A Complete Two-Look Crossing Calculation

Why Separate the Two Looks?

In a group sequential design, a rejection at the second look is possible only if the trial has not already crossed its upper boundary at the first look. Thus, the probability of crossing the second-look boundary is not simply the marginal upper-tail probability at that look. It must be calculated among paths that survived the first look. This tutorial works through that calculation for a two-look design with upper boundaries and shows how to obtain the rejection probability at each look.

The calculation assumes a canonical Gaussian model: information is measured on an increasing time scale, and the standardized statistic has independent Gaussian increments before standardization. The design has fixed look times and fixed boundaries. We consider an upper-boundary rule only: stop and reject at look one if the statistic reaches its boundary; otherwise continue and reject at look two if its boundary is reached. There is no futility stop in this calculation.

Conditional Distribution and Survivor Integral

The two statistics have a bivariate normal distribution. Under the canonical model, the second statistic can be represented as \(Z_2=\rho Z_1+\sqrt{1-\rho^2}\,\varepsilon\), where \(\varepsilon\sim N(0,1)\) is independent of \(Z_1\). Consequently, conditional on the first-look value \(Z_1=z\), the second-look statistic is normal with mean \(\rho z\) and variance \(1-\rho^2\). The conditional probability of crossing the second boundary is therefore a standard normal upper tail.

The first-look survivor density is the original standard normal density restricted to values below the first boundary. It is not renormalized to integrate to one: its total mass is the probability of surviving look one. Multiplying this density by the conditional second-look crossing probability and integrating over survivor values gives the unconditional probability of first rejecting at look two.

The events of first rejection at the two looks are disjoint, so the cumulative probability of rejection by look two is \(p_1+p_2\). Under \(H_0\), this cumulative probability is the design's two-look type I error for the stated boundaries. The probability of rejection at look two conditional on surviving look one is a different quantity: it is \(p_2/\Phi(b_1)\), provided \(\Phi(b_1)>0\).

Worked Calculation Under the Null

The following hypothetical design uses equal increments in information, so \(t_1=0.5\) and \(t_2=1\). The boundaries are \(b_1=2.4\) and \(b_2=2.0\). Under the null model, \(\rho=\sqrt{0.5}=0.7071\), and \(\sqrt{1-\rho^2}=0.7071\). The assumptions are the canonical Gaussian model, fixed looks and boundaries, and an upper-only stopping rule as defined above.

A useful independent check comes from the final-look marginal probability. Every path with \(Z_2\ge b_2\) either crossed at look one or survived look one and crossed at look two. Therefore, \(P(Z_2\ge b_2)=P(Z_1\ge b_1,Z_2\ge b_2)+p_2\). For the example, \(P(Z_2\ge2.0)\approx0.02275\), and the joint probability of crossing both limits is approximately \(0.00377\). Their difference is \(0.02275-0.00377\approx0.01898\), agreeing with the integral. The first-look crossing is counted separately in the cumulative rejection probability; it is not added to the joint crossing probability as a second rejection at look two.

Two Checks on the Calculation

The conditional formulation also provides useful checks when implementing the calculation. The integrand is nonnegative, and \(p_2\) cannot exceed either the probability of surviving look one or the marginal probability of exceeding the final boundary. If these checks fail, the integration limits, conditional tail, or survivor-density normalization may be wrong.

The last identity is often convenient for checking an integral against a bivariate normal probability calculation. It also clarifies why the final-look marginal tail alone is not the stage-two rejection probability: some paths in that tail have already crossed at look one. A second check is to raise \(b_1\) while holding \(b_2\) fixed. More paths then survive the first look, so the look-two rejection probability cannot decrease. The cumulative probability can change as well, because the first-look rejection probability decreases.

Numerical Practice and Common Pitfalls

A one-dimensional integral is often evaluated by Gaussian quadrature or another reliable numerical integration method. The earlier material on Gauss-Legendre and Gauss-Hermite quadrature describes numerical tools; here the important point is to apply them to the correct survivor integral. Compare results at more than one numerical resolution when feasible, and check that the stage-specific probabilities sum to the same cumulative probability as the disjoint-event calculation.

Direct bivariate-normal probability calculations can provide an independent check, while simulation can approximate the crossing probabilities by generating many correlated pairs and counting the disjoint events. Simulation is useful as a diagnostic, but its estimates vary from run to run and can be imprecise for small probabilities. Numerical integration is deterministic up to numerical error and is generally more efficient for this two-look calculation.

• Renormalizing the survivor density. Dividing \(\phi(z)\) by \(\Phi(b_1)\) produces a conditional density among survivors. If using that density, its integral gives a conditional look-two probability and must be multiplied by \(\Phi(b_1)\) to recover \(p_2\). The displayed integral instead uses the unnormalized survivor density directly.
• Using only the final marginal tail. \(1-\Phi(b_2)\) includes paths that already crossed at look one. Subtract the joint early-and-final crossing probability, or integrate only over \(z<b_1\).
• Using the wrong conditional variance or tail. The conditional standard deviation is \(\sqrt{1-\rho^2}\), not one. The integrand is the upper-tail probability, \(1-\Phi(\cdot)\), because rejection occurs above \(b_2\).
• Mislabeling the stage probability. \(p_2\) is an unconditional probability of first rejection at look two. A conditional probability among survivors is \(p_2/\Phi(b_1)\).

This calculation is limited to two fixed looks, an upper-only boundary rule, and the stated Gaussian model. It does not by itself calibrate boundaries to a target error rate, address unequal information increments, or handle additional looks; those require their own calculations. In practice, check the model and the probability accounting before interpreting any numerical output.