Recursive Calculation of Stopped-Outcome Probabilities

Why Recursion Is Useful

A group-sequential outcome has two parts: the look at which the trial stopped and the statistic observed at that look. Under the null hypothesis, the statistics at successive looks are dependent. Calculating a stopped-outcome probability therefore requires accounting for both the dependence and the rule that earlier boundary crossings prevent later observations from being reached.

The earlier material on decomposing stagewise tail probabilities by look establishes the basic sum: a stagewise tail probability is the sum of the relevant stopped-outcome probabilities across looks. Recursion provides a systematic way to calculate each term without repeatedly evaluating a high-dimensional multivariate normal probability. It is especially useful when there are several looks or when the statistic regions defining the ordering vary by look.

Model and Recursive Distribution

Consider a one-sided efficacy design with \(K\) planned looks and increasing information levels \(0<I_1<\cdots<I_K\). Under \(H_0\), let the cumulative score at look \(k\) be \(S_k\), with independent increments \(S_k-S_{k-1}\sim N(0,I_k-I_{k-1})\), where \(S_0=0\). Define \(Z_k=S_k/\sqrt{I_k}\). Thus \(Z_k\sim N(0,1)\) under \(H_0\), but the \(Z_k\)'s are not independent.

For \(k\geq2\), the independent-increment model gives the transition \(Z_k\mid Z_{k-1}=x\sim N(\sqrt{I_{k-1}/I_k}\,x,\;1-I_{k-1}/I_k)\). This model assumes the stated information levels and score-increment distribution are appropriate, the increments are independent under \(H_0\), and the stopping boundaries are fixed in advance. Write \(c_k\) for the upper efficacy boundary at look \(k\). The trial stops at the first \(k\) for which \(Z_k\geq c_k\); if no boundary is crossed, it reaches the final look and stops there.

For \(k\geq2\), propagate the subdensity through the normal transition, then retain only paths that survived the preceding looks. At the current look, \(f_k\) includes values both below and above \(c_k\): values above the current boundary are stopped outcomes at look \(k\), not paths to be carried forward. The probability of reaching look \(k\) is the integral of \(f_k\) over the real line. For \(k<K\), the probability of continuing beyond look \(k\) is its integral below \(c_k\).

The integration limit enforces survival through look \(k-1\). The transition density then distributes that surviving mass across possible values \(y\) at look \(k\). If \(B_k\) is a specified statistic region for outcomes stopped at look \(k\), its stopped probability is \(P_0(\text{stop at }k,\ Z_k\in B_k)=\int_{B_k\cap[c_k,\infty)} f_k(y)\,dy\) for \(k<K\). At the final look, the same expression applies if the trial has an efficacy boundary there; if the design always stops at the final look, use \(B_K\) itself, since there is no later look to continue to.

From Stopped Regions to a Stagewise P-Value

A p-value must be tied to an ordering of stopped outcomes. For an observed outcome at look \(m\) with statistic \(z_{\mathrm{obs}}\), suppose the specified one-sided stagewise ordering identifies a region \(B_j(z_{\mathrm{obs}})\) at each stopping look \(j\): outcomes in that region count as at least as strong evidence against \(H_0\) as the observed outcome. The regions are part of the ordering; they need not be identical at every look.

The null probability of the ordered tail is the stagewise p-value. Since the events “stop at \(j\)” are disjoint, their probabilities add. This computation is exact with respect to the stated null model, stopping rule, and ordering; numerical integration may still introduce approximation error.

Here \(C_j=[c_j,\infty)\) at a look where stopping requires crossing the efficacy boundary. At the final look, \(C_K\) is the whole real line if all remaining trials stop there regardless of crossing. The recursion calculates the look-specific probabilities; the chosen ordering determines which parts of those distributions enter the sum.

Evaluating the Recursion and the Ordered Tail

In practice, the integrals are evaluated using numerical quadrature or a boundary-integration routine. The recursive calculation avoids integrating over every earlier statistic jointly: at each step it carries forward a one-dimensional subdensity, truncated to paths that have not crossed a boundary. For a specified stagewise ordering, the final step integrates only the portions of each stopped distribution that count as at least as extreme as the observation.

Derivation, Checks, and Failure Modes

The recursion follows directly from conditioning on the preceding statistic. A path that reaches look \(k\) must have reached look \(k-1\), with a value \(x<c_{k-1}\). Given that value, the conditional density of \(Z_k=y\) is the normal transition density. Multiplying by the surviving subdensity \(f_{k-1}(x)\) and integrating over all such \(x\) gives the displayed formula for \(f_k(y)\). Integrating \(f_k\) over a stopped region then gives the probability of that region and first stopping at \(k\).

Two mass checks are particularly informative. First, \(\int f_k(y)\,dy\) should equal the probability of reaching look \(k\). Second, for each nonfinal look, the mass below the boundary continues and the mass at or above it stops:

Across all looks, the stopping probabilities and any probability of completing the final look without crossing must sum to 1. A discrepancy can signal truncation of integration limits, an incorrect transition variance, an omitted stopping region, or numerical integration error. These checks diagnose computation; they cannot establish that the assumed normal-increment model or the encoded design matches the actual trial.

A direct multivariate normal calculation is a useful benchmark, particularly for a small number of looks. It evaluates a joint probability over all looks at once, whereas recursion propagates one-dimensional survivor mass look by look. Simulation can also check a calculation approximately, but simulation error means it does not provide the same deterministic probability calculation. For larger designs, recursion can be more convenient than direct high-dimensional integration, but results still depend on accurate quadrature and correct boundary handling.

Common Pitfalls and Limitations

• Propagating paths that already stopped. Integrating over all \(x\) at the next recursion step carries stopped paths forward. Restrict the previous statistic to its continuation region.
• Confusing reaching a look with stopping there. The integral of \(f_k\) over all values is the probability of reaching look \(k\). Stopping at that look requires integrating only over the boundary-crossing region, or over its intersection with the specified statistic region.
• Summing regions that do not match the ordering. A stagewise p-value is defined by its ordering. Use the corresponding \(B_j(z_{\mathrm{obs}})\) at every look; do not silently substitute an ordinary fixed-look tail area.
• Treating numerical precision as model validity. A highly accurate integral can still be wrong for a trial whose increments are not adequately represented by the assumed normal model or whose stopping rule was encoded incorrectly.

The formulas here use one-sided upper efficacy boundaries and independent normal score increments. Other stopping rules require changing the continuation and stopped regions, and other null models require an appropriate transition distribution. A recursive calculation is only as appropriate as those assumptions.