Why Condition on the Stopping Look?
In a group sequential trial, the observed statistic is selected by a stopping rule: the trial reports the statistic at the look where it stops, not necessarily at the final look. A p-value conditional on the stopping look asks a specific question: among trials that stop at this look under the null, how unusual is an outcome at least as extreme as the observed one? That is different from asking how unusual the observed stopped outcome is among all possible trial outcomes.
The distinction matters because the null distribution at a look can be truncated by the stopping rule. For example, if a trial stops for efficacy at its first interim look only when a Z-statistic exceeds a boundary, then observing that it stopped at that look already tells us the statistic exceeded the boundary. A conditional tail probability must use that selected distribution, not the untruncated standard normal distribution.
The conditioning event must match the question being asked. Conditioning on \(T=k\) alone combines every reason the trial could stop at look \(k\); conditioning on \(T=k,C=c\) restricts attention to a particular stopping category. If the design has only one look, \(T=1\) is certain and conditioning on it changes nothing. If look 1 is an interim look in a multi-look trial, however, \(T=1\) occurs only for outcomes that trigger the specified stopping rule.
Model and Calibration
For the standard group sequential Z-statistic setting, suppose the planned looks are \(1,\ldots,K\), the null hypothesis is \(H_0\), and the standardized statistics have their prespecified joint null distribution. In the canonical known-variance model, \(Z_k\sim N(0,1)\) under \(H_0\), and for \(i<j\), \(\operatorname{Corr}(Z_i,Z_j)=\sqrt{I_i/I_j}\), where \(I_k\) is the information at look \(k\). The stopping boundaries and any futility rules are fixed as part of the design. These assumptions let us determine the probability of each stopping event under \(H_0\).
A conditional p-value is calibrated within its specified stratum. For a continuous evidence variable, its conditional tail probability is uniform under the null conditional on that stratum. For discrete outcomes, using the probability of outcomes tied with or more extreme than the observation gives a conservative conditional p-value: its conditional probability of being at most \(a\) is no greater than \(a\). Thus, with the same correctly specified conditioning strata, conditional p-values are also superuniform unconditionally by averaging over strata. This validity does not make a conditional p-value interchangeable with a stagewise p-value: they summarize different orderings and answer different questions.
Worked Example: An Interim Efficacy Stop
Consider a hypothetical two-look trial. At look 1, the trial stops for efficacy if \(Z_1\ge2.00\); otherwise, it continues to look 2. Under \(H_0\), assume \(Z_1\sim N(0,1)\). The observed value is \(Z_1=2.40\), and the trial stops for efficacy at look 1. Calculate the upper-tail p-value conditional on this stop.
The stopping event at the first look is \(T=1\) with efficacy as the stopping category, which here is exactly \(Z_1\ge2.00\). Therefore, $$ p_{\mathrm{cond}} = P_0(Z_1\ge2.40\mid Z_1\ge2.00) = \frac{P_0(Z_1\ge2.40)}{P_0(Z_1\ge2.00)}. $$
Using standard normal upper-tail probabilities, \(P_0(Z_1\ge2.40)\approx0.00820\) and \(P_0(Z_1\ge2.00)\approx0.02275\). Hence $$ p_{\mathrm{cond}} \approx \frac{0.00820}{0.02275} = 0.3604. $$
Among trials that cross the first-look efficacy boundary, about 36.0% would have a first-look statistic at least 2.40 under this null model. The ordinary, unconditioned upper tail is 0.00820, but it answers a different question. In this particular example, \(0.02275\times0.3604\approx0.00820\): that product recovers the probability of exceeding 2.40 at the first look, not a full stagewise p-value that ranks stopped outcomes across all looks.
This is a multi-look calculation: trials not crossing at look 1 continue to look 2. In a genuine one-look design every trial has \(T=1\), so the conditional upper-tail probability would simply be \(P_0(Z_1\ge2.40)=0.00820\), not the truncated-tail ratio above.
Deriving the Conditional Tail
The formula follows directly from conditional probability, but its numerator and denominator have distinct meanings. The numerator is the null probability of both stopping in the specified stratum and attaining evidence at least as extreme as the observed value. The denominator is the null probability of stopping in that stratum, whether or not the evidence reaches the observed value.
