What a Computed Profile Represents
A likelihood profile asks how well the model can fit the data when one parameter is required to take a specified value. The other parameters are not generally held at their full-model estimates: they are allowed to adjust to fit as well as possible under that restriction. In an NLME model, this adjustment can involve fixed effects, random-effects parameters, and residual-error parameters, subject to the model's constraints.
Let the parameter of interest be \(\psi\), and collect all remaining parameters into \(\lambda\). Write the observed-data log likelihood as \(\ell(\psi,\lambda)\). For an NLME model, this is the likelihood for the observed data under the specified model, including the treatment of the random effects and any numerical approximation used to evaluate the likelihood. The set of allowed nuisance-parameter values may depend on the fixed value of \(\psi\); denote it by \(\Lambda(\psi)\).
The supremum notation describes the mathematical target; in computation, an optimizer returns an approximation to it. A successful status message alone does not establish that the returned value is the global maximum over the nuisance parameters. The model, dataset, likelihood definition, and numerical settings should remain the same at every grid point. Only the target parameter is deliberately fixed to a different value.
Why the Nuisance Parameters Must Be Reoptimized
Suppose the full-model maximum occurs at \((\hat\psi,\hat\lambda)\). At a different fixed value \(\psi_j\), the best-fitting nuisance parameters may not be \(\hat\lambda\). Holding them fixed would evaluate \(\ell(\psi_j,\hat\lambda)\), which is a likelihood slice, not the profile likelihood. Since \(\hat\lambda\) is one possible nuisance value, the profile must satisfy
provided \(\hat\lambda\) is admissible at \(\psi_j\). This inequality explains why a slice can make the fit look worse than the actual profile does. The profile permits the remaining parameters to compensate, as far as the model and data allow.
There is also a useful check at the full-model estimate. If \((\hat\psi,\hat\lambda)\) is a global maximum of the same likelihood and \(\hat\lambda\in\Lambda(\hat\psi)\), then
But no value of the likelihood can exceed its global maximum, so \(\ell_p(\hat\psi)\leq\ell_{\max}\). Therefore \(\ell_p(\hat\psi)=\ell_{\max}\). A computed profile should reproduce the full-model maximum, up to numerical tolerance, when the grid includes \(\hat\psi\). This is a basic consistency check, not a test of whether the model itself is appropriate.
Worked Example: An Analytic Profile on a Small Grid
Consider a hypothetical, dimensionless likelihood with target \(\psi\) and nuisance parameter \(\eta\):
For each fixed \(\psi\), the second squared term is minimized at \(\eta^*(\psi)=1+0.2(\psi-5)\). Because the log likelihood is the negative of one-half the sum of the two squared terms, this choice maximizes it over \(\eta\). Substitution gives \(\ell_p(\psi)=-\frac{1}{2}(\psi-5)^2\).
At \(\psi=4\), the nuisance optimum is \(\eta^*=1+0.2(4-5)=0.8\), so \(\ell_p(4)=-\frac{1}{2}[(-1)^2+0^2]=-0.5\). At \(\psi=5\), \(\eta^*=1\) and \(\ell_p(5)=0\). At \(\psi=6\), \(\eta^*=1.2\) and \(\ell_p(6)=-\frac{1}{2}[1^2+0^2]=-0.5\). If instead \(\eta\) were incorrectly held at 1 for \(\psi=6\), the log likelihood would be \(-\frac{1}{2}[1^2+(-0.2)^2]=-0.52\), below the profile value. The difference arises because the nuisance parameter was not allowed to adjust.
This simple calculation illustrates the optimization definition; it is not an NLME data analysis. In a real fit, the nuisance optimum usually has to be found numerically.
A Practical Grid-and-Optimization Procedure
Before calculating a profile, verify that the chosen likelihood can be evaluated consistently and that each parameter has the intended domain. For example, a positive clearance must remain positive, and a covariance matrix must remain admissible. Specify the target parameter on its model scale, the nuisance parameters to be optimized, and any bounds or transformations used to enforce constraints. If the likelihood is approximated numerically, use settings accurate enough that changes in the approximation do not dominate changes across the profile.
Choose grid points that cover the scientifically relevant range and include the full-model estimate when practical. A grid need not be equally spaced on the raw scale: for a positive parameter varying across orders of magnitude, spacing on the log scale may be more informative. Record the scale used, because equally spaced log values are not equally spaced raw values. At each point, fix the target exactly, optimize all remaining parameters, and save the optimized objective, nuisance estimates, convergence status, and useful optimizer diagnostics.
