Reparameterization and Profile-Likelihood Endpoints

Why Parameter Scale Matters

A clearance estimate may be reported in its natural positive units, such as L h\(^{-1}\), while the model is fitted using log-clearance to enforce positivity. A confidence interval calculated on one scale does not keep the same numerical endpoints on the other. The essential point is that a one-to-one transformation changes the coordinates used to describe the parameter, not the likelihood evidence attached to each corresponding model.

This tutorial considers how a profile-likelihood confidence set transforms under a one-to-one parameter transformation. The central example is \(\eta=\log(CL/c_0)\), where \(CL\) is clearance and \(c_0=1\ \mathrm{L\,h^{-1}}\) is a fixed reference unit. Using the reference unit makes the logarithm dimensionless. Thus \(CL=c_0\exp(\eta)\).

Definition: Let \(\psi\) be a scalar parameter of interest and let \(\lambda\) denote all remaining model parameters (the nuisance parameters). For log-likelihood \(\ell(\psi,\lambda)\), the profile log-likelihood is \(\ell_p(\psi)=\sup_{\lambda}\ell(\psi,\lambda)\), where the supremum is taken over the allowed nuisance-parameter values while \(\psi\) is fixed. A likelihood-ratio profile confidence set is the set of \(\psi\) values satisfying \(2\{\ell(\hat{\psi},\hat{\lambda})-\ell_p(\psi)\}\leq q\), for a specified cutoff \(q\).

The Transformation Result

Suppose the parameter of interest is represented instead by \(\phi=g(\psi)\), where \(g\) is one-to-one on the allowed parameter space. For any fixed \(\phi\), there is exactly one corresponding \(\psi=g^{-1}(\phi)\). The constrained models used to calculate the two profiles are therefore the same models, expressed with different coordinates. This gives the profile relationship

\ell_{p,\phi}(\phi)=\ell_{p,\psi}\!\left(g^{-1}(\phi)\right).

The unrestricted maximum likelihood is also unchanged: the transformation simply relabels the parameter values over which the maximum is taken. Consequently, the likelihood-ratio statistic at corresponding values is identical. If \(C_\psi\) denotes the profile confidence set on the \(\psi\) scale, then the set on the \(\phi\) scale is

C_\phi=\{g(\psi):\psi\in C_\psi\}=g(C_\psi).

This result does not require the confidence set to be symmetric or even connected. The transformed set is its image under \(g\). If \(C_\psi\) is one interval and \(g\) is strictly increasing, its endpoints map in the usual order. If \(g\) is strictly decreasing, the mapped endpoints must be swapped to retain lower-to-upper order. For a disconnected set, transform each component rather than replacing the set with a single interval.

Conditions: The transformation must be one-to-one over the parameter space being considered, and corresponding constrained models must have the same likelihood and nuisance-parameter space. The usual likelihood-ratio cutoff also requires its own calibration assumptions; the familiar chi-square cutoff is an asymptotic result under regularity conditions, including an identifiable interior parameter and a sufficiently well-behaved likelihood. At a boundary or under nonidentifiability, the cutoff may need different justification.

For log-clearance, the inverse transformation is \(CL=c_0\exp(\eta)\), which is strictly increasing. Therefore a profile confidence interval \([\eta_L,\eta_U]\) maps to \([c_0\exp(\eta_L),c_0\exp(\eta_U)]\). There is no Jacobian multiplier in this likelihood calculation. A Jacobian appears when transforming a probability density, such as a Bayesian posterior density; a likelihood compares the same data models at corresponding parameter values.

Worked Transformations

Worked Example: Mapping Log-Clearance Endpoints to Clearance

Suppose a hypothetical NLME analysis uses \(\eta=\log(CL/c_0)\), with \(c_0=1\ \mathrm{L\,h^{-1}}\), and gives the profile-likelihood interval \([-0.1177,\ 0.4823]\) for \(\eta\). The maximum-likelihood clearance estimate is \(1.200\ \mathrm{L\,h^{-1}}\). Transform the interval and describe its width on the clearance scale.

