Binary Emax Model in Dose–Response Analysis

• Binary Emax model is a nonlinear dose–response model for binary endpoints that uses an Emax-shaped predictor to capture the saturating relationship between dose and efficacy.
• It employs a nonlinear GLM framework, typically with a logit link, enabling estimation of baseline effect, maximal increment, and potency (ED50) from Bernoulli outcomes.
• Advanced techniques like Firth’s correction and Jeffreys-prior MPLE are implemented to reduce bias and improve stability in small Phase II dose-ranging studies.

Searching arXiv for the cited papers and related Binary Emax work. The Binary Emax model is a nonlinear dose–response model for binary endpoints in which the probability of response is linked to dose through an Emax-shaped predictor. In Phase II dose-ranging studies, it is used to represent a monotone, saturating relationship between dose and efficacy, thereby supporting estimation of response probabilities, characterization of potency through $ED_{50}$, and selection of doses for subsequent confirmatory trials (Zhang et al., 22 Sep 2025, Zhang et al., 2024). In the binary setting, the Emax structure is typically embedded in a generalized linear model, most commonly with a logit link, although probability-scale and probit formulations are also used (Zhang et al., 22 Sep 2025, Han et al., 28 Sep 2025).

1. Model specification

The standard three-parameter Binary Emax model used for binary outcomes specifies

$$\log\left(\frac{\pi_i}{1-\pi_i}\right) = E_0 + \frac{E_{\max}\,\mathrm{Dose}_i}{ED_{50}+\mathrm{Dose}_i},$$

where $\pi_i=P(y_i=1\mid \mathrm{Dose}_i)$, $E_0$ is the placebo logit, $E_{\max}$ is the maximal increment on the logit scale, and $ED_{50}$ is the dose that produces half-maximal effect on that scale (Zhang et al., 2024, Zhang et al., 22 Sep 2025). The corresponding probability is

$$\pi(d)=\operatorname{logit}^{-1}\!\left( E_0+\frac{E_{\max}d}{ED_{50}+d} \right).$$

This formulation is widely used in Phase II dose–response analysis because it combines nonlinearity with a low-dimensional parameterization (Zhang et al., 2024, Zhang et al., 22 Sep 2025).

A four-parameter sigmoid version introduces a Hill coefficient $\lambda>0$: $$\log\left(\frac{\pi_i}{1-\pi_i}\right) = E_0 + \frac{E_{\max}\,\mathrm{Dose}_i^{\lambda}} {ED_{50}^{\lambda}+\mathrm{Dose}_i^{\lambda}},$$ or, on the probability scale,

$$p(x)=E_0+\frac{E_{\max}x^\lambda}{x^\lambda+ED_{50}^\lambda},$$

with appropriate constraints when $$\log\left(\frac{\pi_i}{1-\pi_i}\right) = E_0 + \frac{E_{\max}\,\mathrm{Dose}_i}{ED_{50}+\mathrm{Dose}_i},$$0 is modeled directly rather than through a link (Zhang et al., 22 Sep 2025, Han et al., 28 Sep 2025). Meta-analytic evidence is reported to suggest that $$\log\left(\frac{\pi_i}{1-\pi_i}\right) = E_0 + \frac{E_{\max}\,\mathrm{Dose}_i}{ED_{50}+\mathrm{Dose}_i},$$1 often suffices, which motivates the frequent use of the three-parameter specification in practice (Zhang et al., 2024, Zhang et al., 22 Sep 2025).

For binary outcomes, the sampling model is Bernoulli: $$\log\left(\frac{\pi_i}{1-\pi_i}\right) = E_0 + \frac{E_{\max}\,\mathrm{Dose}_i}{ED_{50}+\mathrm{Dose}_i},$$2 with doses $$\log\left(\frac{\pi_i}{1-\pi_i}\right) = E_0 + \frac{E_{\max}\,\mathrm{Dose}_i}{ED_{50}+\mathrm{Dose}_i},$$3 and $$\log\left(\frac{\pi_i}{1-\pi_i}\right) = E_0 + \frac{E_{\max}\,\mathrm{Dose}_i}{ED_{50}+\mathrm{Dose}_i},$$4 patients per dose (Han et al., 28 Sep 2025). The binary Emax model is therefore a nonlinear logistic regression, or more generally a nonlinear GLM, in which the Emax curve governs the systematic effect of dose on the transformed probability scale (Zhang et al., 22 Sep 2025, Aletti et al., 2023).

