Robust Variance for Pooled Logistic Survival MSMs

Why the Person Is the Variance Unit

A pooled logistic survival model represents follow-up using person-interval records. A person may contribute several rows, but those rows are not separate, independently sampled people. They share a history, a treatment trajectory, and the same underlying outcome process. In particular, an individual can experience the event only once and leaves the risk set after that event. Treating each interval as an independent unit can therefore give an incorrect standard error even when the fitted coefficient is unchanged.

The relevant variance unit is the independent person (or other independent sampling cluster), not the row. The person-clustered sandwich estimates uncertainty from each person's total score contribution, allowing the interval-specific contributions within that person to be dependent. This tutorial treats the treatment weights as fixed in the outcome fit. It does not include uncertainty from estimating those weights.

The vector \(x_{ij}\) can include the baseline-hazard terms and treatment terms used in the marginal structural model. The interval weights are those assigned to the relevant records under the established treatment-weight timing. A person who contributes multiple intervals has one cluster score: the sum of the weighted score contributions across those intervals.

With fixed weights, the negative derivative of the score is \(\sum_j R_{ij}w_{ij}p_{ij}(\theta)\{1-p_{ij}(\theta)\}x_{ij}x_{ij}^{\mathsf T}\). Adding these matrices across people gives the usual weighted logistic “bread.” The sandwich “meat,” by contrast, is built from person-level score totals, not separate rows.

The Clustered Sandwich and Its Justification

The result below is stated for a general estimating equation, so it also covers a weighted score whose model-based variance is not correct. Write \(\theta_0\) for the parameter solving \(E[U_i(\theta_0)]=0\). Under a correctly specified marginal model, this is the target parameter; under a misspecified mean model, it is the population root of the estimating equation, if that root is well defined.

These regularity conditions require, in particular, that the relevant derivatives and score second moments be integrable and that the estimator remain in a neighborhood where the score expansion is valid. They also require independent people; dependence within a person is allowed. For weighted pooled logistic scores, one commonly uses the empirical bread and the outer products of fitted person-level scores:

Here \(\hat{p}_{ij}=p_{ij}(\hat{\theta})\). Because \(\hat{A}\) and \(\hat{B}\) are sums, no additional factor of \(1/n\) appears in this expression. Equivalently, define the empirical averages \(\hat{A}/n\) and \(\hat{B}/n\); the covariance is then \(n^{-1}(\hat{A}/n)^{-1}(\hat{B}/n)(\hat{A}/n)^{-\mathsf T}\). A finite-sample correction may be applied by software, but it does not change the defining cluster-level construction.

Derivation. Set \(\bar U_n(\theta)=n^{-1}\sum_iU_i(\theta)\). The estimating equation gives \(\bar U_n(\hat{\theta})=0\), while \(E[U_i(\theta_0)]=0\). Apply the fundamental theorem of calculus along the line segment from \(\theta_0\) to \(\hat{\theta}\). The resulting matrix is an average Jacobian along that segment; in general, it is not the Jacobian at a single intermediate point.

Call the bracketed matrix \(A_n^*\). By consistency and the stated uniform law of large numbers and continuity conditions, \(A_n^*\xrightarrow{p}A\). The multivariate central limit theorem for independent person-level scores gives \(n^{-1/2}\sum_iU_i(\theta_0)\xrightarrow{d}N(0,B)\). Since \(A\) is nonsingular, Slutsky's theorem yields

For the pooled logistic score, differentiating \(Y_{ij}-p_{ij}(\theta)\) gives the stated bread. To see why the meat must be clustered, expand a person's outer product:

The second sum contains cross-interval terms. Adding separate row-level outer products omits them. This is not a harmless bookkeeping choice: those terms represent the within-person dependence that the variance estimator must allow.

Worked Calculations

Interpretation, Alternatives, and Diagnostics

The usual inverse-Hessian variance is appropriate when the likelihood's independent-observation variance assumptions describe the data. For pooled logistic survival records, that calculation treats rows as independent contributions. The person-clustered sandwich keeps the same fitted estimating equation and bread but replaces the model-based meat with person-level score outer products. A person-level bootstrap is another option: resample people, retaining all of each selected person's intervals. Resampling individual rows would break the observed person histories and is not an appropriate substitute.

The sandwich does not repair a biased estimating equation. It cannot fix incorrect treatment weights, a misspecified marginal mean model, positivity problems, or an incorrect risk-set construction. Nor does the fixed-weight sandwich above account for estimating the weights; that is a separate source of uncertainty. It addresses variance misspecification due to dependence and other departures from the working variance, provided the person-level estimating equation and regularity conditions are suitable.

Check that the cluster identifier is the person, that all of each person's at-risk records enter one cluster score, and that the number of independent people is adequate for the intended approximation. Inspect fitted coefficients and standard errors for sensitivity to highly influential people, and compare clustered with model-based standard errors as a diagnostic—not as a rule for choosing whichever is smaller. Sparse events, extreme weights, or a nearly singular bread can make either variance estimate unstable. With few clusters, ordinary sandwich standard errors can be biased; small-sample corrections or carefully designed person-level resampling may be considered, but neither creates information absent from the data.