Confounding Between EC50 and the Hill Coefficient

Why EC50 and the Hill Coefficient Can Trade Off

In a sigmoid Emax model, EC50 locates the concentration at which half the maximum effect is reached, while the Hill coefficient, $\gamma$, controls how sharply the response changes around that concentration. When observed concentrations span a broad range, the data may contain information about both the location and steepness of the curve. When they cluster in a narrow range, several combinations of EC50 and $\gamma$ can produce similar predictions there. The practical consequence is weak identification: the data may fit well while leaving the separate parameter estimates uncertain.

The issue is not simply whether a fitting procedure returns numerical estimates. A numerical optimizer can return a particular EC50 and Hill coefficient even when many nearby combinations fit almost as well. A profile plot makes that lack of separation visible by showing how the fit changes as one parameter is fixed across a range and the other is re-optimized.

Definition: For a baseline-subtracted response expressed as a fraction of its maximum, the sigmoid Emax response is $$ R(C)=\frac{C^\gamma}{EC_{50}^\gamma+C^\gamma}, \qquad C>0,\ EC_{50}>0,\ \gamma>0. $$ Here $R(C)$ is dimensionless, $C$ and EC50 have the same concentration units, and $\gamma$ is dimensionless. EC50 is the concentration giving $R=0.5$; $\gamma$ controls the steepness of the response near that concentration.

This tutorial focuses on confounding between EC50 and $\gamma$. To isolate it, the response is treated as normalized, with the baseline and maximum effect already accounted for. In an application, measurement error and uncertainty in those other quantities can add further uncertainty, but they are not the focus here.

The Trade-Off on the Log-Concentration Scale

Taking the odds of the fractional response and then its logarithm transforms the model into a straight-line expression in log concentration:

\log\left(\frac{R(C)}{1-R(C)}\right) =\gamma\left(\log C-\log EC_{50}\right).

This identity shows the source of the trade-off. At one concentration $C_*$, a fixed response $R_*$ determines the product of $\gamma$ and the distance between $\log C_*$ and $\log EC_{50}$, not necessarily the two quantities separately. Rearranging gives

EC_{50} =C_*\left(\frac{1-R_*}{R_*}\right)^{1/\gamma}.

For a single response away from 0.5, every positive value of $\gamma$ can be paired with an EC50 from this equation to reproduce that response exactly. At $R_*=0.5$, the equation gives $EC_{50}=C_*$ for every $\gamma$: a response measured exactly at the midpoint locates EC50 but provides no information about steepness by itself.

With several concentrations, the parameters are not generally exactly interchangeable. But if the concentrations are close together, the observed responses mainly describe a small portion of the curve. Modest changes in location and steepness can leave predictions over that portion similar, especially when measurement noise is appreciable. The log-scale equation makes this geometry precise. If the logit response at a representative concentration $C_0$ is held fixed, then candidate pairs satisfy

\gamma_1\left(\log C_0-\log EC_{50,1}\right) = \gamma_2\left(\log C_0-\log EC_{50,2}\right).

This relation is a local description of a trade-off, not a claim that all concentrations will have identical predictions under both pairs. Departures from the representative concentration, particularly over a broad range, provide the information needed to distinguish curve shapes.

Worked Example: Many Parameter Pairs Fit One Concentration

Suppose a hypothetical normalized response is $R=0.60$ at $C=2$ mg/L. Find the EC50 values that give this response for $\gamma=1$, $2$, and $4$.

Using $EC_{50}=C[(1-R)/R]^{1/\gamma}$, the ratio is $(1-0.60)/0.60=0.40/0.60=2/3$.

For $\gamma=1$, $EC_{50}=2(2/3)=1.333$ mg/L. For $\gamma=2$, $EC_{50}=2\sqrt{2/3}=1.633$ mg/L. For $\gamma=4$, $EC_{50}=2(2/3)^{1/4}=1.807$ mg/L.

Each pair predicts a response of 0.60 at 2 mg/L, despite having different EC50 and steepness. One concentration cannot tell us which pair is correct. Additional concentration coverage is needed.

What a Profile Plot Shows

To profile $\gamma$, fix it at a sequence of candidate values. At each value, find the EC50 that gives the smallest residual sum of squares (SSE), keeping the observations and the assumed error model the same. Plot the resulting minimized SSE against the fixed $\gamma$. The corresponding EC50 estimates can be plotted as a second curve or shown alongside the SSE profile. If the response model includes other estimated parameters, those are re-optimized at each fixed $\gamma$ as well.

