Sandwich Variance From Individual Score Processes

Why the Individual Score Matters

An event-history model can assign a score contribution to each observed subject or independent cluster. The total score is the sum of these contributions, but its variance depends on how scores vary between independent units. This distinction matters when a subject contributes repeated events or several event types: the subject’s entire history must first be combined into one score vector. Treating that history as several independent records can discard dependence and produce an incorrect variance.

The sandwich estimator combines two quantities. The bread measures how quickly the estimating score changes with the parameter. The meat measures empirical variation in the independent units’ scores. The resulting covariance estimate is useful alongside model-based information, which describes score variation implied by the specified intensity model.

Individual Martingale Scores and the Sandwich Formula

Suppose \(n\) independent subjects are followed over a specified observation interval. For subject \(i\), let \(N_{ik}(t)\) count events of type \(k\), and let \(Y_{ik}(t)\) indicate whether that event type is at risk. Under a multiplicative intensity model with parameter vector \(\beta\), the intensity is \(Y_{ik}(t)\lambda_{ik}(t;\beta)\). Define the compensated counting process \(M_{ik}(t)=N_{ik}(t)-\int_0^t Y_{ik}(u)\lambda_{ik}(u;\beta)\,du\), and the log-intensity derivative \(S_{ik}(t;\beta)=\partial\log\lambda_{ik}(t;\beta)/\partial\beta\).

This representation presumes the specified intensity is the conditional event rate given the observed past, the covariate and risk processes used in the intensity are predictable, and the compensator is well defined. The martingale score has mean zero under the model when the relevant integrability conditions hold. The subjects’ complete observed histories are the independent units; repeated events within a subject do not create new independent units.

Define the normalized sensitivity, or bread, by \(A(\beta)=-n^{-1}\partial U(\beta)/\partial\beta^\mathsf{T}\), and let \(\hat A=A(\hat\beta)\). Define the empirical score-variation matrix, or meat, by \(\hat B=n^{-1}\sum_i U_i(\hat\beta)U_i(\hat\beta)^\mathsf{T}\). The sandwich covariance estimator is

The transpose on the final bread is essential when the sensitivity matrix is not symmetric. Equivalently, if \(H=-\partial U(\hat\beta)/\partial\beta^\mathsf{T}=n\hat A\) and \(J=\sum_i U_i(\hat\beta)U_i(\hat\beta)^\mathsf{T}=n\hat B\), the same estimator is \(H^{-1}J(H^{-1})^\mathsf{T}\). The two expressions differ only in whether the factors of \(n\) are included in the bread and meat.

Why the Sandwich Has This Form

Assume independent subjects, a consistent interior estimator, differentiable scores, a nonsingular limiting sensitivity, finite score second moments, and a central limit theorem for the summed scores. Also assume the fitted estimating equation is the sum of the subject scores. A first-order expansion around the true parameter \(\beta_0\) gives

Rearranging and multiplying by \(\sqrt n\) yields \(\sqrt n(\hat\beta-\beta_0)\approx A(\beta_0)^{-1}n^{-1/2}\sum_i U_i(\beta_0)\). If the individual scores have covariance matrix \(B\), independence gives covariance \(B\) for their normalized sum. Thus the limiting covariance of \(\sqrt n(\hat\beta-\beta_0)\) is \(A^{-1}B(A^{-1})^\mathsf{T}\); the covariance of \(\hat\beta\) is this matrix divided by \(n\). Replacing the unknown sensitivity and score covariance by their sample estimates gives the sandwich estimator.

This derivation explains why the meat must be formed from independent-unit scores. It also shows the assumptions are substantive: a singular or poorly estimated bread makes the result unstable, and dependence between supposed independent units invalidates the simple sum of outer products.

Model-Based Information Versus Empirical Score Variation

The model-based alternative uses the information implied by the intensity model. In the common case where distinct event types have no simultaneous jumps, the predictable variation of the individual martingale score is

The corresponding normalized model-based information is the sum of these predictable variation matrices divided by \(n\). Under the correctly specified likelihood and the usual regularity conditions, expected score variation agrees with expected information. This is the information identity. The empirical meat instead uses the outer products of the realized, whole-subject scores. With a large sample under the information identity, both approaches target the same asymptotic covariance. In a finite sample, their numerical estimates need not agree.

If event types can jump together, the predictable variation must include the relevant cross-variation terms; simply adding separate type-specific terms is not generally sufficient. The score outer product automatically retains cross-component variation within each subject. Model-based information can be more stable in small samples, while the empirical meat directly estimates variation across independent units. Neither calculation excuses a wrong independence structure or an ill-conditioned bread.

Implementation Checks, Comparisons, and Limitations

A common error is to treat every event contributed by one subject as an independent score. That reasoning uses the wrong independent units and can omit within-subject covariance. The correction is to sum all of that subject’s score-process contributions first, then take one outer product of the resulting vector. Another error is to omit the \(1/n\) factor while using normalized bread and meat; the unnormalized formula \(H^{-1}J(H^{-1})^\mathsf{T}\) is equivalent, but mixing the two conventions is not.

The sandwich estimates uncertainty around the fitted parameter; it does not change the point estimate, repair a singular model, or establish that the intensity specification is appropriate. A few influential scores can dominate the meat, and a poorly conditioned bread can inflate or destabilize the covariance. These diagnostics should prompt examination of the data structure and model calculations, not automatic removal of influential subjects. For dependent clusters, the independent cluster—not the individual event—must be the unit used in the empirical meat, and enough independent clusters are needed for reliable large-sample inference.

The central comparison is therefore between empirical variation of whole-unit scores and information implied by the model. Under the stated regularity and information-identity conditions they agree asymptotically; in finite samples they can differ. Reporting the estimator, independent unit, and assumptions makes the variance calculation interpretable.