Identifiability of Emax and EC50

Why Concentration Coverage Matters

An Emax curve can pass close to every observed concentration-effect point and still leave its parameters uncertain. The key issue is whether the data show enough of the curve’s shape to distinguish its maximum effect from the concentration scale at which that maximum is approached. If observations cover only a narrow region, different combinations of Emax and EC50 may predict nearly the same effects there.

This is a question of identifiability: whether the observed data can determine a parameter uniquely or with useful precision under a specified model. It is important to separate structural identifiability, which concerns ideal noiseless data, from practical identifiability, which concerns finite, noisy observations. A parameter can be structurally identifiable but practically very uncertain.

How the Emax Curve Encodes Its Parameters

Let \(C\) denote concentration and let \(E_0\) be a known baseline effect. For a stimulatory response, write the baseline-relative effect as \(z(C)=E(C)-E_0\). The model is

Here, \(E_{\max}\) is the maximum increase above baseline, and \(EC_{50}\) is the concentration at which that increase is half of \(E_{\max}\). At concentrations far below \(EC_{50}\), the curve is approximately linear:

Thus, low-concentration observations mainly reveal the ratio \(E_{\max}/EC_{50}\), the initial slope, rather than revealing each parameter well on its own. At concentrations far above \(EC_{50}\), the response approaches \(E_{\max}\), but the small remaining curvature that carries information about \(EC_{50}\) diminishes. Observations around the transition, especially around \(EC_{50}\), can help distinguish the parameters because the curve’s bend is visible.

A Profile View of the Problem

A simple way to examine the trade-off is to fix a candidate \(EC_{50}=K\), then find the value of \(E_{\max}=A\) that best fits the responses at that \(K\). Suppose the baseline is known, the responses are \(z_i\) at concentrations \(C_i\), and fitting uses unweighted least squares. Define \(f_i(K)=C_i/(K+C_i)\). The residual sum of squares at fixed \(K\) is

Differentiating with respect to \(A\) gives the unconstrained least-squares stationary point. Provided \(\sum_i f_i(K)^2>0\), it is

Because this model requires \(A>0\), that expression is an allowed minimizing value only when its numerator is positive. If the numerator is nonpositive, there is no attained minimum on the strict domain \(A>0\): the infimum is approached as \(A\) decreases toward zero. If the model instead allowed \(A\geq 0\), the constrained minimizer would be \(\max\{0,\sum_i f_i(K)z_i/\sum_i f_i(K)^2\}\). This boundary detail matters when responses are inconsistent with a positive stimulatory effect.

The profile RSS at \(K\) is the infimum of RSS over the allowed \(A>0\), and is a minimum when the numerator in \(\widehat{A}(K)\) is positive. Repeating this over candidate \(K\) values shows which EC50 values remain compatible with a close fit. A broad, shallow profile indicates that quite different EC50 values have similar fit; a sharply defined minimum indicates stronger information about EC50 under the assumed model and error criterion. This diagnostic does not fix a wrong model or establish that the baseline and error assumptions are correct.

Reading the Concentration Range

The local sensitivities make the range-dependent information explicit:

At low \(C\), the two sensitivities are approximately \(C/EC_{50}\) and \(-E_{\max}C/EC_{50}^2\). Across low concentrations, the changes caused by altering either parameter are nearly proportional. This makes it hard to tell which parameter produced a small change in response. At high \(C\), sensitivity to Emax approaches 1, while sensitivity to EC50 approaches 0. Consequently, a high-concentration plateau can anchor Emax but may leave EC50 weakly determined. Around the transition, both parameters can affect predictions appreciably, and their effects are less alike.

These are tendencies, not guarantees. Precision also depends on the number of observations, their variability, whether the baseline is known, and whether the model and residual-error assumptions are reasonable. Concentration coverage should therefore be judged alongside uncertainty and the pattern of the observed responses.

Diagnostics, Alternatives, and Limits

• Inspect the profile. If a wide range of EC50 values gives similar profiled RSS, report that EC50 is weakly identified rather than presenting a precise estimate without qualification.
• Compare parameter pairs. Evaluate predicted effects across the observed concentrations for plausible alternative Emax–EC50 combinations. Similar predictions reveal a trade-off that a single best-fit pair can hide.
• Check the data’s shape. Low-only data with no visible curvature mainly support an initial slope. High-only data may show a plateau but little transition. Sparse measurements or substantial response variability can make either pattern less informative.
• Check model assumptions. The examples assume a known baseline, positive stimulatory Emax, and unweighted least-squares fitting. If the baseline is estimated, the error variance changes across observations, or the response is inhibitory, the calculations and interpretation must match that model instead.

A parameter profile is an alternative to relying only on the optimizer’s reported point estimate: it exposes whether fit worsens as a parameter moves away from that estimate. It is not, by itself, a confidence interval or a formal test. Formal uncertainty calculations require an appropriate statistical error model and assumptions; a later treatment of profile likelihood addresses that inferential extension. Likewise, this tutorial explains how coverage affects identifiability, not how to select an optimal concentration design.