Choosing Concentrations to Estimate Emax Parameters

Why Concentration Placement Matters

A concentration-effect study can include many observations and still provide weak estimates of the parameters that describe the curve. The issue is not only how many observations are collected, but where the concentrations fall relative to \(EC_{50}\). Observations in different regions of the curve contribute different information: low concentrations help describe the rising response, concentrations near \(EC_{50}\) are sensitive to changes in \(EC_{50}\), and high concentrations help reveal the plateau associated with \(E_{\max}\).

The design question here is how to choose concentration levels to estimate \(E_{\max}\) and \(EC_{50}\) in the standard direct-effect Emax model. The calculations use a local approximation: they assess the information a proposed design would provide if plausible parameter values were close to the values used in the calculation. They are not a guarantee that the eventual estimates will be precise if the model or those values are wrong.

Model and assumptions: Let the expected response at concentration \(C\) be \(\mu(C)=E_{\max}C/(EC_{50}+C)\). For the design calculations below, the baseline is known to be zero, the Hill coefficient is fixed at 1, and the residuals are independent, have mean zero, and have common variance \(\sigma^2\). Concentrations are treated as fixed design values. The information calculations describe local precision near specified values of \(E_{\max}\) and \(EC_{50}\); they do not account for uncertainty in the baseline, Hill coefficient, or concentration measurements.

How Each Concentration Informs the Parameters

A useful way to compare concentrations is to ask how much the predicted mean response changes when a parameter changes slightly. Write \(\eta=\ln(EC_{50})\), so that changes in \(\eta\) represent proportional changes in \(EC_{50}\). At a design concentration \(C_i\), the two local sensitivities are

a_i=\frac{\partial \mu(C_i)}{\partial E_{\max}} =\frac{C_i}{EC_{50}+C_i}, \qquad b_i=\frac{\partial \mu(C_i)}{\partial \eta} =-E_{\max}\frac{C_iEC_{50}}{(EC_{50}+C_i)^2}.

The sensitivity \(a_i\) is the fraction of the plateau response expressed at that concentration. At low concentrations it is small; at concentrations well above \(EC_{50}\) it approaches 1. The sensitivity \(b_i\) is largest in magnitude around \(EC_{50}\), and becomes small at both very low and very high concentrations. Thus, a plateau observation can be especially useful for estimating \(E_{\max}\), while an observation near the bend of the curve is more responsive to \(EC_{50}\).

For independent residuals with common variance \(\sigma^2\), the local information matrix is the sum of the sensitivity contributions from each observation. If \(\theta=(E_{\max},\eta)\), then

I(\theta)=\frac{1}{\sigma^2} \sum_i \begin{pmatrix} a_i^2 & a_ib_i\\ a_ib_i & b_i^2 \end{pmatrix}.

When the matrix is nonsingular, the approximate covariance matrix of the parameter estimates is \(I(\theta)^{-1}\). This approximation makes two design features visible: repeated observations add information, while observations with different relative sensitivities help distinguish the parameters. The off-diagonal terms represent their local dependence.

Design takeaway: Aim to include concentrations in more than one part of the curve. Low concentrations, concentrations near the plausible \(EC_{50}\), and concentrations above it have complementary roles. A single region may be informative about a response feature while leaving the parameters that control other features difficult to separate.

Worked Examples of Concentration Placement

Worked Example: Placing Levels Below, Near, and Above EC50

Suppose a hypothetical planning model has \(E_{\max}=100\) response units and \(EC_{50}=10\) mg/L. Calculate the expected responses at \(0.5\), \(10\), and \(100\) mg/L, and identify the role of each level.

Use \(\mu(C)=100C/(10+C)\). At \(C=0.5\) mg/L, \(\mu(0.5)=100(0.5)/(10+0.5)=50/10.5=4.762\) response units. This is a low-concentration observation, where the response is still far below the plateau.

At \(C=10\) mg/L, \(\mu(10)=100(10)/(10+10)=100/2=50.00\) response units. This is the \(EC_{50}\) concentration: the expected response is half of \(E_{\max}\).

At \(C=100\) mg/L, \(\mu(100)=100(100)/(10+100)=10000/110=90.91\) response units. This is above \(EC_{50}\) and helps show how close the response is to its plateau. It does not reach the plateau exactly: under this model, a finite concentration gives a response below \(E_{\max}\).

