What a Profile-Likelihood Curve Shows
A profile-likelihood plot shows how well the data support different fixed values of one model parameter after the other parameters have been reestimated. It is useful because uncertainty in an NLME parameter may not be well summarized by a symmetric interval around its estimate. A curve can reveal unequal support on the two sides of the optimum, a broad region with little change in likelihood, or more than one region that satisfies a confidence cutoff.
Let \(\psi\) be the scalar parameter of interest and let \(\lambda\) collect the other parameters, including any nuisance parameters. Write the log likelihood as \(\ell(\psi,\lambda)\). In an NLME analysis, this will generally be the likelihood for the observed data under the specified model, with the random effects handled according to that model’s likelihood formulation.
Thus, the profile curve is not usually the likelihood evaluated while all other parameters are held at their full-model estimates. It traces the best fit available at each fixed value of \(\psi\). The earlier material on reoptimizing nuisance parameters in a profile develops that optimization step; here the emphasis is on reading the resulting curve.
Plots use different vertical-axis conventions. A log-likelihood plot is highest at the optimum. A plot of the likelihood-ratio statistic \(D(\psi)=2\{\ell_{\max}-\ell_p(\psi)\}\), sometimes called a profile deviance, is lowest at the optimum. These display the same information with opposite vertical orientation. Always check the axis label before deciding whether a peak or a trough is the best-supported value.
Reading a Confidence Threshold
For a scalar parameter with regular likelihood behavior, a likelihood-ratio confidence set at confidence level \(1-\alpha\) contains the values satisfying \(D(\psi)\leq \chi^2_{1,1-\alpha}\), where \(\chi^2_{1,1-\alpha}\) is the corresponding quantile of a chi-square distribution with one degree of freedom. For a 95% confidence set, the usual asymptotic cutoff is approximately 3.841. On a log-likelihood plot this is equivalent to retaining values for which the log likelihood is no more than \(3.841/2=1.9205\) below \(\ell_{\max}\).
The endpoint or endpoints are found where the profile crosses the cutoff. On a deviance plot, they are the intersections with the horizontal line at 3.841 for a 95% set. On a log-likelihood plot, they are intersections with the line \(\ell_{\max}-1.9205\). If the profile is above the deviance cutoff between two crossings, that intervening region is excluded from the confidence set.
This cutoff is an asymptotic likelihood-ratio approximation, not a guarantee of exact coverage in every NLME data set. Its interpretation relies on an adequately specified likelihood, a well-defined optimum, and regularity conditions for the parameter and model. An endpoint read from a plot is also only as reliable as the optimization and interpolation used to construct the curve.
Worked Example: Reading a Clearance Profile Interval
Suppose a hypothetical NLME analysis profiles clearance \(CL\), measured in L/h. The maximized log likelihood is \(\ell_{\max}=-120.00\) at \(\widehat{CL}=5.0\) L/h. The fitted profile has \(D(CL)=3.84\) at approximately 3.2 and 8.1 L/h, with values below 3.84 between those points and above 3.84 just outside them.
For a 95% profile-likelihood set, the deviance cutoff is 3.841. In log-likelihood units, its threshold is \(\ell_{\max}-3.841/2=-120.00-1.9205=-121.9205\). The stated crossing values correspond to log likelihoods of about \(-121.92\).
The resulting approximate 95% profile interval is 3.2 to 8.1 L/h. It is asymmetric about the estimate: the lower distance is \(5.0-3.2=1.8\) L/h, while the upper distance is \(8.1-5.0=3.1\) L/h. Report the estimate and endpoints with their units; do not replace the interval with a symmetric margin merely because the maximum lies between the endpoints.
This reading assumes the profile was computed by optimizing nuisance parameters at each fixed clearance, that the reported optimum is the relevant maximum, and that the regular one-parameter cutoff is reasonable for this setting. In practice, the crossings may be interpolated between grid points, so their precision should not exceed what the profile computation supports.
Asymmetry, Scale, and Absorption Parameters
The two sides of a profile need not have the same shape. If the log likelihood falls sharply on one side of its maximum and gradually on the other, the confidence interval will have unequal distances from the estimate. This may occur in nonlinear models because parameter effects are nonlinear or because nuisance parameters compensate differently at different fixed values. It is a feature of the likelihood’s support, not by itself proof of a model defect.
How a curve looks also depends on its horizontal axis. For a positive absorption-rate parameter \(k_a\), a plot against \(k_a\) displays equal absolute changes equally, while a plot against \(\log(k_a)\) displays equal ratios equally. The same underlying profile can therefore look more or less symmetric under different axis choices. The interval endpoints must still be stated on the parameter scale of interest, and any transformation used for plotting should be made clear.
Worked Example: Interpreting an Asymmetric Absorption Profile
In a hypothetical oral NLME model, suppose the profile for the absorption rate constant \(k_a\), measured in h\(^{-1}\), is maximized at \(\widehat{k_a}=1.2\) h\(^{-1}\), with \(\ell_{\max}=-85.40\). The profile crosses the 95% cutoff at approximately 0.55 and 3.6 h\(^{-1}\); it is inside the cutoff between these crossings.
The log-likelihood threshold is \(-85.40-3.841/2=-87.3205\). Thus the approximate profile interval is 0.55 to 3.6 h\(^{-1}\). Its distances from the estimate are \(1.2-0.55=0.65\) h\(^{-1}\) and \(3.6-1.2=2.4\) h\(^{-1}\), so the interval is markedly asymmetric on the original scale.