SSE_{\mathrm{profile}}(\gamma) =\inf_{EC_{50}>0}\sum_{i=1}^{n} \left[R_i-\frac{C_i^\gamma}{EC_{50}^\gamma+C_i^\gamma}\right]^2.

This SSE formulation is appropriate for an illustrative least-squares analysis in which residuals are treated as independent, mean-zero, constant-variance errors on the response scale. It is an objective-function diagnostic: lower SSE means closer fitted responses for these data under this criterion. It does not, by itself, establish that a parameter value is scientifically true. If an error model is specified through a likelihood instead, the analogous profile uses the likelihood objective.

A broad, shallow profile means that a range of fixed Hill coefficients can be paired with re-optimized EC50 values that fit nearly as well. This is a warning that the data do not precisely separate the parameters. A narrow, well-defined minimum indicates stronger information about $\gamma$ under the chosen model and error assumptions, though it does not guarantee that the model is otherwise appropriate.

Worked Example: Comparing Fits Over a Narrow Concentration Range

Consider hypothetical normalized responses $0.442,\ 0.500,\ 0.551$ measured at $C=0.9875,\ 1.0000,\ 1.0125$ mg/L. Compare two candidate curves with $EC_{50}=1$ mg/L, one with $\gamma=1.5$ and one with $\gamma=2$. These calculations are fixed-EC50 comparisons, not yet the minimized profile values.

For $EC_{50}=1$ mg/L, the predictions are $R(C)=C^\gamma/(1+C^\gamma)$. At the middle concentration, both curves predict 0.5000. At the two other concentrations, the predictions are approximately 0.4953 and 0.5047 for $\gamma=1.5$, and 0.4937 and 0.5062 for $\gamma=2$.

For $\gamma=1.5$, using the unrounded predictions, the SSE is approximately

(0.442-0.4953)^2+(0.500-0.5000)^2+(0.551-0.5047)^2 \approx 0.00499.

Using the displayed rounded predictions alone gives 0.00498458, or 0.00498 to three significant figures; unrounded model predictions give approximately 0.00498664 (0.004987 to six decimal places). The displayed-prediction calculation rounds to 0.00498 at three significant figures, while the unrounded-model calculation rounds to 0.00499.

For $\gamma=2$, the corresponding calculation is

(0.442-0.4937)^2+(0.500-0.5000)^2+(0.551-0.5062)^2 \approx 0.00468.

Substitution of the displayed rounded predictions gives 0.00467993, and unrounded predictions give approximately 0.004680. Both round to 0.00468. The $\gamma=2$ candidate has the lower SSE at this fixed EC50, but the difference is small. A profile analysis would re-optimize EC50 separately at each $\gamma$; these fixed-EC50 calculations are useful comparisons, not substitutes for that profiling step.

The concentrations lie very close to 1 mg/L, so the predicted responses of the two curves are also close. The observations depart more from the fitted predictions than the two candidate curves differ from each other. That pattern makes it important to examine the full profile rather than treating the slightly lower SSE for one fixed pair as decisive evidence about steepness.

Reading the Shape and Checking the Data

A practical profile analysis should report or display both the objective function and the paired parameter values. The SSE curve alone shows which fixed values of $\gamma$ can be accommodated; the paired EC50 curve shows how potency shifts as steepness changes. A long, gently curving valley in a plot of SSE against $\gamma$ and EC50 is another way to visualize the same issue. The valley’s direction makes the trade-off visible.

Choose a grid of Hill coefficients.
Use positive values spanning the scientifically plausible range. A grid that is too narrow can conceal the fact that the objective continues to change slowly beyond its endpoints.

Optimize EC50 at each grid value.
Keep the data and error criterion fixed. Record both the minimized SSE and the EC50 that achieved it, and check that the numerical search actually found a minimum.

Plot and inspect the profile.
Look for a distinct minimum or a broad, flat region. Inspect the associated EC50 values to see whether they change substantially while the fit barely changes.

Check the concentration coverage.
Compare the observed concentrations with the fitted midpoint and with the range over which the response changes. A narrow cluster around one part of the curve provides limited leverage on shape.

Diagnostics should include verifying that the search over EC50 covers a sufficiently wide range and that different starting values do not produce unexamined solutions. A profile that ends at the edge of the parameter grid without turning upward may mean the grid is too narrow or the parameter is only weakly bounded by these data. Also inspect the measured concentrations and response values for data-entry errors and check whether the assumed error scale is reasonable. A profile cannot rescue a misspecified model or unreliable measurements.

