Profile Likelihood for Weakly Identified Emax Parameters

Why Profile Likelihood Helps

An Emax curve can fit the observed concentration-effect data reasonably well even when the data do not pin down its parameters. If observations cover only the low-concentration, nearly linear portion of the curve, a larger $E_{\max}$ paired with a larger $EC_{50}$ can produce almost the same fitted responses. The optimizer may still return a precise-looking pair of estimates, but the apparent precision can be an artifact of the estimation method or parameter bounds.

Profile likelihood examines this problem directly. It asks how well the model can fit when one parameter is held at a specified value and all other estimable parameters are re-optimized. Repeating this over a range of fixed values reveals which values remain compatible with the data, including values far from the optimum or beyond the range where a symmetric standard-error approximation is useful.

Definition: For a parameter of interest $\theta$, the profile likelihood at $\theta$ is the maximum likelihood achievable with that parameter fixed and all nuisance parameters optimized: $$ L_p(\theta)=\max_{\boldsymbol{\eta}}L(\theta,\boldsymbol{\eta}). $$ Here $\boldsymbol{\eta}$ denotes the other model parameters. For an Emax model, $\theta$ may be $EC_{50}$ or $E_{\max}$; nuisance parameters may include the other Emax parameter, a baseline, and residual-error parameters.

Model, Assumptions, and the Profile Statistic

For a direct stimulatory Emax model, the expected response at concentration $C_i$ can be written

\mu_i=E_0+E_{\max}\frac{C_i}{EC_{50}+C_i}.

Here $E_0$ is the baseline response, $E_{\max}$ is the maximum change from baseline, and $EC_{50}$ is the concentration producing half of that change. In the examples below, $E_0=0$ is known, concentrations are nonnegative, and $E_{\max}$ and $EC_{50}$ are positive. In an application, the baseline might instead be estimated and re-optimized at each profile point. The residual distribution and error model must also be specified; the formulas below assume independent normal errors with constant variance.

Let $RSS_{\min}$ be the minimum residual sum of squares from the unconstrained fit, and let $RSS(\theta)$ be the smallest RSS when $\theta$ is fixed. If the residual standard deviation $\sigma$ is known, the likelihood-ratio statistic is

D(\theta)=-2\log\left\{\frac{L_p(\theta)}{L_{\max}}\right\} =\frac{RSS(\theta)-RSS_{\min}}{\sigma^2}.

If instead the common variance is unknown and estimated by maximum likelihood, the corresponding statistic is

D(\theta)=n\log\left\{\frac{RSS(\theta)}{RSS_{\min}}\right\}.

Under regularity conditions, a likelihood-ratio confidence set with approximate confidence level $1-\alpha$ contains the values satisfying $D(\theta)\leq\chi^2_{1,1-\alpha}$. For a 95% confidence set, the usual reference cutoff is approximately 3.84. This is an asymptotic approximation, not a guarantee of exact coverage, and it can be unreliable with very weak identification, boundary estimates, or small data sets.

Computing and Reading a Profile

To profile $EC_{50}$, fix it at a grid of positive values and optimize $E_{\max}$ and any other nuisance parameters at each value. To profile $E_{\max}$, fix that parameter and re-optimize $EC_{50}$ and the remaining nuisance parameters. Plot $D(\theta)$ against the fixed parameter, often on a logarithmic parameter axis when values span orders of magnitude. The confidence set is the part of the curve at or below the selected cutoff.

An asymmetric curve produces an asymmetric confidence set; there is no requirement that its endpoints be equally distant from the point estimate. If the profile remains below the cutoff as $EC_{50}$ approaches zero or becomes arbitrarily large, the corresponding confidence set reaches that boundary or is unbounded. A profile that stays nearly flat over a wide range signals that the data provide little information about that parameter.

Worked Example: Profiling EC50 Across a Sparse Concentration Range

Suppose hypothetical responses are observed at concentrations $C=1,2,4$, in concentration units, and are $1,1.5,2$, in response units. Use the model $\mu=E_{\max}C/(EC_{50}+C)$, with known $\sigma=0.5$ response units. The observations lie exactly on the curve with $E_{\max}=3$ response units and $EC_{50}=2$ concentration units, so $RSS_{\min}=0$.

