Why Profile the Likelihood of a Target Dose?
A dose that attains a specified response is usually a function of several fitted model parameters. For example, in an Emax model the dose attaining a target response depends on the baseline, maximum effect, potency, and possibly the Hill coefficient. Uncertainty in those parameters does not generally translate into a symmetric uncertainty interval for the dose. The target may be close to a plateau, or the fitted model may allow a wide range of doses to give nearly the same likelihood.
Profile likelihood addresses this by asking, for each proposed target dose, how well the model can fit the data after all other parameters have been re-estimated. Unlike a first-order approximation, this procedure uses the likelihood shape along the target-dose dimension. The resulting interval can be asymmetric, and it can reveal when the data do not support a finite endpoint.
The estimate \(\hat d^*\) maximizes \(\ell_p(d)\), and it agrees with the target dose calculated from the unrestricted maximum-likelihood estimate when that estimate exists and is unique. A likelihood-ratio confidence set retains proposed doses whose profile log likelihood is not too far below its maximum:
Here \(\chi^2_{1,\,1-\alpha}\) is the \(1-\alpha\) quantile of a chi-square distribution with one degree of freedom. For a nominal 95% interval, the common asymptotic cutoff is approximately 3.841. Under regularity conditions, the likelihood-ratio statistic is approximately chi-square with one degree of freedom when the proposed target dose is the true dose. Thus the set is an approximate confidence interval or confidence set, not a probability statement about the fixed dose after the data have been observed.
Construction and Assumptions
The target must first be defined precisely. Let \(f(d;\theta)\) be the model-predicted response at dose \(d\), and let \(r^*\) be the prespecified target. The target dose solves \(f(d^*;\theta)=r^*\). For a given proposed dose \(d\), profiling imposes \(f(d;\theta)=r^*\), then maximizes the likelihood subject to that equality. The profile is therefore a sequence of constrained fits, not a likelihood evaluated by merely plugging in the original parameter estimates.
The usual chi-square cutoff relies on regular likelihood theory: the model is correctly specified, observations follow the stated sampling model, the true parameters are identifiable and interior to the parameter space, and the target dose is a locally smooth, identifiable scalar function of those parameters. The likelihood should have a well-behaved maximum, and the constraint should not put the fit on a boundary or make the target undefined. Sparse data, weak identification, a target near an asymptote, or a target outside the response range can undermine these conditions. In such cases the profile itself remains informative, but nominal coverage from the standard cutoff may be inaccurate.
State the response scale, the target value \(r^*\), and the dose range and units. Check whether the fitted response curve has a unique dose attaining that target.
Find the maximum log likelihood and calculate the corresponding target-dose estimate \(\hat d^*\), if it exists.
At each candidate \(d\), impose \(f(d;\theta)=r^*\), maximize over the remaining parameters, and record \(\ell_p(d)\).
Retain doses with likelihood-ratio statistic below the selected cutoff. Check the profile shape, the numerical optimizer, and whether either interval endpoint is unbounded or outside the meaningful dose range.
Worked Examples
Worked Example: Profiling a Target Dose in a Gaussian Log-Dose Model
Consider a hypothetical experiment with two independent response measurements, one at 0.368 mg and one at 2.718 mg. Suppose the target response is \(R^*=50\) response units, and the observed responses are 47 and 51 units, respectively. Assume the model \[ Y_i\sim N\bigl(50+\beta(z_i-\eta),\,1\bigr),\qquad z_i=\log(d_i/1\text{ mg}),\quad \eta=\log(d^*/1\text{ mg}), \] with independent observations and known residual standard deviation of 1 response unit. The target dose is \(d^*=1\text{ mg}\times e^\eta\). The dose-response curve is linear in log dose, with unknown slope \(\beta\); the model assumptions include a common known variance and normally distributed errors.
The log doses are approximately \(-1\) and \(1\). Subtracting the target response from the observations gives \(-3\) and \(1\). In the unrestricted fit, a line through these two points has slope \(\hat\beta=2\) response units per log-dose unit and crosses the target response at \(\hat\eta=0.5\). Thus \(\hat d^*=e^{0.5}\text{ mg}=1.649\text{ mg}\). The unrestricted residual sum of squares is zero for these two observations.
For a proposed \(\eta\), the model predicts deviations from target of \(\beta(-1-\eta)\) and \(\beta(1-\eta)\). Profiling over \(\beta\) is a least-squares projection of the observed deviation vector \((-3,1)\) onto \((-1-\eta,1-\eta)\). The minimized residual sum of squares is \[ 10-\frac{(4+2\eta)^2}{2+2\eta^2} =\frac{2(2\eta-1)^2}{1+\eta^2}. \] Because the residual standard deviation is known to be 1, twice the log-likelihood loss from the unrestricted fit is exactly this residual sum of squares. The 95% likelihood-ratio set therefore satisfies \[ \frac{2(2\eta-1)^2}{1+\eta^2}\leq 3.841. \]
Rearranging gives \(4.159\eta^2-8\eta-1.841\leq 0\). The roots are approximately \(-0.2077\) and \(2.1315\), so the accepted log-dose values lie between them. Transforming back to mg gives \[ [\,e^{-0.2077},\,e^{2.1315}\,]\text{ mg}\approx[\,0.8125,\,8.427\,]\text{ mg}. \] This interval is asymmetric around the estimated dose on the dose scale. The calculation assumes a fixed, known residual standard deviation; estimating the variance or changing the sampling model changes the profile likelihood and its reference distribution.
