Why Profile-Likelihood Intervals Matter
A nonignorable response parameter describes how response depends on an outcome that may be missing. In selection models, such a parameter can be weakly identified: quite different values may yield similar observed-data likelihoods after the outcome-model parameters are adjusted. A standard-error-based interval can then suggest more precision than the likelihood supports. A profile-likelihood interval instead asks which values of the MNAR parameter remain sufficiently compatible with the data after nuisance parameters are re-estimated at each value.
The construction relies on the observed-data likelihood for the specified selection model. It does not make the MNAR parameter identified by the observed data in every application, nor does it remove dependence on model assumptions. Its value is that it displays the shape of the likelihood relevant to the parameter of interest, rather than summarizing that shape only by its curvature at the maximum.
Definition and Likelihood-Ratio Cutoff
Let \(\psi\) be a scalar MNAR parameter of interest, such as the coefficient linking the current outcome to response. Let \(\lambda\) denote all nuisance parameters, including outcome-model and other response-model parameters. Write the observed-data log likelihood as \(\ell(\psi,\lambda)\). For each fixed \(\psi\), maximize over \(\lambda\); this gives the profile log likelihood. Its maximum over \(\psi\) is the unrestricted maximum log likelihood.
For a 95% interval, the one-degree-of-freedom cutoff is \(\chi^2_{1,0.95}=3.841\). Equivalently, retain parameter values for which the profile log likelihood is no more than \(3.841/2=1.9205\) below its maximum. The interval endpoints, when finite, are the values where \(D(\psi)\) crosses 3.841. The set need not be symmetric about \(\hat{\psi}\); if the profile does not cross the cutoff in one direction, the confidence set may be unbounded in that direction.
This cutoff is an asymptotic result. It is justified when the model is correctly specified for the inferential target, the true parameter is identifiable and lies in the interior of its parameter space, the likelihood is sufficiently smooth, and the usual large-sample likelihood approximation applies. The nuisance parameters must be estimable well enough that profiling is meaningful. In an MNAR model, weak identification can undermine these conditions even if an optimizer returns a numerical maximum. An interval computed with the standard cutoff should therefore be described as an asymptotic likelihood-ratio interval, not as a guarantee of accurate finite-sample coverage.
How to Construct the Interval
Find the maximum observed-data log likelihood \(\hat{\ell}\) and its corresponding estimate \(\hat{\psi}\), while estimating all model parameters.
At each fixed value, maximize the log likelihood over every nuisance parameter. Record the resulting \(\ell_p(\psi)\), not the likelihood from holding nuisance estimates fixed.
Calculate \(D(\psi)=2\{\hat{\ell}-\ell_p(\psi)\}\) and retain values for which \(D(\psi)\leq 3.841\) for a nominal 95% interval.
Find where \(D(\psi)=3.841\) on each side of the maximum. Extend the search range if the profile has not crossed the cutoff; do not treat the end of a computational grid as an interval endpoint.
The defining feature of profiling is re-optimization. If nuisance parameters are held at their unrestricted estimates while \(\psi\) changes, the resulting curve is not \(\ell_p(\psi)\). It can fall more sharply and produce an artificially narrow interval.
Worked Example: An Asymmetric Interval for a Response Coefficient
Suppose a hypothetical trial is fitted with a selection model in which \(\psi\) is the log-odds change in response per one-standard-deviation increase in the current outcome. Thus \(\psi\) is dimensionless, and \(\exp(\psi)\) is the response odds ratio for that increase. After profiling all nuisance parameters, the maximum occurs at \(\hat{\psi}=0.80\). Suppose the profile drop is well approximated by the following asymmetric curve near the maximum:
For a 95% interval, the endpoint condition is \(\hat{\ell}-\ell_p(\psi)=1.9205\). On the lower side, \[ \frac{(\psi-0.80)^2}{2(0.20)^2}=1.9205, \qquad |\psi-0.80|=\sqrt{2(1.9205)}(0.20)=1.96(0.20)=0.392. \] The lower endpoint is therefore \(0.80-0.392=0.408\). On the upper side, \[ \frac{(\psi-0.80)^2}{2(0.30)^2}=1.9205, \qquad |\psi-0.80|=1.96(0.30)=0.588, \] so the upper endpoint is \(0.80+0.588=1.388\). The approximate 95% profile-likelihood interval is \([0.408,1.388]\).
