Exposure-Response Models for Adaptive Decisions

From Exposure to an Interim Decision

An adaptive trial may measure drug exposure before all clinical outcomes are available. A useful exposure-response model connects that measured exposure to a clinical outcome and expresses uncertainty in the connection. At an interim analysis, the model can help assess whether an exposure range is promising enough to continue studying, while also checking whether toxicity appears unacceptably high.

Exposure-response modeling is not the same as comparing randomized dose groups. Participants assigned the same dose can have different exposures, and exposure may be associated with characteristics that also relate to outcome. A fitted exposure-response curve is therefore a model for the outcome conditional on exposure and any included covariates; it is not automatically a causal effect of increasing an individual's exposure. Conclusions depend on the design, the measured data, the model, and the range of exposure actually observed.

Nonlinear Efficacy Curves

A common model for a continuous efficacy endpoint is the Emax curve. Let \(C\) denote exposure in a stated unit, and let \(Y\) be an efficacy outcome measured so that larger values indicate greater benefit. The mean response at exposure \(C\) is

Here, \(E_0\) is the mean response at zero exposure, \(E_{\max}\) is the additional response approached as exposure becomes large, and \(EC_{50}\) is the exposure at which half of that additional response is reached. If \(Y\) is measured in points and \(C\) in mg/L, then \(E_0\) and \(E_{\max}\) are in points and \(EC_{50}\) is in mg/L. The model presumes nonnegative exposure, a positive \(EC_{50}\), and a saturating response of this specified form.

The Hill version permits the curve's steepness to vary:

The Hill coefficient \(h\) changes how sharply response rises around \(EC_{50}\); the other parameters retain their interpretations. These curves are useful when a straight line is implausible, for example when response increases at low exposure but levels off at higher exposure. They should not be chosen merely because a nonlinear curve can fit a small interim data set more closely.

Modeling Toxicity Probabilities

For a binary toxicity endpoint, a standard choice is logistic regression. Let \(T=1\) indicate that a participant experiences the defined toxicity during the chosen observation window, and let \(p(C)=P(T=1\mid C)\). A logit model with log exposure is

This specification requires positive exposure. The intercept \(\beta_0\) is the log odds at \(C=1\) in the chosen exposure unit; \(\beta_1\) is the change in log odds for a one-unit increase in \(\log(C)\). The probability is obtained by the inverse-logit transformation \(p(C)=1/\{1+\exp[-(\beta_0+\beta_1\log C)]\}\). A positive \(\beta_1\) corresponds to increasing modeled toxicity probability with exposure. Changing the exposure unit changes the intercept, so the unit must be stated and used consistently.

Both model types require assumptions. Outcomes should be defined consistently, and observations must be appropriately independent conditional on the model and covariates, or their dependence must be modeled. The chosen mean curve or probability link must adequately describe the relationship over the decision-relevant range. Measurements need to be sufficiently reliable, and the observed exposure range must support the predictions being used. For a continuous outcome, uncertainty calculations also depend on the assumed or estimated residual variability. For binary outcomes, the toxicity window and event definition must be fixed; a participant whose window is incomplete cannot simply be treated as event-free.

Uncertainty and Interim Gates

An interim decision should be tied to a prespecified clinical criterion, not just to whether a fitted curve looks favorable. For example, a protocol could require a lower confidence limit for mean efficacy to exceed a clinically relevant threshold and an upper confidence limit for toxicity probability to remain below a safety threshold. The limits make uncertainty visible: a favorable point estimate alone may be too imprecise to justify continuing with an exposure range.

For nonlinear predictions, the delta method provides a first-order approximation to their standard errors. Let \(\hat{\theta}\) be a vector of fitted model parameters, with estimated covariance matrix \(\widehat{\Sigma}\), and let \(g(\hat{\theta})\) be the model prediction at a fixed exposure. A first-order Taylor expansion around the true parameter vector gives the approximate variance

The gradient contains the partial derivatives of the prediction with respect to each fitted parameter. The covariance terms matter: adding parameter variances while ignoring covariance can materially misstate prediction uncertainty. For a probability modeled on the logit scale, one can first form a confidence interval for the linear predictor and then transform both endpoints; for a continuous mean, the gradient is taken for that mean prediction. These are approximate procedures, especially when samples are small, estimates are near parameter boundaries, or the model is weakly identified.

Justifying the Uncertainty Calculation

The delta-method expression follows from approximating a smooth prediction function by its linear change near the fitted parameters. Write the estimation error as \(\hat{\theta}-\theta\). A first-order Taylor expansion gives \(g(\hat{\theta})\approx g(\theta)+\nabla g(\theta)^{\mathsf T}(\hat{\theta}-\theta)\). If \(\hat{\theta}\) is approximately unbiased with covariance matrix \(\Sigma\), the constant \(g(\theta)\) contributes no variance, and the variance of the linear term is

Replacing the unknown gradient and covariance matrix with their fitted estimates yields the stated approximation. The approximation is most credible when the estimator is approximately normal, the prediction function is smooth near the estimate, and the data identify the parameters. It can be unreliable if a parameter estimate is on a boundary, the curve's plateau is not observed, or the estimate of \(EC_{50}\) is extremely uncertain.

Diagnostics, Alternatives, and Limits

Before using a curve for adaptation, examine whether the data support its shape and predictions. Plot individual exposure and outcome information alongside the fitted curve, inspect residuals for systematic patterns, and check whether predicted toxicity probabilities are compatible with observed event frequencies across exposure ranges. Look for sparse exposure regions, influential observations, convergence warnings, implausible parameter estimates, and wide or asymmetric uncertainty. A smooth fitted line does not repair a lack of data at the exposure where the decision will be made.

The Emax model is compact and interpretable when a saturating response is plausible. A Hill curve is more flexible but requires estimating an additional shape parameter and can be unstable with limited data. A linear model is simpler and may be a reasonable local approximation over a narrow observed range, but it can predict continuing benefit or harm beyond that range even when the true relationship levels off. Splines can represent more varied shapes, but their predictions depend on knot and boundary choices and may be difficult to interpret in sparse regions. For toxicity, the logistic model is common, while other links can be considered if justified and prespecified; changing the link after seeing interim outcomes can make the decision procedure difficult to interpret.

A key failure mode is extrapolation: using a model fitted at low exposures to declare a high, unobserved exposure safe or effective. Another is treating all observed exposure differences as if they arose from random assignment. Exposure can reflect adherence, metabolism, disease severity, or other participant characteristics, so conditional associations need not equal the effect of an intervention that changes exposure. At an adaptive interim, the available data and the choice to continue may also depend on earlier decisions. The analysis plan should specify the model, interim timing, endpoints, and decision rule in advance, and the operating characteristics of the full adaptive procedure should be assessed under plausible data-generating conditions.

The comparison is not simply “nonlinear is better than linear.” A model should be as flexible as the scientific question and data can support, with uncertainty and diagnostics reported at the exposures that matter. Model-based gates can structure an interim discussion, but they do not remove uncertainty, prove causal effects, or guarantee that later data will confirm the interim pattern.