Quantifying Target-Dose Uncertainty With Profile Likelihood

Why Target-Dose Uncertainty Needs Its Own Interval

An estimated target dose can look precise even when the dose-response data do not determine it precisely. In an Emax model, the dose achieving a specified response depends on several fitted parameters. Uncertainty in those parameters—and their correlation—must be carried through to the dose estimate. A profile-likelihood interval does this by asking which fixed target doses remain compatible with the observed likelihood after the other model parameters are re-estimated.

The earlier material on inverting Emax models gives the target-dose relationship. Here, the focus is uncertainty around that derived dose, rather than only its point estimate. The resulting interval can be asymmetric, and its width can reveal that the data provide little information about the target.

Definition: Let $\psi$ be the target dose and let $\lambda$ denote all other model parameters, called nuisance parameters for this calculation. The profile log-likelihood at a fixed $\psi$ is $$ \ell_p(\psi)=\max_{\lambda}\ell(\psi,\lambda), $$ where $\ell(\psi,\lambda)$ is the log-likelihood for the full model. The profile-likelihood ratio statistic is $$ \Lambda(\psi)=2\left\{\ell(\hat{\psi},\hat{\lambda})-\ell_p(\psi)\right\}, $$ where $(\hat{\psi},\hat{\lambda})$ maximizes the full likelihood.

Target Dose and Profile-Likelihood Interval

For a standard Emax mean model, let the expected response at dose $D$ be $E_0+E_{\max}D/(ED_{50}+D)$. Here, $E_0$ is the baseline response, $E_{\max}$ is the maximum response increment, and $ED_{50}$ is the dose producing half of that increment. Suppose the specified target is an absolute expected response $Y_{\text{tar}}$, and define the target increment $r=Y_{\text{tar}}-E_0$. When $0<r<E_{\max}$, solving for the dose gives $$ \psi=D_{\text{tar}}=\frac{ED_{50}r}{E_{\max}-r}. $$

This expression also indicates a key source of uncertainty: as $E_{\max}$ approaches $r$ from above, the denominator approaches zero and the target dose grows rapidly. If plausible parameter values have $E_{\max}\leq r$, the target may not be attainable under the model. A finite point estimate alone does not rule out that possibility.

To profile the target dose, fix $\psi$ at a candidate value. Constrain the Emax parameters to combinations that imply that target dose, then maximize the likelihood over the remaining free parameters. Repeat for candidate doses. Under regularity conditions, a 95% likelihood-ratio interval contains doses satisfying $\Lambda(\psi)\leq 3.841$, the 95th percentile of a chi-square distribution with one degree of freedom.

Assumptions and conditions: The likelihood must represent the sampling model adequately, and the maximum likelihood estimates should be identifiable and interior to the parameter space. The target should be a locally identifiable scalar function of the parameters. The chi-square cutoff is an asymptotic result; it may be inaccurate with very small samples, parameters on boundaries, weak identification, or a target that is not attainable for some parameter values. In those settings, the profile itself remains informative, but the nominal interval coverage needs additional assessment.

Worked Example: Calculating an Interval From a Profile Curve

In a hypothetical Emax dose-ranging analysis, suppose the fitted target dose is 10 mg. The maximized profile log-likelihood over the dose range is summarized by $$ \ell_p(\psi)=\ell_{\max}-\frac{1}{2}\left(\frac{\ln(\psi/10)}{0.25}\right)^2, $$ where $\psi$ is measured in mg. This is a hypothetical fitted profile curve, not a claim about a real study.

