Two Views of Uncertainty Around One Fit
A fitted QSP model can produce both a local standard error from inverse Fisher information and an interval from the profile likelihood. These quantities are related, but they do not summarize uncertainty in the same way. The standard error describes curvature close to the optimum; the profile interval follows the likelihood until it reaches a chosen likelihood-ratio cutoff. When the likelihood is nearly quadratic and the estimate is well inside its allowed range, the two descriptions may agree. Curvature, skewness, or a parameter boundary can make them differ substantially.
The comparison is most useful when it is made for the same parameter, likelihood, fitted model, and nuisance-parameter treatment. A discrepancy is not automatically evidence that one calculation is wrong. It can reveal that a local summary is being asked to describe a region that is not locally symmetric, or that the regular conditions behind a standard cutoff are not met.
An approximate profile-likelihood confidence set contains the values of \(\theta\) satisfying \(D(\theta)\leq c\), where \(c\) is a cutoff chosen for the intended coverage and calibration. Under standard regularity conditions, a common large-sample choice for a one-parameter 95% interval is the 0.95 quantile of a chi-square distribution with one degree of freedom, approximately \(3.841\). When those conditions do not hold, the cutoff may need a different calibration; a boundary is one important case.
The local comparison starts from the observed information matrix, the negative Hessian of the log-likelihood at an interior maximum. Its inverse is a local covariance approximation. For a scalar parameter, the square root of the relevant diagonal element of that inverse is the local standard error. This standard error is commonly used to form a symmetric Wald interval, \(\hat{\theta}\pm z_{0.975}SE\), for a nominal 95% interval. That interval is a local approximation, not a profile-likelihood interval.
Why the Local and Profile Summaries Can Agree
Suppose the profile log-likelihood is well approximated near its maximum by a quadratic:
Then the profile statistic is approximately \(D(\theta)\approx I_{\mathrm{prof}}(\theta-\hat{\theta})^2\). The likelihood-ratio interval therefore has approximate endpoints \(\hat{\theta}\pm\sqrt{c}\,SE_{\mathrm{local}}\). With \(c=3.841\), \(\sqrt{c}\) is approximately \(1.96\), giving the same endpoints as the usual 95% Wald interval. This agreement follows from the quadratic approximation; it is not a guarantee that the two methods agree for an actual fitted model.
Worked Comparisons
Worked Example: A Quadratic Profile Gives Matching Intervals
Suppose a hypothetical QSP calibration has an interior maximum at a pathway rate \(\hat{\theta}=2.00\ \mathrm{day}^{-1}\). Near the maximum, the profile log-likelihood is exactly \(\ell_p(\theta)=\ell_{\max}-(\theta-2.00)^2/(2(0.50)^2)\). Use the 95% cutoff \(c=3.841\) and compare the profile interval with the local interval.
The profile statistic is \(D(\theta)=(\theta-2.00)^2/(0.50)^2\). The local profile curvature is \(1/(0.50)^2=4.00\ (\mathrm{day}^{-1})^{-2}\), so the local standard error is \(1/\sqrt{4.00}=0.500\ \mathrm{day}^{-1}\).
For the profile interval, set \(D(\theta)=3.841\): \[ |\theta-2.00|=0.50\sqrt{3.841}=0.980\ \mathrm{day}^{-1}. \] The resulting interval is \([1.020,\ 2.980]\ \mathrm{day}^{-1}\). The Wald interval is \(2.00\pm1.96(0.500)=[1.020,\ 2.980]\ \mathrm{day}^{-1}\), to the displayed precision.
The agreement is expected because the profile is exactly quadratic, the estimate is interior, and the same regular 95% cutoff is used. The interval describes parameter values not rejected by this likelihood-ratio rule; it does not assign a 95% probability to the fixed parameter being inside these particular endpoints.
