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Likelihood-Ratio Cutoffs for Profile Intervals

You will learn to calibrate a one-parameter NLME profile interval using the appropriate chi-square likelihood-ratio cutoff and to recognize when its asymptotic justification may be unreliable.

Advanced 11 min read

What You'll Learn

  • Define the profile likelihood-ratio statistic for a scalar parameter.
  • Derive the confidence-set cutoff from Wilks' theorem and inversion of a likelihood-ratio test.
  • Convert a chi-square cutoff into the corresponding log-likelihood drop.
  • Calculate profile interval endpoints for symmetric and asymmetric profiles.
  • Check the assumptions and numerical behavior that support or undermine the cutoff.
  • Distinguish likelihood-ratio calibration from Wald and bootstrap interval approaches.

From a Profile to a Confidence Set

A computed profile describes how well the model can fit at each fixed value of a parameter of interest. To turn that curve into an interval, we need a rule for deciding which values remain compatible with the observed data at a chosen confidence level. The likelihood-ratio rule compares the profile log-likelihood at a candidate value with the maximum log-likelihood over all parameter values.

For an NLME model, the likelihood used here is the likelihood of the observed data under the model, with the random effects integrated out. Subjects may have repeated measurements, but the likelihood represents their dependence through the specified model; the independent sampling units are typically subjects, not individual records. The profile for one scalar parameter is obtained by fixing that parameter and maximizing over the other model parameters.

Definition: Let \(\psi\) be a scalar parameter of interest, let \(\lambda\) denote all remaining parameters, and let \(\ell(\psi,\lambda)\) be the log-likelihood. The profile log-likelihood is \(\ell_p(\psi)=\max_{\lambda}\ell(\psi,\lambda)\). Write \(\hat{\psi}\) for the unrestricted maximum-likelihood estimate and \(\hat{\ell}=\ell(\hat{\psi},\hat{\lambda})\) for the maximized log-likelihood. The profile likelihood-ratio statistic at a candidate value \(\psi\) is \(D(\psi)=2\{\hat{\ell}-\ell_p(\psi)\}\).

The factor of 2 is part of the likelihood-ratio statistic. Larger values of \(D(\psi)\) mean that fixing the parameter at \(\psi\) costs more log-likelihood relative to the unrestricted fit. Under suitable regularity conditions, the cutoff for \(D(\psi)\) comes from a chi-square distribution with one degree of freedom.

Why the Cutoff Is Chi-Square

Suppose the candidate value \(\psi_0\) is the true value. Comparing the model with \(\psi=\psi_0\) fixed against the unrestricted model is a likelihood-ratio test of one scalar restriction. Wilks' theorem states that, under regular conditions and as the amount of independent information increases, the likelihood-ratio statistic has an approximate chi-square distribution with degrees of freedom equal to the number of independent restrictions. Here that number is one.

A confidence set can be constructed by inverting this test: retain each candidate value that would not be rejected by the likelihood-ratio test at significance level \(\alpha\). Let \(\chi^2_{1,1-\alpha}\) denote the \(1-\alpha\) quantile of a chi-square distribution with one degree of freedom. Then the approximate \(100(1-\alpha)\%\) profile likelihood confidence set is:

$$ \left\{\psi: D(\psi)\leq \chi^2_{1,1-\alpha}\right\}. $$

If this set is connected, its lower and upper endpoints are the values where \(D(\psi)\) reaches the cutoff. Equivalently, since \(D(\psi)\) is twice the log-likelihood drop, an endpoint is where the profile log-likelihood has fallen by half the chi-square cutoff:

$$ \hat{\ell}-\ell_p(\psi) = \frac{1}{2}\chi^2_{1,1-\alpha}. $$

For a 95% interval, \(\alpha=0.05\) and \(\chi^2_{1,0.95}\approx 3.841\). The corresponding log-likelihood drop is about \(3.841/2=1.9205\). The cutoff is an approximation to repeated-sampling coverage, not a statement that there is a 95% probability the fixed parameter lies in the interval.

