Why Refit After Deleting a Subject?
A casewise score describes how one subject's observed-data likelihood contributes to the fit at the current estimate. Exact case deletion asks a different, concrete question: what estimate would the same model produce if this subject had not been included? The answer provides a direct benchmark for assessing changes in fitted selection-model parameters and treatment contrasts. It is not a test of whether a subject is erroneous, nor does a large change establish that the model is wrong.
For longitudinal data, a subject can contribute both an observed outcome trajectory and information about its missingness pattern. Deleting that subject removes the entire observed record, including its contribution to the selection mechanism. Consequently, deletion can shift outcome-model estimates, selection-model estimates, or both. A treatment contrast may change even when its component coefficients move in opposite directions.
Definition and Assumptions
Suppose \(N\) independent subjects have longitudinal outcomes and missingness indicators. Write \(O_i\) for subject \(i\)'s observed record, including observed outcomes, treatment, covariates, and the missingness pattern. Let \(\theta\) collect the parameters of a specified selection model. Its observed-data log likelihood is the sum of the subject contributions, \(\ell_i(\theta;O_i)\). A contribution includes any integration over missing outcomes required by the observed-data likelihood.
This definition relies on several conditions for a meaningful comparison. The subjects are treated as independent under the model, the observed-data likelihood correctly accounts for each missingness pattern, and the same parameterization, covariates, constraints, and likelihood calculation are used for every fit. Both the full and deleted-data likelihoods must have usable maxima. If removing a subject leaves a treatment group empty or makes a parameter unidentifiable, the deleted estimate may not exist or may not be comparable.
An exact deletion fit is a new optimization problem. It should use the same model and numerical standards as the full fit, with sufficient checks for convergence and alternative maxima. Reusing the full-data estimate as a starting value can be computationally convenient, but it does not make the result an exact deletion estimate: the deleted-data likelihood must actually be optimized.
What Changes, and Why?
The deleted log likelihood is the full log likelihood with one subject contribution removed:
For a differentiable likelihood and an interior maximum, the full estimate satisfies the score equation \(\sum_{j=1}^{N}\nabla\ell_j(\hat{\theta};O_j)=0\), while the deleted estimate satisfies \(\sum_{j\ne i}\nabla\ell_j(\hat{\theta}_{(-i)};O_j)=0\). These equations justify why deletion can matter: the removed contribution may help determine the balance of the likelihood's gradients. But the score equation alone does not give the deleted estimate. The exact change depends on optimizing the entire deleted-data likelihood, which may be nonlinear or have multiple maxima.
For a treatment contrast, compute the contrast from each fitted parameter vector before taking the difference. For instance, if the model-based visit-2 mean difference is \(g(\theta)=\beta_A+\beta_{At}\), then \(\Delta_i g=(\hat{\beta}_{A,(-i)}+\hat{\beta}_{At,(-i)})-(\hat{\beta}_A+\hat{\beta}_{At})\). This reports the change in the estimand in its original units, rather than treating a coefficient change as though it were automatically the treatment-effect change.
Worked Example: Deleting One Subject From a Two-Visit Model
Consider a hypothetical two-arm study with visit 1 observed for everyone and some visit-2 outcomes missing. The specified selection model has a joint outcome distribution with visit-2 mean \( \beta_0+\beta_A A+\beta_t+\beta_{At}A\), where \(A=1\) denotes treatment, and a logistic model for observing visit 2 that can depend on the outcomes. The treatment contrast at visit 2 is \(g(\theta)=\beta_A+\beta_{At}\). The selection coefficient \(\gamma\) is the log-odds change in visit-2 observation probability per one-unit increase in the relevant outcome predictor.
Four hypothetical records are: subject 1, control, outcomes \(10, 12\), complete; subject 2, control, visit 1 outcome \(11\), visit 2 missing; subject 3, treatment, outcomes \(10, 13\), complete; subject 4, treatment, visit 1 outcome \(12\), visit 2 missing. Suppose maximization of the same observed-data selection likelihood gives \(\hat{\beta}_A=-0.40\), \(\hat{\beta}_{At}=1.10\), and \(\hat{\gamma}=0.55\). These are stipulated hypothetical fit results, not estimates calculated from the four records by a closed-form formula.
The full-data visit-2 contrast is
Now delete subject 3, whose treatment-group visit-2 outcome is observed, and maximize the likelihood for the remaining three records. Suppose the converged deleted-data fit gives \(\hat{\beta}_{A,(-3)}=-0.20\), \(\hat{\beta}_{At,(-3)}=0.60\), and \(\hat{\gamma}_{(-3)}=0.22\). Then
Thus, deleting subject 3 lowers the fitted visit-2 treatment contrast by \(0.30\) outcome units, from \(0.70\) to \(0.40\). The selection coefficient changes by \(\Delta_3\gamma=0.22-0.55=-0.33\) log-odds units per outcome unit. The contrast change is an association implied by this fitted model; it is not, by itself, evidence that the subject caused a treatment effect.
This comparison assumes that both optimizations converged to comparable maxima and that the reduced data still support the specified parameters. In a real analysis, those checks are part of interpreting the numbers, not optional details.
Compare Subjects on the Target of Interest
An influence assessment should name its target. A deletion can have a substantial effect on a selection parameter but little effect on the treatment contrast, or the reverse. Raw changes in different parameters also have different units and scales, so their magnitudes should not be compared as though they were on a common scale.
