From Event Data to an Interim Decision
In an event-driven trial, an interim analysis is triggered when a planned number of endpoint events has accrued. The event count tells the monitoring team when to look; the logrank score and its information determine the statistic to compare with the boundary. At each look, the calculation uses all eligible events observed so far, not only the events since the preceding look.
This tutorial uses a sign convention in which a positive score favors the treatment arm: it means fewer treatment-arm events than expected under equal hazards. A negative score points toward harm. The convention must be stated because many software packages report the opposite sign. Reversing the sign does not change a two-sided test, but it does reverse the labels “benefit” and “harm.”
With one event at a time, the variance contribution reduces to \(p_j(1-p_j)\). Tied-event calculations should use the tie handling specified by the analysis plan or software; the formula above is the usual hypergeometric correction for tied events. In either case, score and information are accumulated from the event-time risk sets. Information is not generally identical to the event count.
The standard assumptions include randomized treatment assignment, correctly recorded event and censoring times, and censoring that is independent of the event time within the analysis groups under the null. Risk sets must reflect the observed data cutoff and the planned endpoint definition. The usual null calibration also relies on a sufficiently accurate large-sample normal approximation. Proportional hazards supports a straightforward common-hazard-ratio interpretation of the logrank direction, but is not required to calculate the statistic. If hazards change over time, the ordinary logrank score can summarize different time periods in ways that partly cancel.
How the Sequential Boundary Rule Works
At look \(i\), compare the cumulative \(Z_i\) with the benefit and harm boundaries specified for that look. Write the boundaries as \(b_i^+>0\) and \(b_i^-<0\). A benefit boundary is crossed if \(Z_i\geq b_i^+\); a harm boundary is crossed if \(Z_i\leq b_i^-\). If neither condition holds, there is no efficacy or harm crossing at that look, and monitoring proceeds according to the protocol. A separately specified futility rule, if any, is assessed separately.
Sequential boundaries are calibrated for repeated looks and their dependence. The examples below use the same illustrative boundary table: \((b_i^+,b_i^-)=(2.90,-2.90)\), \((2.40,-2.40)\), and \((2.00,-2.00)\) at looks 1, 2, and 3. Treat these as boundaries already supplied by a design, not as values derived here or as a universally valid schedule. In an actual trial, use the prespecified boundary corresponding to the realized look and information, not a newly chosen ordinary single-look cutoff.
The reason is that the cumulative score is the sum of its event-time contributions, while its variance is the sum of their variance contributions under the independent-increment conditions. Under the null, the canonical approximation gives \(\operatorname{Cov}(U_i,U_k)=V_i\) for \(i\leq k\). Thus later cumulative statistics reuse earlier information; they are not independent new tests. Standardizing each cumulative score gives the statistic that the sequential boundary was designed to assess.
Worked Event-Driven Decisions
Worked Example: A Benefit Boundary Is Crossed at the Final Look
Suppose a hypothetical randomized trial has three planned event-driven looks, at 40, 80, and 120 total events. The logrank program has calculated the following increments from the risk sets since the preceding look. Assume the trial follows the prespecified analysis plan, censoring is non-informative under the null, and the design’s illustrative boundaries are those stated above.
| Look | Total events | \(\Delta U_i\) | \(\Delta V_i\) | Cumulative \(U_i\) | Cumulative \(V_i\) |
|---|---|---|---|---|---|
| 1 | 40 | 8 | 20 | 8 | 20 |
| 2 | 80 | 5 | 20 | 13 | 40 |
| 3 | 120 | 4 | 20 | 17 | 60 |
At look 1, \(U_1=8\), \(V_1=20\), so
This is below the benefit boundary 2.90 and above the harm boundary \(-2.90\), so neither boundary is crossed.
At look 2, accumulate the increments rather than using the second-look increment alone: \(U_2=8+5=13\) and \(V_2=20+20=40\). Therefore,
The statistic is below 2.40, so there is still no boundary crossing. At look 3, \(U_3=13+4=17\) and \(V_3=40+20=60\), giving
Since \(2.194\geq2.00\), the benefit boundary is crossed at the third look. The protocol’s stopping and reporting rules now apply. The positive sign favors fewer events in the treatment arm; it does not, by itself, establish a particular clinical effect size.
Worked Example: An Early Harm Boundary Crossing
In a second hypothetical trial with the same three-look boundary schedule, suppose the risk-set calculations yield score increments \(-8,-8,\) and \(2\), and information increments of 20 at each look. Negative score increments favor harm under the stated convention. The cumulative values are \(U_1=-8\), \(U_2=-16\), and \(U_3=-14\), while \(V_1=20\), \(V_2=40\), and \(V_3=60\).
