Applying Efficacy Boundaries To Logrank Z Statistics

Why the Information Scale Matters

In an event-driven trial, interim analyses are often triggered by accumulated events rather than by a fixed number of enrolled participants. But an event count alone does not say how much statistical information the logrank analysis has accumulated. At each look, the efficacy decision therefore uses two linked quantities: the cumulative logrank Z statistic and the information available at that look. The statistic is compared with the efficacy boundary specified for that information time.

The boundary is not a fresh critical value selected after seeing the result. It is part of a sequential design, calibrated in advance to control the chance of a false efficacy declaration across the planned monitoring path. Applying the correct boundary means using its prespecified value at the look’s information time, then following the design’s stopping rule.

Definition: Let \(U_j\) be the cumulative logrank score through look \(j\), and let \(I_j\) be its cumulative information. The cumulative logrank Z statistic is \(Z_j=U_j/\sqrt{I_j}\). Use the score orientation in which positive \(Z_j\) favors the treatment benefit under consideration (for example, a lower hazard in the treatment group). Let \(b_j\) be the prespecified upper efficacy boundary for look \(j\), or the boundary value assigned to that look’s information time by the prespecified design. An efficacy crossing occurs when \(Z_j\ge b_j\).

The score’s sign convention must be checked before interpreting a result. If the analysis software defines positive scores as favoring control, a positive statistic cannot be interpreted as evidence of treatment benefit using an upper boundary intended for treatment benefit. The sign can be reversed for presentation, but the reversal must be made consistently with the protocol and boundary definition.

Assumptions Behind the Comparison

The boundary rule is meaningful only when the statistics and information values correspond to the design assumptions used for calibration. In the usual canonical approximation for a one-sided group sequential logrank test, the cumulative statistics under the null have mean zero, variance one, and correlations determined by the information times:

\operatorname{Corr}_0(Z_i,Z_j)=\sqrt{\frac{I_i}{I_j}},\qquad i\le j.

This model reflects the shared events and risk sets contributing to successive cumulative analyses; the statistics at successive looks are not independent. It is an approximation whose adequacy depends on the trial and analysis. Relevant conditions include correctly defined risk sets and event coding, appropriate handling of strata where applicable, independent participants under randomization, and censoring that is non-informative for the event process under the analysis assumptions. The information values must also be calculated on the scale used by the design.

Under the null hypothesis of no treatment effect on the event hazard, the sequential boundary schedule is calibrated for the joint path of statistics, not for a single isolated Z statistic. A proportional-hazards alternative can give a convenient interpretation of the direction and drift of the logrank statistic, but the boundary comparison itself is a test of the null based on the prespecified logrank analysis. If hazards are not proportional, a crossing remains a sequential test result; it does not by itself summarize the time pattern or size of the treatment effect.

This tutorial applies boundaries that have already been specified. Their derivation or construction is a separate design task. For application, the analyst needs the correct boundary values, the information definition, and the rule for associating a boundary with each analysis time.

The First-Crossing Rule

At each planned or design-authorized analysis, calculate the cumulative statistic using the cumulative score and information through that cutoff. Compare it with the corresponding upper efficacy boundary. If the statistic reaches or exceeds the boundary, the efficacy criterion is met and the protocol’s efficacy stopping action applies. If it is below the boundary, there is no efficacy crossing at that look; that alone does not establish lack of benefit or determine whether the trial continues, since other stopping rules may apply.

Identify the analysis cutoff.
Confirm which events, participants, and risk-set information are included in the cumulative logrank analysis.

Use cumulative quantities.
Obtain the cumulative score \(U_j\) and cumulative information \(I_j\), not an increment alone, and calculate \(Z_j=U_j/\sqrt{I_j}\).

Match the boundary.
Use the prespecified boundary for this look or for its realized information time, according to the design’s stated mapping.

Apply the decision rule.
If \(Z_j\ge b_j\), record an efficacy crossing at the first such look. Otherwise, record no efficacy crossing at this look and follow the remaining protocol rules.

The first-crossing language matters: once a boundary is crossed and the trial stops for efficacy, later statistics are not collected as if the trial had continued unchanged. The stopping decision is based on the observed path through the crossing look.

