Local Influence for Missingness-Parameter Estimates

Why Focus on the Missingness Parameters?

A selection model describes response through a probability that depends on the outcome and possibly on covariates. A record can therefore affect the fitted response mechanism in two ways: through information about response itself, and through the joint likelihood’s connection between outcome-model and response-model parameters. The second route matters even when the record’s direct score for the response parameters is zero.

Local influence quantifies the first-order change in fitted parameters when the likelihood contribution of a subject is perturbed slightly. Here the target is the missingness-mechanism parameter vector, denoted by \(\gamma\). The calculation is local: it describes sensitivity near the fitted model, not the exact effect of deleting a subject or changing the model specification.

Observed-Data Likelihood and Perturbation

For a simple illustration, let \(R_i=1\) indicate that subject \(i\)'s outcome \(Y_i\) is observed, and let \(X_i\) denote observed covariates. Let \(f(y\mid x;\beta)\) be the outcome density and let \(\pi(y,x;\gamma)=P(R=1\mid Y=y,X=x;\gamma)\) be the response probability. The parameters \(\beta\) describe the outcome model, while \(\gamma\) describes the selection model.

The observed-data likelihood contribution includes the observed outcome for a respondent and integrates over the unobserved outcome for a nonrespondent:

L_i(\beta,\gamma)= \begin{cases} f(Y_i\mid X_i;\beta)\,\pi(Y_i,X_i;\gamma), & R_i=1,\\[4pt] \displaystyle\int f(y\mid X_i;\beta)\{1-\pi(y,X_i;\gamma)\}\,dy, & R_i=0. \end{cases}

Write \(\ell_i(\theta)=\log L_i(\theta)\), where \(\theta=(\beta',\gamma')'\) collects all model parameters. For more complicated response patterns, the observed-data contribution is obtained by integrating the complete-data model over the missing components. The influence calculation below uses these observed-data log-likelihood contributions; it does not substitute unobserved outcomes as if they had been recorded.

Definition: Let \(\omega_i\) be the likelihood weight for subject \(i\), and let \(\hat\theta\) maximize the unperturbed log likelihood \(\ell(\theta)=\sum_i\ell_i(\theta)\). Define the observed information \(J=-\partial^2\ell(\hat\theta)/\partial\theta\partial\theta'\). If subject \(i\)'s weight is perturbed to \(1+\varepsilon a_i\), its weighted score perturbation is \(g_i=a_i\,\partial\ell_i(\hat\theta)/\partial\theta\). When \(J\) is nonsingular and the likelihood is differentiable, the first-order parameter change is \(\dot\theta_i=J^{-1}g_i\). Its selection-parameter component is \(\dot\gamma_i=E_\gamma'J^{-1}g_i\), where \(E_\gamma'\) selects the \(\gamma\) coordinates.

The factor \(a_i\) specifies the perturbation scale for that subject. For comparable subject diagnostics, use the scaling chosen for the perturbation scheme; otherwise, differences in \(g_i\) may reflect incomparable perturbation sizes rather than differences in sensitivity. The derivative is with respect to \(\varepsilon\) at \(\varepsilon=0\), so it describes an infinitesimal change.

Why the Formula Gives a First-Order Change

The fitted parameter under perturbation, \(\hat\theta(\varepsilon)\), satisfies the weighted score equation. Differentiating that equation at the unperturbed fit gives the influence formula.

0=\frac{\partial}{\partial\theta} \left\{\ell(\theta)+\varepsilon a_i\ell_i(\theta)\right\} \bigg|_{\theta=\hat\theta(\varepsilon)}.

At \(\varepsilon=0\), the score is zero. Differentiate with respect to \(\varepsilon\), using the chain rule. The derivative of the full score with respect to \(\theta\) is \(-J\), and the direct derivative of the perturbed score is \(g_i\). Thus

-J\dot\theta_i+g_i=0 \quad\Longrightarrow\quad \dot\theta_i=J^{-1}g_i.

This argument requires a differentiable score, a locally interior solution, and nonsingular \(J\). A well-conditioned, positive-definite observed information matrix at a local maximum is especially useful in practice. Nonsingularity makes the derivative unique; it does not by itself guarantee that a large finite perturbation will be well approximated by the derivative.