Regularity conditions: The chi-square approximation is justified when the model is correctly specified, the parameter is identifiable, the true parameter is in the interior of its allowed parameter space, and the likelihood is sufficiently smooth near the true value. The information matrix must be nonsingular, and the likelihood maximization must find the relevant maximum. For NLME data, the asymptotic argument generally relies on an increasing number of independent subjects and a correctly specified marginal likelihood. The approximation is not guaranteed to be accurate in a small sample or when these conditions fail.

These conditions matter in practice. A variance parameter at zero, a parameter that is weakly identified, or a likelihood with multiple separated maxima can violate the standard approximation or make it unreliable. The cutoff does not repair an incorrect likelihood or a failed optimization. Even when the conditions are plausible, the result is an approximate confidence interval whose finite-sample performance should be assessed in context.

Worked Cutoff Calculations

Worked Example: A Symmetric Profile for Clearance

Hypothetical data: An NLME analysis of 80 subjects estimates typical clearance as \(\hat{\psi}=5.00\) L/h. Suppose the computed profile has the following quadratic form over the range of interest:

$$ \ell_p(\psi)=\hat{\ell}-\frac{(\psi-5.00)^2}{2(0.90)^2}. $$

Assume the fitted model is the working data-generating model, clearance is identifiable and interior, and the optimization is reliable. These assumptions support using the one-parameter chi-square approximation, although 80 subjects do not make that approximation exact.

For a 95% interval, the profile endpoint condition is a log-likelihood drop of \(1.9205\). Substituting the stated profile gives:

$$ \frac{(\psi-5.00)^2}{2(0.90)^2}=1.9205 \quad\Longrightarrow\quad |\psi-5.00|=0.90\sqrt{2(1.9205)} =0.90\sqrt{3.841} \approx 1.764. $$

Thus the approximate profile interval is \(5.00\pm1.764\), or \([3.236,\ 6.764]\) L/h. The values are clearance values that satisfy the likelihood-ratio cutoff under the model and approximation; the interval does not assign a probability distribution to the fixed true clearance.

Worked Example: An Asymmetric Profile for Clearance

Hypothetical data: A second NLME analysis estimates typical clearance as \(4.00\) L/h. Suppose the profile drop is steeper below the estimate than above it, with:

$$ D(\psi)= \begin{cases} (\psi-4.00)^2/(0.50)^2, & \psi\leq 4.00,\\ (\psi-4.00)^2/(1.00)^2, & \psi>4.00. \end{cases} $$

Assume the estimate and the candidate endpoints lie in the interior of the parameter space, the parameter is identifiable, and the profile maximizations are reliable. Applying the 95% cutoff \(D(\psi)=3.841\), the lower endpoint solves:

$$ \frac{(\psi-4.00)^2}{0.50^2}=3.841 \quad\Longrightarrow\quad \psi=4.00-0.50\sqrt{3.841} \approx 3.020\ {\rm L/h}. $$

The upper endpoint solves the upper-branch equation:

$$ \frac{(\psi-4.00)^2}{1.00^2}=3.841 \quad\Longrightarrow\quad \psi=4.00+1.00\sqrt{3.841} \approx 5.960\ {\rm L/h}. $$

The resulting interval is approximately \([3.020,\ 5.960]\) L/h. It is asymmetric around the estimate because the profile falls at different rates on its two sides. A symmetric interval based only on local curvature would not reproduce this shape.

Worked Example: Is a Candidate Value Inside the 95% Set?

Hypothetical profile output: An NLME fit has unrestricted log-likelihood \(\hat{\ell}=-120.00\). At a candidate clearance of \(6.50\) L/h, the maximized profile log-likelihood is \(-121.70\); at \(6.70\) L/h, it is \(-122.05\). Suppose the likelihood is correctly specified, the candidate values are interior, and the relevant maximizations have converged.