On the response-odds-ratio scale, exponentiating the endpoints gives \([\exp(0.408),\exp(1.388)]\), approximately \([1.50,4.01]\). Under this model, the interval is compatible with response odds between about 1.50 and 4.01 times as high per one-standard-deviation increase in the current outcome. It is not symmetric around the estimate on the log-odds scale, and it becomes still less symmetric after exponentiation.
Finding Endpoints From a Computed Profile
In practice, a profile is usually evaluated numerically. A plot helps reveal its shape, but endpoint calculation should use the likelihood values, not visual judgment alone. A root-finding procedure can solve \(D(\psi)-3.841=0\) on each side of the maximum. If only a coarse grid is available, interpolation gives an approximation that should be identified as such; refining the grid or optimizing directly near the crossing improves accuracy.
Worked Example: Approximate Endpoints From Profile Values
In a second hypothetical selection-model analysis, \(\psi\) is again a response log-odds coefficient per one-standard-deviation increase in outcome. The maximized profile occurs at \(\hat{\psi}=0.40\). The table gives profile likelihood-ratio values, each calculated after nuisance parameters were re-optimized. The 95% cutoff is 3.841.
| \(\psi\) | \(D(\psi)\) | Position relative to cutoff |
|---|---|---|
| 0.00 | 4.50 | Above |
| 0.10 | 2.70 | Below |
| 0.40 | 0.00 | Profile maximum |
| 0.80 | 3.20 | Below |
| 1.00 | 4.40 | Above |
The lower crossing lies between 0.00 and 0.10. Linear interpolation gives \[ \psi_L\approx 0.00+ \frac{4.50-3.841}{4.50-2.70}(0.10-0.00) =0.0366. \] The upper crossing lies between 0.80 and 1.00: \[ \psi_U\approx 0.80+ \frac{3.841-3.20}{4.40-3.20}(1.00-0.80) =0.907. \] Thus the grid-based approximate interval is \([0.0366,0.907]\). The profile remains below the cutoff at the maximum and crosses it on both sides, so the table supports a bounded interval. These interpolated endpoints are only approximate; a denser profile calculation or a root finder would be preferable for final reporting.
Unbounded Intervals and Weak Information
An interval can extend indefinitely when the profile likelihood never falls far enough from its maximum in one direction. This is not a computational convention that licenses reporting the edge of a grid as an endpoint. It says that, under the chosen model and cutoff, the data do not exclude arbitrarily large values in that direction. Such a result is especially important for MNAR parameters: a small change in the response mechanism may be hard to distinguish from a compensating change in the outcome model.
Worked Example: A One-Sided Unbounded Confidence Set
Consider a hypothetical fitted profile with maximum at \(\hat{\psi}=0.30\), where \(\psi\) is a response log-odds coefficient per one-standard-deviation increase in outcome. Suppose its likelihood-ratio statistic is described by
On the lower side, solve \(D(\psi)=3.841\): \[ \frac{(\psi-0.30)^2}{0.16}=3.841, \qquad |\psi-0.30|=\sqrt{3.841(0.16)}=0.784. \] The lower endpoint is \(0.30-0.784=-0.484\). On the upper side, put \(u=\psi-0.30>0\). Then \(D(\psi)=3u^2/(u^2+0.25)\), which increases toward 3 as \(u\) increases. Since \(3<3.841\), it never reaches the 95% cutoff. The approximate confidence set is therefore \([-0.484,\infty)\).