That asymmetry warns against assuming that a symmetric interval around 1.2 h\(^{-1}\) captures the same likelihood support. If the graph instead uses \(\log(k_a)\) on the horizontal axis, equal distances on the graph represent equal multiplicative changes, not equal changes in h\(^{-1}\). The estimate and interval should therefore be interpreted on the stated parameter scale, not inferred from visual distances on an unspecified transformed axis.
Flat Regions and Multiple Crossings
A broad, nearly level part of a profile means the likelihood changes little as the parameter moves across that region, after nuisance parameters have been reoptimized. If the profile does not reach the confidence threshold within the evaluated range, the plotted range has not established a bounded endpoint there. The appropriate response is to check the calculation and range, and, if justified, extend the profile—not to report the edge of the plotting window as a confidence limit.
A profile can also cross the cutoff more than twice. In that case, the confidence set is every parameter value satisfying the cutoff, and it may consist of separated regions. Do not automatically report the first two crossings as a single interval: that would include parameter values whose profile is outside the confidence set. Multiple crossings merit checks for adequate grid resolution, convergence at each fixed value, successful nuisance-parameter optimization, and whether the apparent crossings persist under a more careful calculation. A jagged curve or isolated spike may reflect numerical optimization rather than a stable likelihood feature.
Worked Example: A Flat Profile and an Unreached Cutoff
Consider a hypothetical clearance profile with \(\widehat{CL}=6.0\) L/h and deviance values \(D(2)=0.4\), \(D(6)=0\), and \(D(12)=0.5\), with no value above 0.5 anywhere on the computed range from 2 to 12 L/h. The 95% cutoff is 3.841.
Every evaluated value is below 3.841, so the profile does not cross the cutoff in the plotted range. The calculation has not produced finite endpoints within 2 to 12 L/h; it does not justify reporting a 95% interval of 2 to 12 L/h. The very small deviance changes indicate that this profile supplies little discrimination among the evaluated clearance values. First verify the profile computation and that the range and model constraints were handled as intended. If the result is stable, report that the cutoff was not reached over the evaluated range and investigate the uncertainty rather than presenting the plotting limits as confidence limits.
Worked Example: Interpreting Multiple Cutoff Crossings
Suppose a hypothetical deviance profile for a model parameter has \(D=0\) at its global minimum and the 95% cutoff of 3.841 is crossed, in increasing parameter order, at 1.0, 2.0, 4.0, and 5.0 units. Suppose the profile is below 3.841 from 1.0 to 2.0 and from 4.0 to 5.0, and above it between 2.0 and 4.0.
The approximate confidence set is the union of two regions, 1.0 to 2.0 and 4.0 to 5.0 units, subject to the accuracy of the profile and cutoff approximation. Reporting 1.0 to 5.0 as one interval would incorrectly include the intervening values, for which \(D>3.841\). Before interpreting the separated regions substantively, check that the global maximum and all nuisance fits are reliable and that the crossings are not artifacts of a coarse grid or failed optimizations.
Why a Profile May Differ From Other Intervals
A Wald interval uses a local standard error and is typically symmetric on the scale where the approximation is made. It can be convenient when the likelihood is well behaved and approximately quadratic near its maximum, but it may not describe a profile with substantial curvature differences on either side. A profile interval instead uses the likelihood-ratio change while reoptimizing nuisance parameters. A bootstrap interval uses a distribution of estimates from resampled or simulated data and answers uncertainty through that repeated-sampling procedure. These methods can differ because they use different approximations or constructions; a discrepancy is a reason to examine assumptions and computation, not to select an interval solely because it is narrower.
Profile plots also have limitations. They are conditional on the chosen model and likelihood, may be computationally demanding, and can be affected by local optima, parameter constraints, approximation error, or a sparse evaluation grid. A smooth-looking curve is not proof of correct optimization, and a cutoff crossing is not meaningful if the likelihood values used to produce it are unreliable.
Checks Before Reporting
Confirm whether the vertical axis is log likelihood or profile deviance and whether the parameter axis is transformed.
Check that the maximum log likelihood (or minimum deviance) occurs at the reported estimate and corresponds to the full optimization.
For a regular scalar parameter and an asymptotic 95% set, use \(D=3.841\), or a log-likelihood drop of 1.9205.
Record all regions inside the cutoff, and check whether the curve reaches the cutoff at the edge of the computed range.
Inspect optimization convergence, grid resolution, and whether suspicious features persist when the profile is recomputed carefully.
Give the parameter estimate, confidence set or interval, units, confidence level, and any important qualification about the evaluated range or asymmetry.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A profile has \(\ell_{\max}=-50.0\). What log-likelihood value is the approximate 95% threshold for a regular scalar parameter?
- A deviance curve crosses 3.841 at 4 and 9 mg/L and is below the cutoff between them. What is the approximate 95% profile interval?
- An absorption profile is much flatter from 1 to 6 h\(^{-1}\) than near its maximum. What does this say about the profile’s support in that range, and what should be checked?
- A curve crosses the cutoff four times and is below it in two separated regions. Should the entire range from the first to the last crossing be reported as one interval?
Solutions. (1) The threshold is \(-50.0-3.841/2=-51.9205\). (2) The approximate interval is 4 to 9 mg/L, because the curve is inside the cutoff between those crossings. (3) The likelihood changes relatively little across that range after nuisance parameters are reoptimized, so the profile distinguishes those values weakly; check optimization, grid range, and numerical stability before drawing conclusions. (4) No. Report the separated regions satisfying \(D\leq3.841\), not the span that includes values above the cutoff.