The three levels cover different curve regions. If the observations instead clustered close to 10 mg/L, they might describe the bend well but provide less direct evidence about the plateau. These expected responses are planning values under the stated model, not observed data.

Comparing a Narrow Range with a Broader Design

The next example compares local precision for two hypothetical designs. Both use the same planning values, \(E_{\max}=100\) response units and \(EC_{50}=10\) mg/L, and assume \(\sigma=5\) response units. The baseline is fixed at zero and the Hill coefficient at 1, as specified above. The approximate standard errors are conditional on these assumptions and parameter values.

Worked Example: Narrow Levels Versus Broad Coverage

Design A has four observations at 8 mg/L and four at 12 mg/L. Design B has four observations each at 1, 10, and 100 mg/L. For each design, calculate the sensitivity sums and compare the approximate standard errors.

Design A. At 8 mg/L, \(a=8/18=0.4444\) and \(b=-100(8)(10)/18^2=-24.6914\). At 12 mg/L, \(a=12/22=0.5455\) and \(b=-100(12)(10)/22^2=-24.7934\). Summing the four replicates at each level gives

A=\sum a_i^2\approx1.9802,\qquad B=\sum a_ib_i\approx-97.99,\qquad D=\sum b_i^2\approx4897.50.

The determinant of the unscaled sensitivity cross-product matrix is \(AD-B^2\approx95.94\). For parameters \((E_{\max},\eta)\), the approximate covariance matrix is \(\sigma^2\begin{pmatrix}A&B\\B&D\end{pmatrix}^{-1}\). Therefore,

SE(E_{\max})\approx \sqrt{\frac{25D}{AD-B^2}} =\sqrt{\frac{25(4897.50)}{95.94}} \approx35.72\ \text{response units}, \qquad SE(\eta)\approx \sqrt{\frac{25A}{AD-B^2}} =\sqrt{\frac{25(1.9802)}{95.94}} \approx0.7183.

A delta-method approximation for \(EC_{50}=\exp(\eta)\) multiplies the standard error of \(\eta\) by the planning value of \(EC_{50}\). Thus \(SE(EC_{50})\approx10(0.7183)=7.183\) mg/L. The large approximate errors reflect the difficulty of estimating both parameters from levels clustered in a narrow range.

Design B. At 1 mg/L, \(a=1/11=0.09091\) and \(b=-1000/121=-8.2645\). At 10 mg/L, \(a=0.5\) and \(b=-25.00\). At 100 mg/L, \(a=10/11=0.9091\) and \(b=-1000/121=-8.2645\). With four observations at each level, the sensitivity sums are approximately \(A=4.3388\), \(B=-83.06\), and \(D=3046.41\), with \(AD-B^2\approx6319.3\). Hence

SE(E_{\max})\approx \sqrt{\frac{25(3046.41)}{6319.3}} =3.472\ \text{response units}, \qquad SE(\eta)\approx \sqrt{\frac{25(4.3388)}{6319.3}} =0.1310.

The corresponding delta-method approximation is \(SE(EC_{50})\approx10(0.1310)=1.310\) mg/L. Under this local calculation, the broader design provides substantially smaller approximate errors for both parameters. It includes a level at the planning \(EC_{50}\) and levels on either side, including one that approaches the plateau. The comparison is specific to the stated model, error assumptions, and planning values; it is not a universal ranking of every possible design.

Why Replicating at the Plateau Is Not Enough

Adding high concentrations can help estimate the plateau, but observations only at high concentrations do not generally identify both \(E_{\max}\) and \(EC_{50}\). At a single concentration, each observation has one sensitivity vector \((a_i,b_i)\). Replicating that same concentration increases information in that direction, but does not create a second, differently oriented sensitivity vector.

Worked Example: A Plateau-Only Design

Suppose a hypothetical study collects 12 observations at 100 mg/L, with the same planning values \(E_{\max}=100\) response units and \(EC_{50}=10\) mg/L. At this concentration, \(a=100/110=0.9091\) and \(b=-100(100)(10)/110^2=-8.2645\). The unscaled cross-product matrix is

12 \begin{pmatrix} 0.9091^2 & (0.9091)(-8.2645)\\ (0.9091)(-8.2645) & (-8.2645)^2 \end{pmatrix}.

