Tracing Profile Curves and Finding Interval Endpoints

Why Trace a Profile Curve?

A profile likelihood interval is defined by comparing the best fit at each fixed target value with the unrestricted best fit. In practice, this means evaluating the constrained likelihood over a range of target values and finding where the likelihood-ratio curve crosses its cutoff. A grid reveals the curve’s shape and helps locate crossings; a root-finding method then refines their locations.

Numerical results need checks. An optimizer can stop without finding the constrained maximum, and a coarse grid can miss a crossing or make an endpoint look more precise than it is. We develop these steps in a Poisson regression where the calculations can be checked directly.

Poisson Regression Example and Profile Definition

Suppose there are two covariate groups, \(x=0\) and \(x=1\), with 10 observations in each group and unit exposure for every observation. The independent outcomes follow \(Y_i\sim\operatorname{Poisson}(\mu_i)\), with log mean \(\log(\mu_i)=\alpha+\beta x_i\). The observed group totals are 12 at \(x=0\) and 20 at \(x=1\). Write \(T_0=12\), \(T_1=20\), and \(T=T_0+T_1=32\).

Here the likelihood, omitting terms that do not depend on \(\alpha\) or \(\beta\), has log-likelihood

At fixed \(\beta\), differentiating with respect to the nuisance intercept gives \(32-10e^\alpha(1+e^\beta)=0\). The second derivative is \(-10e^\alpha(1+e^\beta)\), which is strictly negative for every finite \(\alpha,\beta\). Thus this score equation identifies the unique constrained maximum:

The implied fitted group totals are \(\hat\mu_0(\beta)=32/(1+e^\beta)\) and \(\hat\mu_1(\beta)=32e^\beta/(1+e^\beta)\). Without the constraint on \(\beta\), the fitted totals equal the observed totals, so \(\hat\mu_0=12\), \(\hat\mu_1=20\), \(\hat\alpha=\log(1.2)\), and \(\hat\beta=\log(20/12)\approx0.510826\).

Calculating and Reading the Profile Curve

Substituting the constrained fitted totals into the Poisson likelihood-ratio statistic yields the deviance form

For a convenient equivalent formula, let \(p(\beta)=e^\beta/(1+e^\beta)\). Then \(\hat\mu_1=32p(\beta)\), \(\hat\mu_0=32\{1-p(\beta)\}\), and

The fractions 0.375 and 0.625 are the observed proportions of the total count in the two groups. A grid is a practical diagnostic: it shows whether the curve has the expected minimum, helps identify brackets for roots, and can expose a failed constrained fit. It does not, by itself, determine an endpoint precisely.

Continuity, Monotonicity, and Endpoint Roots

In this model, \(p(\beta)\) is continuous and strictly between 0 and 1 for every finite \(\beta\). The displayed formula for \(R\) is therefore a composition of continuous functions with positive logarithm arguments, so \(R\) is continuous on the real line. Differentiating the profile log-likelihood, or differentiating the deviance expression, gives

Since \(p(\beta)\) increases strictly and equals \(20/32\) at \(\hat\beta=\log(20/12)\), \(R'(\beta)<0\) to the left of \(\hat\beta\) and \(R'(\beta)>0\) to its right. Consequently, \(R\) decreases to its minimum of zero and then increases. Any level \(c>0\) crossed on each side has exactly one root on each side.

The existence claim is the intermediate value theorem applied to \(f-c\); strict monotonicity prevents two distinct points from having the same function value, proving uniqueness. For profile endpoints, this result justifies using a sign-changing bracket on a monotone side of the curve. It does not justify a root outside the bracket or a root based only on a plotted crossing.

Why Bisection Works and How to Check an Optimizer

Here is the convergence argument for bisection. Let the initial bracket be \([a_0,b_0]\), with opposite signs for \(R-c\), and let its width be \(w_0=b_0-a_0\). At each iteration, continuity ensures a root remains in the half-bracket selected by the sign change. The width after \(k\) iterations is \(w_k=w_0/2^k\). Since \(w_k\to0\), the retained nested intervals shrink to a single point; continuity implies that point has \(R(\beta)=c\). If the root is unique on the bracket, the bisection estimates converge to that root. In particular, to ensure a bracket width no greater than \(\varepsilon>0\), it suffices to take \(k\geq\lceil\log_2(w_0/\varepsilon)\rceil\) halvings.

When constrained maximization is numerical, several checks matter. At each fixed target value, verify that the nuisance-parameter score is near zero and that the fit has not stopped at an inadmissible value. Check the optimizer’s convergence flag, gradient tolerance, and stability from more than one starting value when appropriate. Compare the reported constrained log-likelihood with the unrestricted log-likelihood: the constrained maximum cannot exceed the unrestricted maximum, so \(R(\beta)\) should be nonnegative apart from small numerical error. Finally, evaluate nearby grid points and confirm that the profile changes smoothly. A sudden jump can signal a failed optimization or an implementation error rather than a genuine feature of the profile.

Common Pitfall

A common mistake is to call the first grid value with \(R(\beta)>c\) an endpoint. That value is merely outside the confidence set; the endpoint is the crossing between it and a neighboring inside value. Reporting the grid value also makes the answer depend arbitrarily on grid spacing. The correction is to retain a sign-changing bracket and solve \(R(\beta)-c=0\), checking continuity and constrained-fit convergence before trusting the root. A second warning is that a visually smooth curve is not proof that every nuisance optimization succeeded: numerical diagnostics remain necessary.