Worked Applications

Worked Example: Crossing at an Interim Look

Hypothetical data. A randomized event-driven trial uses a one-sided upper efficacy boundary. Its design has supplied the following boundaries for this example and is assumed to have calibrated them in advance: \(b_1=2.80\), \(b_2=2.30\), and \(b_3=2.00\). The observed cumulative information and scores are \(I=(80,160,320)\) and \(U=(18,31,39)\), respectively. Positive scores favor treatment.

At the first look,

Z_1=\frac{18}{\sqrt{80}}=\frac{18}{8.9443}=2.012.

Since \(2.012<2.80\), there is no efficacy crossing at look 1. At look 2,

Z_2=\frac{31}{\sqrt{160}}=\frac{31}{12.6491}=2.451.

Now \(2.451\ge 2.30\). The trial crosses the upper efficacy boundary at look 2, so the first crossing is at this look and the efficacy stopping rule applies. The score and information are cumulative quantities, as required; the second-look statistic is not calculated by dividing only the increment in score by the square root of the incremental information. A third-look value is not needed for the decision after stopping.

Worked Example: A Large-Looking Statistic That Does Not Cross

Hypothetical data. A different design supplies prespecified boundaries \(2.80\), \(2.35\), and \(2.10\) at information values \(100\), \(225\), and \(400\). Its cumulative scores at those looks are \(20\), \(34\), and \(39.6\), with positive values favoring treatment.

The cumulative Z statistics are

Z_1=\frac{20}{\sqrt{100}}=2.000,\qquad Z_2=\frac{34}{\sqrt{225}}=\frac{34}{15}=2.267,\qquad Z_3=\frac{39.6}{\sqrt{400}}=\frac{39.6}{20}=1.980.

The comparisons are \(2.000<2.80\), \(2.267<2.35\), and \(1.980<2.10\). There is no efficacy crossing at any of these looks. The final statistic may appear notable when viewed as a single fixed-look test: its one-sided standard normal tail area is about \(0.02385\). That tail area is not the sequential decision rule. The prespecified design requires comparing the whole monitoring path with its efficacy boundaries; here, none is crossed.

Worked Example: Matching a Boundary to Realized Information

Hypothetical data. A design specifies that the efficacy boundary is to be evaluated as a function of the realized information fraction \(t=I/I_{\mathrm{target}}\). The protocol gives \(b(0.50)=2.50\) and \(b(0.90)=2.05\). At the first analysis, \(I=150\), \(I_{\mathrm{target}}=300\), and \(U=22\). At the next authorized analysis, the realized information is \(I=270\), with cumulative score \(U=34.0\). The design’s specified mapping gives the boundary at the realized fraction.

At the first analysis, \(t=150/300=0.50\), so

Z_1=\frac{22}{\sqrt{150}}=\frac{22}{12.2474}=1.796<2.50.

There is no crossing at this look. At the next analysis, the realized information fraction is \(t=270/300=0.90\), and

Z_2=\frac{34.0}{\sqrt{270}}=\frac{34.0}{16.4317}=2.069\ge b(0.90)=2.05.

The statistic crosses the boundary assigned by this design at the realized information fraction. The comparison is justified here because the hypothetical protocol explicitly specifies how the boundary is assigned at that fraction. If a design supplies only boundaries for particular planned looks and gives no rule for off-schedule information, an analyst must not invent an interpolation or select a more favorable boundary after observing the data.

Why Repeated Looks Need a Joint Rule

At each look, a nominal fixed-look test would compare \(Z_j\) with a one-look critical value. Repeating that test without accounting for the sequence can inflate the probability of at least one false positive. The group sequential boundary is designed for the probability of crossing anywhere along the monitored path, using the dependence among the cumulative statistics.

To see the structure of the calibration, define \(A_j\) as the event that look \(j\) is the first efficacy crossing. For upper boundaries, this is the event that every earlier statistic is below its boundary and the current statistic reaches or exceeds its boundary:

A_j=\{Z_1<b_1,\ldots,Z_{j-1}<b_{j-1},\,Z_j\ge b_j\}.

These first-crossing events are disjoint, so the probability of at least one efficacy crossing under the null is

P_0(\text{efficacy crossing})= P_0\!\left(\bigcup_{j=1}^{K}A_j\right) =\sum_{j=1}^{K}P_0(A_j).