Worked Example: Why Broader Coverage Helps

For a hypothetical curve with $EC_{50}=1$ mg/L, compare predictions for $\gamma=1.5$ and $\gamma=2$. Near the midpoint, at $C=0.9875$ and $1.0125$ mg/L, the predictions are about 0.4953 and 0.5047 for $\gamma=1.5$, and 0.4937 and 0.5062 for $\gamma=2$. The differences between the curves are only about 0.0016 at either concentration.

Now consider concentrations farther from the midpoint. At $C=0.25$ mg/L, the $\gamma=1.5$ prediction is $0.25^{1.5}/(1+0.25^{1.5})=0.125/1.125=0.1111$. For $\gamma=2$, it is $0.25^2/(1+0.25^2)=0.0625/1.0625=0.05882$.

At $C=4$ mg/L, the $\gamma=1.5$ prediction is $8/(1+8)=0.8889$, while the $\gamma=2$ prediction is $16/(1+16)=0.9412$. These predictions differ much more than they did near 1 mg/L. This does not guarantee precise estimates in real data—measurement error and the rest of the design still matter—but it shows why a narrow concentration cluster may fail to distinguish the curve shapes.

Alternatives, Limits, and a Common Mistake

A local standard error or covariance matrix from a fitted model can summarize uncertainty near one optimum. It is useful when the objective surface is reasonably well behaved around a distinct minimum, but a local approximation can understate uncertainty when the surface is elongated or asymmetric. A profile plot instead follows the fit as a parameter is moved away from its optimum and the other parameter is re-optimized. The two views complement one another: a local summary is compact, while a profile makes non-quadratic or weakly identified behavior easier to see.

The main remedy for a broad profile is often more informative concentration coverage, not a more elaborate optimizer. Measurements that span the rising portion of the curve and, where feasible, extend toward low- and high-concentration regions can help distinguish location from shape. The design must still be scientifically and ethically appropriate. If collecting additional data is not possible, report the weak separation and avoid presenting one parameter estimate as precise merely because software returned several decimal places.

Common mistake: “The optimizer returned EC50 = 1.00 mg/L and $\gamma=1.8$, so both parameters are well identified.” A converged numerical solution establishes only that the procedure found a candidate optimum under its settings. It does not show that the objective rises sharply when either parameter changes. The correction is to examine the profile and paired parameter values: if a broad range of $\gamma$ values can be offset by EC50 changes with little increase in SSE, the data weakly identify the pair.

Profiles also depend on the assumed response model, error criterion, and parameter constraints. A broad profile is evidence of limited information under those choices, not proof that the biological system has no meaningful Hill coefficient. Conversely, a narrow profile does not establish that the sigmoid Emax model is correct. Parameter uncertainty and model adequacy are distinct questions.

Key takeaway: When concentrations cluster narrowly, EC50 and $\gamma$ can compensate for one another over the observed range. Profile $\gamma$ by re-optimizing EC50, inspect the paired estimates, and interpret a flat profile as weak separation—not as evidence for a uniquely determined curve shape.

Check Your Understanding

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

At one concentration $C_*=3$ mg/L, the normalized response is 0.75. What EC50 gives this response when $\gamma=2$?
Why does a response of exactly 0.5 at a single concentration identify EC50 at that concentration but not identify $\gamma$?
When constructing an SSE profile for $\gamma$, what is done to EC50 at each fixed value of $\gamma$?
A profile is nearly flat across a wide range of $\gamma$, while the paired EC50 estimates vary. What does this suggest, and what does it not establish?
Why can concentrations farther from the midpoint help distinguish two Hill coefficients?

Solutions. (1) $EC_{50}=3[(1-0.75)/0.75]^{1/2}=3\sqrt{1/3}=1.732$ mg/L. (2) The model gives $R=0.5$ when $C=EC_{50}$, regardless of positive $\gamma$, so that observation locates the midpoint but gives no steepness information. (3) EC50 is optimized to minimize SSE for that fixed $\gamma$, with other estimated parameters also re-optimized if present. (4) The data weakly separate the parameters under the fitted model and error criterion; this does not prove that any one parameter value is biologically correct or that the model is adequate. (5) The predicted responses can diverge more away from the midpoint, providing greater information about curve steepness if measurement error and the design permit those differences to be observed.