2. Parameter interpretation and dose–response geometry

The defining feature of the Binary Emax model is pharmacologically plausible saturation. On the logit scale, $$\log\left(\frac{\pi_i}{1-\pi_i}\right) = E_0 + \frac{E_{\max}\,\mathrm{Dose}_i}{ED_{50}+\mathrm{Dose}_i},$$5 is the baseline effect at placebo, $$\log\left(\frac{\pi_i}{1-\pi_i}\right) = E_0 + \frac{E_{\max}\,\mathrm{Dose}_i}{ED_{50}+\mathrm{Dose}_i},$$6 is the maximum achievable increment, and $$\log\left(\frac{\pi_i}{1-\pi_i}\right) = E_0 + \frac{E_{\max}\,\mathrm{Dose}_i}{ED_{50}+\mathrm{Dose}_i},$$7 is the potency parameter locating the dose at which half of $$\log\left(\frac{\pi_i}{1-\pi_i}\right) = E_0 + \frac{E_{\max}\,\mathrm{Dose}_i}{ED_{50}+\mathrm{Dose}_i},$$8 is attained (Zhang et al., 2024, Zhang et al., 22 Sep 2025). On the response-probability scale, this translates into a curve that rises with dose and approaches an asymptote (Zhang et al., 22 Sep 2025).

This parameterization is central in dose selection because it yields interpretable summaries of baseline response, maximal attainable response, and the location of the clinically informative transition region. Once $$\log\left(\frac{\pi_i}{1-\pi_i}\right) = E_0 + \frac{E_{\max}\,\mathrm{Dose}_i}{ED_{50}+\mathrm{Dose}_i},$$9 is obtained, the fitted curve can be used to estimate success rates at candidate doses, identify minimally effective and optimal doses, support go/no-go decisions, and inform Phase III dose selection (Zhang et al., 2024).

In the continuous Emax literature, the mean function

$\pi_i=P(y_i=1\mid \mathrm{Dose}_i)$0

is described as the concave branch of a rectangular hyperbola on the dose interval $\pi_i=P(y_i=1\mid \mathrm{Dose}_i)$1 (Aletti et al., 2023). A plausible implication is that the same geometric constraints govern the binary version once the Emax curve is used as the linear predictor: the model encodes monotonicity and concavity on the link scale, and therefore fits best when the empirical dose–response is compatible with that shape (Aletti et al., 2023, Zhang et al., 22 Sep 2025).

The four-parameter sigmoid extension adds $\pi_i=P(y_i=1\mid \mathrm{Dose}_i)$2, which controls steepness. In that version, $\pi_i=P(y_i=1\mid \mathrm{Dose}_i)$3 retains its role as the midpoint of the Emax transition, while $\pi_i=P(y_i=1\mid \mathrm{Dose}_i)$4 governs how sharply the curve moves from baseline toward plateau (Han et al., 28 Sep 2025, Zhang et al., 22 Sep 2025).

3. Likelihood-based estimation and existence issues

Under independent Bernoulli sampling, the Binary Emax log-likelihood is

$\pi_i=P(y_i=1\mid \mathrm{Dose}_i)$5

with score

$\pi_i=P(y_i=1\mid \mathrm{Dose}_i)$6

where

$\pi_i=P(y_i=1\mid \mathrm{Dose}_i)$7

for the three-parameter model (Zhang et al., 22 Sep 2025). Maximum likelihood estimation typically proceeds through Newton–Raphson or quasi-Newton optimization (Zhang et al., 22 Sep 2025, Zhang et al., 2024).

Asymptotically, the usual regularity results are invoked: consistency, asymptotic normality, and Wald-type inference based on the inverse observed or expected information (Zhang et al., 22 Sep 2025). However, the supplied studies repeatedly emphasize that these large-sample approximations may be questionable in the small or moderate sample sizes typical of Phase II dose-ranging trials (Zhang et al., 22 Sep 2025, Zhang et al., 2024).

