Recursive Calibration of Efficacy Boundaries

Why Boundary Calibration Must Track Surviving Paths

An efficacy boundary at an interim analysis is not calibrated in isolation. A path that crossed an earlier boundary has already stopped, so it must not be counted again when calculating the probability of crossing at a later look. Recursive calibration accounts for this by integrating the later statistic only over paths that have remained below every earlier efficacy boundary.

The goal is to match the design's cumulative alpha-spending target at each planned analysis. The boundary at look $j$ is chosen so the probability of having crossed by that look equals the target $A_j$, under the null model. This differs from setting the marginal tail probability at look $j$ equal to $A_j$: the latter ignores earlier crossings.

Definition: Consider a one-sided group sequential design with $K$ analyses. Let $Z_j$ be the standardized test statistic at look $j$, $t_j$ its information fraction, and $b_j$ the upper efficacy boundary. Crossing at look $j$ means $Z_j \ge b_j$, provided no earlier boundary was crossed. If $A_j=A(t_j)$ is the prespecified cumulative alpha-spending target by look $j$, calibration requires $$ P_0\!\left(\bigcup_{k=1}^{j}\{Z_k\ge b_k\}\right)=A_j, $$ where $P_0$ denotes probability under the null hypothesis.

The calibration described here uses the canonical joint-normal model under the null: each $Z_j$ has a standard normal marginal distribution, and $$ \operatorname{Corr}(Z_i,Z_j)=\sqrt{\frac{t_i}{t_j}},\qquad i<j. $$ Information fractions satisfy $0<t_1<\cdots<t_K=1$. Under this model, the sequence has a Markov transition: given $Z_{j-1}=x$, $Z_j$ is normal with mean $\sqrt{t_{j-1}/t_j}\,x$ and variance $1-t_{j-1}/t_j$. This construction assumes the planned information times and test-statistic model are appropriate, and that the efficacy rule is an upper-boundary rule. A two-sided or futility design needs its own crossing regions and calibration.

The Recursive Crossing Calculation

Write $\phi(z)$ for the standard normal density and $\Phi(z)$ for its cumulative distribution function. Define $q_j(z)$ as the density at $Z_j=z$ among paths that have not crossed an efficacy boundary through look $j$. It is a subdensity: its integral is the probability of surviving the stopping rule, not necessarily one.

q_1(z)=\phi(z)\,I(z<b_1),

where $I(\cdot)$ is 1 when its condition holds and 0 otherwise. For $j\ge 2$, let $\rho_j=\sqrt{t_{j-1}/t_j}$. Propagate the surviving density through the next conditional-normal transition, then remove paths at or above the new boundary:

q_j(z)=I(z<b_j)\int_{-\infty}^{b_{j-1}} q_{j-1}(x)\, \frac{1}{\sqrt{1-\rho_j^2}}\, \phi\!\left(\frac{z-\rho_j x}{\sqrt{1-\rho_j^2}}\right)\,dx.

The probability of a first crossing at look $j$ is the mass that enters the new upper-tail region from the surviving paths:

\Delta_j(b_j)= \int_{b_j}^{\infty}\int_{-\infty}^{b_{j-1}} q_{j-1}(x)\, \frac{1}{\sqrt{1-\rho_j^2}}\, \phi\!\left(\frac{z-\rho_j x}{\sqrt{1-\rho_j^2}}\right)\,dx\,dz.

For the first look, $\Delta_1(b_1)=1-\Phi(b_1)$. At every later look, choose $b_j$ so that $\Delta_j(b_j)=A_j-A_{j-1}$, with $A_0=0$. The earlier material on numerical integration of crossing probabilities supplies the numerical integration tools; the essential calibration step is to carry forward only the probability density of paths that survived.

Calibration principle: Once $b_1,\ldots,b_{j-1}$ have been fixed, compute the surviving-path distribution at look $j-1$. Select $b_j$ so the probability of crossing $b_j$ at look $j$, conditional only through integration over those surviving paths, equals the new spending increment $A_j-A_{j-1}$. Then retain only $Z_j<b_j$ to continue to the next look.

A Short Justification of the Calibration Equation

The event of having crossed by look $j$ is the disjoint union of first-crossing events at looks $1,\ldots,j$. Therefore its probability is the sum of their probabilities. If the first $j-1$ boundaries have already achieved cumulative spending $A_{j-1}$, achieving $A_j$ at the next look requires a new first-crossing probability of exactly $A_j-A_{j-1}$. The transition density in the recursive integral is the conditional density of $Z_j$ given $Z_{j-1}$, while $q_{j-1}$ includes only paths that have not crossed before. Integrating over $z\ge b_j$ consequently counts precisely the new crossings, not repeated crossings of stopped paths.

