Stagewise Rejection Probabilities Under an Alternative

Why Separate Power by Look?

A group sequential design can reject the null hypothesis at more than one scheduled analysis. Under a specified alternative, total power is the probability of rejecting at any look. Knowing how that power is distributed across looks is useful for understanding when rejection occurs and for checking a calculation. The key is to count each rejecting path once: a path that crosses a boundary early is not a new first rejection if its statistic also crosses a boundary later.

The distinction is between a marginal crossing probability, such as the probability that the statistic is above the final upper boundary at the final look, and a stagewise first-rejection probability, the probability that the first rejection occurs at that look. The latter accounts for whether the path survived all earlier looks.

The stagewise probability is an unconditional probability under the alternative: it is the probability, among all repetitions of the planned trial under that alternative, of first rejecting at the specified look. It is not the probability of rejection at that look conditional on having reached it. A conditional probability can be reported too, but it must be multiplied by the probability of surviving to that look to obtain the unconditional stagewise contribution.

Survivor-Weighted Calculation

Use the canonical Gaussian information model. At look \(j\), let \(t_j\) be cumulative information, with \(0<t_1<\cdots<t_K\). Under the alternative with drift \(\theta\), write \(Z_j=B(t_j)/\sqrt{t_j}\), where \(B(t)\) has independent Gaussian increments, mean \(\theta t\), and variance \(t\). Thus \(Z_j\sim N(\theta\sqrt{t_j},1)\). Conditional on the preceding statistic, the transition is

This model assumes the specified drift and information times are appropriate, the increments follow the stated Gaussian model, and the rejection boundaries are fixed as specified for the calculation. It is the joint dependence across looks—not a collection of independent marginal tests—that determines the probability of first rejection.

Suppose look \(j\) has lower and upper rejection boundaries \(\ell_j\) and \(u_j\). A path survives look \(j\) when its statistic lies in \(S_j=(\ell_j,u_j)\). Define \(g_j(z)\) as the density at \(z\) at look \(j\), multiplied by the indicator that the path has survived every look through \(j\). It is a survivor density, so its integral is the probability of survival, not necessarily one. At the first look, \(g_1\) is the alternative density restricted to \(S_1\). At a later look it is obtained by integrating the preceding survivor density against the Gaussian transition density and restricting the result to \(S_j\).

where \(f_{\theta}(y\mid z)\) is the conditional Gaussian density from the transition model. The stagewise upper and lower probabilities at look \(j\) are the mass that leaves the survivor region through the corresponding boundary:

At look 1, use the alternative marginal distribution directly rather than a preceding survivor density. If the design has only an upper efficacy boundary, the lower-boundary contribution is zero. The total power is the sum of the upper and lower first-rejection probabilities over all looks.

Why the Sum Gives Total Power

The events \(R_1,\ldots,R_K\) cannot overlap: a trial that first rejects at look \(j\) has not rejected at any earlier look, so it cannot also first reject at look \(k\ne j\). The event of rejection by the end of the design is exactly their union. Additivity of probability for disjoint events therefore gives the sum of the stagewise probabilities. This also explains why adding marginal crossing probabilities is generally wrong: a path may cross a boundary at several looks.

Reconciliation With a Marginal Probability

A useful check for the preceding calculation is to compare the look-2 first-crossing probability with the marginal probability of being above the look-2 boundary. Under the alternative, \(Z_2\sim N(0.5\sqrt{2},1)\), so

This marginal probability includes paths that had already crossed the upper boundary at look 1. The probability of crossing the upper boundary at both looks is approximately 0.0191. Subtracting that overlap gives the look-2 first-crossing probability:

Then total power can also be viewed as the final-look marginal crossing probability plus the probability of an early crossing that is not above the final boundary. Here that additional early-only probability is about \(0.1076-0.0980=0.0096\). This identity is a check, not a replacement for tracking which paths have already rejected.

Two-Sided Boundaries and Numerical Checks

For a two-sided design, calculate the lower and upper first-crossing contributions separately at each look. For example, at look \(j\), a surviving path can cross below \(\ell_j\) or above \(u_j\); neither contribution should include paths that already crossed either boundary earlier. Sum all these disjoint contributions to obtain the probability of rejecting in either direction. If the scientific alternative favors one direction, the upper and lower contributions may differ substantially; symmetry should not be assumed unless the model, alternative, and boundaries actually imply it.

In numerical integration, the survivor density’s total mass is an important diagnostic. Its integral at a look should equal the probability of reaching that look without rejection. At each update, that mass plus the lower and upper crossing masses at that look should equal the preceding survivor mass, up to numerical error. A noticeable discrepancy can indicate a truncated integration range, a grid too coarse near a boundary, or an incorrect transition mean or variance. Refining the grid or quadrature and checking convergence helps distinguish numerical error from a genuine design result.

The main alternatives for computing the probabilities are recursive survivor-density integration, direct multivariate Gaussian probability calculations, and simulation under the specified alternative. Recursive integration directly produces stagewise probabilities and is useful when boundaries vary by look. A direct multivariate calculation can provide an independent check, but the event must encode survival at every earlier look. Simulation is also a check, with Monte Carlo uncertainty: a simulated stagewise estimate is the number of trials first rejecting at that look divided by the total simulated trials. These approaches should agree within their numerical or simulation error when they represent the same model and events.

A frequent mistake is to report \(P_{\theta}(Z_j\ge u_j)\) as the stagewise upper probability. That is a marginal probability, and it counts paths that may have rejected earlier. The correction is to restrict to paths that survived all earlier looks, either by integrating the survivor density or by subtracting the appropriate earlier-crossing overlap. Another mistake is to add marginal crossing probabilities across looks; overlapping events make that sum larger than the probability of rejection at any look.