At a fixed $EC_{50}=k$, define $f_i=C_i/(k+C_i)$. Because the model is linear in $E_{\max}$ at fixed $k$, the best-fitting value and profile RSS are

\widehat E_{\max}(k)=\frac{\sum_i f_i y_i}{\sum_i f_i^2}, \qquad RSS(k)=\sum_i y_i^2-\frac{(\sum_i f_i y_i)^2}{\sum_i f_i^2}.

At $k=1$, the $f_i$ values are $1/2,2/3,4/5$. Thus $\sum f_i y_i=3.1$, $\sum f_i^2=1201/900$, and $\sum y_i^2=7.25$. Therefore $\widehat E_{\max}=2.323$ response units and $RSS(1)=7.25-3.1^2/(1201/900)=233/4804=0.04850$ response-units squared. The likelihood-ratio statistic is $D(1)=0.04850/0.5^2=0.1940$.

At $k=4$, the $f_i$ values are $1/5,1/3,1/2$. This gives $\sum f_i y_i=1.7$, $\sum f_i^2=361/900$, and $\widehat E_{\max}=4.238$ response units. The profile RSS is $7.25-1.7^2/(361/900)=65/1444=0.04501$ response-units squared, so $D(4)=0.1801$. Both profile statistics are below 3.84.

The limiting profile RSS as $k$ approaches zero is $0.5$: the fitted curve approaches a constant, whose best value is the response mean, $1.5$. Thus $D$ approaches $0.5/0.25=2.00$. As $k$ tends to infinity, the model approaches a straight line through the origin. Its best slope is $\sum C_i y_i/\sum C_i^2=12/21$, giving $RSS=7.25-144/21=11/28=0.3929$ and $D=1.571$. Both limiting values are below 3.84. Consequently, the profile confidence set includes values arbitrarily close to zero and arbitrarily large $EC_{50}$; the data do not give finite lower and upper limits at this cutoff. The exact best fit at $EC_{50}=2$ does not make that parameter well identified.

When a Point Estimate Is Not Supported

A finite optimizer output is not by itself evidence that the concentration range supports a parameter estimate. If all observed concentrations are low relative to $EC_{50}$, then

E_{\max}\frac{C}{EC_{50}+C}\approx \frac{E_{\max}}{EC_{50}}C \quad\text{when }C\ll EC_{50}.

The data may then inform the ratio $E_{\max}/EC_{50}$ much more strongly than either parameter separately. Conversely, data concentrated near a plateau may be informative about $E_{\max}$ but weak about $EC_{50}$. Profile likelihood makes such trade-offs visible by allowing the nuisance parameter to compensate while the parameter of interest changes.

Worked Example: An Emax Estimate Driven by Extrapolation

Consider hypothetical responses $1,2,4$ response units measured at concentrations $1,2,4$ concentration units, with known $\sigma=0.2$ response units and $E_0=0$. The responses are exactly proportional to concentration, with slope 1 response unit per concentration unit. For a sequence of increasingly large $E_{\max}=M$, choose $EC_{50}=M$ in the corresponding concentration units. The model predictions are $M/(M+1)$, $2M/(M+2)$, and $4M/(M+4)$, which approach $1,2,4$ as $M$ increases.

For example, at $M=100$, the predictions are $100/101=0.9901$, $200/102=1.961$, and $400/104=3.846$ response units. The RSS is approximately $(1-0.9901)^2+(2-1.961)^2+(4-3.846)^2=0.0251$ response-units squared. As $M$ and $EC_{50}$ increase together, the predictions approach the observations and the RSS approaches zero. No finite Emax curve reproduces all three points exactly, because the curve is strictly concave for positive concentration while these points lie on a straight line. The likelihood therefore improves toward an infinite-parameter limit rather than attaining a finite maximum.

