Why the Final Look Needs a Sequential Interpretation
When a group sequential trial reaches its planned final analysis, the observed statistic is not simply an ordinary fixed-sample statistic in isolation. The trial has already passed earlier opportunities to stop for efficacy. A p-value that respects the trial’s stopping rule therefore uses the null distribution of the whole stopped-outcome process, not just the marginal null distribution of the final Z-statistic.
This tutorial considers a one-sided efficacy design. The trial stops at an earlier look if the Z-statistic reaches that look’s efficacy boundary; otherwise it proceeds to the final look. The observed outcome is a final-look statistic \(Z_K=z\), with no efficacy crossing at any earlier look. The question is how to measure its evidence while accounting for the earlier stopping opportunities.
This ordering gives a direct calculation. The null probability of any earlier efficacy stop counts as stronger evidence, and the final-look tail counts the outcomes at the final look with \(Z_K\) at least as large as the observed value. The earlier-stop probability is included even though the actual trial did not stop early: it is part of the null reference distribution used to evaluate the observed outcome.
Model, Assumptions, and Calculation
Use the canonical group sequential normal model. Let \(Z_j\) be the standardized test statistic at look \(j\), with information \(I_j\), where \(0<I_1<\cdots<I_K\). Under the one-sided null hypothesis \(H_0\), assume the statistics have mean zero, variance one, and the canonical covariance structure
This model follows from independent increments of information, with the usual normal or large-sample approximation and a prespecified information schedule. Assume the efficacy boundaries \(b_1,\ldots,b_K\) are fixed in advance, efficacy stopping occurs when \(Z_j\ge b_j\), and there is no other stopping rule that changes which outcomes can reach the final look. The one-sided alternative is toward larger Z-statistics. These conditions matter: changing the analysis plan or the stopping rule changes the null distribution used in the p-value.
Write \(E_j\) for the event that the trial first stops for efficacy at look \(j\). For a final-look observation \(z\), the event ranked at least as strongly is the union of all earlier efficacy stops and final-look paths with \(Z_K\ge z\) that did not stop earlier. Thus
The subscript 0 indicates probabilities calculated under \(H_0\). If the observed trial reached the final look, its actual earlier statistics satisfy the survival conditions. The p-value is nevertheless an unconditional null probability: it includes paths that would have stopped earlier under the null.
Calculate the null probability that the first efficacy crossing occurs before the final look. Recursive boundary integration can supply this probability.
Calculate the null probability of reaching the final look and observing \(Z_K\ge z\), enforcing all earlier noncrossing conditions.
The result is the one-sided final-look stagewise p-value for the stated ordering and stopping plan.
Worked Examples in a Two-Look Design
For concrete calculations, suppose a hypothetical trial has two looks, with \(I_1/I_2=1/4\). Under \(H_0\), \(Z_1\) and \(Z_2\) are standard normal with correlation \(0.5\). The efficacy boundaries are \(b_1=2.4\) and \(b_2=2.0\). Thus, the probability of an efficacy stop at the first look is
For any final-look value \(z\), the surviving final-look tail is
because \(Z_2\mid Z_1=x\) is normal with mean \(0.5x\) and variance \(1-0.5^2=0.75\). The following tail probabilities are rounded outputs of this null-model integration. They are hypothetical calculations for this specified design.
Worked Example: A Final Statistic Above the Efficacy Boundary
Suppose the trial did not cross the first boundary and reaches the final look with \(Z_2=2.1\). This exceeds the final efficacy boundary \(2.0\), so the trial stops for efficacy at the final look. The ordinary one-sided terminal Z-test p-value is
For the stagewise p-value, the null integration gives \(q(2.1)=0.0162\). Add the earlier efficacy-stop probability:
The stagewise value is larger than the ordinary terminal value because it also counts null paths that would have stopped for efficacy at the first look. Under this ordering and null model, the observed final-look outcome has a one-sided sequential p-value of about \(0.0244\). For example, it is below \(0.05\), but the p-value is not the probability that \(H_0\) is true.
Worked Example: A Final Statistic Below the Efficacy Boundary
Now suppose the same hypothetical design reaches its final look with \(Z_2=1.6\), again with no first-look crossing. The terminal Z-test gives
For the sequential calculation, the surviving final-look tail is \(q(1.6)=0.0516\). Therefore,
Here too, the stagewise p-value exceeds the ordinary terminal value. Under the specified stagewise ordering, the statistic does not reach the final efficacy boundary, and the sequential p-value is about \(0.0598\). The ordinary terminal p-value alone omits the probability of outcomes that would have stopped at the first look.