The unbounded upper side means this likelihood-ratio rule does not rule out arbitrarily large positive response coefficients. It does not establish that such values are plausible in every substantive sense, nor that the selection model is correct. It reports the information contained in this fitted likelihood relative to the stated cutoff. In a real analysis, numerical accuracy, parameter constraints, and the validity of the regular asymptotic cutoff also need assessment.
Interpretation, Diagnostics, and Alternatives
The interval describes parameter values not rejected by the corresponding likelihood-ratio procedure at its nominal level, under the specified model and its calibration assumptions. In repeated samples satisfying those assumptions, the procedure is designed to cover the true parameter about 95% of the time asymptotically. It is not a 95% probability statement about the fixed parameter after observing the data.
Before interpreting endpoints, inspect whether the profile is maximized where expected and whether nuisance optimization has succeeded across the full parameter range. In particular:
- Check that the profile has been computed by re-optimizing nuisance parameters at every fixed \(\psi\), and that those optimizations have converged.
- Extend the grid far enough to establish whether each side crosses the cutoff. A profile that is still below the cutoff at the grid boundary does not have a finite endpoint established by that grid.
- Look for abrupt jumps, irregular numerical noise, or unexplained discontinuities. These may reflect integration or optimization problems rather than genuine likelihood features.
- Compare results across reasonable starting values and numerical settings. A curve that changes materially with these choices is not a reliable basis for precise endpoints.
- Ask whether the fitted model is weakly informative about \(\psi\). A flat or slowly declining profile can yield very wide or unbounded intervals, even when a single point estimate is reported precisely by software.
A Wald interval uses the estimate and an estimated standard error, typically \(\hat{\psi}\pm1.96SE\) for a nominal 95% interval. It is quick and can be adequate when the likelihood is locally close to quadratic and the estimate is well inside the parameter space. A profile interval instead follows the likelihood away from its maximum, so it can reflect skewness and weakly informed directions. It costs more computation and still depends on model specification and asymptotic calibration; it is not automatically reliable in small samples or under weak identification. The earlier material on profile score and curvature concerns local behavior at the maximum; endpoint construction requires examining the profile over a wider range.
The usual one-parameter chi-square cutoff should not be assumed valid if regularity conditions fail, for example when the estimate is on a parameter-space boundary or the parameter is not identified under the relevant model. Those situations require suitable calibration rather than mechanically applying 3.841. In addition, an interval conditional on one selection model does not account for uncertainty about whether that model is appropriate. Profile likelihood quantifies sampling uncertainty within the specified model; it does not resolve sensitivity to untestable assumptions about missing outcomes.
Common Mistake
Correction: A Wald interval uses local curvature, while a profile-likelihood interval uses the points where \(2\{\hat{\ell}-\ell_p(\psi)\}\) reaches the specified chi-square cutoff. The profile can be asymmetric farther from the maximum, and it may not cross the cutoff on one side. Compute and inspect the profile to determine its actual crossings.
Check Your Understanding
Answer each question, then check the solutions below.
- For a scalar MNAR parameter, what must be maximized at each fixed value of the parameter to obtain its profile log likelihood?
- At a 95% confidence level, what value of \(D(\psi)\) defines the boundary under the usual one-parameter asymptotic cutoff?
- A profile remains below the cutoff as \(\psi\) increases across the entire grid. Can the grid maximum be reported as a finite upper endpoint?
- Why can a profile-likelihood interval be asymmetric even if the estimate and a standard error are available?
- State one condition under which the usual chi-square cutoff may not be justified.
Solutions. (1) Maximize the observed-data log likelihood over all nuisance parameters while holding the chosen MNAR parameter fixed. (2) \(D(\psi)=3.841\) for a nominal 95% interval with one parameter of interest. (3) No. The search must be extended or otherwise checked; if the profile never crosses, the upper side is unbounded under that cutoff, while an insufficient grid alone leaves the endpoint unresolved. (4) The profile can decline at different rates on its two sides, so its likelihood-ratio cutoff crossings need not be equally distant from the maximum. (5) For example, the parameter may lie on the boundary of its parameter space, or it may be weakly identified; then the standard regular asymptotic approximation may fail.