Binary-Outcome Score Statistics

Why a Binary Score Process Has a Canonical Form

In a group sequential trial with a binary outcome, each participant either experiences the event or does not. At each planned look, the treatment and control groups therefore contribute two event counts. The statistical challenge is not just to compare the two observed proportions at one look: the comparisons across looks use overlapping participants and are correlated. A useful starting point is the score statistic for equality of the two risks. With nested samples and a stable allocation ratio, its cumulative numerator can be expressed as a sum of stage contributions. That structure gives an approximate canonical joint normal distribution after standardization.

The result is an approximation, not a guarantee for every binary trial. It relies on independent participants, adequate event and non-event counts, and a consistent sampling design. In particular, a fixed allocation ratio across stages makes the efficient score contributions line up in a simple way. The earlier material on score statistics in regular models motivates standardizing a score by its information; here we derive what that means for two binary risks.

Model, Assumptions, and Information

For each participant \(i\) in group \(g\), write the binary outcome as \(Y_{gi}\), where \(Y_{gi}=1\) indicates an event and \(Y_{gi}=0\) otherwise. Assume outcomes are independent across participants, with \(Y_{1i}\sim\operatorname{Bernoulli}(p_1)\) and \(Y_{0i}\sim\operatorname{Bernoulli}(p_0)\). Within each group, the event probability is constant over participants and looks. Participants counted at an earlier look remain in the cumulative sample at later looks; later looks add new participants rather than replacing earlier ones. We assume no clustering, outcome-dependent enrollment, or unaccounted-for missingness.

The score test estimates the common risk under the null by pooling the groups. If the common risk \(p\) were known, the efficient score for the treatment-control contrast at look \(k\) would be proportional to \(\hat p_{1k}-\hat p_{0k}\). Define the effective sample size \(q_k=n_{1k}n_{0k}/(n_{1k}+n_{0k})\). Under the null, the variance of the risk difference is \(p(1-p)(1/n_{1k}+1/n_{0k})=p(1-p)/q_k\), so its reciprocal is the information for that contrast:

In practice, the pooled estimate replaces the unknown \(p\) in the score statistic. For large samples with probabilities away from 0 and 1, this replacement has a negligible first-order effect on the standardized statistic. A stable treatment-to-control allocation ratio is also important: if every stage adds participants in the same ratio, then \(q_k\) is proportional to cumulative enrollment and the efficient score increments have a common scale. If allocation changes substantially from stage to stage, the simple information-time argument below needs modification.

From Nested Score Contributions to a Joint Normal Distribution

Suppose each stage adds the same number of treatment and control participants. Let \(m\) denote the number added to each group at a stage, and let \(D_j\) be the difference between the treatment and control event proportions within stage \(j\). If the null risk \(p\) is known, the stage contrast has mean zero and variance \(p(1-p)(2/m)\). Equivalently, the unstandardized efficient score contribution is \(mD_j/2\), with variance \(mp(1-p)/2\). Contributions from distinct stages are independent because they involve disjoint participants.

Let \(U_k\) be the sum of these score contributions through look \(k\). Then \(Z_k=U_k/\sqrt{I_k}\) when \(p\) is known, where \(I_k\) is cumulative information. Under the null, the covariance of \(U_i\) and \(U_j\) for \(i\leq j\) is \(I_i\): the first \(i\) looks’ worth of contributions are shared, and the later contributions are independent. Standardizing each cumulative score gives the canonical correlation.

For sufficiently large stage samples, the central limit theorem makes the vector of standardized score contributions approximately multivariate normal. Consequently, under \(H_0\), the cumulative statistics are approximately jointly normal with mean zero, unit variances, and the correlations above. Under a local alternative, the same covariance approximation applies to first order, while the means shift. If \(\Delta=p_1-p_0\) is small enough that the variance is well approximated by its null value, the mean of the score statistic is approximately \(\Delta\sqrt{I_k}\). Writing \(\delta\) for the effect on the information scale gives the canonical mean \(\delta\sqrt{I_k}\).

The approximation is especially natural under local alternatives, where the risk difference shrinks as the sample size grows in a way that keeps \(\Delta\sqrt{I_k}\) bounded. For a fixed, non-small difference, the pooled-null denominator need not equal the actual sampling standard deviation under the alternative. In that setting, the canonical variance-one statement is not automatically exact: actual group-specific variances can matter. The drift formula remains a useful planning approximation when the difference is modest and the null variance is a good approximation, but it should not be presented as an exact fixed-alternative result.

Observed Statistics Across Three Looks

A practical calculation can show how the pooled score statistic changes as observations accumulate. The next example uses nested cumulative counts. It evaluates the score statistic separately at each look; it does not treat those values as independent tests.

Score Versus Wald Calculations, Diagnostics, and Limits

The score statistic uses a pooled estimate of the common risk when constructing its null standard error. A Wald statistic for the risk difference instead commonly uses separate estimated variances, of the form \((\hat p_1-\hat p_0)/\sqrt{\hat p_1(1-\hat p_1)/n_1+\hat p_0(1-\hat p_0)/n_0}\). The two statistics can be close in large samples, especially near the null, but they are not interchangeable in small samples or when event rates are near 0 or 1. The score formulation is the natural one here because its null information gives a clear cumulative information scale. Exact methods may be considered when counts are sparse, but they do not automatically yield this same normal score process.

Before relying on the canonical approximation, examine the design and the counts. Check that the actual sample is nested, the allocation ratio is approximately stable across increments, and the event probabilities have not shifted over time. At each look, very small numbers of events or non-events weaken the normal approximation; a pooled estimate near 0 or 1 also makes the estimated variance unstable. Large departures from the planned enrollment pattern, clustering, or a change in outcome ascertainment can undermine the assumed score variance or independence of stage contributions.

A common mistake is to calculate each look’s \(Z_k\) correctly and then treat the sequence as independent because each statistic is standardized to have variance about one. Standardization does not remove the shared participants. The correction is to use the cumulative information: for nested looks \(i\leq j\), the approximate correlation is \(\sqrt{I_i/I_j}\), not zero. Another mistake is to claim that a pooled-null standard error forces variance one under every alternative. It does so approximately under the null and suitable local alternatives; under a substantial fixed risk difference, the actual Bernoulli variances may differ from the null variance.