Worked Example: A Very Large Final Statistic
Suppose the trial reaches the final look with \(Z_2=3.0\). The ordinary one-sided terminal p-value is small:
The null-model integration gives \(q(3.0)\approx0.0011\). Hence the stagewise value is approximately
The stagewise value remains above the early-stop probability of \(0.00820\). This is a feature of the stated ordering: all first-look efficacy stops rank at least as strongly as any final-look outcome. It does not mean that an extreme final statistic contains little evidence; it means that evidence is being summarized using a particular ordering of the complete stopped outcomes.
Why the Calculation Is a Valid P-Value
The justification is the defining tail-probability property of a p-value. Under \(H_0\), assign each possible stopped outcome its rank in the chosen evidence ordering, and let its p-value be the null probability of outcomes ranked at least as strongly. Outcomes with smaller p-values occupy the most extreme part of that ordering. Therefore, for every \(u\) between zero and one,
This is superuniformity. With a continuous statistic within each stopping stage, equality holds over the continuous portions of the ordering; discreteness or tied outcomes can make the probability smaller. The result depends on using the correct null distribution for the prespecified design and ordering. It does not validate a calculation based only on the paths that happened to survive.
They consist of early efficacy stops and outcomes that survive to the final look.
For a final-look observation \(z\), every earlier efficacy stop ranks at least as strongly, as does every surviving final outcome with \(Z_K\ge z\).
The set’s probability is exactly the sum in the final-look formula. Since it is the upper tail of the outcome ordering, its p-value is superuniform under \(H_0\).
Comparison, Diagnostics, and Limitations
| Quantity | What it calculates | What it omits or conditions on |
|---|---|---|
| Ordinary terminal p-value | \(1-\Phi(z)\) under a standard normal null for the terminal statistic | Does not account for the full stopped-outcome ordering |
| Final-look stagewise p-value | Earlier efficacy-stop probability plus the surviving final-look tail | Uses the prespecified sequential design and the stated ordering |
| Surviving-path tail alone | \(P_0(\text{survive earlier looks}, Z_K\ge z)\) | Leaves out the earlier-stop outcomes ranked above a final-look result |
The ordinary p-value remains a familiar summary of the terminal statistic under its marginal null distribution, but it is not automatically a valid summary for the sequential ordering. The surviving-path tail alone is also not the stagewise p-value defined here: it omits earlier stops. A tail probability conditional on survival is a different, conditional quantity and should not be substituted for the unconditional stagewise calculation.
In practice, obtain both terms from the same null model and stopping rules. Recursive integration or direct multivariate normal calculation can evaluate them. Check that the probabilities are nonnegative, that the earlier-stop and final-survival events are disjoint, and that their combined probability is no greater than one. Compare numerical calculations with simulation or an independent integration method when practical. Probability-mass leakage, a mistaken boundary inequality, or failure to enforce earlier noncrossing conditions can materially distort the result.
This result is specific to a one-sided efficacy-only design and the stated stagewise ordering. Futility stopping, two-sided evidence orderings, different boundary structures, or a different ordering require their own null-probability calculation. The introductory material on stagewise p-values gives the broader setting; here the essential interpretation is that a final-look p-value uses the full trial’s null distribution, including earlier efficacy stops.
Common Mistake
Check Your Understanding
Use the definitions and two-look calculations above. Each solution is included after its question.
- For the stated two-look design, write the formula for the stagewise p-value when the observed final statistic is \(z\).
Solution: \(p_{\mathrm{seq}}(z)=P_0(E_1)+P_0(Z_1<2.4,Z_2\ge z)=0.00820+q(z)\). - Why does the final-look calculation include \(P_0(E_1)\), even though the observed trial did not stop at look 1?
Solution: The p-value is an unconditional null tail probability over stopped outcomes. Under the stagewise ordering, every first-look efficacy stop ranks at least as strongly as a final-look outcome. - For \(Z_2=2.1\), which is larger, the ordinary or stagewise p-value, and what are their approximate values?
Solution: The ordinary value is \(0.0179\); the stagewise value is \(0.00820+0.0162=0.0244\), which is larger. - Does a final statistic of \(Z_2=3.0\) make the stagewise p-value arbitrarily close to zero in this design? Explain.
Solution: No. As the surviving final-look tail shrinks, the stagewise p-value approaches the earlier-stop probability, about \(0.00820\), because earlier efficacy stops remain in the ordered tail. - Name one condition needed for the displayed numerical calculation to apply.
Solution: Examples include the canonical null joint normal model with correlation \(0.5\), the fixed boundaries \(2.4\) and \(2.0\), the prespecified two-look schedule, or the absence of another stopping rule that alters the paths reaching the final look.