Choosing a Dose When the Efficacy Curve Plateaus

Why a Plateau Changes the Dose Question

When efficacy rises with exposure and then approaches a ceiling, the decision is not simply whether a higher dose can produce a larger model-predicted response. The practical question is whether the additional efficacy is large enough to justify the additional safety risk, burden, and uncertainty. A flat-looking curve is evidence to investigate, not proof that all higher doses are equivalent.

The case below uses a hypothetical dose-ranging study. Participants receive one of four daily doses, and exposure is summarized by a steady-state concentration $x$, measured in ng/mL. The study’s efficacy endpoint is a binary clinical response measured at a prespecified time. An adverse event is also recorded as a binary outcome. These models describe response probabilities at the study time; they do not establish that the same relationship applies at other times or in other populations.

Models, Assumptions, and Dose-to-Exposure Mapping

Definition: A saturating exposure-response model describes an outcome whose expected response approaches a limiting value as exposure increases. For a binary efficacy endpoint, one possible model is $$ p(x)=P(Y=1\mid x)=\frac{1}{1+\exp[-(\beta_0+\beta_1x)]}, $$ where $Y=1$ denotes clinical response, $p(x)$ is the response probability at exposure $x$, and $\beta_0,\beta_1$ are fitted parameters. When $\beta_1>0$, the model increases with exposure and approaches 1 as $x$ becomes large.

For this illustration, the fitted efficacy model has $\hat{\beta}_0=-3.5747$ and $\hat{\beta}_1=0.0100$ per ng/mL. The fitted slope is positive, so the predicted response probability increases toward its upper limit. A separate model describes the probability of an adverse event:

q(x)=P(A=1\mid x)=\frac{1}{1+\exp[-(\gamma_0+\gamma_1x)]}, \qquad \hat{\gamma}_0=-4.000,\quad \hat{\gamma}_1=0.00400\text{ per ng/mL}.

Here $A=1$ denotes an adverse event. These models assume that the chosen exposure summary is an appropriate predictor, the dose groups are comparable for the estimands being contrasted, the logit-linear forms are adequate over the observed exposure range, and observations are independent conditional on the model. They also assume that exposure and outcomes are measured consistently. If the trial uses repeated observations per participant or has important baseline predictors, the fitted analysis must account for those features rather than treating all records as independent.

The hypothetical dose-exposure mapping is $x=2d$, where $d$ is daily dose in mg and $x$ is in ng/mL. This mapping is an assumption for the case, not a universal rule. It must be supported by pharmacokinetic evidence; if exposure varies substantially among participants at a given dose, a dose-level average can conceal that variation.

Worked Example: Fit the Curve and Locate the Apparent Plateau

Suppose a hypothetical randomized study assigns 500 participants to each of four dose groups. The observed counts are rounded to illustrate the data pattern; the parameter estimates below are illustrative fitted-model estimates.

Dose (mg)	Exposure (ng/mL)	Clinical responses / 500	Adverse events / 500
100	200	86	20
200	400	302	42
400	800	494	155
800	1,600	500	458

At exposure 800 ng/mL, the fitted efficacy probability is $p(800)=1/[1+\exp(-(-3.5747+0.0100(800)))]$. The linear predictor is $4.4253$, giving $p(800)\approx0.9882$. At 1,600 ng/mL, the linear predictor is $12.4253$, giving $p(1600)\approx0.999996$. Thus the model predicts an efficacy increase of about $0.0118$, or 1.18 percentage points, when moving from 400 mg to 800 mg. That small point-estimated gain is the plateau question: it does not, by itself, show that the gain is exactly zero or precise.

For safety, $q(800)=1/[1+\exp(-(-4+0.004(800)))]\approx0.3100$, whereas $q(1600)\approx0.9168$. The fitted adverse-event probability therefore rises by about $0.6068$, or 60.68 percentage points, for the same dose increase. These predictions make the safety concern apparent, but their uncertainty and the model fit still require review.