The log-clearance interval is ordered from lower to upper, and the exponential transformation is increasing, so the endpoints stay in that order:

CL_L=c_0\exp(-0.1177) =(1\ \mathrm{L\,h^{-1}})(0.8889627) =0.8890\ \mathrm{L\,h^{-1}},

CL_U=c_0\exp(0.4823) =(1\ \mathrm{L\,h^{-1}})(1.6198) =1.620\ \mathrm{L\,h^{-1}}.

Thus the mapped interval is approximately \([0.8890,\ 1.620]\ \mathrm{L\,h^{-1}}\). Relative to the estimate, its lower-side width is \(1.200-0.8890=0.3110\ \mathrm{L\,h^{-1}}\), while its upper-side width is \(1.620-1.200=0.420\ \mathrm{L\,h^{-1}}\) using the rounded endpoint. The equal distances of 0.3000 on the log scale have not remained equal distances on the clearance scale.

Worked Example: Checking an Endpoint Against the Likelihood-Ratio Cutoff

Consider another hypothetical profile for log-clearance. Suppose its maximum log-likelihood is \(-245.200\), and a 95% profile interval uses the one-parameter asymptotic cutoff \(q=3.841\). The profile curve reaches \(-247.1205\) at a proposed endpoint \(\eta_L=-0.1177\). Check whether this point meets the endpoint criterion and state its clearance value.

At this candidate point, the likelihood-ratio statistic is

2\{\ell_{\max}-\ell_p(\eta_L)\} =2\{-245.200-(-247.1205)\} =2(1.9205) =3.841.

It is therefore on the specified cutoff, subject to rounding, and is an endpoint under the stated profile and cutoff. Its corresponding clearance is

CL_L=(1\ \mathrm{L\,h^{-1}})\exp(-0.1177) =0.8890\ \mathrm{L\,h^{-1}}.

The endpoint is not obtained by taking a new likelihood-ratio cutoff on the clearance scale. The same fitted model at the corresponding clearance has the same likelihood-ratio statistic.

Worked Example: A Decreasing Transformation Reverses Endpoint Order

Using the interval \([0.8890,\ 1.620]\ \mathrm{L\,h^{-1}}\), define the reciprocal parameter \(R=1/CL\). This is a one-to-one decreasing transformation for positive clearance. Find the mapped interval in \(\mathrm{h\,L^{-1}}\).

Because the transformation is decreasing, the image of the original upper endpoint is the new lower endpoint:

R_L=\frac{1}{1.620\ \mathrm{L\,h^{-1}}} =0.6173\ \mathrm{h\,L^{-1}}, \qquad R_U=\frac{1}{0.8890\ \mathrm{L\,h^{-1}}} =1.125\ \mathrm{h\,L^{-1}}.

The transformed interval is \([0.6173,\ 1.125]\ \mathrm{h\,L^{-1}}\). Writing the reciprocals without reversing the order would give a lower endpoint greater than the upper endpoint, a clear sign that the interval has been reported incorrectly.

Why Endpoint Widths Change

For a smooth, increasing transformation \(g\), a small change \(\Delta\psi\) near an estimate \(\hat{\psi}\) produces approximately \(g'(\hat{\psi})\Delta\psi\) change on the transformed scale. This follows from a first-order Taylor expansion:

g(\hat{\psi}+\Delta\psi) =g(\hat{\psi})+g'(\hat{\psi})\Delta\psi +o(\Delta\psi).

For \(CL=c_0\exp(\eta)\), the derivative is \(c_0\exp(\eta)=CL\). Thus a fixed log-scale change corresponds approximately to a larger absolute clearance change at larger clearance. For substantial endpoint distances, the exact exponential mapping, not this local approximation, determines the transformed endpoints. This explains why equal log-scale distances around an estimate generally produce unequal absolute distances in clearance units.

The mapped interval remains a profile-likelihood interval with the same likelihood-ratio evidence under the transformation. Its asymmetry on the clearance scale is not evidence of a calculation error. It reflects the nonlinear relationship between log-clearance and clearance, and is often more faithful to the shape of the likelihood than a symmetric interval built directly around the clearance estimate.