Worked Example: Ranking Deletions by Contrast Change
Suppose the full fit in a hypothetical longitudinal selection-model analysis gives a visit-2 treatment contrast of \(1.20\) points. Three separate exact refits give the following results. Each row reports the estimate after deleting that subject, not a simultaneous deletion.
| Deleted subject | Deleted-fit contrast (points) | Deleted-fit selection coefficient |
|---|---|---|
| 5 | 1.05 | 0.48 |
| 8 | 1.31 | 0.21 |
| 11 | 0.94 | 0.51 |
For subject 5, \(\Delta_5 g=1.05-1.20=-0.15\) points. For subject 8, \(\Delta_8 g=1.31-1.20=0.11\) points. For subject 11, \(\Delta_{11}g=0.94-1.20=-0.26\) points. Ranking by absolute change in the treatment contrast gives subject 11 (\(0.26\) points), subject 5 (\(0.15\) points), then subject 8 (\(0.11\) points).
The selection-coefficient changes are also informative but answer a different question. If the full-fit selection coefficient was \(0.45\), the changes are \(0.03\), \(-0.24\), and \(0.06\), respectively. Subject 8 has the largest absolute change in that coefficient, while subject 11 has the largest absolute change in the treatment contrast. A report should therefore state which estimate is being compared, give its units, and avoid describing a subject as simply “most influential” without specifying the target.
These hypothetical results assume all four fits are successful and use the same likelihood and contrast definition. The table alone cannot verify that condition; convergence and likelihood checks are required.
Diagnostics and Failure Modes
A large exact deletion change can identify a fit that is sensitive to one record, but the size of the change is not a diagnostic verdict. Check whether the full and deleted fits used comparable optimization settings, attained plausible maxima, and retained enough information to estimate the parameters. In particular, inspect whether deletion removes a rare dropout pattern, eliminates observations from a treatment-by-time cell, or leaves the selection equation close to separation. Such changes may make the deleted fit unstable or weakly identified.
A numerical failure is not a large influence estimate. If the deleted fit does not converge, has materially different likelihood values across starts, or runs to a boundary, report that the deletion comparison is unresolved under the specified fitting procedure. Do not use an unstable output as though it were an ordinary estimate of \(\Delta_i\theta\). The earlier material on multiple local maxima and validating quadrature accuracy is relevant when a deletion changes the optimization landscape or the integration burden.
Exact deletion differs from two common alternatives. A casewise score evaluates a subject's contribution at the full-data estimate; it is a local diagnostic, not the result of removing that subject and refitting. A first-order deletion approximation, considered separately in the next stage of local-influence analysis, estimates the change without carrying out the complete refit. Exact deletion is valuable as a benchmark for those diagnostics, especially when changes are large or nonlinear, but it requires repeated optimization and can itself be unstable.
Worked Example: When a Deleted Fit Is Not Interpretable
In a hypothetical study, the full model has 12 subjects in each treatment arm and estimates a visit-2 contrast of \(0.80\) points. Subject 7 is the only control subject with a particular dropout history. On deletion, an optimizer returns a contrast of \(1.25\) points and a selection coefficient of \(4.8\), but different starting values return selection coefficients of \(2.1\) and \(5.0\) with substantially different log-likelihoods.
The arithmetic difference from the first returned fit is \(1.25-0.80=0.45\) points. However, it is not justified to report \(0.45\) as a reliable exact deletion change: the deleted likelihood has not been shown to have a stable, comparable maximum. The selection-coefficient estimates are also highly start-dependent, and deletion removed the only record for one dropout history. These facts indicate a fitting or information problem that must be investigated before interpreting a treatment-contrast shift.
The appropriate conclusion is that deletion of subject 7 exposes instability or weak support for part of the model, not that the subject has a confirmed influence of \(0.45\) points. Further optimization checks or a careful assessment of model estimability are needed. Even a successfully computed deletion difference describes sensitivity of this fitted model to the record; it does not establish that the record is incorrect or that deleting it is substantively appropriate.
Practical Interpretation
Exact case deletion is most useful when reported as a transparent set of comparisons: the full estimate, the deleted estimate, their difference, and the fit diagnostics supporting the comparison. For a treatment contrast, include the contrast's units and direction. For selection parameters, explain the parameter scale—for example, a log-odds change per outcome unit—rather than interpreting its numerical change as a probability difference.
Deletion is inherently a one-subject-at-a-time perturbation. It does not measure the effect of removing several subjects together, establish robustness to every missingness assumption, or account for sampling uncertainty in the deletion difference. The same data-generating and selection-model assumptions remain in force after deletion. A stable exact deletion benchmark is informative about sensitivity to that record under the specified model; it is not a substitute for uncertainty analysis or substantive assessment of the missingness assumptions.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- If the full-data treatment contrast is \(0.60\) points and the exact deleted-data contrast is \(0.25\) points, what is the deletion change, and how should it be interpreted?
- Why is a casewise score at the full-data estimate not itself an exact case-deletion estimate?
- A deleted-data optimizer gives different parameter estimates and likelihood values from different starting values. Should the difference from the full fit be reported as a reliable exact deletion change? Explain.
- If deleting one subject changes a selection coefficient substantially but barely changes the treatment contrast, which quantity indicates greater sensitivity to that deletion?
Solutions. (1) \(\Delta_i g=0.25-0.60=-0.35\) points: the fitted contrast decreases by \(0.35\) points after deletion. This is a model-based sensitivity comparison, not a causal claim about the subject. (2) A casewise score evaluates one contribution at the full-data estimate; an exact deletion estimate instead maximizes the likelihood formed from all the remaining subjects. (3) No. The deleted fit has not been shown to be stable, so its difference is unresolved rather than a reliable exact deletion result. (4) Sensitivity depends on the target: the selection coefficient is more sensitive on its own scale, while the treatment contrast is less sensitive. State the scale and units for each rather than ranking unlike parameters by raw numerical changes.