Worked Example: Selecting a Dose With an Emax Model

From an Emax Fit to a Dose Decision

A fitted dose-response curve is not itself a dose recommendation. A curve describes the estimated average response at each dose, but the fitted parameters are uncertain, and individual outcomes may vary around that average. When a decision depends on achieving a clinical target, the relevant question is not just whether the estimated mean response exceeds the target. It is also how uncertain that mean is, and how likely an outcome at the chosen dose is to attain the target.

This worked example uses a saturating Emax model to connect those steps. The response is a baseline-adjusted efficacy score, in points, and dose is measured in mg. The numerical data and fit results below are hypothetical. The analysis focuses on efficacy attainment; it does not include a toxicity constraint or a utility penalty for higher doses.

The baseline-adjusted form sets the response at zero dose to zero. It is appropriate only when the response has been defined relative to baseline in a way that makes this constraint reasonable. If the outcome has a nonzero expected value at zero dose, a baseline parameter \(E_0\) may be needed, giving \(\mu(d)=E_0+E_{\max}d/(ED_{50}+d)\). The examples here use the simpler form.

The model assumes that the average response rises with dose and approaches a plateau, with no downturn at high doses. Its parameters are positive, and the response observations are assumed to be adequately described by this curve plus random error. These are substantive assumptions, not automatic consequences of a good numerical fit. In particular, this model does not describe individual covariate effects, time-varying responses, or toxicity.

Fitting and Checking the Model

For observations \(y_i\) at doses \(d_i\), a common fitting method is nonlinear least squares: choose \(E_{\max}\) and \(ED_{50}\) to minimize the sum of squared residuals between each observation and \(\mu(d_i)\). A likelihood-based fit can instead specify an error distribution, such as independent normal errors with a common variance. In either case, the fitting procedure should respect positive parameter constraints and should account for the actual study design and error structure.

The table gives hypothetical response means from a dose-ranging study. Suppose a nonlinear least-squares fit to the underlying study data, summarized here by these dose-group means, gives rounded estimates \(E_{\max}=100\) points and \(ED_{50}=50\) mg. Suppose a bootstrap calculation from the study data gives the following approximate covariance matrix for the two fitted parameters, in the order \(E_{\max}, ED_{50}\). The covariance matrix is part of the hypothetical fit output; it describes joint uncertainty, not the variability of individual responses.

The diagonal entries correspond to approximate parameter variances, so the associated standard errors are 8 points for \(E_{\max}\) and 10 mg for \(ED_{50}\). The off-diagonal entry is positive: in the fitted uncertainty distribution, larger values of one parameter tend to occur with larger values of the other. Keeping that dependence matters when calculating uncertainty in a response, because both parameters determine the curve together.

Useful checks include plotting observations and fitted means together, examining residuals by dose, checking whether residual spread changes with dose, and checking whether the fitted curve is driven by only one or two dose groups. Systematic residual patterns, an apparent decline at high doses, or strong sensitivity to starting values can indicate that the model form or fitting procedure is unsuitable. A plausible-looking curve is not proof that the Emax assumptions hold.

Propagating Parameter Uncertainty

The fitted mean response is a function of two uncertain parameters. A first-order approximation, called the delta method, uses the local slope of that function to translate parameter uncertainty into response uncertainty. Write the parameter vector as \(\theta=(E_{\max},ED_{50})^\mathsf{T}\), and let \(\widehat{\Sigma}\) be its estimated covariance matrix. At a fixed dose \(d\), the gradient is

The first component describes how much the response changes with \(E_{\max}\); the second describes its change with \(ED_{50}\). The approximate variance of the fitted mean response is the gradient multiplied by the parameter covariance matrix and by the transposed gradient.

This follows by a first-order Taylor expansion around the fitted parameter estimates: a small parameter deviation \(\Delta\theta\) changes the fitted response by approximately \(\nabla_\theta\mu(d)^\mathsf{T}\Delta\theta\). The variance of that linear combination is the gradient-covariance-gradient expression above. The approximation is most useful when parameter uncertainty is not so large that substantial curvature or parameter-boundary effects dominate.

Mean-Response Uncertainty and Target Attainment

A dose-attainment probability requires an outcome distribution, not just an interval for the model mean. For this example, assume that a response at dose \(d\) is the model mean plus an independent normal residual with standard deviation 5 points. Also use the approximate normal distribution for the uncertain fitted mean obtained from the delta method. Under these assumptions, the approximate variance of a future response combines parameter uncertainty and residual variance:

This calculation does not include between-person heterogeneity beyond the residual term specified here. If the decision concerns a particular population, covariate mix, or individual variability structure, that distribution must be represented in the model rather than silently omitted. A predictive probability is conditional on its model and uncertainty assumptions.

For comparison, the approximate 90% interval for the mean response at 50 mg is \(50\pm1.645\sqrt{29}\), or 41.09 to 58.91 points; at 200 mg it is \(80\pm1.645\sqrt{38.91}\), or 69.74 to 90.26 points. These are intervals for the dose-specific model mean. The target-attainment calculations instead use predictive standard deviations that also include residual variation. For instance, at 100 mg the approximate 90% predictive interval is \(66.67\pm1.645(7.679)\), or 54.04 to 79.30 points. Confusing these intervals changes the quantity being described.

Why the Joint Uncertainty and Model Checks Matter

The covariance term can materially affect the propagated variance. At 100 mg, its contribution is negative because the gradient with respect to \(ED_{50}\) is negative while the parameter covariance is positive. Omitting that term would give \(28.44+19.75=48.20\) points squared instead of 33.98. Treating the parameters as independent when they are not can therefore misstate uncertainty and, in turn, the estimated probability of meeting the target.

The normal approximation also has limits. The Emax curve is nonlinear, the parameters are constrained to be positive, and a parameter distribution with substantial skewness may not be well summarized by a covariance matrix. The delta method may produce poor intervals when uncertainty is large, the dose is far outside the range supported by observations, or the fitted parameters are near a boundary. In those cases, propagate joint bootstrap or posterior parameter draws through the Emax equation and calculate response and attainment summaries from the resulting draws. If the response residual is non-normal or heteroscedastic, simulate from an appropriate residual model as well.

A common mistake is to select a dose because its estimated mean exceeds the target, then describe that as a high probability of individual target attainment. The correction is to state the event and the distribution being used. A mean above 60 points does not imply that most outcomes exceed 60: at 100 mg the fitted mean is 66.67 points, while the estimated predictive attainment probability is about 0.807 under the assumptions above. Conversely, an uncertainty interval for the mean is not a predictive interval for outcomes.

A further limitation is that the candidate-dose rule optimizes only the efficacy criterion stated here. A clinically appropriate decision may also need to consider a safety limit, dose burden, or consequences of missing the target. Those considerations require explicit decision criteria; they cannot be inferred from the Emax fit alone. The fit provides a model-based summary, while the decision rule states how that summary is used.