Write the exact event, such as \(T=k\) and efficacy stopping at look \(k\). Include earlier noncrossing requirements when they are needed to characterize that event.
State which outcomes count as at least as extreme as the observed outcome within the stratum. For a one-sided upper-tail ordering, these are outcomes with a statistic at least as large as the observed statistic.
Evaluate the joint probability in the numerator and divide by the probability of the stopping stratum. Use the joint distribution of the sequential statistics and the actual stopping rules.
If \(R\) has a continuous conditional distribution in the stratum, the probability integral transform gives the calibration result. Let \(F_{k,c}(r)=P_0(R\le r\mid T=k,C=c)\). Then the upper-tail value \(1-F_{k,c}(R)\) is uniform on \([0,1]\) under the conditional null distribution. When the distribution has atoms, including the observed value in the tail prevents anti-conservatism; the resulting p-value can be larger than a uniform random variable would be, but not systematically smaller.
Conditional and Stagewise P-Values
An unconditional stagewise p-value ranks outcomes in the full stopped-trial sample space using the prespecified stagewise ordering. Its tail probability includes qualifying outcomes from every look. In contrast, a conditional p-value restricts the probability calculation to one stopping stratum and renormalizes by the probability of entering that stratum. The denominator is the central difference; it is not a correction that can be omitted.
Worked Example: Ranking Outcomes Across Looks
Consider a hypothetical discrete two-look design. Suppose the null probabilities of six mutually exclusive stopped-outcome categories are as shown. The design’s evidence ordering, from strongest to weakest, is \(D,A,B,E,C,F\). An observed outcome is category \(B\), which stopped at look 1.
| Outcome | Stopping look | Null probability | Evidence rank |
|---|---|---|---|
| D | 2 | 0.10 | Strongest |
| A | 1 | 0.04 | Second |
| B (observed) | 1 | 0.06 | Third |
| E | 2 | 0.20 | Fourth |
| C | 1 | 0.10 | Fifth |
| F | 2 | 0.50 | Weakest |
The probabilities sum to \(0.10+0.04+0.06+0.20+0.10+0.50=1.00\). Within look 1, the outcomes at least as extreme as \(B\) are \(A\) and \(B\), and the total probability of stopping at look 1 is \(0.04+0.06+0.10=0.20\). Thus $$ p_{\mathrm{cond}}(T=1;B) = \frac{0.04+0.06}{0.20} = 0.500. $$
For the stated full-sample-space evidence ordering, the stagewise p-value for \(B\) includes \(D,A,\) and \(B\), including the more extreme outcome \(D\) at look 2. It is $$ p_{\mathrm{stage}}(B) = 0.10+0.04+0.06 = 0.200. $$
The two values differ because they use different reference sets. The conditional value asks for the relative rank of \(B\) among look-1 stops; the stagewise value counts all outcomes at least as extreme in the full stopped-trial space. Neither number can be substituted for the other without changing the interpretation.
For continuous group sequential statistics, the corresponding calculations generally require the joint null distribution of the looks, since \(T=k\) includes survival of every earlier stopping opportunity. The simple first-look example required only a univariate normal tail because there were no earlier looks. At a later look, calculating \(P_0(T=k,C=c)\) and the joint tail can require multivariate normal probabilities or an equivalent recursive calculation.
Worked Example: Conditioning on the Look Versus the Stop Reason
Suppose a hypothetical trial has an interim rule at look 1: stop for efficacy if \(Z_1\ge2.00\), stop for futility if \(Z_1\le-2.00\), and otherwise continue. Assume \(Z_1\sim N(0,1)\) under \(H_0\). The observed value is \(Z_1=2.40\). Calculate the upper-tail conditional p-value first given only \(T=1\), and then given both \(T=1\) and efficacy stopping.