Worked Example: Solving a Profile-Likelihood Boundary Equation
Suppose a hypothetical dose-response analysis uses data from 12 participants, a prespecified target response, and a fitted likelihood model whose profile calculation is summarized on the log-dose scale \(x=\log(d/1\text{ mg})\). At the 95% likelihood-ratio cutoff, the boundary equation from the constrained fits is \[ 1.28975x^2-3x+0.3598333333=0. \] This is a hypothetical numerical profile output; the equation represents the points where the profile likelihood-ratio statistic equals 3.841.
Using the quadratic formula, \[ x=\frac{3\pm\sqrt{(-3)^2-4(1.28975)(0.3598333333)}}{2(1.28975)}. \] The discriminant is approximately \(9-1.856380=7.143620\), whose square root is approximately 2.67276. Therefore the roots are approximately \[ x_L=\frac{3-2.67276}{2.57950}=0.1269,\qquad x_U=\frac{3+2.67276}{2.57950}=2.1992. \]
If the profile likelihood is above the cutoff between these roots and below it outside, the accepted values are between the roots. Since \(d=1\text{ mg}\times e^x\), the endpoints are \[ [\,1\text{ mg}\times e^{0.1269},\,1\text{ mg}\times e^{2.1992}\,] \approx[\,1.135,\,9.018\,]\text{ mg}. \] This illustrates why root calculations should be checked by substitution: inserting the rounded upper root \(2.1992\) into the quadratic gives a value close to zero, whereas an inaccurately rounded root can noticeably shift the dose endpoint after exponentiation. The interval is symmetric on the log-dose scale in this quadratic example, but its dose-scale endpoints are not equally distant from the point estimate.
Worked Example: Recognizing an Unbounded Upper Endpoint
Suppose a hypothetical Emax analysis targets a response very close to the estimated maximum effect. A profile-likelihood scan on log dose gives a 95% accepted region extending from \(x=0.2\) through \(x=3\), with likelihood-ratio values 2.1 at \(x=3\) and 1.4 at \(x=5\); the cutoff is 3.841. Further constrained fits remain below 3.841 as \(x\) increases, and the fitted response approaches the target only as dose tends to infinity.
At \(x=3\) and \(x=5\), the likelihood-ratio statistic is still below the cutoff, so neither is an upper confidence limit. The accepted set continues to arbitrarily large log doses, and exponentiating it gives an unbounded upper dose endpoint. In this situation the correct report is a finite lower limit and no finite upper limit, together with the reason: the target is near the plateau and the data do not distinguish among sufficiently large doses. Reporting the largest grid dose as the upper endpoint would confuse a computational search limit with an inferential limit.
Interpreting Shape, Diagnostics, and Alternatives
A profile interval need not be symmetric about \(\hat d^*\), or even about \(\log(\hat d^*)\). The likelihood can decline at different rates on either side of the estimate. On the dose scale, exponentiating an interval that is symmetric in log dose naturally produces unequal additive distances. A profile may also have multiple separated high-likelihood regions if several doses can attain the target under distinct parameter combinations. In that case the confidence set can be disconnected; it should not automatically be replaced by one interval that includes low-likelihood doses between the regions.
Inspect the profile over a sufficiently broad range and verify that constrained optimizations converge. A suspicious kink, abrupt jump, or isolated spike may indicate optimizer failure, a branch change in the target-dose solution, or a model boundary. Refit starting from multiple parameter values around such points. Check that the unrestricted maximum is at least as large as every constrained maximum, and verify that the likelihood-ratio statistic is nonnegative. If a proposed dose cannot attain the target for any allowed parameter values, its constrained likelihood is unavailable and it is not part of the confidence set.
The delta method, covered in the earlier material on Delta-Method Uncertainty for Target-Effect Doses, approximates uncertainty by linearizing the target-dose function near the parameter estimate. It can be convenient and efficient when the likelihood is regular and the function is nearly linear over the relevant parameter region. Profile likelihood instead re-optimizes nuisance parameters at each target dose and retains the likelihood's curvature and asymmetry. Its costs are computational: many constrained fits are needed, and the usual chi-square cutoff is still an approximation. Neither approach repairs model misspecification or creates information absent from the data.
A common mistake is to stop the profile scan at the largest dose considered, then report that dose as a finite upper confidence limit. That conclusion is not justified unless the likelihood-ratio statistic crosses the cutoff there. Continue the scan, assess the limiting behavior, and report an unbounded endpoint if the accepted set does not close. Another mistake is to interpret a 95% likelihood-ratio interval as a 95% probability that the fixed target dose lies inside it. Its nominal coverage refers to the long-run performance of the interval procedure under the model and regularity assumptions.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- What is maximized when constructing the profile log likelihood at a proposed target dose?
- For a 95% likelihood-ratio interval with one target-dose parameter, what is the usual asymptotic cutoff, and what does it not mean?
- A profile scan reaches its largest evaluated dose with a likelihood-ratio statistic of 2.0. Is that dose necessarily the upper confidence limit?
- Why can a confidence interval that is symmetric on the log-dose scale be asymmetric on the dose scale?
Solutions. (1) Maximize the full log likelihood over all model parameters satisfying the constraint that the proposed dose attains the response target. (2) The usual cutoff is \(\chi^2_{1,0.95}\approx3.841\); it is not a 95% probability statement about the fixed dose after observing the data. (3) No. Since 2.0 is below 3.841, the endpoint has not been reached; the scan must continue or the upper limit must be reported as unbounded if the accepted set never closes. (4) Exponentiation is nonlinear: equal distances in log dose become multiplicative rather than equal additive distances in dose units.