Partition the information and the perturbation score into outcome and selection blocks. When \(J_{\beta\beta}\) is invertible, the block equations show directly how the outcome score can enter the selection estimate:

\begin{aligned} J_{\beta\beta}\dot\beta_i+J_{\beta\gamma}\dot\gamma_i&=g_{\beta i},\\ J_{\gamma\beta}\dot\beta_i+J_{\gamma\gamma}\dot\gamma_i&=g_{\gamma i},\\ \left(J_{\gamma\gamma}-J_{\gamma\beta}J_{\beta\beta}^{-1}J_{\beta\gamma}\right) \dot\gamma_i &=g_{\gamma i}-J_{\gamma\beta}J_{\beta\beta}^{-1}g_{\beta i}. \end{aligned}

The final line follows by solving the first equation for \(\dot\beta_i\) and substituting into the second. If the cross-information blocks are nonzero, \(g_{\beta i}\) can contribute to \(\dot\gamma_i\), even when \(g_{\gamma i}=0\). The matrix multiplying \(\dot\gamma_i\) is the Schur complement; when the full \(J\) is positive definite, it is also positive definite.

Worked Example: A Subject with a Direct Selection Score

Suppose a fitted two-parameter selection model has \(\theta=(\beta,\gamma)'\), observed information \(J=\begin{pmatrix}10&2\\2&4\end{pmatrix}\), and a scaled subject score perturbation \(g_i=(0.6,1.2)'\). The first coordinate is the outcome parameter and the second is the response-mechanism parameter. The information is positive definite because its leading diagonal element is 10 and its determinant is \(10(4)-2(2)=36>0\).

The local parameter change is

\dot\theta_i=J^{-1}g_i =\frac{1}{36} \begin{pmatrix}4&-2\\-2&10\end{pmatrix} \begin{pmatrix}0.6\\1.2\end{pmatrix} =\frac{1}{36} \begin{pmatrix}0\\10.8\end{pmatrix} = \begin{pmatrix}0\\0.3000\end{pmatrix}.

Thus the first-order change in \(\gamma\) is \(0.3000\) parameter units per unit \(\varepsilon\). For a small perturbation of \(\varepsilon=0.01\), the linear approximation gives a change of \(0.003000\) in \(\gamma\). It is a local approximation, not a claim that the refitted value must change by exactly that amount.

Worked Example: Zero Selection Score Does Not Necessarily Mean Zero Selection Influence

Use the same information matrix, and suppose a subject has scaled score perturbation \(g_i=(1,0)'\). The subject’s selection-score component is zero, but its full score vector is not. The information remains positive definite by the determinant check above.

The selection-parameter derivative is the second coordinate of \(J^{-1}g_i\):

\dot\gamma_i = \left[\frac{1}{36} \begin{pmatrix}4&-2\\-2&10\end{pmatrix} \begin{pmatrix}1\\0\end{pmatrix}\right]_2 =\frac{-2}{36} =-0.05556.

The outcome-score perturbation is transmitted to \(\gamma\) through the nonzero cross-information. The negative sign means that, for a small positive increase in this subject’s weight, the fitted \(\gamma\) initially moves downward. This is a statement about the local fitted parameter, not a causal effect on response.

By contrast, if the full score perturbation were \(g_i=(0,0)'\), then \(J^{-1}g_i=(0,0)'\). With nonsingular \(J\), that subject would have zero first-order influence on every fitted parameter. Zero in the selection-score block alone and zero in the full score vector are different conditions.

Comparing Subjects on the Selection-Parameter Scale

For each subject, the vector \(\dot\gamma_i\) describes both the direction and size of local movement in the response-mechanism parameters. The components should be interpreted in their parameter units. For example, a logit coefficient for a covariate measured in years is measured in log-odds per year; a coefficient for a binary covariate is measured per change from its reference category.

If the selection model has multiple parameters, a scalar ranking requires a stated scale or metric. For a chosen positive-definite matrix \(W_\gamma\), one option is

I_i^{(\gamma)}=\sqrt{\dot\gamma_i'W_\gamma\dot\gamma_i}.

Taking \(W_\gamma\) as the identity compares Euclidean lengths in the raw parameter units, which can be misleading if parameters use very different scales. A metric that standardizes by prespecified parameter scales gives a more interpretable comparison. The metric and the subject perturbation scaling should be stated; neither is determined by the derivative formula alone.

