient when the true curve is strongly nonlinear, such as sigmoidal or saturating [2509.23777].

SEMAP-curvature differs by treating a sigmoidal, saturating curve as the baseline. The paper characterises the contrast in conceptual terms: LiMAP treats a straight line as the default and regularises toward linearity, whereas SEMAP treats a sigmoid Emax curve as the default and regularises toward pharmacologically plausible nonlinear patterns.

The comparison with MCP-Mod is sharper. MCP-Mod requires prespecification of a finite set of parametric candidate models, conducts model-based multiple contrast tests, and then fits the selected model. The paper states that its performance is sensitive to model misspecification. SEMAP-curvature, by contrast, does not require a discrete set of candidate models and works directly on the dose-specific means \(\mu_i\) with a curvature penalty relative to a flexible default model.

The principal comparisons reported in the source are summarized below.

| Method | Default structure | Stated strengths |
|---|---|---|
| LiMAP-curvature | Linear | Effective for approximately linear or mildly curved responses |
| SEMAP-curvature | Sigmoid Emax | Better for concave downward, saturating, pharmacologically realistic shapes |
| MCP-Mod | Finite candidate model set | Standard parametric benchmark; sensitive to misspecification |

In simulations, SEMAP-curvature generally outperforms LiMAP-curvature and MCP-Mod in detecting dose-response signals and estimating MED for concave downward shapes resembling the sigmoid Emax model, including emax, quadratic, power, betaMod, and sigEmax. The paper also reports exceptions: for models with initial concave upward curvature, such as exponential1, logistic1, and exponential2, SEMAP can underperform LiMAP in power. That pattern is attributed to priors favouring sigmoidal saturation together with limited low-dose information. This suggests that SEMAP-curvature is not a universally dominant smoother, but one whose regularisation bias is intentionally aligned with a particular pharmacological prior geometry.

## 5. Historical borrowing extension

The MAP-curvature framework is extended to incorporate one historical trial using the hierarchical model of Han et al. (2024), allowing borrowing across arbitrary dose patterns and accounting for prognostic and predictive heterogeneity [2509.23777].

Let \(x_0,\dots,x_M\) be all distinct doses used in either the current or the historical trial. Let \(\mathcal{I}^{(c)}\subset\{0,\dots,M\}\) denote the current-trial dose indices and \(\mathcal{I}^{(h)}\subset\{0,\dots,M\}\) the historical-trial dose indices. The current and historical outcomes are modelled as
\[
Y_{ij}^{(c)}\mid \mu_i,r\sim N(\mu_i+r,\sigma^2),
\qquad i\in\mathcal{I}^{(c)},\ j=1,\dots,N_i^{(c)},
\]
and
\[
Y_{ij}^{(h)}\mid \mu_i,a,r\sim N(a\mu_i-r,\sigma^2),
\qquad i\in\mathcal{I}^{(h)},\ j=1,\dots,N_i^{(h)}.
\]

Here \(r\) captures prognostic heterogeneity as a baseline shift, with prior
\[
r\sim N(0,\rho^2),
\]
and \(a\) captures predictive heterogeneity as a multiplicative treatment-effect change, with prior
\[
a\sim N_{[b,1/b]}(1,\eta^2).
\]
The paper notes that they often choose \(b=1/3\), and in simulations use
\[
a\sim N_{[1/3,3]}(1,0.2^2),\qquad r\sim N(0,0.5^2).
\]

The extended joint prior is
\[
p(\boldsymbol{\mu},\gamma,\boldsymbol{\theta},a,r)
=
p(a)p(r)p(\gamma)p(\boldsymbol{\theta})p(S_{\boldsymbol{\mu}}\mid \gamma,\boldsymbol{\theta})\prod_{i=0}^{M}p(\mu_i),
\]
and the posterior is
\[
p(\boldsymbol{\mu},\gamma,\boldsymbol{\theta},a,r\mid \boldsymbol{Y}^{(c)},\boldsymbol{Y}^{(h)})
\propto
p(\boldsymbol{\mu},\gamma,\boldsymbol{\theta},a,r)\,
p(\boldsymbol{Y}^{(c)},\boldsymbol{Y}^{(h)}\mid \boldsymbol{\mu},\gamma,\boldsymbol{\theta},a,r).
\]
MAP estimation is then performed over \(\boldsymbol{\mu},\gamma,\boldsymbol{\theta},a,r\).

The paper emphasises three properties of this construction. First, it borrows across arbitrary dose patterns, not only perfectly matched dose grids. Second, it adjusts automatically for between-trial differences through \(a\) and \(r\). Third, if the current data conflict with the historical data, the posterior for \(a\) and/or \(r\) moves away from \((1,0)\), effectively down-weighting historical information. A plausible implication is that the borrowing mechanism functions as a robust commensurability device rather than as fixed pooling.