Quantifying Contrasts and Their Uncertainty

A dose decision should compare clinically meaningful contrasts, not just inspect a curve. Define the efficacy gain for a move from exposure $x_L$ to $x_H$ as $r=p(x_H)-p(x_L)$, and define the safety cost as $s=q(x_H)-q(x_L)$. Positive $r$ indicates higher predicted response probability; positive $s$ indicates higher predicted adverse-event probability. These are absolute probability differences, so $r=0.02$ means a 2-percentage-point gain, not a 2% relative increase.

For the higher-dose contrast, $x_L=800$ corresponds to 400 mg and $x_H=1600$ corresponds to 800 mg. The fitted efficacy difference is approximately $r=0.0118$. The safety difference is approximately $s=0.6068$. These point estimates favor neither dose in isolation: one expresses possible benefit and the other possible harm. A decision requires clinical thresholds or an explicit valuation of benefit and risk, as well as uncertainty.

A useful check is to calculate the local sensitivity of a contrast to its model parameters. For $p(x)$, the derivative with respect to $\beta_1$ is $x p(x)[1-p(x)]$. Thus, for the contrast $p(400)-p(200)$,

\frac{\partial r}{\partial\beta_1} =400p(400)[1-p(400)]-200p(200)[1-p(200)] \approx67.19\text{ ng/mL}.

This is the sensitivity for exposure 400 versus 200 ng/mL, which corresponds under the stated mapping to 200 mg versus 100 mg—not 400 mg versus 100 mg. Using the fitted probabilities $p(200)\approx0.17155$ and $p(400)\approx0.60475$, the corresponding efficacy contrast is about $0.4332$. The derivative describes how that contrast changes locally as the fitted slope changes; it is not itself an efficacy gain.

Worked Example: Propagate Uncertainty for the High-Dose Efficacy Gain

Suppose the fitted covariance matrix for $(\hat{\beta}_0,\hat{\beta}_1)$, in the parameter units specified above, is approximately diagonal with variances $0.25$ and $1.00\times10^{-6}$. For $r=p(1600)-p(800)$, the gradient is

\nabla r= \begin{pmatrix} p(1600)[1-p(1600)]-p(800)[1-p(800)]\\ 1600p(1600)[1-p(1600)]-800p(800)[1-p(800)] \end{pmatrix} \approx \begin{pmatrix} -0.0117\\ -9.345 \end{pmatrix}.

The delta method approximates the variance of a smooth fitted contrast by its gradient multiplied by the parameter covariance matrix and the transposed gradient. Here this gives

\widehat{\operatorname{Var}}(r) \approx (-0.0117)^2(0.25)+(-9.345)^2(1.00\times10^{-6}) \approx0.0001215, \qquad SE(r)\approx\sqrt{0.0001215}=0.0110.

Using a normal approximation, a 95% interval is $0.0118\pm1.96(0.0110)$, or approximately $(-0.0098,\,0.0334)$. In context, the data and this fitted model are compatible with a small decrease as well as an efficacy gain of about 3.34 percentage points for 800 mg versus 400 mg. This interval is not a 95% probability statement about the fixed true contrast. It reflects the repeated-sampling uncertainty represented by the model and the delta-method approximation.

The interval’s inclusion of zero does not prove equivalence. To claim equivalence, a clinically justified margin and an analysis designed to evaluate that margin would be needed. The normal approximation may also be poor for contrasts near a probability boundary; simulation or a more suitable interval can be considered if the approximation is doubtful.

From Model Output to a Dose Decision

A practical decision rule should be specified in clinical units before reviewing the final contrast. For illustration only, suppose the team would consider a higher dose when its expected efficacy gain is at least 3 percentage points and its additional adverse-event risk is no more than 5 percentage points. Under that rule, 800 mg versus 400 mg does not qualify: the fitted efficacy gain is about 1.18 percentage points, while the fitted safety increase is about 60.68 percentage points. The uncertainty interval for efficacy also includes zero.