Save the maximized log likelihood, the target estimate, the nuisance estimates, and the convergence information. This provides a reference fit for checking the profile.
For a chosen \(\psi_j\), impose \(\psi=\psi_j\) as a fixed constraint. Do not merely add a penalty that permits the optimizer to move away from \(\psi_j\).
Maximize \(\ell(\psi_j,\lambda)\) over the allowed nuisance-parameter space. Use feasible starting values and, where needed, several starts.
Retain a profile value only after checking convergence, constraint satisfaction, and agreement across suitable starts. Save the nuisance estimates and the optimizer's objective as well as its status.
Proceed through the grid, then compare the value at the full-model estimate with the unconstrained maximum. Investigate failed or inconsistent points rather than silently dropping them.
Warm starts can make a grid calculation faster: use the nuisance estimates from a nearby successful grid point to initialize the next optimization. They should not be the only check. A path of warm starts can track a local optimum, and results can depend on the direction in which the grid is traversed. At selected points, use additional starts, including starts based on the unconstrained fit, and compare the resulting optimized objective values.
Worked Example: Comparing Constrained Optimizer Starts
Suppose a hypothetical NLME analysis uses concentration data from 24 subjects, with clearance \(CL\) as the target parameter and volume, between-subject variability, and residual error among the nuisance parameters. The unconstrained fit gives \(CL=5.2\) L/h and a minimum \(-2\ell=16.80\). At the grid point \(CL=4.5\) L/h, two nuisance-parameter starting sets are tried. The target is fixed at 4.5 L/h in both runs.
| Starting set | Final \(-2\ell\) | Maximum scaled gradient | Convergence and constraints |
|---|---|---|---|
| A | 18.42 | 0.0008 | Converged; all constraints satisfied |
| B | 18.42 | 0.0011 | Converged; all constraints satisfied |
The runs agree on the objective value and give small scaled gradients, so they provide stronger evidence for the constrained optimum than a single run would. The profile log likelihood at 4.5 L/h is \(-18.42/2=-9.21\), while the unconstrained maximum is \(-16.80/2=-8.40\). Thus the profile log likelihood is \( -9.21-(-8.40)=-0.81\) relative to the maximum at this grid point. Equivalently, the increase in \(-2\ell\) is \(18.42-16.80=1.62\). These are profile-computation summaries; no inferential cutoff is applied here.
If another start had produced \(-2\ell=17.95\), that smaller value would indicate a better likelihood because the optimizer minimizes \(-2\ell\). The discrepancy would be a warning that the reported value 18.42 is not yet established as the best constrained fit. The analyst should investigate starts, constraints, scaling, and convergence before accepting a profile value.
Grid Scale, Diagnostics, and Failure Modes
The profile is a collection of constrained optimization results, so its reliability depends on both grid design and optimizer behavior. A coarse grid can miss sharp changes or make the location of important features difficult to resolve. A grid concentrated too narrowly may fail to show how the fit behaves over the range of interest. Use an initial grid to map the range, then add points where the objective changes rapidly, optimization becomes unstable, or greater resolution is needed. Interpolation can summarize a computed curve, but it does not replace constrained fits at points requiring greater accuracy.
Several checks are particularly useful:
- Check the fixed value. Confirm that the saved target equals the requested grid value to the model's numerical precision. A penalty approximation or an incorrectly specified bound may leave it free to move.
- Check feasibility. Confirm that every nuisance estimate respects its parameter bounds and that derived quantities, such as covariance matrices, remain admissible.
- Check the objective and convergence. Inspect the objective value, gradient or comparable optimizer diagnostics, and termination message. A success flag is helpful, but not sufficient by itself.
- Compare starts and directions. At difficult points, compare independent starts and, for warm-start calculations, runs traversing the grid in both directions. Meaningfully different final objective values require investigation.
- Check continuity cautiously. Abrupt changes in nuisance estimates or profile values may reflect a real change in the best-fitting solution, but can also signal a local optimum, numerical instability, or an inadequate grid. Do not smooth away a discrepancy without diagnosing it.
- Check the full-fit anchor. Include or evaluate the target at its unconstrained estimate and confirm agreement with the unconstrained objective within a stated numerical tolerance.