At the first look,
which does not reach the harm boundary \(-2.90\). At look 2,
Because \(-2.530\leq-2.40\), the harm boundary is crossed at look 2. Under the specified stopping rule, the trial stops for harm at that look. The third increment is not used to undo the decision: later data are not part of the information available at the time the boundary was crossed.
Worked Example: A Large Early Statistic Does Not Cross Its Boundary
Consider a third hypothetical trial with information increments of 16 and score increments \(10,-3,\) and \(4\). The cumulative scores are \(10,7,\) and \(11\), and cumulative information is \(16,32,\) and \(48\). The changing score increments indicate that later events have partly offset the earlier treatment-favoring score.
The statistics are
At look 1, 2.500 is less than the benefit boundary 2.90, so there is no crossing despite the apparently large positive statistic. Neither later statistic reaches its corresponding benefit boundary, and none reaches a harm boundary. The decision at all three looks is to continue, subject to any separate protocol-specified rule. Comparing \(Z_1\) with the final boundary, or treating the positive cumulative score alone as a stopping criterion, would give the wrong decision.
Calculation Checks, Alternatives, and Limitations
A reliable boundary decision is a short chain of checks, not just a comparison of one printed number. Confirm that each look uses the right data cutoff; that the risk sets, event definition, and censoring rules match the protocol; and that score and information increments have been added to the previous cumulative totals. Then verify the score sign, the realized information, and the boundary for that look. If the boundary is information-dependent, use the boundary corresponding to the information actually reached rather than assuming the planned event count guarantees the planned information.
The main alternative to a group-sequential boundary rule is a single final logrank test. A final-only test avoids interim boundary decisions but cannot provide the same planned early stopping opportunities. Conversely, repeatedly applying an ordinary fixed-sample significance cutoff at each look ignores the dependence and multiplicity created by repeated monitoring. A properly calibrated sequential boundary accounts for those repeated looks. It is not enough that each individual comparison looks conventional.
The arithmetic above also depends on the analysis being appropriate for the endpoint and design. Unequal allocation, stratification, and tied events affect how the risk-set score and variance are calculated; the table of increments must come from the prespecified version of the logrank analysis. Heavy censoring or limited information can make normal approximations less reliable. Departures from proportional hazards do not invalidate the mechanical boundary comparison automatically, but they can complicate interpretation: a single ordinary logrank score may combine early and late treatment effects, possibly with opposing signs.
Confirm the event cutoff, eligible patients, risk sets, and analysis specification used to produce the score and information increments.
Add the new \(\Delta U_i\) and \(\Delta V_i\) to the preceding \(U\) and \(V\); do not substitute increments for cumulative values.
Calculate \(Z_i=U_i/\sqrt{V_i}\), use the boundary for this look and information, and identify benefit, harm, or no crossing.
Record the look and direction of any crossing, and follow the prespecified stopping, continuation, and reporting rules.
Common mistake: “The look-specific score increment is positive and large, so the benefit boundary has been crossed.” This is wrong because a boundary is compared with the standardized cumulative statistic, not the latest increment or the unstandardized score. Add all score increments, add all information increments, divide cumulative score by the square root of cumulative information, and only then compare with the look-specific boundary.
Check Your Understanding
For each problem, assume the benefit-positive convention and use the illustrative boundary pairs \((2.90,-2.90)\), \((2.40,-2.40)\), and \((2.00,-2.00)\), unless stated otherwise.
- At look 2, cumulative score is 12 and cumulative information is 25. Calculate \(Z_2\) and state whether either look-2 boundary is crossed.
- At look 1, \(U_1=6\) and \(V_1=9\). Since the score increment at look 2 is \(-10\) and the information increment is 16, calculate \(Z_2\) and state the decision.
- Explain why adding \(Z_1\) and \(Z_2\) is not a valid way to obtain a cumulative statistic at look 2.
- A reported look-3 statistic is \(Z_3=-2.10\). What direction does it favor, and does it cross either illustrative look-3 boundary?
Solutions. (1) \(Z_2=12/\sqrt{25}=2.400\), which meets the look-2 benefit boundary \(2.40\); the benefit boundary is crossed. (2) \(U_2=6-10=-4\) and \(V_2=9+16=25\), so \(Z_2=-4/\sqrt{25}=-0.800\). Neither boundary is crossed. (3) Sequential looks share earlier event information, so their standardized statistics are dependent and are not additive. Add the score increments and information increments, then standardize the cumulative totals. (4) A negative statistic favors harm, but \(-2.10\) is greater than the look-3 harm boundary \(-2.00\); it does not cross that boundary, and it is not a benefit crossing either.