Why Profile \(k_{e0}\)?
The equilibration rate constant \(k_{e0}\) controls how quickly the effect-site concentration approaches the plasma concentration in an effect-compartment model. Its estimate can be strongly linked to response parameters: changing \(k_{e0}\) changes the predicted effect-site trajectory, while changes to parameters such as baseline, maximum effect, or concentration sensitivity can partly compensate. A single best-fitting value therefore does not show how much the data distinguish among plausible \(k_{e0}\) values.
A profile likelihood measures fit across a range of fixed \(k_{e0}\) values while allowing the other model parameters to be refitted at each value. It can reveal uncertainty that a symmetric standard-error interval may obscure, including a long tail, a broad region of nearly equivalent fits, or a confidence set that reaches a practical bound.
For example, the model may use the effect-compartment equation \(dC_e/dt=k_{e0}(C_p-C_e)\), with a response function relating \(C_e\) to the observed response. At each trial value of \(k_{e0}\), fit the response parameters anew; do not hold their estimates from the overall fit fixed. The earlier material on identifiability of \(k_{e0}\) and response parameters motivates this joint fitting: a profile makes the resulting trade-offs visible.
Likelihood-Ratio Profile and Its Cutoff
Define the profile likelihood-ratio statistic as \(D(k)=-2\log\{L_p(k)/L_p(\hat{k}_{e0})\}\). It is zero at the profile maximum and is nonnegative elsewhere. Larger values indicate that fixing \(k_{e0}=k\) produces a worse fit relative to the best fit, after accounting for the best possible nuisance-parameter values at that \(k\).
A common setting is a Gaussian response model with independent residuals of constant variance. Let \(y_i\) be observed responses and \(m_i(k,\psi)\) the corresponding predictions. Define \(RSS(k)=\min_{\psi}\sum_{i=1}^n[y_i-m_i(k,\psi)]^2\), where the minimization is performed with \(k\) fixed. If the residual variance is unknown and is estimated by maximum likelihood, the profile deviance has a simple form.
This expression assumes the same observations and Gaussian residual model are used at every candidate \(k\), and that variance is profiled as well as the response parameters. If the residual variance is known and equal to \(\sigma^2\), the corresponding statistic is \([RSS(k)-RSS(\hat{k}_{e0})]/\sigma^2\), not the logarithmic expression. For other likelihood models, calculate \(D(k)\) directly from their maximized likelihoods.
The cutoff follows from the 95th percentile of the chi-square distribution with one degree of freedom. The confidence set is the collection of parameter values not rejected by this likelihood-ratio rule; it need not be symmetric around \(\hat{k}_{e0}\), and it may consist of separated ranges if the profile has multiple minima. The approximate coverage relies on the stated regularity conditions. A practical fitting bound, sparse information, or non-identifiability can make the approximation unreliable.
Why the RSS Formula Holds
For independent normal residuals with common variance \(\sigma^2\), the likelihood at fixed \(k,\psi,\sigma^2\) is proportional to \((\sigma^2)^{-n/2}\exp[-RSS(k,\psi)/(2\sigma^2)]\). At fixed \(k\), the best \(\psi\) minimizes RSS. Substituting that profiled RSS into the likelihood and maximizing over \(\sigma^2\) gives \(\hat{\sigma}^2(k)=RSS(k)/n\).
Subtract the corresponding quantity at the overall optimum. The constants and the factor \(n\) inside both logarithms cancel, giving \(D(k)=n\log\{RSS(k)/RSS(\hat{k}_{e0})\}\). This derivation explains why the RSS ratio must be dimensionless and why the same sample size and response observations must be used throughout the profile.
