Incremental Information Between Analyses

Why Incremental Information Is a Risk-Set Calculation

At an interim analysis, the logrank information accumulated so far is determined by the variance contributions of the event risk sets—not simply by how many participants have enrolled or how many events have occurred. The next analysis adds information as newly incorporated event records contribute additional risk-set variance. Calculating that increment explicitly makes the change in information auditable and helps detect problems such as inconsistent risk sets, mishandled ties, or accidental double-counting.

The focus here is the calculation from one analysis to the next. The risk-set contribution is the event-level variance of the logrank score under the null hypothesis \(H_0\). The earlier material on deriving information from risk-set variances establishes how to obtain that contribution; the essential formulas are stated below so the incremental calculation is self-contained.

Definition and Assumptions

For an event time \(j\), let \(Y_{1j}\) and \(Y_{0j}\) be the numbers at risk immediately before that time in treatment groups 1 and 0. Write \(Y_j=Y_{1j}+Y_{0j}\). With one event at that time, the logrank variance contribution is

If \(d_j\) events are tied at the same time, the usual conditional hypergeometric variance contribution is

This tied-event formula applies when \(Y_j>1\), and the event count satisfies \(0\le d_j\le Y_j\). If \(d_j=1\), it reduces to the untied-event formula. A tied event should be handled as a tied set using the selected logrank tie convention; treating its events as a sequence of untied events can give a different variance.

Let \(\mathcal{J}_k\) denote the newly incorporated event times at analysis \(k\). Then

These expressions assume that the treatment-group risk sets, event counts, and tie handling are defined consistently at both analyses. The standard logrank calculation also relies on independent participants and censoring that is non-informative for the event process, at least conditional on the factors used in the analysis. In particular, within the risk sets, censoring must not selectively remove participants in a way that distorts the event comparison. If the analysis is stratified, calculate contributions within each stratum and add them; do not replace the stratum-specific risk sets with a pooled risk set.

Why the Contributions Add

The addition follows from the variance structure of the sequential logrank score. Under \(H_0\), the score contribution at each event time has conditional mean zero given the risk set and the preceding observed history. The risk-set variance \(V_j\) is its conditional variance. The event-time score increments are orthogonal under the logrank martingale formulation, so the unconditional variance of their sum is the sum of their unconditional variances. The sum of the predictable variance contributions, \(\sum_{j\in\mathcal{J}}V_j\), is the observed information.

Here \(U_j\) is the logrank score increment for event time \(j\), and \(\mathcal{J}\) is the set of event times included. If the later analysis contains all the earlier event increments unchanged, its cumulative variance is the earlier cumulative variance plus the variances of the new increments. Subtracting the two cumulative values therefore gives the sum over newly incorporated events. This justification depends on consistent data and risk-set construction: if an earlier event's risk set is revised, a simple sum of only the newly recorded event contributions need not equal the change in cumulative information.

Tied Events and Stratified Increments

When multiple events occur at the same recorded time, the risk set is the set immediately before that tied time, and the variance contribution uses the tied-event formula. In a stratified analysis, each tied or untied contribution is calculated within its stratum; the total increment is then the sum across newly incorporated events and strata.

Checking the Calculation and Choosing a Comparison

A practical calculation should retain the event-level contributions, not just the final sum. For each newly incorporated event record, verify the event time, the analysis stratum, \(Y_1\), \(Y_0\), the number tied \(d\), and the tie formula used. Then verify that the reported increment is the sum of exactly those records not already included at the preceding look. When possible, compare that sum with the difference between the two cumulative information values, using unrounded values for the check.

Incremental information is a risk-set variance measure. A raw event count is easier to report, but it does not show how the events were distributed across treatment and control risk sets. For example, a risk set with participants in both groups contributes positive information, whereas an event in a risk set with no participants in one group has zero contribution under the untied formula. Thus, equal numbers of events need not imply equal information. The direct sum of new contributions is the primary calculation; subtracting cumulative values is a useful cross-check when the underlying records are nested and unchanged.

• Check for double-counting. Label which event records were already included at the prior look. Do not sum all records at the later look as if they were all new.
• Check tie handling. Use the tied-event formula for tied records under that convention, rather than repeating an untied formula using one unchanged risk set.
• Check risk-set timing. Counts must be immediately before the event time and must reflect the stated analysis population and stratum.
• Check data revisions. If earlier event times, censoring times, treatment assignments, or stratum labels are corrected, recompute affected historical risk sets and contributions before interpreting the difference as information from new events alone.
• Check precision. Sum unrounded contributions and round the reported total at the end.

A useful diagnostic is whether the increment agrees with both the event-level sum and the difference in cumulative information. A discrepancy can indicate a missing or duplicated event, inconsistent tie handling, a change in the analysis population, or premature rounding. Another warning sign is a risk set with \(Y_1=0\) or \(Y_0=0\): its untied contribution is zero, so an unexpectedly positive contribution suggests a coding or calculation error.

Common Mistake and Limitations

The additivity calculation is only as reliable as the data and risk-set construction behind it. Non-informative censoring, correct event and censoring times, consistent eligibility rules, and appropriate treatment of ties matter. The sum is the logrank information under the stated analysis and variance convention; it does not by itself establish that the proportional-hazards interpretation is suitable or that a planned analysis has achieved its intended operating characteristics. If historical risk sets change between looks, a simple “new events only” sum is not enough: the changed contributions must also be recalculated.