Nuisance-Adjusted Curvature for a Target Parameter

Why Adjust Curvature to the Target?

A perturbation to a selection-model likelihood can move several fitted parameters at once. Yet a scientific question may concern only one target, such as a treatment-effect coefficient or a contrast. Curvature for the full parameter vector describes the total local displacement of the fit; it does not, by itself, say how strongly the target moves relative to its own information scale. Nuisance-adjusted target curvature isolates that movement while allowing nuisance parameters to respond.

The distinction matters when a perturbation changes nuisance estimates substantially but barely affects the target, or when it moves the target in a direction where the target is estimated precisely. In both situations, full-vector curvature and target-specific curvature can tell different stories. The calculation below uses the observed selection likelihood at its fitted value and a smooth, local perturbation. It does not require choosing a new missingness perturbation scheme.

Assumptions and the Local Response

The local calculation requires a twice differentiable log likelihood in a neighborhood of the fitted point, an interior stationary fit satisfying \(\ell_\theta(\hat{\theta},\omega_0)=0\), and a nonsingular positive-definite observed information matrix \(M\). The perturbation must be small enough that a nearby fitted solution exists and varies smoothly. For the target calculation, \(c\) must be nonzero and \(c'M^{-1}c>0\). These are local regularity conditions: they do not establish that the selection model is correctly specified or that a global maximum has been found.

Differentiate the fitted score equation with respect to a small change \(h\) in \(\omega\). The derivative of the score with respect to \(\theta\) is \(-M\), while its derivative with respect to \(\omega\) is \(\Delta\). Thus the first-order parameter response is \(M^{-1}\Delta h\). Applying the chain rule to \(g(\hat{\theta})\) gives the target response. The inverse information is essential: it accounts for the joint adjustment of all fitted parameters, not just the target coordinate.

The likelihood-displacement approximation uses the same response. Define the likelihood displacement as twice the loss in maximized log likelihood relative to the reference fit. A second-order Taylor expansion around the fitted point gives \(\mathrm{LD}(h)\approx -2\ell_\omega'h-h'\ell_{\omega\omega}h-h'\Delta'M^{-1}\Delta h\). The first two terms are direct perturbation effects. Separately, under the common normal-curvature convention, the full-vector fitted-response curvature is twice the information-metric squared response. It measures total fitted-parameter displacement and is not, in general, the curvature of the likelihood displacement.

For a scalar target, a useful target-specific curvature standardizes the squared target response by the local variance scale \(c'M^{-1}c\). This scale is the usual inverse-information variance expression for the target under the local likelihood approximation. The result is not a probability that the target changed; it is a dimensionless local influence measure. Its factor of 2 follows the normal-curvature convention used for the full vector.

How Nuisance Adjustment Appears

The information-matrix formula can be made explicit by separating a scalar target parameter \(\psi\) from nuisance parameters \(\lambda\), so that \(\theta=(\psi,\lambda')'\). Partition the information and perturbation derivative into matching blocks. Solving the nuisance score equations shows how nuisance parameters move in response to both the target change and the perturbation. Substitution into the target equation leaves an adjusted information and an adjusted perturbation derivative.

Here \(I_{\mathrm{eff}}\) is the Schur complement of the nuisance-information block. The term subtracted from \(\Delta_\psi\) is the part of the target score perturbation associated with movement through the nuisance directions. Solving the block equations gives \(d\psi/d\omega=I_{\mathrm{eff}}^{-1}\Delta_{\mathrm{adj}}\). The target variance scale is \(I_{\mathrm{eff}}^{-1}\), so the target curvature in direction \(h\) is \(2(h'\Delta_{\mathrm{adj}})^2/I_{\mathrm{eff}}\). This is algebraically the same as the general \(M^{-1}\) formula, not a separate adjustment layered on top of it.

Worked Nuisance-Parameter Calculation

General Targets and Interpretation

A target need not be a single coordinate. For a differentiable contrast or other scalar function \(g(\theta)\), use its gradient \(c\) at the fitted value. For example, if \(g(\theta)=\theta_1+2\theta_2\), then \(c=(1,2)'\). The target response is the gradient dotted with the full parameter response; coordinates can move substantially and still cancel in the target. The general formula automatically accounts for that cancellation and for the information scale of the target.

For a scalar target, \(C_g\) is a rank-one matrix (or the zero matrix if every local target response is zero). Under a specified Euclidean scale for perturbations, its largest eigenvalue identifies the direction with the greatest target curvature; that direction is aligned with the vector of target responses across perturbation coordinates. The full-vector curvature matrix \(\Delta'M^{-1}\Delta\) can have higher rank because it records changes in all parameter directions. Rescaling perturbation coordinates changes these matrix summaries, so any comparison of directions requires a stated, consistent perturbation scale.

Diagnostics, Alternatives, and Limitations

A useful calculation check is to compute the target response in two ways: directly as \(c'M^{-1}\Delta h\), and, when the target is a coordinate, from the Schur-complement expression. They should agree up to numerical precision. Also verify that \(M\) is symmetric and positive definite at the fitted point, that \(I_{\mathrm{eff}}>0\), and that the computed target variance scale is positive. Large disagreement between direct and block calculations often signals an indexing, transposition, or nuisance-block error.

Curvature is local. A high value flags a direction in which a small perturbation can have a marked target effect on the chosen scale; it does not guarantee that a finite perturbation will have the same effect. Compare a flagged direction with exact case deletion or other direct refitting benchmarks when those changes are substantively relevant. First-order case-deletion approximations and weighted-likelihood influence calculations answer related questions, but they do not replace this target-specific normalization. The calculation also depends on the fitted model, the perturbation definition, and how perturbation directions are scaled.

A frequent mistake is to report full-vector curvature as though it were curvature for the target. The reasoning is that the target is one component of \(\theta\), so any measure of total parameter movement should describe its influence. The correction is that full curvature includes nuisance movement and ignores how precisely the target is estimated. Use \(C_g(h)\) when the question concerns a scalar target, and report full curvature separately when the question concerns overall fitted-parameter displacement.

Another failure mode is a nearly singular information matrix. Then \(M^{-1}\), \(c'M^{-1}c\), and the target curvature may be highly unstable; large values can reflect weak identification or numerical problems rather than a robust substantive signal. Check convergence, conditioning, and sensitivity to reasonable numerical tolerances before interpreting rankings. Curvature is a diagnostic of local sensitivity, not evidence of causation, model adequacy, or a particular missingness mechanism.