Normal Curvature of the Likelihood Displacement

Why Normal Curvature Matters

A selection-model likelihood can respond differently to small changes in different aspects of the analysis: subject weights, covariates in a selection equation, or other specified perturbations. A local quadratic approximation summarizes that response near the unperturbed analysis. The normal curvature measures how quickly the likelihood displacement grows as the perturbation moves away from its baseline. Its largest value over unit-length perturbations gives a concise measure of the most sensitive local direction.

The derivation is useful because curvature depends on two ingredients with distinct roles: the likelihood's local information about the parameters and the way the perturbation changes the score. This tutorial derives their combination and states carefully which likelihood difference is being approximated. The earlier material on likelihood displacement and its local quadratic approximation motivates the quantity; the focus here is its curvature matrix and interpretation.

Setup and Definition

Let \(\ell(\theta;\omega)\) be a twice continuously differentiable log likelihood for a selection model, where \(\theta\) is a \(p\)-dimensional parameter vector and \(\omega\) is a \(q\)-dimensional perturbation vector. The unperturbed value is \(\omega_0\), and \(\hat{\theta}\) is an interior local maximizer of \(\ell(\theta;\omega_0)\). Thus the baseline score is zero. Assume the observed information matrix at this fit is positive definite:

The mixed derivative matrix describes how the score changes with the perturbation:

For \(\omega\) near \(\omega_0\), let \(\hat{\theta}(\omega)\) denote the local maximizer that continues from \(\hat{\theta}\). We use the positive refitting gain as the likelihood displacement:

Both likelihood values in this definition use the same perturbed likelihood. The first is evaluated at its refitted maximizer and the second at the baseline parameter. This makes the displacement nonnegative when the refitted point maximizes that perturbed likelihood over the parameter region being compared.

Deriving the Curvature Matrix

The derivation uses the score equation for the perturbed fit. Write \(a=\omega-\omega_0\) and \(d=\hat{\theta}(\omega)-\hat{\theta}\). Differentiability and the implicit function theorem give a local smooth solution to the score equation when \(J\) is nonsingular. Expanding that equation around the baseline gives

Since \(J\) is invertible, this first implies \(d=O(\|a\|)\), and then

Now compare the two parameter values under the same perturbed likelihood. At the baseline parameter, the perturbed score is \(\ell_\theta(\hat{\theta};\omega_0+a)=\Delta a+o(\|a\|)\), because the score at \((\hat{\theta},\omega_0)\) is zero. Taylor expansion in the parameter gives

Substituting \(d=J^{-1}\Delta a+o(\|a\|)\), and using the symmetry of \(J\), yields the local improvement:

Multiplication by two gives \(LD(\omega_0+a)=a'\Delta'J^{-1}\Delta a+o(\|a\|^2)\), or equivalently the definition's \(\tfrac12a'Ca\) form. The factor of two in \(C\) therefore corresponds to defining curvature as the second derivative of displacement along a direction.

This result also explains the matrix's key properties. Since \(J\) is positive definite, \(J^{-1}\) is positive definite, so \(a'Ca=2(\Delta a)'J^{-1}(\Delta a)\geq 0\). Thus \(C\) is positive semidefinite. If \(\Delta a=0\), the perturbation direction has zero curvature to this order: it produces no first-order score change, although higher-order effects are not ruled out.

Worked Curvature Calculations

Largest Curvature and Interpretation

For any unit vector \(a\), the directional curvature is the quadratic form \(a'Ca\). The largest possible value over all unit directions is the largest eigenvalue of the symmetric matrix \(C\); a direction attaining it is a locally most sensitive unit perturbation direction. This characterization follows from the Rayleigh-quotient result for symmetric matrices. It is a local statement: it compares perturbations using the specified coordinate scale and approximation near \(\omega_0\), not arbitrary finite changes.

A large maximum curvature means that some small unit-length perturbation can produce relatively large likelihood displacement, to second order. It does not establish that the perturbation is plausible, that the model is wrong, or that a particular treatment-effect contrast changes substantially. Curvature summarizes refitting of the full parameter vector; a target-specific assessment requires a different diagnostic.

The units and scaling of \(\omega\) matter. If one coordinate is measured in days and another in years, a unit Euclidean perturbation treats those numerical units as comparable, which may be scientifically inappropriate. Rescaling coordinates changes \(C\) and its largest curvature. Use a defensible perturbation scale, such as prespecified comparable changes, before interpreting the maximum.

Assumptions, Diagnostics, and Alternatives

The derivation relies on a smooth likelihood near the baseline, an interior stationary fit, and nonsingular positive-definite observed information. It also assumes that the perturbed local maximizer continues smoothly from the baseline one. In selection models, numerical instability, weak identification, boundary estimates, or multiple competing maxima can undermine these conditions. The curvature then describes, at best, one local branch and should not be treated as a global summary.

Useful checks include verifying that the fitted score is close to zero, checking that \(J\) is symmetric and positive definite, and confirming that the perturbation derivatives use the same parameter and perturbation scales as the likelihood. A numerical check compares directly refitted displacements at small \(t\) with \(\tfrac12 t^2a'Ca\); the ratio should approach one as \(t\) decreases when the approximation is working. Large discrepancies, a ratio that does not stabilize, or branch changes in the optimizer may indicate derivative errors, an overly large perturbation, or failure of local regularity.

Normal curvature complements rather than replaces other influence approaches. Exact case deletion assesses a finite change to the observed data contribution, while a first-order case-deletion approximation targets that particular removal. Weighted-likelihood perturbations provide a chosen coordinate system for subject influence; normal curvature summarizes their joint local response over directions. Likelihood-based sensitivity curves can instead track a specified treatment effect as a sensitivity parameter varies. These methods answer different questions, and a large normal curvature should be followed by inspection of the corresponding perturbation scale and practical consequences.

Common Pitfalls and Limitations

• Mixing perturbation values in the likelihood difference. The baseline fit evaluated at \(\omega_0\) and the perturbed fit evaluated at \(\omega\) include direct changes to the likelihood. Their difference is not the refitting gain derived here.
• Reversing the sign. The positive gain is the perturbed maximum minus the baseline parameter's likelihood under that same perturbation. Reversing the order produces a nonpositive quantity.
• Dropping the factor of two. With the convention \(LD=2\) times the gain, the matrix is \(C=2\Delta'J^{-1}\Delta\), and the expansion is \(\tfrac12a'Ca\).
• Ignoring units. A largest curvature is meaningful only relative to the perturbation coordinates and their scaling.
• Overinterpreting local evidence. Small-perturbation curvature does not guarantee accurate predictions for large changes or diagnose causal effects.