Why Influence Must Be Assessed at the Subject Level
A transition regression may contain several transition-specific models, and one subject’s observed path can contribute risk-time and event information to more than one transition. A subject might, for example, move through an intermediate state and later experience another transition. Treating each transition record as if it belonged to an unrelated individual would miss the fact that these contributions came from the same person. Influence diagnostics therefore ask a subject-level question: how much would the fitted transition-regression results change if this subject’s entire observed history were omitted?
A large change does not establish that a subject is erroneous, nor does it prove that the model is wrong. It identifies a case whose data, in combination with the model and the other subjects, materially affect an estimate or its uncertainty. The case may be valid and scientifically important. The diagnostic is a prompt to inspect the path, covariates, event history, and model fit—not an automatic rule for exclusion.
Definition and Assumptions
The subject contributions need not be independent across transitions within a person. The likelihood must instead factor across independent subjects, so that \(\ell(\beta)=\sum_i\ell_i(\beta)\). The definition presumes a fixed model specification and parameterization in the full and deleted fits. The estimator should be an interior maximum with differentiable log-likelihood and nonsingular observed information, at least near the full estimate. The event and censoring process must satisfy the assumptions of the chosen transition model; case deletion does not repair informative censoring or a misspecified transition hazard.
For a multi-state trajectory, \(\ell_i\) includes all of subject \(i\)'s modeled transition contributions, including relevant risk intervals, transition events, and censoring information. Do not delete just one transition record while retaining the rest of that subject’s path when the diagnostic is intended to assess subject-level influence. If subjects are clustered, for example within families or centers, deleting one subject assesses individual influence; it does not replace a separate assessment of cluster influence.
Exact Deletion and the Score Approximation
The most direct diagnostic is to refit the model after deleting each subject. For each refit, record coefficient changes and uncertainty changes. This procedure uses the fitted model itself and does not require a first-order approximation, though a deleted fit can fail to converge or become unidentified. When there are many subjects or model fits are expensive, a one-step score approximation can screen for influential cases.
Write the subject score as \(u_i(\beta)=\partial\ell_i(\beta)/\partial\beta\). Define the full observed information \(J(\beta)=-\partial^2\ell(\beta)/\partial\beta\partial\beta^\mathsf{T}\), and the deleted-sample observed information evaluated at the full estimate as \(J_{(-i)}(\hat{\beta})=-\partial^2\{\ell(\beta)-\ell_i(\beta)\}/\partial\beta\partial\beta^\mathsf{T}\) at \(\beta=\hat{\beta}\). At the full maximum, the full score is zero. Thus the deleted-sample score at that same point is \(-u_i(\hat{\beta})\), which generally is not zero.
This distinction matters most when a subject has unusual covariates, a rare event, or substantial exposure to a transition. Such a subject may contribute appreciably to the information as well as to the score. In that situation, the full-information approximation can be inaccurate even though the algebraic form looks convenient. The one-step result is also only a local approximation: if deletion moves the estimate a long way, the likelihood’s curvature may change and an exact refit is preferable.
Worked Example: A Scalar Coefficient and Deleted Information
Suppose a hypothetical transition-regression fit has a scalar coefficient \(\hat{\beta}=0.50\), full observed information \(J=20\), and subject \(i\)'s score contribution \(u_i(\hat{\beta})=1.5\). The subject’s observed information contribution is 2, so \(J_{(-i)}(\hat{\beta})=20-2=18\). Assume the deleted likelihood is locally well approximated by its quadratic expansion.
Using the deleted information—not the full information—the approximate change is $$ \hat{\beta}-\hat{\beta}_{(-i)}\approx \frac{1.5}{18}=0.08333. $$ Therefore \(\hat{\beta}_{(-i)}\approx 0.50-0.08333=0.4167\). The full-information shortcut would instead give \(1.5/20=0.07500\); that is a further approximation and gives a smaller estimated change here.
If this is a log hazard-ratio coefficient, the full fitted hazard ratio is \(\exp(0.50)=1.649\), while the approximate deleted-fit hazard ratio is \(\exp(0.4167)=1.517\). These are model-based transition hazard ratios, not overall state-occupancy probabilities. The approximation says the subject shifts this fitted coefficient upward; it does not say the subject caused a change in hazard.