The rule is a decision aid, not a universal standard. The 3- and 5-percentage-point thresholds must be justified for the disease, endpoint, adverse event, and treatment context. Some adverse events are more consequential than others, and a risk difference alone does not represent severity, duration, reversibility, or patient preference. Where benefits and harms are valued differently, a transparent utility or benefit-risk framework may be more appropriate.

Define the contrast.
Name both doses and their corresponding exposures. State whether the target is a response difference, an adverse-event difference, or another estimand.

Check the fitted response.
Evaluate efficacy and safety at each exposure, and inspect whether the dose-exposure mapping and model predictions are supported by observed data.

Quantify uncertainty and apply clinical criteria.
Report intervals as well as point estimates, then compare them with prespecified clinical thresholds without treating a nonsignificant difference as proof of no effect.

Diagnostics, Alternatives, and Limitations

The apparent plateau is credible only if the model describes the data adequately where the decision is being made. Plot observed response proportions by exposure alongside fitted probabilities and uncertainty bands. Look for systematic departures, influential dose groups, or sparse data at the upper end. Check whether the fitted slope has the expected direction and whether estimates are stable when plausible model forms or influential observations are examined. A model that predicts almost certain response at high exposure should be scrutinized especially closely if the observed response rate is materially lower.

The logistic curve has an upper asymptote of 1 and a particular shape on the log-odds scale. A flexible alternative, such as a four-parameter Emax curve, can estimate a maximum response below 1 and a concentration associated with half of that maximum. A categorical-dose analysis avoids imposing a smooth curve but may be less precise between tested doses and does not naturally interpolate. Model choice should follow the scientific question and observed support, not which curve makes a preferred dose look best.

All these comparisons have limits. A dose-level exposure summary can hide individual variability, and the assumed mapping may be inaccurate. Unmeasured differences between groups, missing outcomes, adherence, or a changing exposure over time can undermine interpretation. The models do not prove causality merely because exposure and response are associated. Nor does a precise fit within the studied range guarantee reliable extrapolation beyond it. The earlier material on model-based dose selection provides the broader decision context; here the essential case-specific safeguard is to connect the plateau claim to both its uncertainty and its safety cost.

Common Pitfall

Key takeaway: Do not equate “the curve looks flat” with “higher doses have no benefit” or “the doses are equivalent.” State the dose and exposure contrast, quantify the efficacy and safety differences, examine uncertainty, and use a clinically justified decision rule. Always translate exposure contrasts back to dose correctly before describing the result.

A common error is to read a small high-dose point estimate as proof that the true efficacy gain is negligible, while ignoring that its interval may include larger gains or losses. The correction is to report the estimate and interval together, distinguish lack of evidence from evidence of equivalence, and weigh the safety contrast on its own clinical scale. Another frequent error is to label an exposure contrast as the wrong dose contrast; under the case mapping, 200 versus 400 ng/mL means 100 versus 200 mg.

Check Your Understanding

Answer each question using the models and assumptions in this case.

Under the stated mapping, which doses correspond to exposures of 200 and 400 ng/mL?
What does an efficacy contrast of $0.0118$ mean in percentage-point terms?
Does a 95% interval for the efficacy difference that includes zero establish equivalence? Why or why not?
Name one diagnostic that could reveal a poor fit near the apparent plateau.

Solutions. (1) They correspond to 100 and 200 mg, respectively, because $x=2d$. (2) It is a predicted increase of 1.18 percentage points in response probability for the specified higher-versus-lower contrast. (3) No. Inclusion of zero means the interval is compatible with no difference under the model, but equivalence requires a prespecified clinically acceptable margin and an appropriate analysis. (4) One option is to compare observed response proportions at the highest exposures with fitted probabilities and uncertainty bands; systematic disagreement would call the fitted curve into question.