Thus a calibrated design controls this joint probability at the intended one-sided significance level (or below it, depending on the design). This is why the boundary values must be treated as a schedule rather than as independent per-look cutoffs. The calculation also explains why a non-crossing result at an early look is not itself a final test decision: it identifies which first-crossing events remain possible if monitoring continues.

Diagnostics, Alternatives, and Limitations

Before reporting a boundary decision, check that the analysis follows the design on the same information scale. Useful diagnostics include:

Sign orientation: Confirm that positive \(Z_j\) points toward the treatment effect represented by the upper efficacy boundary.
Cumulative versus incremental quantities: Verify that \(U_j\) and \(I_j\) include all information through look \(j\). Incremental scores are not the numerator of the cumulative statistic.
Boundary indexing: Verify that the boundary corresponds to the correct look or realized information fraction under the prespecified rule.
Information calculation: Check that event cutoffs, censoring, strata, and risk sets are handled consistently with the design’s information definition.
Monitoring schedule: Confirm that the analysis was authorized by the protocol. Extra unplanned looks cannot automatically be treated as if the original boundary schedule covered them.

The main alternative is a single final logrank analysis with one fixed-look critical value. It is simpler when no interim efficacy monitoring is planned, but it does not provide the same decision rule as a group sequential design. Another tempting alternative is to calculate a nominal p-value at every look and stop whenever one is below a conventional threshold. That approach ignores the joint monitoring path unless the threshold rule has itself been calibrated for sequential use.

Approach	What is compared	Important limitation
Prespecified group sequential efficacy rule	Cumulative \(Z_j\) against the boundary for its look or information time	Requires correct boundary mapping and the design’s assumptions
Single final logrank test	One final statistic against a fixed-look criterion	Does not authorize repeated unadjusted efficacy testing before the final look
Repeated nominal fixed-look tests	Each interim statistic against a conventional one-look cutoff	Generally does not control the probability of a false crossing over the full sequence

Even a correctly applied boundary has limits. The conclusion concerns the prespecified one-sided efficacy test, not the probability that the null hypothesis is true, the clinical importance of the effect, or the shape of the hazard difference over time. Departures from planned information, censoring assumptions, or the canonical dependence approximation can affect operating characteristics. A crossing should therefore be reported together with the look, realized information, cumulative Z statistic, boundary, and the direction of the logrank score.

Key takeaway: At each authorized event-information look, calculate \(Z_j=U_j/\sqrt{I_j}\) from cumulative logrank quantities and compare it with the efficacy boundary assigned by the prespecified design. The first statistic reaching or exceeding its boundary is the efficacy crossing; nominal per-look p-values do not replace this rule.

Check Your Understanding

Use the stated upper-boundary rule in each question. Assume the hypothetical designs have supplied valid boundaries and positive scores favor treatment.

At a look, \(U=24\), \(I=144\), and \(b=2.10\). Calculate \(Z\) and decide whether an efficacy crossing occurs.
At two successive looks, \(Z_1=2.30\), \(b_1=2.40\), and \(Z_2=2.50\), \(b_2=2.45\). At which look, if any, does the first crossing occur?
A planned boundary is specified at information fraction \(0.75\), but a data cutoff yields fraction \(0.70\). The protocol gives no mapping for this off-schedule fraction. What should the analyst do?
Why is comparing each interim statistic with the same nominal fixed-look critical value generally not equivalent to applying a group sequential efficacy boundary?

Solutions. (1) \(Z=24/\sqrt{144}=24/12=2.000\). Since \(2.000<2.10\), there is no crossing. (2) Look 1 does not cross because \(2.30<2.40\). Look 2 crosses because \(2.50\ge2.45\), so the first crossing is at look 2. (3) The analyst should not choose or interpolate a boundary after seeing the statistic. The analysis and boundary assignment must follow a prespecified rule; if none covers this information fraction, the issue requires resolution under the trial’s governance and statistical plan before treating it as a calibrated efficacy look. (4) The cumulative statistics are dependent, and repeated comparisons can raise the null probability of at least one false crossing. A group sequential boundary is calibrated for that joint path, whereas an ordinary fixed-look cutoff is not generally calibrated for repeated use.