Influence Across Multiple Transitions
With transition-specific parameters, \(\beta\) is a vector. The subject score has one component for each coefficient, while the information matrix can have off-diagonal elements. These arise when parameters are estimated jointly or their score contributions are statistically associated. Therefore, a subject whose direct score for one transition is modest can still affect another coefficient through the inverse information matrix. Inspect both transition-level changes and an overall summary.
Worked Example: A Subject Contributes Across Two Transitions
Consider a hypothetical model with coefficients \(\beta_A\) and \(\beta_B\) for two transitions. For one subject, suppose the score at the full estimate is \(u_i=(0.8,-1.2)^\mathsf{T}\), and the deleted-sample information evaluated at the full estimate is $$ J_{(-i)}= \begin{pmatrix} 16 & 4\\ 4 & 9 \end{pmatrix}. $$ Its determinant is \(16(9)-4(4)=128\), so $$ J_{(-i)}^{-1}=\frac{1}{128} \begin{pmatrix} 9 & -4\\ -4 & 16 \end{pmatrix}. $$
The approximate coefficient change is $$ \hat{\beta}-\hat{\beta}_{(-i)} \approx J_{(-i)}^{-1}u_i =\frac{1}{128} \begin{pmatrix} 9(0.8)-4(-1.2)\\ -4(0.8)+16(-1.2) \end{pmatrix} = \begin{pmatrix} 0.09375\\ -0.1750 \end{pmatrix}. $$
Thus deletion is expected to lower the first coefficient by about 0.09375 and raise the second by about 0.1750. The off-diagonal information terms affect both changes; calculating each as its score divided by a diagonal information element would not reproduce this result. The signs describe changes in the fitted coefficients, not whether either transition effect is statistically significant. Exact deletion refits would be needed to check the accuracy of these local approximations.
A single scalar summary can help rank cases, but should not replace transition-specific inspection. For example, a model-based generalized Cook-type measure is $$ D_i=\frac{(\hat{\beta}-\hat{\beta}_{(-i)})^\mathsf{T}J(\hat{\beta}-\hat{\beta}_{(-i)})}{q}, $$ where \(q\) is the number of fitted coefficients and \(J\) is the full observed information. This measures the coefficient displacement relative to the curvature of the full likelihood. It is dimensionless under a consistent parameterization, but there is no universal cutoff that makes a case “influential” in every application. A large value should lead to review of the component-wise changes and their scientific relevance.
Coefficient and Standard-Error Changes
For every exact deletion refit, report the coefficient difference in its original units and, when useful, a standardized change such as \((\hat{\beta}_j-\hat{\beta}_{j,(-i)})/SE(\hat{\beta}_j)\). This standardization measures the change relative to the full-fit standard error; it is not a test statistic with a universal reference distribution. For coefficients on the log hazard scale, also consider translating the full and deleted coefficients to hazard ratios by exponentiation, while retaining the transition and covariate contrast in the interpretation.
Worked Example: Interpreting an Exact Deletion and Its Standard Error
In a hypothetical transition model, an exact full fit gives a coefficient of 0.40 with standard error 0.20. Removing one subject and refitting gives a coefficient of 0.25 with standard error 0.18. The coefficient change is \(0.40-0.25=0.15\), or \(0.15/0.20=0.75\) full-fit standard errors. The deleted-fit standard error is lower by \(0.20-0.18=0.02\), which is a 10% decrease relative to the full-fit standard error: \(0.02/0.20=0.10\).
If the coefficient is a log transition hazard ratio, the corresponding estimates are \(\exp(0.40)=1.492\) and \(\exp(0.25)=1.284\). These hypothetical calculations show that the subject affects both the point estimate and its estimated precision. They do not determine whether the scientific conclusion changes: that requires examining the full and deleted estimates, intervals, model context, and the prespecified inferential question.
Standard errors can increase or decrease after deletion. A subject may add information, or may create instability through an unusual covariate pattern or sparse event contribution. Compare the full and deleted covariance estimates, not just the standard errors for one coefficient. With a sandwich variance estimator, recompute the sandwich covariance for the deleted subject set; the inverse observed information alone is not the sandwich standard error. If deletion changes which parameters are estimable, a simple numerical comparison may no longer be meaningful.
