First-Crossing Probabilities Under Classical Boundaries

Why First Crossings Matter

A group sequential design permits efficacy testing at several planned looks. Under the null hypothesis, a test statistic may cross a boundary at more than one look along a hypothetical sequence of results. But a trial that stops at the first crossing does not proceed to later looks. Therefore, the probability of stopping at look \(k\) is not simply the probability that the statistic crosses the boundary at look \(k\). It is the probability of crossing there and not having crossed earlier.

This distinction is central to interpreting the design’s Type I error. The first-crossing events cannot occur together: if the trial stops at look 1, it cannot also stop at look 2. Their probabilities therefore add directly to give the probability of rejecting the null at any planned look. By contrast, the events of crossing a boundary at each look, considered without regard to earlier crossings, can overlap.

Model, Events, and Notation

Consider a design with \(K\) planned looks and an upper efficacy boundary \(b_k\) at look \(k\). Let \(Z_k\) denote the test statistic at that look. The null hypothesis is \(H_0\), and the overall Type I error is the probability that the design rejects \(H_0\) at least once when \(H_0\) is true. This tutorial concerns efficacy stopping; it does not include a futility stopping rule.

For the standard canonical group sequential model under \(H_0\), the vector of statistics is jointly normal with mean zero and unit variances. If \(I_k\) is the information at look \(k\), with \(0<I_1<\cdots<I_K\), its correlations are \(\operatorname{Corr}(Z_i,Z_j)=\sqrt{I_i/I_j}\) for \(i<j\). This model reflects accumulated information: results at later looks are related to earlier results, not independent new tests. The boundary vector is treated as fixed when calculating the null crossing probabilities.

Write \(R_{k-1}=A_1^c\cap\cdots\cap A_{k-1}^c\) for the event that the design reaches look \(k\) without an earlier efficacy crossing. In particular, \(R_0\) is the certain event. When \(P(R_{k-1}\mid H_0)>0\), define the conditional crossing probability \(q_k=P(A_k\mid R_{k-1},H_0)\). This is the probability of crossing at the current look among paths that have survived the earlier looks.

Thus, the probability of stopping at a look is the probability of reaching it multiplied by the probability of crossing there conditional on reaching it. For look 1, this reduces to \(p_1=q_1=P(Z_1\ge b_1\mid H_0)\). For later looks, \(q_k\) accounts for the fact that the earlier statistics did not cross.

Derivation and Overall Alpha

The first-crossing events \(F_1,\ldots,F_K\) are pairwise disjoint: if \(F_i\) occurs for some \(i<j\), then \(A_i\) has occurred, so \(F_j\), which requires no earlier crossing, cannot occur. Their union is the event that the design crosses at least one efficacy boundary. By adding probabilities of disjoint events, the overall Type I error is

The probability of reaching each look can itself be written in terms of the earlier conditional crossing probabilities. Since surviving look \(j\) has conditional probability \(1-q_j\), the chain rule gives \(P(R_{k-1}\mid H_0)=\prod_{j=1}^{k-1}(1-q_j)\). Therefore,

where the empty product for \(k=1\) is 1. This expression is a probability decomposition, not an assumption that the \(Z_k\) are independent. Each \(q_k\) is conditional on survival to that look, and its value must reflect the joint null distribution of the statistics and the specified boundaries.

The same conclusion follows by tracking survival. The probability of reaching look 1 is 1; after look 1, the probability of survival is \(1-q_1\). After look 2, it is \((1-q_1)(1-q_2)\), and so on. The probability of ever crossing is one minus the probability of surviving every look:

Expanding the right-hand side yields the same sum of first-crossing probabilities. This identity provides a useful check on arithmetic and on a probability calculation obtained by another method.

Marginal Crossings Are Not First Crossings

The marginal crossing probability \(P(A_k\mid H_0)\) asks whether the statistic at look \(k\) crosses its boundary, whether or not the path crossed earlier. It is generally not equal to \(p_k\). In fact, \(F_k\) is a subset of \(A_k\), so \(p_k\le P(A_k\mid H_0)\). A marginal crossing at a later look can occur on a path that already crossed and stopped; that path contributes to the marginal probability but not to the probability of stopping at that later look.

Using the Decomposition to Check Overall Alpha

The identity also shows how conditional probabilities can be used to check a target overall error rate. Once the earlier conditional crossing probabilities are known, the final conditional probability needed to reach a specified overall alpha can be solved algebraically. This is a probability-accounting step; it does not, by itself, establish that a particular boundary produces that conditional probability. The boundary must still be evaluated under the appropriate joint null distribution.

Assumptions, Diagnostics, and Limitations

These calculations assume a prespecified look schedule, fixed efficacy boundaries, and a correctly specified null joint distribution for the test statistics. In the canonical model, the information values and induced correlations must correspond to the planned analyses. The interpretation of \(\alpha_{\mathrm{overall}}\) is conditional on \(H_0\): it is the probability of at least one efficacy rejection when the null is true, not the probability that \(H_0\) is true.

For a classical design with a different boundary geometry, the bookkeeping is unchanged: only the \(b_k\), and hence the conditional crossing probabilities, change. The probabilities may be calculated by multivariate normal integration or estimated by simulation under the null. Either way, the calculation must account for survival to each look. A simulation should apply the stopping rule to each simulated path and record the first crossing, rather than count every look at which a path crosses.

Useful checks include verifying that each \(p_k\) is nonnegative, that the survival probability never increases across looks, and that \(\sum_k p_k\) agrees with \(1-P(R_K\mid H_0)\). A discrepancy can indicate that a later crossing probability was treated as marginal rather than conditional, that overlapping events were counted more than once, or that rounding occurred too early. When numerical integration or simulation is used, its numerical error should also be considered; a small discrepancy may reflect approximation rather than a conceptual error.

First-crossing probabilities describe where null paths stop, not the probability of stopping at each look under an alternative. They also do not, on their own, describe expected sample size or the operating characteristics of futility rules. Those questions require additional quantities and assumptions. The main limitation here is that the probability decomposition is exact, but numerical values depend on the validity of the model, the information schedule, and the boundaries used.

Common Mistake