Stagewise Error Spending Under Unequal Arm Counts

Why Unequal Arm Counts Change Calibration

In a multi-arm, multi-stage trial, treatment arms need not enroll the same number of participants, and their information may accrue at different rates. This affects both the precision of each treatment-control contrast and the dependence among contrasts that share a control group. A stagewise error allocation that ignores either feature can use the wrong marginal boundaries or the wrong joint crossing probability.

The task here is to specify an error budget and verify that the probability of at least one false rejection stays within the familywise error rate (FWER) target. The calculations below use one-sided tests for benefit; reversing the direction of the hypotheses changes the tail, not the principles. We first use a normal model with known variances to make the dependence structure explicit. With estimated variances, the corresponding joint distribution or calibration must reflect that estimation rather than simply treating the statistics as independent standard normals.

Write \(\theta_i\) for the treatment-minus-control mean difference in the benefit direction. At stage \(j\), let \(n_{ij}\) and \(n_{0j}\) be the cumulative sample sizes in treatment arm \(i\) and the shared control, respectively. Under independent normally distributed outcomes with known, stable variances \(\sigma_i^2\) and \(\sigma_0^2\), the contrast estimate and its variance are

Under \(H_i:\theta_i=0\), \(Z_{ij}\) has a standard normal distribution in this model. The information for the contrast is \(I_{ij}=1/V_{ij}\). Thus, an information fraction is a ratio of information values, not necessarily a ratio of calendar times or total participant counts. Different arm allocations can give different information fractions at the same scheduled look.

Shared-Control Correlation With Unequal Counts

Suppose treatment arms are independent of one another and of the control, and all participants observed by an earlier look are included at later looks. At one look, the covariance of two distinct treatment-control contrasts comes from their shared control mean. Consequently,

The correlation is generally not the same for every pair: unequal treatment sample sizes or variances change the denominators. It is also generally not zero, even though the treatment groups themselves are independent. At different looks, the covariance must reflect which observations are shared across time. Under nested sampling, for example, the covariance of two control means at looks \(j\) and \(k\) is \(\sigma_0^2/\max(n_{0j},n_{0k})\). The analogous treatment contribution is present only when the contrasts refer to the same treatment arm.

Allocating Error Across Arms and Stages

A transparent option is to allocate the total error budget in pieces. Let \(\gamma_{ij}\) be the one-sided significance level assigned to testing arm \(i\) at look \(j\), with every \(\gamma_{ij}\geq 0\). If

then testing each arm-look hypothesis at level \(\gamma_{ij}\) controls strong FWER at \(\alpha\), provided each test has its stated marginal size under its null. At a given arm-look, the normal-model boundary is \(c_{ij}=\Phi^{-1}(1-\gamma_{ij})\), and rejection occurs when \(Z_{ij}\geq c_{ij}\). This union-bound guarantee does not require independent arms, equal sample sizes, or equal inter-arm correlations. Its cost is potential conservatism because it does not exploit the dependence among tests.

One may express the allocation as stage increments \(\Delta_j\) and arm weights \(w_{ij}\), where \(\sum_i w_{ij}=1\) and \(\gamma_{ij}=w_{ij}\Delta_j\). The stage increments must themselves satisfy \(\sum_j\Delta_j\leq\alpha\). These are local test levels with a direct Bonferroni guarantee. They are not automatically the same as the boundaries obtained from a conventional cumulative group-sequential spending function, which accounts for repeated looks when solving for its boundaries.

Correlation-Aware Spending and Its Verification

A less conservative calibration can use the joint distribution of the stagewise statistics. For each relevant intersection of null hypotheses, specify the joint covariance matrix using the arm counts, variances, and overlap of participants across looks. Then choose boundaries so the cumulative probability of crossing by look \(j\) equals the prespecified cumulative spending for that intersection. In generic terms, if \(E_j\) is the event of at least one rejection by look \(j\), the target is \(P(E_j)=A(t_j)\) under the relevant intersection null, where \(A(t_j)\) is the cumulative error spent by that information time. The new boundary is chosen to make the probability of a first crossing at look \(j\) equal to the increase in cumulative spending, while accounting for paths that did not cross earlier.

This calculation is a multivariate normal probability (or the appropriate joint probability under the selected test statistic). Unequal information accrual affects both the marginal variances and the cross-look correlations; unequal arm counts affect the shared-control correlations. It is therefore not generally valid to substitute a single common information fraction or an equal-correlation matrix merely for convenience. Numerical integration or a validated boundary-calibration algorithm is commonly used, followed by checks that the target crossing probabilities are achieved.

Derivation, Diagnostics, and Limitations

The reason the weighted guarantee survives arbitrary correlation is the union bound. Let \(I\) be the true-null set, and let \(E_{ij}\) be rejection of arm \(i\) at look \(j\). A false rejection must lie in at least one event \(E_{ij}\) with \(i\in I\). Therefore,

This argument explains both the strength and the limitation of the method: it remains valid without knowing the correlations, but it can count overlapping rejection events more than once. Joint calibration uses the dependence to reduce that conservatism, but requires a correctly specified joint distribution and a strong-control justification.

Before using either approach, check that cumulative counts and variances match the actual allocation, that the shared control is represented in the covariance matrix, and that the information fractions are calculated from contrast information. With staged accrual, verify that the covariance matrix is symmetric and positive semidefinite and that its diagonal entries are one after standardization. A useful numerical diagnostic is to compare marginal tail probabilities with their assigned levels and to recompute joint crossing probabilities using a second numerical method or tighter integration tolerances.

Calibration can fail if outcome variances change across stages but are treated as constant, if observations are not nested but a nested-sample covariance is used, or if a common correlation is imposed despite unequal allocations. Estimated-variance statistics also need their appropriate joint distribution; marginal \(t\) calibration does not by itself guarantee that a multivariate normal max boundary is exact. Finally, correlation-aware calibration is not automatically more robust: a misspecified covariance can understate crossing probabilities. Conservative allocations avoid reliance on the correlation for the error bound, though their power may be lower.