Given only \(T=1\), the stopping event is \(|Z_1|\ge2.00\). For an upper-tail ordering, the numerator is \(P_0(Z_1\ge2.40)\), while the denominator includes both kinds of early stops: $$ P_0(T=1) = P_0(Z_1\ge2.00)+P_0(Z_1\le-2.00) \approx0.02275+0.02275=0.04550. $$
Therefore, conditional on stopping at look 1 for either reason, $$ P_0(Z_1\ge2.40\mid T=1) \approx \frac{0.00820}{0.04550} = 0.1802. $$ If we instead condition specifically on efficacy stopping, the denominator is \(P_0(Z_1\ge2.00)\), giving $$ P_0(Z_1\ge2.40\mid T=1,C=\mathrm{efficacy}) \approx \frac{0.00820}{0.02275} = 0.3604. $$
Both calculations are coherent, but they condition on different events. Reporting “conditional on stopping at look 1” requires saying whether the stopping category is included in that condition, especially when the rule permits more than one stop reason at the same look.
Diagnostics, Alternatives, and Limitations
Before calculating a conditional p-value, verify that the event in the denominator matches the trial’s actual stopping rule. For a later look, it normally includes all required prior noncrossings as well as the stopping condition at the current look. Confirm also that the null distribution reflects the planned information times, correlations, boundaries, and any futility rule. A boundary from a different design changes the conditional distribution and therefore changes the p-value.
The main alternatives serve different purposes. An ordinary fixed-look Z-test uses the marginal distribution of the statistic at a specified look and ignores selection by the stopping rule; it is generally not the appropriate unconditional inference after data-dependent stopping. A conditional p-value explicitly conditions on a stopping stratum. A stagewise p-value instead uses an ordering over the full stopped-outcome space, so it retains information about outcomes at other looks and is the appropriate comparison when that full-sample-space ordering is the chosen convention. The introductory material on stagewise p-values establishes that their calibration is assessed over the stopped-trial outcomes, rather than by treating the observed look as if it had been fixed in advance.
Conditional calculations also have limits. A conditional p-value can be large even when the stopping event itself was rare under \(H_0\), as the first example showed. That is not a contradiction: the conditional value deliberately removes the probability of entering the stopping stratum. Conversely, a very small conditional p-value does not by itself quantify the full-sample-space evidence under a stagewise ordering. For composite null hypotheses or nuisance parameters, the null distribution used for the conditional probability must be justified for the inferential claim; a calculation under one specified point null does not automatically guarantee validity uniformly over a composite null.
Common Mistakes
- Treating an interim stop as certain. In a multi-look trial, \(T=1\) is not automatic. If look 1 is an interim efficacy look, \(T=1\) for efficacy means the boundary was crossed, while noncrossers continue.
- Using the untruncated tail as the conditional tail. The ordinary normal tail does not condition on having crossed the boundary. Divide the joint tail probability by the probability of the specified stopping stratum.
- Confusing conditional and stagewise p-values. The former restricts to one stratum; the latter ranks outcomes across the full stopped-trial space according to its ordering.
- Leaving the stopping category ambiguous. If efficacy and futility can both cause a stop at the same look, say whether the conditioning event is \(T=k\) alone or \(T=k\) with a specified cause.
- Using the wrong design distribution. Later-look conditioning depends on prior survival and the joint distribution of sequential statistics, not just the marginal distribution at the observed look.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In a multi-look trial, look 1 is an interim efficacy look with boundary \(Z_1\ge2.00\). Why is \(T=1\) not the same event as \(Z_1\ge2.00\) in a one-look design?
- Suppose under \(H_0\), \(Z_1\sim N(0,1)\), the efficacy boundary is 1.96, and the observed first-look value is 2.33. Write the conditional upper-tail formula. Use \(P(Z\ge2.33)=0.00990\) and \(P(Z\ge1.96)=0.02500\) to calculate it.
- In one sentence, distinguish the reference set used by a conditional p-value from that used by a stagewise p-value.
- A trial can stop at look 1 for efficacy if \(Z_1\ge2\) or futility if \(Z_1\le-2\). What event belongs in the denominator when conditioning on the look alone?
Solutions. (1) In a multi-look design, noncrossers continue, so \(T=1\) for efficacy is the boundary-crossing event; in a one-look design, every trial stops at look 1 regardless of \(Z_1\). (2) The conditional value is \(0.00990/0.02500=0.3960\). (3) A conditional p-value ranks outcomes within the specified stopping stratum, whereas a stagewise p-value ranks outcomes in the full stopped-trial space. (4) The denominator is \(P_0(Z_1\ge2)+P_0(Z_1\le-2)\), because either stop reason results in stopping at look 1.