For a 95% interval, retain candidate doses whose likelihood-ratio statistic is no greater than 3.841: $$ 2\{\ell_{\max}-\ell_p(\psi)\} =\left(\frac{\ln(\psi/10)}{0.25}\right)^2 \leq 3.841. $$ Taking square roots gives $\left|\ln(\psi/10)\right|\leq 0.25\sqrt{3.841}$. Since $\sqrt{3.841}$ is approximately 1.96, the log-dose limits are $-0.490$ and $0.490$. Transforming back to mg, $$ \psi_{\text{lower}}=10e^{-0.490}=6.126\text{ mg},\qquad \psi_{\text{upper}}=10e^{0.490}=16.32\text{ mg}. $$

The profile-likelihood interval is therefore approximately 6.13 to 16.3 mg. Under the stated model and regularity conditions, this procedure has approximately 95% repeated-sampling coverage. It does not mean there is a 95% probability that this fixed target dose lies inside this particular interval.

Why Parameter Correlation Changes the Interval

Profiling is important because Emax parameters are often correlated. For instance, a larger $E_{\max}$ can sometimes be offset by a different $ED_{50}$ while preserving similar predicted responses over the observed doses. Since the target dose depends on both parameters, treating one as fixed can understate uncertainty.

A local quadratic log-likelihood makes this point explicit. Let $\theta=\ln(\psi/10\text{ mg})$ represent the log target dose relative to 10 mg, and let $\eta$ be a standardized nuisance parameter. Suppose their fitted values are both zero, their respective standard errors are 0.25 and 0.20, and their correlation is $\rho=0.80$. Write $x=\theta/0.25$ and $z=\eta/0.20$. The local likelihood is $$ \ell(\theta,\eta)=\ell_{\max}-\frac{1}{2}\frac{x^2-2\rho xz+z^2}{1-\rho^2}. $$

For a fixed $x$, profiling means choosing $z$ to maximize this log-likelihood, equivalently to minimize the quadratic expression. Differentiating that expression with respect to $z$ gives $2(z-\rho x)/(1-\rho^2)=0$, so the maximizing nuisance value is $z=\rho x$. Substitution then gives $$ \frac{x^2-2\rho x(\rho x)+(\rho x)^2}{1-\rho^2}=x^2. $$

Thus, the profile deviance is $x^2$: it reflects variation in the target dose while allowing the nuisance parameter to move with it. If the nuisance parameter were incorrectly held at its fitted value $z=0$, the deviance would instead be $x^2/(1-\rho^2)$, which increases more rapidly and produces an artificially narrow interval.

Worked Example: Profiling a Correlated Nuisance Parameter

Use the hypothetical likelihood above, with $\rho=0.80$. From the profiling calculation, the 95% limits satisfy $|x|\leq1.96$, so $|\theta|\leq0.25(1.96)=0.490$. On the dose scale, the interval is $10e^{-0.490}$ to $10e^{0.490}$, or approximately 6.13 to 16.3 mg.

If instead the nuisance parameter were fixed at its estimate, the conditional deviance would be $x^2/(1-0.80^2)=x^2/0.36$. The corresponding limits would satisfy $|x|\leq1.96\sqrt{0.36}=1.176$. Therefore $|\theta|\leq0.25(1.176)=0.294$, giving $10e^{-0.294}=7.45$ mg and $10e^{0.294}=13.4$ mg.

The fixed-nuisance interval, 7.45 to 13.4 mg, is narrower than the profiled interval because it excludes the correlated parameter changes that also fit the data. It is not the appropriate interval for uncertainty in the target dose when that nuisance parameter is estimated from the same data.

Weak Dose-Ranging Information and Diagnostic Clues

If observed doses do not adequately distinguish the slope and plateau of the response curve, many parameter combinations can fit almost equally well. The target-dose profile may then be broad, asymmetric, or nearly flat over a substantial dose range. For a target near the estimated plateau, small changes in $E_{\max}$ can imply large changes in the dose required to reach the target. If the likelihood supports parameter values for which the target is unattainable, the profile may not close at a finite upper dose.