Worked Example: Ranking Three Scaled Subject Perturbations

Suppose \(\theta=(\beta,\gamma)'\), \(J=\begin{pmatrix}12&3\\3&6\end{pmatrix}\), and the three subjects have already been put on the same perturbation scale, with score vectors \(g_A=(0.2,0.1)'\), \(g_B=(-0.1,0.5)'\), and \(g_C=(0.4,0)'\). The matrix is positive definite: its leading diagonal element is 12 and its determinant is \(12(6)-3(3)=63>0\). Use absolute change in the scalar \(\gamma\) as the ranking measure.

The inverse is \(J^{-1}=\frac{1}{63}\begin{pmatrix}6&-3\\-3&12\end{pmatrix}\), so for a score vector \((g_\beta,g_\gamma)'\), the selection component is \(\dot\gamma=(-3g_\beta+12g_\gamma)/63\). Substitution gives

\begin{aligned} \dot\gamma_A&=\frac{-3(0.2)+12(0.1)}{63} =\frac{0.6}{63}=0.009524,\\ \dot\gamma_B&=\frac{-3(-0.1)+12(0.5)}{63} =\frac{6.3}{63}=0.1000,\\ \dot\gamma_C&=\frac{-3(0.4)+12(0)}{63} =\frac{-1.2}{63}=-0.01905. \end{aligned}

On this common scale, subject B has the largest absolute first-order effect on \(\gamma\), followed by C and then A. The signs distinguish the directions of movement; the ranking by absolute value does not. These are hypothetical local calculations based on the specified score vectors and information matrix.

Diagnostics, Alternatives, and Limitations

A large value of \(I_i^{(\gamma)}\) identifies a subject whose small, specified weight perturbation is associated with substantial local movement in the fitted selection parameters. Before treating a ranking as meaningful, check that the perturbations use the intended common scale, the information matrix is nonsingular and not nearly singular, and the selected parameter metric matches the scientific question. If small changes in the data or fitting setup produce large changes in the result, the local approximation may be unstable.

Weak identification is a particularly important warning in selection models. If the observed data provide little information about how response depends on unobserved outcomes, the likelihood can be relatively flat in some directions of \(\gamma\). Then \(J^{-1}\) can amplify score perturbations, making derivatives large and sensitive to numerical or modelling choices. Large local influence in this setting is not, by itself, evidence that the subject is erroneous; it may also signal that the fitted mechanism is weakly determined by the observed data.

Subject deletion asks a different question: it refits after removing a record, which is a finite change in its weight. A score-based diagnostic or a case-weight local influence calculation is cheaper and gives a first-order description, but may be inaccurate for a large change. Direct refitting after deletion can assess that finite change, although it can also alter the fitted solution more substantially. Neither procedure resolves sensitivity to different selection-model assumptions; that is a model-comparison question rather than a subject perturbation.

Do not confuse blocks. A zero selection-score component does not guarantee zero influence on \(\gamma\); cross-information can transmit outcome-score changes.
Do not over-interpret a large derivative. It is local to the fitted model and the stated perturbation direction.
Do not rank unscaled perturbations as if they were equivalent. The size of a derivative depends on how the perturbation is defined.
Do not treat the fitted response model as established fact. Influence measures sensitivity conditional on the specified model; it does not establish that the selection model is correct.

The central practical value of this analysis is its separation of two questions: which subjects can move the fitted missingness parameters under a specified local perturbation, and whether the assumed selection model is substantively adequate. Local influence addresses the first; it does not answer the second.

Check Your Understanding

Answer each question, then compare with the solutions below.

In a selection model, what part of the observed-data likelihood contribution for a nonrespondent accounts for the missing outcome?
If a subject’s full weighted score perturbation is zero and \(J\) is nonsingular, what is its first-order influence on the fitted parameters?
If only the subject’s selection-score component is zero, can its first-order influence on \(\gamma\) still be nonzero? Explain the condition that allows this.
Why should a scalar ranking of a multidimensional \(\dot\gamma_i\) state its parameter metric?
What is one reason a large local derivative may be unreliable as a prediction of the effect of deleting a subject?

Solutions. (1) The nonrespondent’s likelihood contribution integrates \(f(y\mid X_i;\beta)\{1-\pi(y,X_i;\gamma)\}\) over the unobserved outcome. (2) It is zero for every parameter, since \(J^{-1}0=0\). (3) Yes. A nonzero outcome-score component can affect \(\dot\gamma\) when the information has cross-block terms connecting outcome and selection parameters. (4) Components can have different units or scales, so a length or ranking depends on how those components are combined. (5) Deletion is a finite weight change, whereas the derivative describes only the first-order response at the unperturbed fit; the linear approximation may not hold over a large change.