A central result from the Emax estimation literature is that failures of maximum likelihood are not merely computational accidents; they can reflect genuine non-existence of the MLE (Aletti et al., 2023). For three-point designs in the continuous model, exact analysis identifies two problematic scenarios: non-increasing concave data and convex data (Aletti et al., 2023). The paper attributes these failures to the geometry of the Emax family, whose limiting shapes include strictly increasing straight lines, horizontal lines, and a horizontal line with a jump at the lower boundary (Aletti et al., 2023). This suggests that, in binary Emax fitting, non-convergence or extreme estimates can arise because the observed binary pattern is better approximated by a limit shape than by any finite-parameter Emax curve.

The same concern appears empirically in the binary setting. In simulations with five dose arms and $\pi_i=P(y_i=1\mid \mathrm{Dose}_i)$8, MLE fails or is unstable in approximately $\pi_i=P(y_i=1\mid \mathrm{Dose}_i)$9 to $E_0$0 of runs, with ill-conditioned information and extreme parameter estimates; even when finite, $E_0$1 can have large bias and enormous Wald intervals (Zhang et al., 22 Sep 2025). The problem is especially pronounced when observed dose–response patterns are non-monotonic, convex, or nearly flat over the studied dose range (Zhang et al., 22 Sep 2025).

4. Separation, missingness, and instability

Binary Emax models inherit the pathologies of logistic-type models. Separation and quasi-separation occur when dose, or a function of dose, nearly or perfectly predicts response class; in such cases, likelihood maximization may drive parameters to infinity, produce flat likelihood regions, or fail numerically (Zhang et al., 2024, Zhang et al., 22 Sep 2025). In the Emax setting, these failures can manifest as very large estimates of $E_0$2, extreme $E_0$3, or non-convergence (Zhang et al., 2024).

A second complication is missing binary outcome data. One paper formulates a joint selection-model factorization

$E_0$4

where the missingness model is logistic and may depend on the unobserved outcome $E_0$5, thus allowing nonignorable missingness (Zhang et al., 2024). If the coefficient on $E_0$6 in the missingness model is nonzero, missingness is nonignorable or MNAR; if it is zero, the mechanism is ignorable given covariates (Zhang et al., 2024).

The same study argues that non-responder imputation, although often recommended by regulatory agencies, can induce substantial bias when missingness depends on unobserved response (Zhang et al., 2024). According to the reported simulations, non-responder imputation yields severe mean bias error and mean squared error, while complete-case and standard multiple-imputation analyses also remain biased under MNAR (Zhang et al., 2024). Explicit joint modeling of response and missingness is therefore presented as necessary when nonignorable missingness is plausible (Zhang et al., 2024).

These issues are not only theoretical. In the TURANDOT Phase II ulcerative colitis study, remission status at Week 12 is binary, missingness is treated as nonignorable, and the missingness model shows that remission response is statistically significant in predicting missingness under both IL and FIL estimation (Zhang et al., 2024). The same application also exhibits non-monotone dose–response, with the highest dose showing lower efficacy than intermediate doses, illustrating how missingness, model misspecification, and Emax nonlinearity can interact in practice (Zhang et al., 2024, Zhang et al., 22 Sep 2025).

5. Bias reduction and penalized likelihood

Three bias-reduction strategies are compared in recent binary Emax work: Cox–Snell bias correction, Firth-score modification, and maximum penalized likelihood estimation using Jeffreys prior (Zhang et al., 22 Sep 2025). Their common objective is to improve estimation when finite-sample bias, separation, or model instability undermine ordinary ML.

Cox–Snell correction is post-hoc. It uses a first-order bias approximation of the form

$E_0$7

and defines the corrected estimator by

$E_0$8

(Zhang et al., 22 Sep 2025). The method requires successful and stable ML estimation; consequently, when the information matrix is ill-conditioned or the ML algorithm fails, Cox–Snell correction is unavailable or numerically unstable (Zhang et al., 22 Sep 2025). In the reported simulations, it frequently produces extreme estimates and very large MSE, and sometimes performs worse than ML (Zhang et al., 22 Sep 2025).