Worked Example: Weak Dose-Ranging Information Produces a Wide Interval

Consider a second hypothetical analysis with sparse dose-ranging information, such as observations at 0, 5, and 10 mg, and a fitted target of 10 mg. Suppose its profile is summarized by $$ \ell_p(\psi)=\ell_{\max}-\frac{1}{2}\left(\frac{\ln(\psi/10)}{0.50}\right)^2. $$ The likelihood-ratio condition becomes $\left|\ln(\psi/10)\right|\leq0.50(1.96)=0.980$. Hence $$ \psi_{\text{lower}}=10e^{-0.980}=3.75\text{ mg},\qquad \psi_{\text{upper}}=10e^{0.980}=26.6\text{ mg}. $$

The interval is much wider than the 6.13 to 16.3 mg interval in the first example because the profile curve is flatter: a larger range of target doses loses little likelihood relative to the maximum. Its upper end also extends well beyond the highest dose in the illustrative design. That is a warning that the estimate depends on model-based extrapolation, not direct observation across the full interval.

Useful diagnostics include plotting the profile log-likelihood against dose and checking whether it has a clear maximum and crosses the likelihood-ratio cutoff on both sides. Inspecting the fitted nuisance parameters along the profile can reveal extreme or unstable compensating values. A missing crossing, a long flat section, or abrupt changes may indicate weak identification, a boundary, an unattainable target, numerical optimization problems, or model inadequacy. These features call for investigation rather than an automatic report of a finite, symmetric interval.

Choosing and Interpreting an Uncertainty Method

A Wald interval uses a local standard error for the target estimate, often obtained by approximating the target function near the fitted parameters. It is computationally convenient, but can be misleading when the target is a nonlinear function, the likelihood is asymmetric, or parameter correlation is substantial. A profile interval instead re-optimizes the nuisance parameters at each candidate target and can display asymmetry or failure to close.

A bootstrap interval is another alternative: repeatedly simulate or resample data under a specified scheme, refit the model, and calculate the target dose in each replicate. It can reflect some nonlinear behavior, but depends on the bootstrap design and can fail frequently when fits are unstable or targets are unattainable. Profile likelihood is not automatically reliable in small samples; its usual chi-square calibration is approximate. The curve, the model fit, and the data’s dose coverage all need review.

The interval quantifies uncertainty under the chosen model and likelihood. It does not by itself establish that the target is clinically worthwhile, safe, or supported by adequate evidence across the dose range. Those judgments require the clinical objective and the broader dose-justification evidence.

Common pitfall: “The target-dose estimate has a small standard error, so its uncertainty is small.” This can be wrong when the target is a nonlinear function of correlated Emax parameters. Fixing nuisance parameters or relying only on a symmetric Wald interval may hide a broad or asymmetric likelihood. Profile the target, inspect the nuisance-parameter path, and check whether the profile closes within a range supported by the data.

Define the target.
State the response target, the dose parameter $\psi$, and the model conditions under which the target is attainable.

Profile each candidate dose.
Fix $\psi$, maximize the likelihood over the remaining parameters, and record $\ell_p(\psi)$.

Find the interval and assess it.
Retain doses with likelihood-ratio statistic at or below the chosen chi-square cutoff, then inspect asymmetry, nuisance estimates, and dose-range support.

Check Your Understanding

Answer each question, then compare with the solutions below.

For a 95% profile-likelihood interval with one target parameter, what likelihood-ratio cutoff is used under regularity conditions?
Why can profiling give a wider interval than calculating uncertainty with a nuisance parameter fixed?
For the first example, what does the interval 6.13 to 16.3 mg say about the target dose?
What could it mean if the profile does not cross the cutoff at a finite upper dose?

Solutions. (1) Use 3.841, the 95th percentile of a chi-square distribution with one degree of freedom. (2) Profiling allows nuisance parameters to adjust with the fixed target dose, accounting for their joint likelihood and correlation; fixing them can make the interval artificially narrow. (3) Under the model and regularity conditions, the procedure has approximately 95% repeated-sampling coverage for the target dose; it does not assign a 95% probability to the fixed dose being in this particular interval. (4) The target may be weakly identified, the likelihood may remain compatible with arbitrarily large doses, or the target may be unattainable for supported parameter values; inspect the fit and model assumptions before reporting a finite interval.