For \(6.50\) L/h, the log-likelihood drop is \(1.70\), so the likelihood-ratio statistic is:

$$ D(6.50)=2\{-120.00-(-121.70)\}=3.40. $$

Because \(3.40<3.841\), this value is retained in the approximate 95% confidence set. For \(6.70\) L/h:

$$ D(6.70)=2\{-120.00-(-122.05)\}=4.10. $$

Here \(4.10>3.841\), so \(6.70\) L/h is outside the set. This comparison classifies candidate values; finding the interval endpoints still requires locating where the profile reaches the cutoff on either side of the maximum.

Calculation, Diagnostics, and Alternatives

A reliable cutoff calculation depends on a correctly computed profile. At each fixed candidate value, the likelihood being compared must be on the same scale as the unrestricted maximum: use the same observations, model, likelihood convention, and numerical accuracy. A constant added to every log-likelihood value cancels in the difference, but inconsistencies across fits do not.

Useful checks include confirming that the unrestricted fit has at least as high a log-likelihood as the constrained profiles, that \(D(\hat{\psi})\) is approximately zero, and that the profile is evaluated on both sides of the estimate far enough to locate crossings. Optimization warnings, sensitivity to starting values, erratic profile changes, or a profile that fails to return to the cutoff are reasons to investigate rather than report an automatic interval. A crossing that is not found within the explored parameter range is not evidence of a finite endpoint.

The likelihood-ratio interval has a different emphasis from common alternatives. A Wald interval uses local curvature near the maximum and is usually symmetric on the parameter scale used for calculation. The profile approach instead uses the likelihood across candidate values, so an asymmetric profile can yield an asymmetric interval. Bootstrap intervals use a resampling distribution to calibrate uncertainty; they can be useful when the likelihood-ratio approximation is doubtful, but their quality depends on the resampling design and successful fits. None of these methods compensates for a misspecified or weakly identified model.

Common pitfall: For a 95% one-parameter interval, comparing the log-likelihood drop directly with \(3.841\) uses the wrong threshold. The chi-square cutoff applies to \(D(\psi)=2\{\hat{\ell}-\ell_p(\psi)\}\), so the log-likelihood drop is compared with \(3.841/2\approx1.9205\). Alternatively, double the drop first and compare the result with \(3.841\).

The cutoff has one degree of freedom because each candidate profile comparison fixes one scalar parameter, while allowing the remaining parameters to vary in the profile. It is not determined by the number of observations, repeated measurements, or nuisance parameters. The relevant degrees of freedom count the restrictions tested in this likelihood-ratio comparison.

1
Choose the confidence level.
For a two-sided \(100(1-\alpha)\%\) profile set, obtain \(\chi^2_{1,1-\alpha}\).
2
Use the profile log-likelihood.
At each candidate value, calculate \(D(\psi)=2\{\hat{\ell}-\ell_p(\psi)\}\).
3
Retain values below the cutoff.
The approximate set contains the candidate values with \(D(\psi)\leq\chi^2_{1,1-\alpha}\); check that the profile and optimization are credible before interpreting its endpoints.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

  1. For a 95% interval for one scalar parameter, what is the approximate chi-square cutoff, and what log-likelihood drop does it represent?
  2. A candidate value has profile log-likelihood \(\hat{\ell}-2.10\). Is it retained by the usual 95% one-parameter cutoff?
  3. Why does the standard cutoff use one degree of freedom even if the NLME model has many nuisance parameters?
  4. Name two conditions or computational problems that could make the usual chi-square approximation unreliable.

Solutions. (1) The cutoff is \(\chi^2_{1,0.95}\approx3.841\); the corresponding log-likelihood drop is \(3.841/2\approx1.9205\). (2) The statistic is \(D=2(2.10)=4.20\), which exceeds \(3.841\), so the candidate is not retained. (3) Each profile comparison imposes one restriction, fixing the one scalar parameter of interest; nuisance parameters are maximized over rather than counted as additional restrictions. (4) Examples include a true parameter on the boundary, weak identification, model misspecification, a small number of independent subjects, a singular information matrix, or failure to find the relevant likelihood maximum.

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