Influence on a Treatment-Effect Contrast

Why Track a Treatment-Effect Contrast?

A subject can have a noticeable effect on a selection-model fit without materially changing the treatment comparison of interest. Conversely, a modest change in the whole parameter vector can matter greatly if it moves a particular treatment-effect contrast. For interpretation, the relevant question is therefore not only whether the fitted model is sensitive, but whether the specified scientific contrast changes under a perturbation.

For example, a longitudinal trial model may estimate an average treatment difference at each visit. A prespecified target might be the difference at the final visit, or the change in treatment difference from an earlier to a later visit. The influence calculation in this tutorial follows that target as the likelihood contribution of one or more subjects is perturbed. It describes local sensitivity within the chosen model; it does not establish that treatment caused an outcome difference.

Weighted Subject Contributions and the Contrast Derivative

Suppose there are \(n\) independent subjects, each with an observed record that includes the observed outcomes and the missingness pattern. Under a specified selection model, write the observed-data log-likelihood as \(\ell(\theta)=\sum_{i=1}^{n}\ell_i(\theta)\), where \(\ell_i(\theta)\) is subject \(i\)'s contribution. This setup presumes that the model is differentiable near the fitted parameter, the maximum is an interior point, and the information matrix is nonsingular. The selection model and its assumptions remain in force throughout: a perturbation diagnoses sensitivity within that model, rather than resolving uncertainty about the missing-data assumptions.

A weighted perturbation replaces the log-likelihood by \(\ell(\theta;t)=\sum_i\omega_i(t)\ell_i(\theta)\), where \(\omega_i(0)=1\). For a direction \(a=(a_1,\ldots,a_n)\), set \(\omega_i(t)=1+t a_i\). A positive \(a_i\) locally increases subject \(i\)'s contribution; a negative one decreases it. The perturbation must be considered only for values of \(t\) for which the weights are admissible, and for which the perturbed fit exists.

Define the subject score \(u_i(\theta)=\partial\ell_i(\theta)/\partial\theta\), the total score \(U(\theta,t)=\partial\ell(\theta;t)/\partial\theta\), and the observed information at the unperturbed estimate \(\hat{\theta}=\hat{\theta}(0)\) by \(I=-\partial U(\theta,0)/\partial\theta^\mathsf{T}\), evaluated at \(\hat{\theta}\). The fitted score equation is \(U\{\hat{\theta}(t),t\}=0\). Differentiating this equation at \(t=0\) gives the local parameter movement. For the chosen direction, let \(b=\sum_i a_i u_i(\hat{\theta})\).

This is a local result: it uses the score and information at the unperturbed fit. For a differentiable contrast, the chain rule then gives

For a linear contrast \(q(\theta)=c^\mathsf{T}\theta\), this becomes \(c^\mathsf{T}I^{-1}b\). Thus the contrast’s influence is a projection of the full parameter movement onto the target contrast. A large parameter movement need not imply a large contrast change, and the reverse can also occur. If the contrast is nonlinear, its gradient is evaluated at the fitted parameter; the derivative still gives only the first-order change.

Worked Contrast Calculations

Checking the Local Approximation With Refits

A derivative is useful for identifying local sensitivity, but it is not a substitute for examining how a fitted contrast behaves at meaningful perturbation sizes. Exact refits recompute the maximum-likelihood estimate at each selected \(t\); they can reveal curvature, instability, or failure to converge that a derivative at zero cannot show. The model, data, and estimation method should be held fixed so the differences can be attributed to the perturbation.

Diagnostics, Alternatives, and Limitations

A useful influence calculation depends on both a well-behaved fit and a clearly defined target. Check the following before interpreting a large or small contrast derivative:

• Convergence and curvature. Confirm that the unperturbed fit satisfies the score equations to numerical tolerance and that the information matrix is nonsingular and appropriately conditioned. Near-singularity can make \(I^{-1}b\) unstable, so small numerical errors may produce large apparent influence.
• Derivative agreement. Compare the analytic derivative with centered finite differences from small positive and negative perturbations. Large disagreement may indicate a coding error, an insufficiently small step, non-smooth behavior, or a fit that is not tracking the same solution.
• Refit behavior. At selected perturbation values, check convergence, parameter estimates, and the target contrast. A jump to a different likelihood maximum, nonconvergence, or strong asymmetry warns that the local derivative does not summarize the path well.
• Target definition. Verify the sign and units of the contrast. For example, week-12 treatment minus control differs in sign from control minus treatment, and a contrast between visits is not the same as a treatment effect at one visit.
• Model dependence. Subject influence is conditional on the fitted selection model and its missingness assumptions. It does not establish that the model is correct or that missingness is ignorable.

This target-specific derivative differs from several useful alternatives. A full-vector parameter influence reports movement in all coordinates but does not by itself identify whether the scientific contrast changes. Exact case deletion refits after removing a subject and is a finite, potentially substantial perturbation; it is a useful benchmark, but it is not the same as the local derivative at weight one. Likelihood displacement summarizes movement in the fitted model more broadly, whereas the contrast derivative projects movement onto the selected treatment comparison. A direct refit path is more informative for sizable changes but costs more computation and can expose multiple solutions rather than a single smooth path.

No one measure settles influence in every setting. A small derivative at zero does not rule out a large change for a substantial perturbation, and a large derivative does not imply that the subject is erroneous or should be excluded. The direction \(a\), scale of \(t\), selected contrast, and assumed selection model all affect the result. Report these choices alongside the estimate and its units. A sensitivity diagnostic is a prompt to investigate data, model fit, and robustness—not a license to remove a subject or to make a causal claim.

A Short Derivation and Common Pitfall

The key derivative follows directly from the perturbed score equation. At the unperturbed maximum, \(U(\hat{\theta},0)=0\). For \(\omega_i(t)=1+t a_i\), the derivative of the score with respect to \(t\), holding \(\theta\) fixed, is \(\sum_i a_i u_i(\hat{\theta})=b\). The derivative of the score with respect to \(\theta\) is \(-I\). Applying the chain rule to \(U\{\hat{\theta}(t),t\}=0\) gives

When \(I\) is invertible, rearrangement yields the parameter derivative, and applying the chain rule to \(q\) yields the contrast derivative above. This argument requires a differentiable score and a locally identifiable solution; it does not guarantee that the likelihood has only one maximum or that a distant perturbed fit remains on the same solution path.

Common mistake: “The subject has a large score, so the treatment-effect contrast must be highly influential.” A large score only describes the subject’s direct score contribution. The information matrix and the contrast direction also matter: the actual first-order target movement is \(\nabla q(\hat{\theta})^\mathsf{T}I^{-1}b\). The correction is to calculate that projection for the prespecified contrast and then check it against refits. Another mistake is to read the derivative as the exact effect of deleting the subject. Deletion is a finite change in weight, not an infinitesimal change; compare an exact deletion refit if that finite change is the question.