Implementation, Diagnostics, and Alternatives

There are two mathematically equivalent workflows: profile \(\eta\) and map the resulting set through \(CL=c_0\exp(\eta)\), or profile \(CL\) directly and solve for the clearance values where the likelihood-ratio statistic reaches the cutoff. In a correctly implemented fit, both workflows give the same set. Direct profiling in clearance may be convenient for reporting, while profiling in log-clearance often makes positivity automatic and can be numerically convenient.

Check the result by confirming that each reported endpoint corresponds to the intended constrained model and that its likelihood-ratio statistic is at the cutoff, within numerical tolerance. Also check that the mapped endpoints are ordered, have the intended units, and are consistent with the parameter domain. If the two workflows disagree materially, investigate optimization convergence, parameter bounds, the nuisance-parameter reoptimization, and whether both calculations used the same likelihood and cutoff. A profile that fails to reach the cutoff on one side does not supply a finite endpoint on that side merely because a transformed estimate can be calculated.

A Wald interval is a different construction: it uses a local estimate and standard error, and its endpoints depend on the scale on which that approximation is made. Transforming a Wald interval from log-clearance does not make it the same as a Wald interval calculated on clearance. Bootstrap intervals also arise from a different procedure; when a bootstrap interval is defined by quantiles and the transformation is monotone, transforming its endpoints maps the quantile interval to the transformed scale (with endpoint order reversed for a decreasing transformation). These alternatives should not be confused with the profile set, whose invariance follows from the likelihood-ratio statistic and corresponding constrained models.

Common Mistakes and Limitations

Keeping equal endpoint distances after exponentiation. A log interval symmetric around its estimate is not generally symmetric in clearance units. Transform each endpoint separately.
Failing to reorder endpoints for a decreasing map. For a reciprocal transformation, take the reciprocal of the original upper endpoint first to obtain the new lower endpoint.
Adding a Jacobian to the likelihood. A density transformation is not needed when the same likelihood is evaluated at corresponding parameter values.
Treating a mapped interval as a remedy for a poor profile. Reparameterization cannot resolve practical nonidentifiability, a flat or irregular profile, an unreliable optimizer, or inappropriate likelihood-ratio calibration.

The transformation result is exact for corresponding likelihood-based sets, but the confidence interpretation still depends on how the cutoff is justified. In particular, mapping a set does not correct a failure of the regularity assumptions behind a chi-square approximation. Nor does mapping an interval make it appropriate to report only one number: the scale and units should be stated so readers can interpret the uncertainty.

Key takeaway: Under a one-to-one reparameterization, transform the entire profile-likelihood confidence set. For log-clearance to clearance, exponentiate both endpoints in order; for a decreasing transformation, reverse their order. The set's likelihood-ratio content is unchanged, although its numerical width and symmetry can change.

Check Your Understanding

Answer each question, then compare with the short solutions below.

A log-clearance profile interval is \([0.10,\ 0.40]\), with \(c_0=1\ \mathrm{L\,h^{-1}}\). Write the clearance interval approximately.
Why does a one-to-one transformation preserve the profile likelihood-ratio statistic at corresponding parameter values?
A clearance interval is \([2,\ 5]\ \mathrm{L\,h^{-1}}\). What is the correctly ordered interval for \(R=1/CL\)?
Does exponentiating a log-clearance Wald interval necessarily produce the same interval as a clearance-scale Wald calculation? Explain briefly.
If a log-clearance profile does not cross the likelihood-ratio cutoff on its upper side, can a finite upper clearance endpoint be obtained by exponentiating a finite log endpoint?

Solutions. (1) Exponentiate each endpoint: \([\exp(0.10),\exp(0.40)]\), approximately \([1.105,\ 1.492]\ \mathrm{L\,h^{-1}}\). (2) Each fixed transformed value corresponds to exactly one original value and the same constrained model; the maximized likelihood and unrestricted maximum are therefore unchanged. (3) The reciprocal map is decreasing, so the interval is \([1/5,\ 1/2]=[0.200,\ 0.500]\ \mathrm{h\,L^{-1}}\). (4) No. The Wald approximation depends on the scale and local standard error; transforming its endpoints does not generally reproduce a separately calculated Wald interval. (5) No. Without a cutoff crossing there is no finite upper endpoint on the log scale to transform; the profile may instead indicate an unbounded confidence set in that direction.