Worked Example: Asymmetric Curvature Produces Unequal Profile Limits
Consider a hypothetical fitted pathway parameter with \(\hat{\theta}=1.00\ \mathrm{day}^{-1}\). Suppose its profile statistic is \(D(\theta)=z^2+0.5z^3+0.1z^4\), where \(z=(\theta-1.00)/(0.200\ \mathrm{day}^{-1})\). Use \(c=3.841\) to find the profile limits and compare them with a local Wald interval.
At the maximum, the quadratic term gives \(D(\theta)\approx z^2\). Thus the local standard error is \(0.200\ \mathrm{day}^{-1}\). The Wald interval is \(1.00\pm1.96(0.200)=[0.608,\ 1.392]\ \mathrm{day}^{-1}\).
The profile limits solve \(z^2+0.5z^3+0.1z^4=3.841\). Numerical solution gives approximately \(z=-3.07\) and \(z=1.42\). Substituting into the parameter transformation gives \(\theta_{\mathrm{lower}}=1.00+(-3.07)(0.200)=0.386\ \mathrm{day}^{-1}\) and \(\theta_{\mathrm{upper}}=1.00+(1.42)(0.200)=1.284\ \mathrm{day}^{-1}\).
The profile interval, approximately \([0.386,\ 1.284]\ \mathrm{day}^{-1}\), is wider on the lower side than on the upper side. The local standard error captures the curvature at the maximum but not the changing curvature farther away. Here the profile shape is asymmetric, so a symmetric interval based on one local standard error conceals an important feature of the likelihood.
Worked Example: A Boundary Changes the Interval Interpretation
Suppose a hypothetical rate parameter is constrained to \(\theta\geq0\), and the constrained fit is at \(\hat{\theta}=0\ \mathrm{day}^{-1}\). For illustration, let the log-likelihood on the permitted range be \(\ell_p(\theta)=\ell_{\max}-\theta^2/[2(0.200\ \mathrm{day}^{-1})^2]\). Compare the local standard error with a boundary-respecting likelihood-ratio set.
The curvature gives a local scale of \(SE=0.200\ \mathrm{day}^{-1}\). A mechanically applied symmetric Wald interval using \(1.96SE\) is \([-0.392,\ 0.392]\ \mathrm{day}^{-1}\). Negative values are impossible under the model, so this interval is not a usable parameter interval without modification.
For a one-sided 95% likelihood-ratio set under the usual asymptotic boundary case, a commonly used reference distribution is a 50:50 mixture of a point mass at zero and a chi-square distribution with one degree of freedom. Its 95th percentile is about \(2.706\). With that cutoff, the permitted set is \[ 0\leq\theta\leq0.200\sqrt{2.706}=0.329\ \mathrm{day}^{-1}. \] This example assumes the boundary asymptotic calibration is appropriate. In a particular model, the cutoff should be selected to match the inferential target and justified by the model’s conditions or a suitable calibration.
The profile calculation respects the constraint and gives a one-sided set. The local curvature remains informative about the likelihood near the boundary, but a symmetric Wald interval ignores the geometry of the permitted parameter space. A boundary estimate also means that standard interior likelihood-ratio results cannot simply be presumed to apply.
Checking the Comparison in Practice
Before interpreting a discrepancy, ensure both summaries refer to the same estimand and fitted objective. In the profile calculation, nuisance parameters must be re-optimized at each fixed value of \(\theta\); holding them at their joint estimates gives a slice through the likelihood, not a profile. In the local calculation, the relevant standard error for \(\theta\) comes from the inverse of the full information matrix when nuisance parameters are estimated jointly. Inverting only the \(\theta\)-to-\(\theta\) entry generally ignores uncertainty and correlation involving nuisance parameters.