A profile value can fail to be trustworthy even when the model fit at the unconstrained estimate is stable. For example, a fixed target value may force the optimizer near a boundary, expose a poorly scaled direction, or make the likelihood approximation less stable. Conversely, a slowly changing objective is not by itself proof of an error; it may reflect weak information in the data. The appropriate response is to inspect the optimization and model constraints, not to infer a cause from the curve alone.
Worked Example: A Positive Parameter on a Log-Spaced Grid
Consider a hypothetical analysis with \(CL>0\) L/h as the target. To cover values from 2 to 8 L/h, use the log-spaced grid \(CL=2,4,8\) L/h; adjacent points differ by a factor of two. At each point, the nuisance parameters are optimized. Suppose the resulting constrained minima of \(-2\ell\) are 52.0, 49.6, and 50.2, respectively. The best value among these grid points is at \(CL=4\) L/h.
Relative to that grid maximum, the increases in \(-2\ell\) are \(52.0-49.6=2.4\) at 2 L/h, \(49.6-49.6=0\) at 4 L/h, and \(50.2-49.6=0.6\) at 8 L/h. Since \(\ell=-(-2\ell)/2\), the corresponding profile log-likelihood differences are \(-1.2\), \(0\), and \(-0.3\). The optimizer's nuisance estimates should be saved at all three points; the target grid alone is not enough to reproduce or assess the calculations.
On the log scale, the grid locations are \(\log(2)\), \(\log(4)\), and \(\log(8)\), with equal successive differences \(\log(2)\). This avoids treating a one-unit change as equally meaningful at very small and very large positive clearances. If an optimizer is parameterized in \(\log(CL)\), fixing \(CL=4\) L/h means fixing \(\log(CL)=\log(4)\), not fixing the raw parameter to 4. The mapping between the reported scale and the optimization scale must be explicit.
Alternatives and Limits of the Computation
A Wald interval uses local curvature near the unconstrained estimate and can be convenient when that approximation is adequate; the earlier material on when Wald intervals fail explains why it can be unreliable in difficult NLME settings. A profile calculation instead refits the nuisance parameters at selected target values, which can reveal behavior that a local approximation does not capture. Bootstrap intervals use resampled or simulated datasets and therefore address uncertainty through a different computational procedure. These approaches are not interchangeable, and none removes the need for a defensible model and reliable numerical fitting.
A computed profile is limited by its grid resolution, the optimizer's ability to find the best constrained fit, and the accuracy of the likelihood evaluation. It does not automatically provide an interval: a later step must specify how profile values are converted into inferential limits. In particular, do not choose a cutoff or declare an endpoint just because a plotted curve appears to cross a visually convenient level. Keep the profile values and their numerical diagnostics separate from that later inferential decision.
A common mistake is to fix the target parameter and also hold all nuisance parameters at their unconstrained estimates, then label the resulting likelihood values a profile. That calculation is a slice. The correction is to maximize over the nuisance parameters at each fixed target value, verify the optimization, and retain enough information to reproduce the result. If the constrained optimizations are not reliable, a smooth-looking curve does not make them reliable.
Check Your Understanding
Answer each question, then compare with the solutions below.
- For \(\ell_p(\psi)=\sup_{\lambda\in\Lambda(\psi)}\ell(\psi,\lambda)\), what is held fixed and what is optimized at one grid point?
- A constrained fit returns \(-2\ell=24.0\), while the unconstrained fit returns \(-2\ell=21.0\). Calculate the difference in profile log likelihood relative to the unconstrained maximum.
- Why should a warm-start profile sometimes be computed in both grid directions?
- A positive clearance is optimized using \(\xi=\log(CL)\). What value of \(\xi\) corresponds to fixing \(CL=6\) L/h?
- One start gives a converged constrained fit with \(-2\ell=31.2\); another gives \(-2\ell=30.7\). Which fit is better, and what should be done before recording the profile point?
Solutions. 1. Fix \(\psi\) exactly at the chosen grid value and maximize the likelihood over admissible \(\lambda\). 2. The log-likelihood difference is \(-24.0/2-(-21.0/2)=-12.0+10.5=-1.5\); the constrained profile value is 1.5 log-likelihood units below the unconstrained maximum. 3. Traversing both directions can reveal path dependence, such as warm starts following different local optima. 4. Fix \(\xi=\log(6)\), approximately 1.792 when the numerical value is expressed in L/h. 5. The fit with \(-2\ell=30.7\) has the larger likelihood because it has the smaller negative twice log likelihood. Investigate starts, constraints, and convergence, and seek a reproducible best constrained fit before recording the point.