Worked Example: A Two-Sided, Asymmetric Profile
Worked Example: A Two-Sided, Asymmetric Profile
Suppose a hypothetical response dataset has \(n=12\) observations. A Gaussian-error effect-compartment model is fitted at each listed \(k_{e0}\), with the response parameters and residual variance re-estimated each time. The minimum RSS is 24.0 response-units-squared at \(\hat{k}_{e0}=0.20\ \mathrm{h}^{-1}\). The table gives selected profile values.
| \(k_{e0}\) (\(\mathrm{h}^{-1}\)) | Profile RSS | \(D(k)\) |
|---|---|---|
| 0.06 | 50.0 | 8.808 |
| 0.10 | 34.0 | 4.180 |
| 0.12 | 31.5 | 3.263 |
| 0.20 | 24.0 | 0 |
| 0.30 | 27.5 | 1.634 |
| 0.40 | 33.0 | 3.821 |
| 0.42 | 35.0 | 4.528 |
For \(k_{e0}=0.12\ \mathrm{h}^{-1}\), the calculation is \(D=12\log(31.5/24.0)=12\log(1.3125)=3.263\). This is below 3.841, so this grid value belongs to the approximate 95% set. At \(0.10\ \mathrm{h}^{-1}\), \(D=12\log(34.0/24.0)=4.180\), which is above the cutoff.
Interpolate linearly between adjacent displayed profile points to approximate the lower crossing: \(0.10+[(4.180-3.841)/(4.180-3.263)](0.02)=0.1074\ \mathrm{h}^{-1}\). For the upper crossing, interpolate between 0.40 and 0.42: \(0.40+[(3.841-3.821)/(4.528-3.821)](0.02)=0.4006\ \mathrm{h}^{-1}\). The resulting approximate interval is \(0.107\) to \(0.401\ \mathrm{h}^{-1}\), subject to the profile-grid and linear-interpolation approximations.
This interval is not centered on the estimate: its lower and upper distances from \(0.20\ \mathrm{h}^{-1}\) differ. Report the estimate and the two endpoints rather than replacing the profile with a symmetric standard-error interval. The interval describes uncertainty in \(k_{e0}\) under the fitted model and likelihood assumptions; it does not establish a physiological mechanism by itself.
Worked Example: A Flat Profile
Worked Example: A Flat Profile
In a second hypothetical dataset, there are \(n=10\) responses and the minimum profile RSS is 20.0 at \(k_{e0}=0.20\ \mathrm{h}^{-1}\). Suppose the refitted RSS values at \(k_{e0}=0.05,\ 0.10,\ 0.40,\) and \(0.80\ \mathrm{h}^{-1}\) are 20.0, 20.5, 20.7, and 21.5, respectively.
At \(0.10\ \mathrm{h}^{-1}\), \(D=10\log(20.5/20.0)=10\log(1.025)=0.247\). At \(0.80\ \mathrm{h}^{-1}\), \(D=10\log(21.5/20.0)=10\log(1.075)=0.723\). Both are well below 3.841; the other listed values are also below it. The profile is nearly flat over this range, so these data do not sharply distinguish among the displayed rates.
The grid alone does not prove that every \(k_{e0}\) value is in the confidence set. Extend the profile over the scientifically and computationally appropriate range and locate any crossings. If the profile remains below the cutoff toward one or both ends, report the corresponding confidence set as unbounded or unresolved over that range rather than inventing an endpoint. A flat profile is evidence of weak information under this model, not evidence that all rates are equally plausible without limit.
Worked Example: An Estimate Near a Practical Boundary
Worked Example: An Estimate Near a Practical Boundary
Consider a third hypothetical fit with \(n=10\), minimum RSS 18.0 at \(\hat{k}_{e0}=0.012\ \mathrm{h}^{-1}\), and a software search restricted to \(k_{e0}\geq0.010\ \mathrm{h}^{-1}\). Suppose the profiled RSS values are 18.02 at 0.010, 19.0 at 0.020, 24.0 at 0.050, and 27.0 at 0.070 \(\mathrm{h}^{-1}\).
At the lower search bound, \(D=10\log(18.02/18.0)=0.0111\), far below 3.841. At \(0.050\ \mathrm{h}^{-1}\), \(D=10\log(24.0/18.0)=2.877\); at \(0.070\ \mathrm{h}^{-1}\), \(D=10\log(27.0/18.0)=4.055\). Interpolating the upper crossing gives \(0.050+[(3.841-2.877)/(4.055-2.877)](0.020)=0.0664\ \mathrm{h}^{-1}\).