Firth’s method is preventive rather than corrective. It replaces the ordinary score by modified equations

$E_0$9

designed to remove the leading $E_{\max}$0 bias term (Zhang et al., 22 Sep 2025). In canonical GLMs this is equivalent to maximizing a Jeffreys-prior penalized likelihood; in nonlinear Emax models, however, the modified score is not the gradient of an ordinary penalized likelihood because its derivative matrix need not be symmetric (Zhang et al., 22 Sep 2025). Even so, the estimator is finite under separation and usually substantially less biased than ML (Zhang et al., 22 Sep 2025, Zhang et al., 2024).

Jeffreys-prior MPLE maximizes

$E_{\max}$1

or equivalently

$E_{\max}$2

(Zhang et al., 22 Sep 2025). The corresponding penalized score is

$E_{\max}$3

In nonlinear binary Emax models this is closely related, but not identical, to Firth’s adjustment (Zhang et al., 22 Sep 2025). The empirical claim made in the paper is that MPLE with Jeffreys prior provides the most stable estimation across sample sizes and misspecification scenarios, always produces finite estimates in the reported simulations, and generally has the smallest MSE and smallest estimated standard errors among the compared methods (Zhang et al., 22 Sep 2025).

The simulation results are especially sharp at small sample size. With $E_{\max}$4, MLE and Cox–Snell have $E_{\max}$5 failure to estimate and $E_{\max}$6 unstable estimates, Firth has $E_{\max}$7 failure and $E_{\max}$8 instability, and MPLE has $E_{\max}$9 failure and $ED_{50}$0 instability (Zhang et al., 22 Sep 2025). In a flat high-$ED_{50}$1 scenario, MPLE again shows $ED_{50}$2 failure and $ED_{50}$3 instability, whereas MLE and Cox–Snell fail in $ED_{50}$4 of runs and Firth has $ED_{50}$5 instability (Zhang et al., 22 Sep 2025). In the TURANDOT analysis, MPLE yields much tighter standard errors for $ED_{50}$6 than ML, Cox–Snell, or Firth, and its bootstrap dose–response probabilities are reported to be more stable than those of the competing estimators (Zhang et al., 22 Sep 2025).

A related penalized-likelihood approach addresses MNAR binary outcomes and separation simultaneously by embedding Jeffreys-prior penalization inside an EM algorithm of Ibrahim–Lipsitz type (Zhang et al., 2024). There the FIL estimator is reported to have the lowest MBE for $ED_{50}$7 and $ED_{50}$8, the lowest MSE and estimated SE across methods and sample sizes, and narrower confidence intervals with coverage close to or slightly above nominal under the studied MNAR scenarios (Zhang et al., 2024).

6. Design, extensions, and related frameworks

Binary Emax methodology is closely tied to design. In the continuous Emax literature, the locally D-optimal three-point design is

$ED_{50}$9

and the same structural logic is presented as directly relevant to binary Emax experiments (Flournoy et al., 2018, Aletti et al., 2023). A plausible implication is that dose placement should target baseline, the curvature region around $$\pi(d)=\operatorname{logit}^{-1}\!\left( E_0+\frac{E_{\max}d}{ED_{50}+d} \right).$$0, and the plateau region if stable estimation of $$\pi(d)=\operatorname{logit}^{-1}\!\left( E_0+\frac{E_{\max}d}{ED_{50}+d} \right).$$1 and $$\pi(d)=\operatorname{logit}^{-1}\!\left( E_0+\frac{E_{\max}d}{ED_{50}+d} \right).$$2 is desired.

Adaptive designs further complicate inference. In a two-stage nonlinear Emax model with Gaussian errors, when the second-stage design is chosen by plugging the stage-one MLE into the locally optimal design, the unconditional asymptotic law of the final MLE is a Gaussian mixture rather than a single normal limit (Flournoy et al., 2018). The paper explicitly treats continuous outcomes, but it argues that the same conceptual issues—random designs, mixture asymptotics, and sensitivity to small-sample interim bias in the potency parameter—should carry over to binary Emax models under GLM regularity conditions (Flournoy et al., 2018).