Its determinant is zero because its rows and columns come from repeated copies of the same sensitivity direction. Equivalently, for any number of observations all at this one concentration, \(AD-B^2=0\). The matrix cannot be inverted to produce separate local standard errors for \(E_{\max}\) and \(\eta\). These data may strongly constrain the response at 100 mg/L, but that response alone does not uniquely determine both curve parameters.

A Short Justification: Why Different Regions Help

The determinant of the unscaled information matrix measures whether the sensitivity directions vary across observations. For two parameters, write each observation's sensitivity vector as \((a_i,b_i)\). Then

\left(\sum_i a_i^2\right)\left(\sum_i b_i^2\right) -\left(\sum_i a_ib_i\right)^2 = \sum_{i<j}(a_ib_j-a_jb_i)^2.

To verify the identity, expand the first product into terms indexed by \(i,j\), and expand the square into the corresponding products. Terms with \(i=j\) cancel. For each pair \(i<j\), the remaining terms are \(a_i^2b_j^2+a_j^2b_i^2-2a_ib_ia_jb_j\), which equal \((a_ib_j-a_jb_i)^2\). The result is nonnegative and is zero when all sensitivity vectors point in the same direction. Thus, repeated observations in one region can increase the amount of information along that direction, but varied sensitivities are needed for a nonsingular two-parameter problem.

Practical Checks and Limitations

A useful design review checks both the range of concentrations and the assumptions behind the precision calculation. The following checks do not replace a study-specific design analysis, but they expose common weaknesses early.

Check coverage relative to plausible \(EC_{50}\). If all concentrations lie below, near, or above the plausible \(EC_{50}\), consider whether the parameters that rely on the missing region are still well supported.
Check whether the high levels actually approach a plateau. A nominally high concentration is not necessarily high relative to \(EC_{50}\). If plausible \(EC_{50}\) values are uncertain, assess coverage across that range rather than relying on a single assumed value.
Check the sensitivity matrix, not only the number of observations. A singular or nearly singular matrix signals that the design may not distinguish the parameters well. A small determinant can indicate weak separation, although its numerical size depends on parameter scaling.
Check model and error assumptions. Unequal residual variance, correlated observations, an uncertain baseline, or model misspecification changes the information calculation. Precision from a fixed-baseline, constant-variance Emax model should not be presented as if it automatically applies to those settings.

The calculations here are local and use the standard Emax model with the Hill coefficient fixed at 1. They do not establish that this curve is appropriate for a particular pharmacologic question, and a design optimized for one pair of parameter values may perform less well if the true values differ substantially. In practice, evaluate candidate designs at several plausible values and consider the scientific and ethical limits on achievable concentrations.

A common mistake is to argue, “The response is almost maximal at my highest concentration, so repeating that concentration will estimate both \(E_{\max}\) and \(EC_{50}\) precisely.” A near-plateau observation can inform \(E_{\max}\), but its sensitivity to \(EC_{50}\) is relatively small; repeating it does not supply the distinct sensitivity direction needed to separate the two parameters. The correction is to include concentrations in complementary parts of the curve and evaluate the resulting parameter information under explicit assumptions.

Key takeaway: Concentration placement is part of parameter estimation, not just a way to sample the response range. For the Emax model, seek complementary information from the rising region and the plateau region; do not infer that a large sample at one concentration necessarily identifies both \(E_{\max}\) and \(EC_{50}\).

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Under the stated Emax model, what fraction of \(E_{\max}\) is the expected response at \(C=EC_{50}\)?
For the sensitivity \(a=C/(EC_{50}+C)\), what value does \(a\) approach as \(C\) becomes very large?
Why can 20 observations at one concentration fail to identify both Emax parameters even though the sample size is substantial?
If a local calculation gives \(SE(\eta)=0.15\) and the planning value is \(EC_{50}=8\) mg/L, what is the approximate delta-method standard error for \(EC_{50}\)?

Solutions. (1) At \(C=EC_{50}\), \(\mu=E_{\max}EC_{50}/(EC_{50}+EC_{50})=E_{\max}/2\), so the response is half the plateau. (2) As \(C\) grows, \(C/(EC_{50}+C)\) approaches 1. (3) Replication at one concentration adds information in the same sensitivity direction; the two-parameter information matrix remains singular, so the parameters cannot be separated by that concentration alone. (4) The delta-method standard error is approximately \(EC_{50}SE(\eta)=8(0.15)=1.20\) mg/L.