The local-profile relationship can be justified with a block Hessian. Partition the observed information at the joint maximum as \[ J= \begin{pmatrix} J_{\theta\theta} & J_{\theta\eta}\\ J_{\eta\theta} & J_{\eta\eta} \end{pmatrix}. \] For a small displacement \(\delta\) in \(\theta\), the quadratic log-likelihood loss is minimized over a small nuisance displacement \(u\). The quadratic form is \[ \frac{1}{2} \begin{pmatrix}\delta&u^\mathsf{T}\end{pmatrix} J \begin{pmatrix}\delta\\u\end{pmatrix}. \] Setting its derivative with respect to \(u\) to zero gives \(J_{\eta\theta}\delta+J_{\eta\eta}u=0\), hence \(u=-J_{\eta\eta}^{-1}J_{\eta\theta}\delta\), when \(J_{\eta\eta}\) is invertible. Substitution gives a profile loss of \[ \frac{1}{2}\delta^2 \left(J_{\theta\theta}-J_{\theta\eta}J_{\eta\eta}^{-1}J_{\eta\theta}\right). \] The bracket is the Schur complement. The \(\theta\)-to-\(\theta\) element of \(J^{-1}\) is its reciprocal, establishing the local relationship. This argument assumes a smooth, nonsingular interior maximum and a valid quadratic approximation; it does not establish that the likelihood remains quadratic over a wide interval.
Useful checks include whether the profile curve is smooth and maximized at the reported estimate, whether each profile optimization converged, and whether the likelihood-ratio statistic rises to the chosen cutoff on both sides. A flat profile, multiple crossings, abrupt jumps, or strong dependence on starting values can indicate weak identification, numerical failure, or multiple modes. The earlier material on numerical diagnostics for profile optimization is relevant when these features appear. Also check parameter units and transformations: a standard error on a log scale is not directly comparable with an interval width on the original scale without transforming it.
| Feature | Local inverse-information summary | Profile-likelihood summary |
|---|---|---|
| What it uses | Curvature at the fitted maximum | Likelihood loss across a range, re-optimizing nuisance parameters |
| Typical interval shape | Symmetric on the parameter scale used for the Wald calculation | Can be asymmetric and can respect parameter limits |
| Main strength | Fast local diagnostic; useful for approximate precision and correlation assessment | Shows how support changes away from the optimum and can expose asymmetry or flat regions |
| Main limitation | Can misrepresent uncertainty when the local quadratic approximation is poor | Requires reliable repeated optimization and a justified likelihood-ratio cutoff |
Neither approach repairs a misspecified observation model or makes an unidentifiable parameter estimable. Inverse information can be unstable when the information matrix is nearly singular; a profile may reveal a broad or open interval, but may also be hard to compute reliably. Profile intervals are not automatically superior in every finite-sample setting: their coverage depends on the likelihood assumptions and cutoff calibration. Use the two summaries as complementary diagnostics, and investigate disagreement rather than selecting whichever interval is more convenient.
Common Mistake
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In one sentence, distinguish the quantity summarized by a local inverse-information standard error from the quantity used to construct a profile-likelihood interval.
- A fitted parameter is far from its bounds, and its profile is nearly quadratic throughout the 95% region. What relationship would you expect between a Wald interval and a profile interval using a regular one-parameter cutoff?
- Why must nuisance parameters be re-optimized at each fixed value of the parameter of interest when constructing a profile?
- A constrained estimate is at zero, but a symmetric Wald interval extends below zero. Name one reason the interval is problematic and one diagnostic that remains useful.
Solutions. (1) The local standard error summarizes curvature at the fitted maximum, whereas the profile interval identifies parameter values whose optimized likelihood is within a chosen likelihood-ratio cutoff. (2) They should be similar because a quadratic profile gives symmetric likelihood-ratio limits that are approximately the Wald limits when the regular cutoff is used. (3) Profiling accounts for nuisance-parameter adjustment at each fixed value; holding nuisance parameters fixed instead evaluates a likelihood slice and can give a different curvature and interval. (4) Negative values violate the constraint, and the interior Wald approximation is inappropriate at a boundary. The local curvature can still describe behavior near the fit, while a boundary-respecting profile and an appropriate cutoff can provide a more suitable interval.