Diagnostics, Alternatives, and Failure Modes
A practical assessment combines exact refits for leading cases with score approximations for screening. For each subject, retain the vector of coefficient changes, standard-error changes, and a summary measure if appropriate. Sort or plot these quantities by subject, and examine which transitions and covariates drive the largest changes. A case with a large aggregate displacement concentrated in one transition can be hidden by a summary alone; conversely, several moderate changes across transitions may matter jointly.
| Approach | What it answers | Main caution |
|---|---|---|
| Exact case deletion | How estimates and uncertainty change after removing the subject and refitting | Computationally demanding; deleted fits may be unstable or unidentified |
| Score-based one-step approximation | Which cases may have substantial local coefficient influence | Can be poor when deletion materially changes information or moves the estimate far |
| Residual diagnostics | Whether a subject’s observed outcome or transition pattern is unusual relative to the fitted model | A large residual is not equivalent to a large effect on fitted coefficients |
| Leverage or design review | Whether covariate values or risk-set representation are unusual | Unusual design values alone do not establish coefficient influence |
Transition-specific Schoenfeld and martingale residual diagnostics address aspects of model fit, including proportional-hazards and covariate-form concerns; they are not substitutes for deletion diagnostics. Conversely, case deletion cannot establish that a model assumption is adequate. The earlier material on Schoenfeld Residual Diagnostics by Transition and Martingale Residuals for Transition Covariate Forms addresses those residual-based questions.
Warning signs include a large discrepancy between the one-step approximation and exact deletion, a deleted fit with very large standard errors, unstable coefficient signs, failure to converge, or a nearly singular deleted information matrix. Sparse events can make these failures more likely: removing one event subject may leave a transition with little or no information for a coefficient. In that case, report the failure as an important fragility of the fitted analysis rather than forcing a numerical influence score.
Common Pitfalls and Limitations
- Using full information without qualification. The Taylor expansion of the deleted score uses \(J_{(-i)}\). Substituting \(J\) requires a separate negligible-information-contribution approximation.
- Deleting only a transition record. For subject-level influence, omit the person’s complete modeled contribution across transitions, not just one row from a long-format data set.
- Treating a diagnostic as an exclusion rule. An influential subject may be a valid observation carrying important information. Investigate data quality and sensitivity; do not remove a case solely to obtain a preferred result.
- Using one cutoff across all models. Influence depends on parameterization, sample size, event distribution, and the scientific scale of the effect. No universal threshold applies to every transition model.
- Confusing influence with fit or significance. A case can change an estimate substantially without having a large residual, and a large residual need not materially change a coefficient. Influence is also not evidence for or against a null hypothesis.
Deletion diagnostics are finite-sample sensitivity analyses conditional on the chosen model. They do not assess all possible perturbations, explain why a case is influential, or guarantee validity when the model is misspecified. Their value is greatest when the subject-level changes are presented transparently and interpreted alongside transition-specific fit checks and the study’s substantive context.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Why is a subject’s entire observed path generally removed for a subject-level influence analysis in a multi-state model?
- A full fit has \(J=30\), and a subject has information contribution 6 and score 2. What are the deleted-information one-step coefficient change and the full-information shortcut? Which is the direct Taylor approximation?
- For two transition coefficients, why can the change in one coefficient depend on both components of the subject’s score?
- A deletion reduces a coefficient’s standard error. Does that imply the subject was harmful to the analysis? Explain.
Solutions. (1) The person contributes linked risk-time and event information across transitions; deleting only one record does not represent removing that subject and can leave an inconsistent partial contribution. (2) The deleted information is \(30-6=24\). The direct one-step change is \(2/24=0.08333\). The full-information shortcut is \(2/30=0.06667\), which additionally assumes deletion has negligible effect on information. (3) The inverse deleted information matrix generally has off-diagonal entries, so multiplying it by the score vector mixes score components in calculating each coefficient change. (4) No. The standard error may decrease because the removed subject contributed relatively little information or affected the fitted information structure. The decrease is a sensitivity result, not a judgment that the subject is harmful or should be excluded.