Reconstructing Cumulative Scores Across Looks

Why Reconstruct the Cumulative Quantities?

A group sequential logrank analysis reports a statistic at each planned look, but the underlying evidence accrues between looks. Keeping the interval-specific score and variance increments separate from their cumulative totals makes it possible to verify what was available at each analysis, reproduce the standardized statistic, and locate inconsistencies in an analysis record. A standardized value at one look cannot generally be obtained by adding standardized values from earlier looks: the score and its variance must be accumulated first.

The reconstruction assumes nested looks: each later look includes all event information available at earlier looks, plus information from the intervening interval. Each event contribution must be assigned once, and the score convention and variance calculation must be consistent across intervals. The earlier material on independent logrank increments supplies the conditions for treating interval increments as independent or, in the usual large-sample formulation, approximately so. Here the emphasis is on reconstructing and checking the quantities at the looks.

For a clear sign convention, take each logrank score contribution to be the observed number of treatment-arm events minus the number expected under the null, conditional on the risk sets and event totals. Thus a positive score indicates more treatment-arm events than expected. When events are undesirable, that direction generally represents harm rather than benefit; conclusions must follow the convention and endpoint in the analysis plan. Score increments are dimensionless contrasts in event counts, and variance increments are in squared score units. If the analysis is stratified, each interval’s score and variance increments are the corresponding sums across strata.

Reconstructing and Standardizing at Each Look

The reconstruction is a cumulative sum, not an average. At look \(k\), add every interval increment through \(k\), then divide the cumulative score by the square root of the cumulative variance. Because a variance increment cannot be negative, the cumulative variance should not decrease. A zero variance increment is possible when an interval contributes no score information; a standardized statistic is undefined if the cumulative variance is zero.

The sums also explain why the method is appropriate under the stated increment assumptions. If the interval scores have mean zero under \(H_0\), are independent (or satisfy the relevant asymptotic independent-increment conditions), and have variances \(\Delta V_j\), then the variance of their sum is the sum of their variances:

The middle equality follows because independent increments have zero covariance, so all cross-covariance terms in the variance of the sum are zero. Under the usual large-sample normal approximation, \(U_k/\sqrt{V_k}\) is therefore the standardized score at look \(k\). The approximation is not a claim that every finite-sample logrank statistic is exactly normal. It relies on adequate information and the applicable event-process assumptions.

Recovering Increments from Cumulative Records

Sometimes a record gives cumulative values at each look rather than the interval increments. In that case, recover an interval increment by subtracting the preceding cumulative value. The initial cumulative values are \(U_0=0\) and \(V_0=0\). This subtraction works only when the reported cumulative values refer to the same score definition, strata, event inclusion rules, and analysis population at successive looks.

Checks, Assumptions, and Failure Modes

A reconstruction is meaningful only if each interval’s quantities belong to the same sequential analysis. In particular, the looks must be nested, interval boundaries must not omit or double-count events, and the score’s treatment-arm convention must remain unchanged. The variance increments must be computed consistently with the logrank score, including the trial’s handling of strata and tied event times. The risk sets and event contributions used to produce the increments must also be valid for the specified analysis cutoffs.

Under the independent-increment conditions, the cumulative variance is the sum of the interval variances. More generally, the exact identity for the variance of a sum includes covariance terms; simply adding interval variances is justified only when those cross-covariances are zero or negligible under the adopted approximation. If increments are correlated in a way the analysis has not accounted for, the sum of their marginal variance estimates need not be the variance of the cumulative score. Reconstructing the arithmetic cannot repair a failure of the underlying assumptions.

A practical audit should check the following before interpreting the resulting \(Z_k\) values:

• Re-add the score increments. Each cumulative score should equal the sum through that look, with the same sign convention throughout.
• Check variance monotonicity. Each \(\Delta V_j\) should be nonnegative, and \(V_k\) should not decrease. A decrease often signals a transcription error or a change in what is being reported.
• Recalculate each standardization. Compare the reported statistic with \(U_k/\sqrt{V_k}\), using the cumulative values, not the latest interval alone.
• Check the analysis record. Confirm that the intervals partition the events used in the looks and that the reported score and variance use a consistent method.
• Distinguish rounding from a substantive mismatch. Small differences may arise from rounding displayed increments. Reconcile with unrounded values when available; do not silently alter an increment to force agreement.

What This Reconstruction Does—and Does Not—Establish

Reconstructing the cumulative quantities is a bookkeeping and verification step, not a test of efficacy by itself. The value \(Z_k\) summarizes the cumulative score relative to its estimated standard deviation at look \(k\). Whether it crosses a prespecified efficacy or futility boundary, and how repeated looks affect error rates, require the monitoring plan and its relevant decision rules. A statistic’s magnitude alone does not supply those rules.

This reconstruction is also not a substitute for recomputing event contributions from the underlying risk sets when those contributions are in doubt. Summing supplied increments verifies their arithmetic consistency, but cannot establish that the increments were calculated from the correct participants, events, strata, or cutoff-specific risk sets. The alternative of computing each look directly from all event-level records can help investigate a mismatch, but it is a distinct calculation that must use the same analysis definitions. When both approaches are valid, cumulative summation and a direct calculation should agree apart from rounding.