Another extension is historical borrowing within a Bayesian framework. The MAP-curvature methodology defines a general model-free approach to dose–response estimation by placing a prior on the total curvature of a transformed dose–response curve (Han et al., 28 Sep 2025). Its sigmoid-Emax instantiation, SEMAP-curvature, uses the sigmoid Emax model as the default curve and penalizes deviations from that shape through the transformed quantity $$\pi(d)=\operatorname{logit}^{-1}\!\left( E_0+\frac{E_{\max}d}{ED_{50}+d} \right).$$3 (Han et al., 28 Sep 2025). The worked examples in that paper use normal outcomes, but the framework is described as likelihood-agnostic, and the details explicitly state that replacing the normal likelihood by a Bernoulli likelihood yields a Bayesian Binary Emax model within the same curvature-penalized structure (Han et al., 28 Sep 2025).

Under this extension, historical borrowing is introduced through a Bayesian hierarchical model with prognostic heterogeneity $$\pi(d)=\operatorname{logit}^{-1}\!\left( E_0+\frac{E_{\max}d}{ED_{50}+d} \right).$$4 and predictive heterogeneity $$\pi(d)=\operatorname{logit}^{-1}\!\left( E_0+\frac{E_{\max}d}{ED_{50}+d} \right).$$5 linking current and historical trials (Han et al., 28 Sep 2025). The reported simulations show that SEMAP-curvature, particularly with historical borrowing, improves proof-of-concept power, curve estimation, and minimum effective dose estimation when the true dose–response is concave downward and saturating, that is, when it resembles a sigmoid Emax shape (Han et al., 28 Sep 2025). This suggests that Binary Emax models can be embedded in broader semiparametric or model-free frameworks without abandoning their pharmacological structure.

7. Applications, limitations, and interpretation

The principal application domain described in the supplied literature is Phase II clinical dose-finding with binary efficacy endpoints such as remission or response (Zhang et al., 2024, Zhang et al., 22 Sep 2025). In that setting, Binary Emax is valued because it encodes saturation, supplies interpretable potency and maximum-effect parameters, and supports practical tasks such as identifying minimum effective or optimal doses (Zhang et al., 2024).

At the same time, the literature is explicit about limitations. The model assumes a monotone concave dose–response on the link scale; when the true relationship is non-monotone, strongly convex, or otherwise outside the Emax family, estimation can be unstable and interpretation becomes approximate rather than structural (Aletti et al., 2023, Zhang et al., 22 Sep 2025). Penalization can stabilize estimation but does not eliminate model misspecification (Zhang et al., 22 Sep 2025). Likewise, explicit MNAR modeling can address outcome-dependent missingness, but only insofar as the missingness model itself is plausible and adequately specified (Zhang et al., 2024).

The TURANDOT study illustrates these tensions. The observed remission rates are non-monotonic, with 225 mg less effective than intermediate doses, so a three-parameter Binary Emax model cannot be interpreted as a literal description of the full dose–response relation (Zhang et al., 22 Sep 2025). In that context, the paper treats Emax as an approximate parametric summary and finds that MPLE delivers more stable inference than ML, Cox–Snell, or Firth, especially for $$\pi(d)=\operatorname{logit}^{-1}\!\left( E_0+\frac{E_{\max}d}{ED_{50}+d} \right).$$6 (Zhang et al., 22 Sep 2025). A plausible implication is that, in applied work, Binary Emax should often be fitted alongside model diagnostics and possibly alternative functional forms rather than being treated as self-validating.

In the broader methodological context, the Binary Emax model occupies the intersection of pharmacodynamic dose–response modeling, nonlinear GLMs for binary data, and bias-reduced or penalized likelihood inference (Zhang et al., 22 Sep 2025). Across the supplied literature, its modern development is characterized by three recurring themes: the need to control finite-sample and separation-induced pathologies, the need to account for nonignorable missingness when present, and the need to preserve enough structural flexibility to remain useful when observed Phase II dose–response patterns deviate from ideal monotone saturation (Zhang et al., 2024, Zhang et al., 22 Sep 2025, Han et al., 28 Sep 2025).