Uncertainty in Emax Predictions and Derived Quantities

Why uncertainty in a curve is not just uncertainty in its parameters

An Emax fit estimates parameters, but many scientific questions concern quantities calculated from them: the expected response at a particular concentration, or the concentration needed to reach a specified response. These derived quantities inherit uncertainty from every estimated parameter. The effect of that uncertainty depends on the model’s local sensitivity to each parameter and on how parameter estimates covary.

This tutorial uses the stimulatory Emax model with a Hill coefficient \(h\). The same uncertainty calculations apply to an inhibitory model after writing its response function and differentiating it. The focus is on uncertainty in the estimated mean curve and derived quantities—not on residual diagnostics or model selection.

Response prediction and its gradient

Write the direct-effect model as

where \(C\) is concentration, \(E_0\) is baseline response, \(E_{\max}\) is the maximum increment above baseline, \(EC_{50}\) is the concentration producing half that increment, and \(h>0\) is the Hill coefficient. For a response prediction, define \(f(C)=C^h/(EC_{50}^h+C^h)\). If \(h\) is estimated, use the parameter vector \(\boldsymbol{\theta}=(E_0,E_{\max},EC_{50},h)^\mathsf{T}\); if it is fixed, omit it from the vector and covariance matrix.

The gradient describes how a small change in each parameter changes the predicted response. Its components are

The third component is the derivative with respect to \(EC_{50}\), and the fourth is the derivative with respect to \(h\). In particular, the Hill-coefficient derivative is zero at \(C=EC_{50}\), since the log ratio is then zero. This does not imply that the entire prediction is known precisely there: uncertainty in \(E_0\), \(E_{\max}\), and their covariance can still contribute.

How the delta approximation works

The method uses a first-order Taylor expansion around the true parameter vector. For a small estimation error \(\widehat{\boldsymbol{\theta}}-\boldsymbol{\theta}\),

If the parameter estimator is approximately unbiased with covariance matrix \(\boldsymbol{\Sigma}\), the variance of the linear term is

Replacing the unknown parameters and covariance with their estimates gives the delta-method variance estimate. The approximation is most credible when parameter estimates are reasonably precise, the derived function is smooth near the estimates, and the likelihood is not strongly asymmetric or truncated by a parameter boundary. A normal-based pointwise interval is the estimate plus or minus \(1.96\) standard errors for an approximate 95% interval. It describes uncertainty in the estimated mean response at that concentration; it is not a prediction interval for a future individual observation.

Concentrations producing a specified response

A derived target concentration is meaningful only if the target lies within the model’s attainable response range. For the increasing Emax model, a target \(R^*\) must satisfy \(E_0<R^*<E_0+E_{\max}\). Define \(q=R^*-E_0\), the target increment above baseline. Solving the model equation for concentration gives

This expression is smooth only away from the limiting responses. Its gradient, when \(h\) is estimated, is

The entries correspond to \(E_0,E_{\max},EC_{50},h\). When \(h\) is fixed, omit the final entry. Notice that the derivatives involving \(E_0\) and \(E_{\max}\) can become large near the response limits: a small parameter change can then produce a large change in the concentration needed to approach the target.

Uncertainty changes across the concentration range

A single parameter covariance matrix induces a different prediction variance at each concentration because the gradient changes along the curve. The resulting intervals are pointwise: each one approximates coverage at its specified concentration. A collection of 95% pointwise intervals is not automatically a 95% simultaneous confidence band for the whole curve.

Assumptions, alternatives, and failure modes

The delta method is a local approximation, not a guarantee that the derived estimate has a normal distribution. It relies on a usable covariance estimate for the fitted parameters and a sufficiently smooth derived function near the estimate. It can be unreliable when parameters are weakly identified, estimates are near constraints, the target approaches the model’s attainable limits, or parameter uncertainty is large enough that the first-order Taylor approximation is poor. For example, a symmetric normal interval for a positive concentration can extend below zero, or an interval for a response can extend beyond plausible values.

The covariance terms must be retained unless independence is genuinely justified. In general, for gradient components \(g_i\), the variance is \(\sum_i g_i^2\widehat{\mathrm{Var}}(\hat{\theta}_i)+2\sum_{i<j}g_i g_j\widehat{\mathrm{Cov}}(\hat{\theta}_i,\hat{\theta}_j)\). Omitting the second sum can either overstate or understate uncertainty. The first worked example shows why covariance cannot be discarded simply because each parameter has a reported standard error.

A common mistake is to report the uncertainty of \(EC_{50}\) as if it were the uncertainty of every predicted response, or to use only the uncertainty of \(E_{\max}\) for a target concentration. The correction is to define the quantity of interest, differentiate it with respect to every estimated parameter, and propagate the full covariance matrix. Another mistake is to interpret a mean-response interval as an interval for future observations. A future observation also varies because of residual error; that additional variability is not included here.

Alternatives are useful when the local approximation is questionable. A parametric bootstrap can simulate datasets under the fitted model, refit the model to each simulated dataset, and recalculate the derived quantity, thereby representing nonlinear and asymmetric uncertainty when the simulation model is adequate. Profile likelihood can assess uncertainty for a derived quantity by constraining that quantity and evaluating the likelihood; it can be informative under weak identification, though it requires an appropriate likelihood and careful computation. These approaches do not remove dependence on model assumptions. When estimates are near parameter boundaries or the target is near a response limit, report the instability clearly rather than treating a symmetric delta interval as definitive.