Realized Information Fractions At Interim Looks

Why Planned and Realized Fractions Can Differ

In an event-driven logrank design, interim analyses are often scheduled by target event counts. Those targets provide a practical schedule, but the statistical time scale for sequential monitoring is information, not simply the number of participants enrolled or events observed. At a data cutoff, the risk sets and event contributions may yield more or less logrank information than planned. Consequently, the actual information fraction at a look can differ from the fraction assigned to that look in the design.

This distinction matters when a sequential boundary is indexed by information time. Applying a boundary associated with the planned fraction without checking the information actually available can misrepresent where the analysis lies on the design’s information scale. The key is to compare like with like: use the observed logrank information at the cutoff and the design’s prespecified maximum information, unless the maximum has been formally revised under a recalibrated design.

For a logrank analysis, information is derived from the event contributions and risk sets; for stratified analyses it accumulates across strata. Thus, the event count may be exactly on target while information is below or above target, or event attainment may itself differ from plan. Earlier material on deriving logrank information and updating risk sets supplies the quantities \(I_j\) used here. The present question is how to place that observed information on the sequential design’s scale.

Which Fraction Governs the Information Scale?

Suppose the maximum information is fixed at \(I_{\max}\). The planned fraction \(t_j^*\) describes where the design intended look \(j\) to occur. The realized fraction \(t_j\) describes where the available information places the actual analysis. When the sequential procedure is defined on an information-time scale, the realized fraction is the relevant location for applying or recalculating the boundary under the prespecified rule. The planned fraction remains important for documenting the departure from schedule, but it does not change the amount of information observed.

This use of realized information follows from the canonical information scale. For two looks \(i<j\), the canonical approximation gives the correlation of their standardized logrank statistics as approximately \(\sqrt{I_i/I_j}\). Dividing each cumulative information value by the same maximum \(I_{\max}\) places the looks at information times \(t_i\) and \(t_j\); their relative position is determined by the information actually accumulated. Replacing \(I_i\) with a planned value when the observed information differs would describe a different information trajectory.

The equation explains why the fraction is not merely a scheduling label. It reflects the relative information available at each look. It does not, by itself, specify a numerical efficacy boundary or establish that a particular boundary procedure is valid; those require the design’s prespecified monitoring rule. If a protocol specifies boundaries only at planned looks, the protocol must also specify how to handle a look that occurs at a different information fraction, such as evaluating a boundary rule at the observed fraction or using an explicitly approved adjustment. Simply relabeling the realized look as the planned look is not a statistical correction.

Worked Comparisons

A Short Justification for Using Observed Information

The justification can be made directly from the information-time representation. Suppose the cumulative information at two actual looks is \(I_1<I_2\), and define \(t_1=I_1/I_{\max}\) and \(t_2=I_2/I_{\max}\), with a common fixed \(I_{\max}>0\). Then

So the information-time locations preserve the relative information that determines the canonical correlation approximation. If one instead inserts planned information \(I_1^*\) when the actual value is \(I_1\ne I_1^*\), the ratio becomes \(\sqrt{I_1^*/I_2}\), which is generally not \(\sqrt{I_1/I_2}\). The planned location may still be used if the procedure explicitly defines a fixed planned-look rule, but it should not be described as the realized information time.

Assumptions, Diagnostics, and Failure Modes

Interpreting a realized fraction as information time relies on a coherent information calculation and a monitoring design based on the relevant sequential model. In particular, information should be calculated from the data available at the cutoff using the prespecified logrank definition, risk-set rules, and stratification. The maximum \(I_{\max}\) must be the same design quantity used to define the monitoring plan. The canonical approximation also depends on conditions supporting the sequential logrank model, including appropriate handling of event contributions and the planned analysis structure.

Before using a fraction, check the following:

• Consistent numerator: \(I_j\) must be cumulative information at the current cutoff, not information added since the preceding look. Incremental information is useful for updating cumulative information, but it is not the cumulative fraction by itself.
• Consistent denominator: \(I_{\max}\) must refer to the prespecified maximum information on the same scale. Do not divide observed interim information by a projected final value unless the design explicitly defines and adopts that denominator.
• Correct cutoff data: event dates, censoring, risk-set membership, strata, and other analysis rules must correspond to the stated cutoff. A data error can make an apparently precise fraction misleading.
• Event counts are only a proxy: a ratio of event counts can be reported as an event fraction, but it is not automatically an information fraction. Compare the actual variance-based information with its target.
• Boundary rule matches the scale: confirm whether the approved procedure uses actual information time, fixed scheduled looks, or another specified rule. Do not choose whichever fraction produces a preferred decision.

Large differences between planned and realized fractions warrant investigation. They may reflect event timing, unequal contributions to information, changing risk-set composition, or an incorrect calculation. A realized fraction above 1 under a fixed \(I_{\max}\) is a warning that the design maximum has been exceeded, the denominator is inconsistent, or the information calculation needs review; it is not automatically permission to extend or redefine the design.

How This Differs From Other Scales

A practical alternative is to define looks by exact information targets rather than approximate event targets, when operationally feasible. Event-driven schedules are often easier to implement, but the realized information should then be calculated at each cutoff. Another option is to retain fixed planned-look boundaries; this is a different design rule and must be specified and calibrated as such. Neither approach justifies treating event count, elapsed time, and information as identical scales.

The realized fraction also has limitations. It is an estimate of the position on the information scale, not a guarantee that all model assumptions hold or that a boundary procedure retains its intended error properties under every departure from plan. If the information calculation is unstable, the event pattern is unusual, or the maximum information is revised, the study statistician should follow the prespecified adaptation and oversight procedures. Recomputing a fraction alone does not repair a misspecified monitoring design.

Common Mistake

Wrong reasoning: “The first look was planned at 40% of maximum events, and 40% of the events have now occurred, so the analysis is at 40% information.”

That conclusion assumes event count is proportional to logrank information. The proportionality can fail because event contributions depend on risk sets and the allocation of events. The correction is to calculate the cumulative logrank information at the cutoff, divide by the applicable maximum information, and report the planned fraction separately. Then apply the boundary rule specified for the resulting scale; do not silently substitute a planned event or information fraction.