## 6. Operating characteristics, practical use, and terminological scope

The main simulation design uses a randomised, double-blind, placebo-controlled parallel-group Phase II trial with current-trial doses
\[
\mathcal{D}_c=\{0,0.15,0.5,0.8,1.0\},
\]
placebo effect fixed at \(0\), maximum treatment effect \(0.5\), total sample size \(200\) with \(40\) per arm, and continuous outcomes with \(\sigma=1\). Twelve true dose-response shapes are examined: linear, emax1, emax2, exponential1, quadratic1, logistic1, exponential2, quadratic2, sigEmax, power, logistic2, and betaMod. Four historical-data scenarios are considered, ranging from full overlap to no historical trial, together with prognostic heterogeneity \(r\in\{0,0.2\}\) and predictive heterogeneity \(a\in\{1,0.8\}\), for \(156\) combinations and \(10{,}000\) virtual trials per combination [2509.23777].

For dose-response signal detection, the test statistic is
\[
T=\max\{\mu_1,\dots,\mu_M\}-\mu_0,
\]
with critical value calibrated by Monte Carlo under \(H_0:\mu_0=\mu_1=\dots=\mu_M\). Without historical data, the reported findings are that SEMAP-curvature consistently yields higher ROC curves than LiMAP and MCP-Mod for concave downward shapes; for emax1, emax2, and quadratic1, it gains about \(3\%\)–\(15\%\) power over LiMAP and \(2\%\)–\(14\%\) over MCP-Mod at \(5\%\) type I error; and for quadratic2 and betaMod, it improves power by \(8\%\)–\(40\%\) compared with MCP-Mod at \(5\%\) type I error. With historical borrowing, the greatest gains occur under full overlap, while partial overlap still outperforms no borrowing in most cases.

For dose-response curve estimation, the paper states that SEMAP-curvature without historical data tends to track complex nonlinear true curves such as emax2, exponential2, logistic2, and sigEmax more closely than LiMAP and MCP-Mod, but that its error bars are generally wider. With historical data, full-overlap borrowing yields mean curves closer to the truth than no borrowing, though variability is not always reduced because heterogeneity through \(a\) and \(r\) introduces additional uncertainty.

For MED estimation, representative results under \(r=0.2\), \(a=0.8\), and threshold \(0.3\) include the following: for emax2 with true MED \(0.083\), SEMAP-curvature S4 has bias \(0.140\) and MSE \(0.066\), compared with LiMAP bias \(0.465\), MSE \(0.275\), and MCP-Mod bias \(0.260\), MSE \(0.121\); for quadratic2 with true MED \(0.257\), SEMAP S4 has bias \(0.063\) and MSE \(0.052\), compared with LiMAP bias \(0.280\), MSE \(0.129\), and MCP-Mod bias \(0.120\), MSE \(0.056\); for betaMod with true MED \(0.075\), SEMAP S4 has bias \(0.052\) and MSE \(0.018\), compared with LiMAP bias \(0.170\), MSE \(0.066\), and MCP-Mod bias \(0.142\), MSE \(0.042\). For linear or mildly curved responses, LiMAP may have smaller bias and MSE.

The practical workflow stated for real Phase II use is to standardise or choose doses on \([0,1]\), specify priors for \(E_{\max}\), \(ED_{50}\), \(\lambda\), and \(\gamma\), optionally specify \(a\) and \(r\) for historical borrowing, fit SEMAP-curvature by MAP optimisation, construct the estimated curve by interpolation, test proof of concept via the statistic \(T\), and define MED as the smallest dose whose interpolated response exceeds \(\mu_0+\Delta\), where \(\Delta\) is a clinically relevant threshold such as \(0.3\). The authors note several limitations: sensitivity to \(\tau\) and Emax priors, restriction to continuous outcomes with known variance, absence of strict monotonicity constraints, reliance on MAP rather than full posterior inference, and focus on a single historical trial in the main article.

A separate terminological point is that the string “SEMAP-curvature” also appears in a very different mathematical context. The paper “Curvature of Metrics on Semple Jet bundles” discusses how one could understand a notion like “SEMAP-curvature” on the Semple tower, but explicitly states that the paper does not use the term; there it would refer to the curvature of an invariant singular Hermitian metric on the Semple jet bundle or associated invariant jet bundles [1612.07847]. The established use of SEMAP-curvature as a named method, however, is the dose-finding construction based on sigmoid Emax regularisation [2509.23777].

Source: https://www.emergentmind.com/topics/semap-curvature