A Four-Look Boundary Calibration Workshop

From Cumulative Targets to Four Boundaries

A four-look design is not calibrated by applying the same single-look cutoff four times. The statistics at successive looks are dependent, and a crossing at an early look prevents later testing. Calibration must therefore account for the probability of reaching each look without having crossed previously. This workshop develops that calculation for a one-sided efficacy design with four upper boundaries.

Assume the canonical Gaussian model under the null hypothesis: the information fractions satisfy \(0<t_1<t_2<t_3<t_4=1\), each \(Z_i\) has a standard normal marginal distribution, and the correlation between \(Z_i\) and \(Z_j\) is \(\sqrt{t_i/t_j}\) for \(i<j\). The model arises when standardized cumulative score statistics have independent increments and information \(I_i\), with \(t_i=I_i/I_4\). The design uses fixed information times, and the null distribution and information schedule are treated as correctly specified. At look \(i\), efficacy is declared if \(Z_i\ge b_i\); otherwise the trial continues if another look is scheduled.

The conditional distribution makes the sequential dependence explicit. For \(i\ge2\), let \(\rho_i=\sqrt{t_{i-1}/t_i}\). Under the null, \(Z_i\mid Z_{i-1}=z\) is normal with mean \(\rho_i z\) and variance \(1-\rho_i^2\). Denote its density by \(g_i(x\mid z)\). If \(f_{i-1}(z)\) is the density of \(Z_{i-1}=z\) among paths that have survived all earlier looks, then

Here \(q_i=P_0(A_i)\) is the probability of a first crossing at look \(i\), and \(f_1(x)=\phi(x)\mathbf{1}(x<b_1)\). Thus \(C_i=\sum_{j=1}^{i}q_j\). These formulas keep the surviving probability mass below each previous boundary; they do not restart the process from an unconditional standard normal distribution at each look.

Why Sequential Root-Finding Works

Calibrate the first boundary from \(P_0(Z_1\ge b_1)=\alpha_1\). At look \(i>1\), keep the already calibrated boundaries fixed and choose \(b_i\) so that \(q_i=\alpha_i-\alpha_{i-1}\). Increasing \(b_i\) decreases the probability of crossing at that look, while leaving all earlier crossing probabilities unchanged. Consequently, the cumulative crossing probability through look \(i\) decreases continuously with \(b_i\). A bracketed root finder can solve the equation, provided the requested increment is feasible: it must lie between zero and the probability mass that survived to look \(i-1\).

The sequential nature of the calculation is important: a change to \(b_2\) changes the survivor distribution passed to looks 3 and 4. Therefore, after a boundary changes, later boundaries must be recalibrated. The construction is unique at each step when the target increment is strictly between zero and the available survivor probability, because the crossing probability falls continuously from that available mass toward zero as the boundary rises from very low to very high values.

Power Under an Alternative Drift

For an alternative with standardized drift \(\theta\), use the same information schedule and covariance, but give \(Z_i\) mean \(\theta\sqrt{t_i}\). In the canonical model, \(Z_i\sim N(\theta\sqrt{t_i},1)\). The survivor recursion is updated using these alternative conditional normal distributions, and the stagewise probabilities \(P_\theta(A_i)\) are integrated using the same boundaries. Overall power is \(\sum_{i=1}^{4}P_\theta(A_i)\), because the first-crossing events are disjoint.

The calculation assumes the alternative is represented by the specified canonical drift and that the information times remain fixed. If the alternative changes the variance model, information schedule, or distributional approximation, the power calculation may need a different joint model. The null calibration and alternative power calculation answer different questions: the former controls the null probability of any rejection, while the latter evaluates rejection probability under a stated alternative.

Numerical Convergence and Failure Modes

A small root-finder residual is necessary but not sufficient evidence of an accurate calibration. A solver may accurately find a root of an inaccurate integral. Numerical validation should therefore include at least three distinct checks: the target residual for each cumulative probability, refinement of the integration grid or quadrature tolerance, and probability-mass balance. For a one-sided design with no futility stopping, the total mass at look \(i\) must satisfy \(P_0(\text{cross by }i)+P_0(\text{survive look }i)=1\), up to numerical error. Under an alternative, the same accounting identity applies.

Common warning signs include a cumulative probability that decreases from one look to the next despite increasing targets, a negative survivor mass, a root that changes materially under refinement, or cumulative crossing plus survivor probability far from one. These often indicate a boundary inequality coded in the wrong direction, survivor density integrated beyond an earlier boundary, omitted or duplicated mass, or an integration domain too coarse near a stopping boundary. Very narrow brackets and excessive rounding can also make a valid root appear unstable.

A useful alternative is direct multivariate normal probability calculation for the joint event of staying below all boundaries. It provides an independent benchmark for a four-look design, while recursive integration is often easier to extend and inspect stage by stage. Simulation can also validate the recursion, but Monte Carlo error can obscure small discrepancies: with \(M\) simulated trials, the standard error of an estimated probability \(\hat p\) is approximately \(\sqrt{\hat p(1-\hat p)/M}\). Neither alternative removes the need to specify the correct joint null model and information times. Grid-based integration can be inaccurate in tails or near a boundary, whereas direct multivariate normal routines can also have numerical tolerances and simulation can be inefficient for rare crossings.

This workshop addresses one-sided upper efficacy boundaries only. Two-sided designs require accounting for both tails, and futility stopping changes the survivor region and the set of paths that can reach later looks. Those are different stopping rules, not small adjustments to the one-sided upper-boundary calculation. The reported boundaries also depend on the chosen cumulative target sequence; a different allocation of the same total type I error generally gives different interim thresholds.

Common Mistake

The distinction is practical: a marginal tail probability describes all hypothetical values at look \(i\), including paths that would already have stopped. Calibration concerns only the paths still under observation. Using marginal tails generally misses the cumulative targets, even when each single-look calculation appears numerically precise.