Person-Level Clustering in Repeated-Interval Survival Fits

Why the Person, Not the Interval, Is the Cluster

A start-stop data set may contain several records for each person: one for each period during which covariates, treatment, or risk status are represented as constant. Those records help express a time-varying survival model, but they do not create new independent people. Repeated intervals from one person share a history and are linked by the same event and censoring process. Treating them as independent units can therefore give an incorrect estimate of uncertainty.

The key operation is to form the score contribution for each interval and then add all contributions belonging to the same person. The sandwich “meat” is built from these person-level sums. For a Cox fit, a useful interval contribution accounts not only for an observed event, but also for the interval’s fitted contribution through the risk set. This distinction matters: assigning the whole event-time score only to the person who experienced the event is not, in general, the right person-level decomposition for robust variance.

Interval Contributions in a Weighted Cox Fit

Let \(i=1,\ldots,n\) index people, and let \(k\) index the start-stop intervals for person \(i\). Write \(Y_{ik}(t)\) for the interval’s at-risk indicator, \(X_{ik}(t)\) for its covariate vector, and \(W_{ik}(t)\) for its weight. The weight is assumed predictable: its value at time \(t\) is determined by information available before the event increment at \(t\). Let \(\beta\) be the Cox regression parameter.

At time \(t\), define the weighted risk-set sums and the corresponding weighted covariate mean by

Here \(X^{\otimes 0}=1\) and \(X^{\otimes 1}=X\). For simplicity, the formulas below use a Breslow-style weighted risk-set denominator; with tied-event handling based on another specified convention, the contributions must follow that same convention.

Let \(N_{ik}(t)\) count events on interval \(ik\), and let \(\widehat{\Lambda}_0(t)\) be the fitted baseline cumulative hazard. At a candidate value \(\beta\), a martingale-style score-residual contribution for the interval is

The first term inside braces records the observed event increment; the second is the fitted event increment. The risk-set centering in \(X_{ik}(t)-\bar X(\beta,t)\) is essential. Define the person-level contribution by summing across that person’s records:

Cluster Sandwich Variance

Let \(\hat\beta\) solve the weighted Cox score equation \(U(\hat\beta)=0\), and let

The person-cluster sandwich estimate of the covariance matrix is

The bread \(\widehat A\) measures the sensitivity of the fitted score to changes in \(\beta\); the meat sums one outer product per independent person. This formula uses the total-score scaling shown above, so the covariance estimate already has the appropriate sample-size scaling. Small-sample software corrections may multiply it by a factor such as \(G/(G-1)\), where \(G\) is the number of independent clusters. Such a correction does not change which records must be clustered together.

To see the argument, write \(U_i(\beta)\) for the person-level estimating contribution, with \(U(\beta)=\sum_i U_i(\beta)\). A first-order expansion of the estimating equation around \(\beta_0\) gives

where \(A_n(\beta)=-\partial U(\beta)/\partial\beta^\mathsf{T}\), and \(\tilde\beta\) lies between \(\hat\beta\) and \(\beta_0\). Under consistency and convergence of the normalized derivative, \(A_n(\tilde\beta)/n\xrightarrow{p} A\) for a nonsingular matrix \(A\). Therefore

The cluster contributions, not the interval records, are the independent terms in this sum. The cluster-level central limit theorem gives a limiting covariance equal to \(A^{-1}BA^{-\mathsf T}\), where \(B\) is the covariance of a person’s score contribution under the chosen normalization. Replacing \(A\) and \(B\) by the empirical bread and the sum of person-level outer products yields the stated sandwich estimator. The argument fails if rows from the same person are incorrectly treated as independent summands. \(\square\)

This derivation concerns the uncertainty calculation conditional on the weights used in the outcome fit. It does not, by itself, account for uncertainty from estimating those weights.

Implementation, Diagnostics, and Alternatives

A reliable implementation keeps the unit of the outcome model separate from the unit of independent sampling. Intervals determine when each person contributes to risk sets and which covariates apply. People determine the independent clusters used in the meat. In practice, verify that the cluster identifier is the person identifier, that every interval has been assigned to the right person, and that the contribution being summed is the appropriate Cox score residual for the selected tie and weighting conventions.

Useful checks include confirming that the sum of person-level residual contributions agrees with the fitted score, that there is one meat contribution per person rather than per interval, and that the number of independent clusters is reported separately from the number of rows. A large change in standard errors after correcting the cluster identifier is a reason to inspect the data structure and score-residual calculation, not a reason to choose whichever result is smaller.

The model-based Cox variance is an alternative when its information-based variance assumptions are appropriate. It is not a substitute for clustering merely because the data are stored in a Cox model. Row-level sandwich calculations likewise do not protect against dependence between intervals of one person. A person-level bootstrap, which resamples whole people and retains all their intervals, is a different way to assess sampling variation; resampling rows independently would break the observed within-person structure.

Cluster-robust uncertainty has limitations. It relies on having enough independent people for the large-sample approximation, and a few highly influential people can make the meat unstable. It does not repair a misspecified causal estimand, invalid weights, an incorrectly formed risk set, or an inappropriate cluster definition. It also does not automatically incorporate estimation of the weights. These issues require separate checks and methods.

Common Mistakes

• Clustering by interval. Multiple records from one person are not independent just because they occupy separate rows. Use the person as the cluster.
• Squaring before summing. The correct scalar meat contribution is \((\sum_k q_{ik})^2\), not \(\sum_k q_{ik}^2\). In vector form, it is \(U_iU_i^\mathsf{T}\), formed after aggregation.
• Giving all event-time score to the event person. The Cox score-residual decomposition also has fitted-risk contributions for people at risk. Use score residuals consistent with the weighted risk-set score.
• Assuming clustering fixes everything. A robust sandwich changes the uncertainty calculation; it does not correct an invalid weighting model, poor positivity, or a wrongly specified outcome model.