Why the Stopped Sampling Law Matters
A confidence interval formed after a sequential trial must reflect how the reported estimate was selected. The trial may stop at an early look when the interim result crosses a boundary, or continue otherwise. Consequently, an estimate reported at an early look comes from a truncated part of its sampling distribution, while a later estimate comes from outcomes that passed the earlier stopping rule. Treating the reported estimate as though it came from an untruncated fixed-sample experiment can therefore misallocate probability to its tails.
Here the target is an equal-tailed interval based on the full stopped sampling law. Its tail probabilities are calculated over all possible stopping stages under the design—not just over repetitions that stop at the observed stage. This differs from conditioning on the observed stopping stage.
Constructing the Stopped-Law CDF
Let \(R_j\) be the set of data histories at which the design stops at stage \(j\). These regions are determined by the stopping rule. They are disjoint, and their union contains every possible history that ends the trial. If \(f_{\theta,j}(x_{1:j})\) is the joint density of the data through stage \(j\), the CDF of \(T\) is
Each term counts outcomes that both stop at stage \(j\) and report an estimate no greater than \(t\). The restriction to \(R_j\) is essential: it represents the stage-specific truncation. The formula does not condition on reaching or stopping at any particular stage. For a design with discrete data, the integrals are replaced by sums of probabilities.
For a continuous stopped distribution whose CDF is strictly decreasing in \(\theta\) at fixed \(t\), an equal-tailed interval with confidence level \(1-\alpha\) is obtained by solving two equations. The lower endpoint \(L(t)\) assigns probability \(1-\alpha/2\) to the lower tail ending at the observed \(t\); the upper endpoint \(U(t)\) assigns probability \(\alpha/2\) to that tail:
The order of the probabilities may initially seem reversed. Increasing \(\theta\) shifts the stopped estimate upward in many standard models, so \(F_\theta(t)\) decreases as \(\theta\) increases. Thus the larger CDF value gives the lower endpoint. Under continuity and the stated monotonicity, the event that the interval covers \(\theta\) is equivalent to \(\alpha/2\le F_\theta(T)\le 1-\alpha/2\). Since \(F_\theta(T)\) is uniform on \((0,1)\) for a continuous distribution, this construction has coverage \(1-\alpha\). If the distribution has atoms, exact equality and exact coverage may not be attainable without randomization; nonrandomized inversion generally gives conservative coverage.
Worked Example: A Two-Stage Stopped Estimate
Consider a hypothetical sequential experiment with a parameter \(\theta\). At the first stage, observe \(S\sim N(\theta,1)\). Stop and report \(S\) if \(S>0\). Otherwise continue, observe an independent \(Y\sim N(\theta,1)\), and report \(Y\). This is a deliberately simple stopping design that makes the truncation visible. Its assumptions are the stated normal distributions, independence of \(S\) and \(Y\), and the specified stopping rule.
The stopped estimate is \(T=S\) when \(S>0\), and \(T=Y\) when \(S\le0\). For \(t\le0\), an early stop cannot produce \(T\le t\), so
At the observed value \(t=0\), this becomes \(F_\theta(0)=\Phi(-\theta)^2\). For a 90% interval, \(\alpha=0.10\). The lower limit solves \(\Phi(-\theta)^2=0.95\), so \(\Phi(-\theta)=\sqrt{0.95}=0.974679\). Therefore \(-\theta\approx1.954\), giving \(L(0)\approx-1.954\). The upper limit solves \(\Phi(-\theta)^2=0.05\), so \(\Phi(-\theta)=\sqrt{0.05}=0.223607\). Therefore \(-\theta\approx-0.759\), giving \(U(0)\approx0.759\).
The resulting equal-tailed 90% interval is approximately \([-1.954,\ 0.759]\) in the units of \(\theta\). The asymmetry is not an arithmetic error: the stopping rule changes the distribution of the reported estimate. A fixed-sample normal interval centred on zero would not represent this stopped law.
Checking the Stage Contributions
A useful calculation separates the probabilities from each stopping stage before combining them. In the two-stage example, when \(t>0\), the event \(T\le t\) can occur either through an early stop with \(0<S\le t\), or through continuation with \(S\le0\) and \(Y\le t\). Independence gives
Worked Example: Computing a CDF Value Across Stages
Use the same hypothetical design and take \(\theta=0\) and \(t=1\). The model assumptions remain \(S,Y\sim N(0,1)\), independent, with stopping at the first stage exactly when \(S>0\). The early-stop contribution is \(P(0<S\le1)=\Phi(1)-\Phi(0)=0.341345\). The continuation contribution is \(P(S\le0,Y\le1)=\Phi(0)\Phi(1)=0.5(0.841345)=0.420672\).