A numerical optimizer may report a very large estimate or stop at an imposed upper bound. That result should not be presented as a data-supported finite $E_{\max}$. The profile for $E_{\max}$ extends toward arbitrarily large values with improving fit; the observed concentration range has not revealed the plateau needed to anchor the maximum effect.

Worked Example: A Flat Emax Profile at One Concentration

Suppose three hypothetical responses, $0.9,1.0,1.1$ response units, are measured at the same concentration, $C=10$ concentration units. Let $E_0=0$, $\sigma=0.2$ response units, and use the Emax model. The sample mean response is $1.0$ and $RSS_{\min}=(-0.1)^2+0^2+(0.1)^2=0.0200$ response-units squared.

Fix any $E_{\max}=M>1$ response units. Choosing $EC_{50}=10(M/1-1)$ concentration units makes the predicted mean response exactly 1, since $M(10)/(10(M-1)+10)=1$. This also gives the minimum possible RSS, $0.0200$, for every such $M$. Thus the profile likelihood is flat for all $M>1$, and no finite upper confidence limit for $E_{\max}$ can be obtained from these observations. Repeated measurements at one concentration estimate the response there, but do not separate the plateau from the concentration scale.

Diagnostics, Alternatives, and Limitations

Inspect the entire profile rather than only the reported interval. A rising curve that never crosses the cutoff indicates an open confidence set in that direction. A flat segment indicates weak information; sharp changes or multiple low regions can indicate distinct parameter combinations that fit similarly. Also check whether the optimizer repeatedly reaches parameter bounds, whether different starting values yield different nuisance-parameter solutions, and whether the profile changes materially when the grid or numerical tolerances change. These are signs to investigate, not reasons to treat a boundary estimate as a precise measurement.

A Wald interval uses an estimate and an approximate standard error, and is often convenient when the likelihood is well behaved and nearly quadratic near its maximum. With a strongly asymmetric or nearly flat profile, that local approximation can miss long tails or produce implausible limits. Profile likelihood is more informative in those cases, but it still depends on the chosen model, residual-error assumptions, and likelihood-ratio approximation. It cannot create information absent from the concentration design. If the likelihood-ratio cutoff is questionable because of weak identification or a boundary, calibrated simulation or other inferential methods may be needed.

A common mistake is to interpret a converged point estimate as proof that a parameter is identified. The correction is to examine how much the optimized likelihood worsens as the parameter moves across a scientifically meaningful range. If it barely worsens, reaches the cutoff only on one side, or does not reach it at all, report the asymmetry or open confidence set and explain that the data do not support a finite bound. Do not imply that the parameter is known merely because software printed digits.

Key takeaway: A profile likelihood measures how well the model can fit at each fixed parameter value after allowing the other parameters to adapt. A finite estimate may coexist with an asymmetric, unbounded, or flat confidence set; that shape is essential evidence about what the concentration data actually identify.

Check Your Understanding

Answer each question, then compare with the concise solutions below.

When profiling $EC_{50}$, which parameters are re-optimized?
With known $\sigma$, $RSS_{\min}=2.0$, $RSS(\theta)=3.0$, and $\sigma=0.5$, calculate the likelihood-ratio statistic.
What does a profile that remains below the 95% cutoff as $EC_{50}$ tends to infinity imply?
Why can low-concentration data estimate $E_{\max}/EC_{50}$ more strongly than either parameter?
Why is a point estimate at an imposed upper bound not evidence that the data support that value?

Solutions. (1) Re-optimize $E_{\max}$, the baseline if it is estimated, and other nuisance parameters, while holding $EC_{50}$ fixed. (2) $D=(3.0-2.0)/0.5^2=4.00$. This exceeds the approximate 95% cutoff of 3.84. (3) The confidence set is unbounded above at that cutoff; the data do not provide a finite upper limit. (4) When $C\ll EC_{50}$, the effect is approximately $(E_{\max}/EC_{50})C$, so many parameter pairs with similar ratios predict similar responses. (5) The optimizer may simply have stopped at the search boundary; inspect the profile and expand the range before making an inference about finite support.