Within the searched range, the approximate set reaches the practical lower bound and ends near \(0.0664\ \mathrm{h}^{-1}\). Because the profile is still well inside the cutoff at 0.010, the data have not established a lower confidence endpoint above that bound. Report that the lower limit is unresolved below the search bound and consider whether the bound should be extended. If the bound is a true parameter-space boundary rather than merely computational, the usual chi-square cutoff may also need boundary-aware calibration; do not automatically interpret the truncated result as a regular two-sided 95% interval.
Diagnostics, Alternatives, and Limitations
A useful profile is not just a table of likelihood values. Plot \(D(k)\) against \(k_{e0}\), mark zero at the fitted estimate, and draw the chosen likelihood-ratio cutoff. Check that the fit at every grid value converged and that nuisance parameters were re-optimized. A sudden jump, irregular spike, or apparent secondary optimum may reflect optimizer failure, a coarse grid, or a genuinely complicated likelihood surface. Repeat fits from multiple starting values and refine the grid around minima and cutoff crossings before interpreting the shape.
A broad, shallow bottom indicates weak information about \(k_{e0}\); a long tail on one side signals asymmetric uncertainty. Multiple dips mean the likelihood-ratio confidence set may be disconnected, in which case reporting only the outer endpoints conceals an important feature. Always inspect nuisance-parameter estimates along the profile as well: large compensating changes can reveal parameter trade-offs. The profile is conditional on the chosen effect-compartment and response models, plasma input, residual distribution, and observation set. It does not diagnose every form of model misspecification.
| Approach | What it provides | Main caution |
|---|---|---|
| Profile likelihood | Likelihood-based uncertainty after refitting nuisance parameters at each fixed \(k_{e0}\) | Can be computationally demanding; cutoff approximation may fail near boundaries or under weak identification |
| Wald interval | A symmetric interval based on an estimate and its local standard error | Can poorly represent skewness, flatness, or a boundary |
| Parametric bootstrap | Sampling-based uncertainty under a specified fitted model | Depends on the simulation model and can be costly; it does not automatically fix model misspecification |
The Wald interval may be adequate when the likelihood is locally quadratic and the estimate is far from constraints, but those conditions should not be presumed for \(k_{e0}\). A parametric bootstrap can be useful when regular likelihood-ratio approximations are doubtful, provided the simulation assumptions are defensible. Neither alternative removes the need to examine fit, identifiability, and boundary sensitivity.
Common Mistakes
- Holding response parameters fixed. This compares a constrained fit with nuisance parameters frozen at their overall estimates, not the profile likelihood. Refit them at each fixed \(k_{e0}\).
- Using the wrong RSS ratio direction. The deviance is \(n\log[RSS(k)/RSS(\hat{k}_{e0})]\); reversing the ratio gives negative values for worse fits.
- Forcing symmetry. A profile interval can have unequal distances from the estimate. Report its actual crossings, not an averaged margin.
- Treating a search bound as a confidence endpoint. If the profile remains below the cutoff at that bound, the endpoint is unresolved there; expand the search or report the limitation.
- Calling the confidence set the probability range for the true value. Under the model and approximation, the procedure has approximately 95% coverage over repeated datasets; the interval is not a 95% probability statement about a fixed parameter.
Check Your Understanding
Work through the questions, then compare your reasoning with the solutions.
- In one sentence, explain what is re-estimated when constructing a profile likelihood for \(k_{e0}\).
- For \(n=8\), \(RSS(\hat{k}_{e0})=16.0\), and \(RSS(k)=20.0\), calculate \(D(k)\) for unknown Gaussian residual variance.
- If the profile remains below the 95% cutoff at the smallest value allowed by the fitting software, what can be concluded about the lower confidence endpoint?
- Why can a Wald interval be misleading when the profile is strongly asymmetric?
Solutions. (1) Fix \(k_{e0}\) at each candidate value and re-optimize all nuisance parameters, including the response parameters and, when unknown, residual variance. (2) \(D=8\log(20.0/16.0)=8\log(1.25)=1.785\), which is below 3.841; this candidate is in the approximate 95% set under the regularity assumptions. (3) The data have not located a lower crossing above the search bound. Extend the profile if appropriate and report the lower endpoint as unresolved at that bound; do not call the bound a data-supported confidence limit. (4) A Wald interval uses local curvature to produce symmetric uncertainty, whereas a strongly asymmetric profile has different likelihood behavior on its two sides, so equal margins need not represent comparable support.