Adding the mutually exclusive contributions gives \(F_0(1)=0.341345+0.420672=0.762017\). The calculation includes the early and continued outcomes in their respective stopping regions. Using only the distribution of \(Y\), or only the early-stop distribution, would omit part of the sampling law and yield the wrong CDF.
Worked Example: A Truncated Tail Probability Is Conditional
Suppose a single interim statistic \(X\sim N(0,1)\) triggers an early efficacy stop when \(X>0\). Among trials that stop early, the conditional probability that \(X\) lies between 0 and 1 is
This is a conditional probability within the early-stop subset; it is not the unconditional probability \(P(0<X<1)=0.341345\). The distinction matters when forming interval limits: the full stopped-law CDF sums probabilities across all stages, while a stage-conditional calculation restricts attention to one stage. The denominator and the probability being reported must match the stated conditioning.
Practical Inversion and Diagnostics
In a realistic group sequential design, the stage-specific regions may involve correlated interim and final statistics. The CDF still follows the same principle: integrate the joint sampling density over each stopping region and the additional condition that the estimate at that stage is at most \(t\). Numerical quadrature or Monte Carlo methods may be needed to evaluate these probabilities. Once \(F_\theta(t)\) is available, a root-finding method can solve the two endpoint equations.
State which estimate is reported at each possible stopping stage and define \(T\) accordingly.
Write the region \(R_j\) for every stage. Check that the regions are disjoint and cover all trial outcomes.
Sum or integrate the stage-specific probabilities, retaining the restriction to each \(R_j\).
Solve the two tail equations, confirm \(L(t)\le U(t)\), and evaluate the CDF at both endpoints to check the intended probabilities.
Several diagnostics can reveal a faulty calculation. The stage contributions to \(F_\theta(t)\) must be nonnegative, and their sum must lie between zero and one. As \(t\) increases, the CDF cannot decrease; at the extremes it must approach zero and one. For the usual location-type model, \(F_\theta(t)\) should decrease with \(\theta\), and numerical solutions should satisfy the endpoint equations to the chosen tolerance. A failure of these checks may indicate omitted stopping regions, double-counted outcomes, incorrect correlation assumptions, or a mistaken tail orientation.
What This Construction Does—and Does Not—Promise
| Approach | Probability calculation | Main distinction |
|---|---|---|
| Stopped-law equal-tailed interval | Uses the full sampling distribution of the reported stopped estimate | Includes the truncation and probability contribution at every stopping stage |
| Fixed-sample interval | Uses the distribution for a prespecified sample size without the stopping rule | May misstate tail probabilities after sequential stopping |
| Stage-conditional interval | Conditions on the observed stopping stage | Answers a conditional-coverage question, not the full-law question here |
The stopped-law interval is exact only when the model and stopping rule used in its CDF match the design, the parameter inversion is well behaved, and the distributional assumptions hold. A misspecified variance, an incorrectly implemented boundary, or an unmodelled adaptation can invalidate the nominal coverage. In discrete settings, jumps in the CDF may prevent the two tail equations from being satisfied exactly. Also, equal tails do not guarantee the shortest interval or the most scientifically useful interval; they specify how the noncoverage probability is allocated, not an optimality criterion.
A common mistake is to compute a fixed-sample interval from the observed stage and then argue that the nominal level still applies because the estimate is normal at that stage. The relevant distribution is not generally an untruncated normal: reaching that stage required passing the earlier stopping rule, and stopping early selected a different set of reported estimates. The correction is to incorporate each stage’s stopping region in the probability calculation, then make clear whether the interval uses the full stopped law or conditions on the observed stage.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In a two-stage design, what does each term in the stopped-law CDF represent?
- For a continuous strictly monotone stopped-law CDF, which CDF probability defines the lower endpoint of a \(1-\alpha\) equal-tailed interval?
- In the two-stage example, calculate \(F_0(0)\). Why is it not \(0.5\)?
- Why is \(P(0<X<1\mid X>0)\) different from the unconditional probability \(P(0<X<1)\) for \(X\sim N(0,1)\)?
Solutions. (1) The \(j\)th term is the probability that the trial stops at stage \(j\) and the estimate reported there is at most \(t\); summing includes every stopping stage. (2) The lower endpoint solves \(F_{L(t)}(t)=1-\alpha/2\), because the CDF decreases with the parameter in the stated monotone location setting. (3) \(F_0(0)=\Phi(0)^2=0.5^2=0.25\): \(T\le0\) requires continuation, which has probability \(0.5\), and then \(Y\le0\), also with probability \(0.5\). (4) The conditional probability divides the joint event probability by \(P(X>0)=0.5\); numerically it is \(0.341345/0.5\approx0.6827\), whereas the unconditional probability is approximately \(0.3413\).