Perturbing Covariates in the Selection Equation

Why Perturb a Selection Covariate?

In a selection model, the probability that an outcome is observed may depend on the outcome itself and on measured covariates. A subject can therefore be influential because their observed outcome is unusual, because their covariate value strongly affects the modeled observation process, or both. A targeted covariate perturbation helps distinguish these sources: it changes the subject’s input to the selection equation while leaving their outcome measurement and the outcome model unchanged.

This is a local diagnostic, not a proposal to alter the recorded data. It asks how fitted estimates would respond to a small, explicitly defined change in the covariate value used by the selection model. The distinction matters when assessing whether conclusions are sensitive to a few covariate values rather than to the outcomes themselves.

For a continuous covariate, \(d_i\) can select one component or a meaningful direction in a vector of covariates. The size and units of \(\omega_i\) must be stated: for example, \(\omega_i=0.2\) might represent a change of 0.2 years of age if \(d_i\) selects age measured in years. An additive perturbation is not automatically meaningful for a categorical covariate; a relevant category contrast or other explicit coding is needed.

Selection Likelihood and the Perturbation

Consider a simple selection model with outcome density \(f(y_i\mid X_i;\beta)\) and response probability \(q_i(y_i,Z_i;\gamma)=P(R_i=1\mid Y_i=y_i,Z_i)\). For illustration, suppose the selection equation is logistic:

Here \(\gamma_z\) is the vector of selection-equation coefficients for \(Z_i\). The observed-data likelihood contribution is \(f(y_i\mid X_i;\beta)q_i(y_i,Z_i;\gamma)\) when \(R_i=1\) and \(Y_i=y_i\) is observed. When \(R_i=0\), the outcome is unobserved, so the contribution is the integral of the outcome density times the probability of nonresponse:

This likelihood form assumes that the outcome model and selection model together specify the joint distribution relevant to observation, and that the covariate being perturbed is available for the subject even when \(Y_i\) is missing. The fitted model must also be identifiable and sufficiently regular for its likelihood and derivatives to be calculated. The logistic equation is an example; the same diagnostic idea can be applied to other differentiable selection probabilities.

Suppose \(Z_i(\omega_i)=Z_i+\omega_i d_i\). Define \(c_i=\gamma_z^\mathsf{T}d_i\). Then the linear predictor changes by \(\omega_i c_i\). In particular, the perturbation does not change \(y_i\), \(f(y_i\mid X_i;\beta)\), or \(X_i\). For a responder, differentiating the log selection contribution at \(\omega_i=0\) gives \(\partial\log q_i/\partial\omega_i=c_i(1-q_i)\). For a nonresponder, the outcome is unobserved and must be integrated out; its derivative is not obtained by substituting an imputed outcome as though it were observed.

For a nonresponder, the ratio is the model-based expectation of \(q_i(Y_i,Z_i;\gamma)\) among outcomes weighted by their probability of nonresponse. Thus increasing a covariate that raises response probability generally lowers the nonresponse contribution to the likelihood. The derivative also shows that a covariate can have little local effect when its selection coefficient is near zero, even if its recorded value is large.

From a Covariate Change to Parameter Sensitivity

Let \(\ell(\theta;\omega)\) be the observed log likelihood under a vector of subject-specific perturbations \(\omega\), with \(\theta\) collecting the model parameters. Write \(\hat\theta\) for the maximum-likelihood estimate at \(\omega=0\). Define the observed information \(I=-\partial^2\ell/\partial\theta\partial\theta^\mathsf{T}\) at \((\hat\theta,0)\), and the mixed-derivative matrix \(\Delta=\partial^2\ell/\partial\theta\partial\omega^\mathsf{T}\) at the same point. If \(I\) is nonsingular and the perturbed maximum remains near the baseline maximum, implicit differentiation of the likelihood score equation gives the local parameter response.

The first equality follows by differentiating the score, which is zero at the fitted estimate. The derivative with respect to \(\theta\) is the Hessian \(-I\); the derivative with respect to \(\omega\) is \(\Delta\). Solving the resulting linear equation yields the sensitivity matrix. For a prespecified perturbation direction \(v\), the approximate parameter change is \(I^{-1}\Delta v\) per unit of perturbation in that direction.

A useful scalar summary of the local change in fitted parameters, measured in the information metric, is \(C(v)=v^\mathsf{T}\Delta^\mathsf{T}I^{-1}\Delta v\). It is nonnegative when \(I\) is positive definite. It measures local parameter sensitivity for the chosen perturbation scale; it is not a probability, a test statistic with a universal reference distribution, or proof that a subject is erroneous. It is comparable across subjects only when their perturbation directions and scales have been made comparable.

Diagnostics, Comparisons, and Limitations

A useful diagnostic begins by specifying exactly which selection covariate is perturbed, its direction \(d_i\), and the units of \(\omega_i\). Then inspect both the subject’s selection-probability change and the resulting parameter or contrast sensitivity. A large covariate value alone does not establish influence: the selection coefficient, fitted response probability, outcome-model integration for missing records, and information in the rest of the data all matter.

• Check the model scale. A one-unit shift in age, a standardized age shift, and a one-unit change in a binary indicator are not interchangeable. Report the perturbation scale alongside any influence ranking.
• Check the selection equation and data availability. The covariate must be defined for the subject whose response status is modeled. A value missing for that subject cannot be perturbed as though it were observed without making additional modeling choices.
• Check local approximation. If a perturbation is not small, refit the model at several values and compare the actual changes with the derivative-based approximation. Large discrepancies indicate curvature or other nonlocal behavior.
• Check fit and numerical stability. A nearly singular \(I\), unstable integration over missing outcomes, or poorly determined selection coefficients can make \(I^{-1}\Delta\) unreliable. Inspect convergence and sensitivity to integration accuracy before interpreting a large diagnostic.
• Check the intended separation. If a covariate also appears in the outcome model, a selection-only perturbation deliberately holds its outcome-model role fixed. This isolates one model component; it is not equivalent to changing the subject’s actual recorded covariate everywhere.

This diagnostic differs from perturbing observed outcomes: that changes an outcome input for an observed record, whereas the present perturbation changes only a selection-equation covariate. It also differs from case deletion or a case-weight perturbation, which changes a subject’s contribution to the likelihood rather than changing the modeled selection predictor. The earlier material on observed-outcome perturbations treats the first of these distinct questions.

The result remains conditional on the selected model and its assumptions. A large local influence may flag sensitivity to a covariate value, but does not demonstrate that the value is incorrect or that the model is misspecified. Conversely, a small local derivative does not validate the selection equation, resolve weak identification, or guarantee robustness to a large change. Local derivatives can also depend strongly on parameterization and perturbation scale. For consequential findings, compare local approximations with actual refits over substantively plausible covariate changes and report the selection-only nature of the diagnostic.