For fixed earlier boundaries, $\Delta_j(b)$ is nonincreasing in $b$: raising the cutoff shrinks the event $Z_j\ge b$, while leaving the surviving-path distribution unchanged. With a nondegenerate normal transition and positive surviving mass, it is strictly decreasing over finite cutoffs. Thus a target increment determines at most one finite cutoff. The increment must also be attainable: it cannot be negative or exceed the probability mass surviving to look $j-1$. If it is zero, a limiting infinite boundary represents no additional rejection; if it exceeds the surviving mass, the proposed cumulative targets cannot be achieved by this design.

Worked Example: Calibrating a Second-Look Boundary

Suppose a hypothetical one-sided design has two looks at $t_1=0.5$ and $t_2=1$. The first-look target is $A_1=0.0100$, and the cumulative target at the second look is $A_2=0.0289$. Under the canonical null model, set the first boundary from the standard normal tail:

1-\Phi(b_1)=0.0100 \quad\Longrightarrow\quad b_1=\Phi^{-1}(0.9900)\approx 2.3263.

The required second-look increment is $A_2-A_1=0.0189$. Here $\rho_2=\sqrt{0.5/1}=0.7071$, and the conditional standard deviation is $\sqrt{1-\rho_2^2}=0.7071$. Integrating over paths with $Z_1<b_1$ gives

\Delta_2(b_2)= \int_{-\infty}^{2.3263} \phi(x)\, \Phi\!\left(\frac{0.7071x-b_2}{0.7071}\right)\,dx.

Numerical quadrature followed by calibration gives $b_2\approx1.990$, for which the integral is approximately $0.0189$. Adding the earlier crossing probability gives $0.0100+0.0189=0.0289$, matching the specified cumulative target to the reported precision. The boundary is a cutoff on the standardized statistic, not an effect estimate in clinical units.

Worked Example: Checking a Rounded Cutoff

Keep the same hypothetical first look, with $b_1\approx2.3263$, but set the second-look cutoff to $b_2=2.000$. The second-look crossing probability is not simply $1-\Phi(2)$, because some paths above 2 at the second look already crossed at the first. The correct increment is

\begin{aligned} \Delta_2(2) &=P_0(Z_2\ge2,\ Z_1<2.3263)\\ &=P_0(Z_2\ge2)-P_0(Z_1\ge2.3263,\ Z_2\ge2)\\ &\approx 0.02275-0.00435\\ &\approx 0.01840. \end{aligned}

Thus cumulative crossing probability is approximately $0.0100+0.01840=0.02840$, not $0.0289$. A cutoff rounded from approximately $1.990$ to $2.000$ misses the desired cumulative target by about $0.00050$. This check illustrates why the displayed boundary precision and the reported achieved error probability should be consistent.

Worked Example: Carrying the Recursion Through a Third Look

Consider a hypothetical three-look design at $t_1=0.25,\ t_2=0.5,\ t_3=1$. Let $b_1=2.3263$ and $b_2=2.000$, and choose $b_3=2.000$ for this illustration. The first-look crossing probability is approximately $0.0100$. From the preceding calculation, the second-look increment is approximately $0.01840$, so cumulative spending through look 2 is about $0.02840$.

At look 2, the surviving subdensity is obtained by propagating the first-look survivors with $\rho_2=\sqrt{0.25/0.5}=0.7071$ and truncating at 2:

q_2(x)=I(x<2)\int_{-\infty}^{2.3263} \phi(u)\, \frac{1}{\sqrt{0.5}}\, \phi\!\left(\frac{x-0.7071u}{\sqrt{0.5}}\right)\,du.

For the third look, $\rho_3=\sqrt{0.5/1}=0.7071$. The new increment at the chosen cutoff is therefore

\Delta_3(2)= \int_{2}^{\infty}\int_{-\infty}^{2} q_2(x)\, \frac{1}{\sqrt{0.5}}\, \phi\!\left(\frac{z-0.7071x}{\sqrt{0.5}}\right)\,dx\,dz.

The cumulative target matched by this illustrative third boundary is $A_3=0.02840+\Delta_3(2)$, with the exact value defined by the displayed integral rather than by a rounded approximation. In an actual design, the specified $A_3$ would be supplied first; the third boundary would then be adjusted until the same integral equals $A_3-A_2$. This step shows why the density entering the third-look calculation must already reflect survival at both earlier looks.

Practical Diagnostics and Alternative Approaches

A calibration should be checked as a sequence, not only by inspecting its final cutoff. At each look, record the target increment, the integrated first-crossing probability, and the cumulative sum of the increments. Confirm that the cumulative values reproduce $A_j$ within the intended numerical tolerance. Also check that surviving mass decreases as boundaries are applied and that each proposed increment is no larger than the mass available to cross.

Marginal-tail substitution: using $1-\Phi(b_j)$ as the new increment ignores paths that crossed earlier. Use it only at the first look, or in a setting where the earlier crossing restrictions genuinely do not apply.
Summing marginal tails: adding $P_0(Z_j\ge b_j)$ across looks double-counts paths that cross more than once in the unconstrained statistic sequence. Sum first-crossing probabilities instead.
Lost truncation: if the recursion does not restrict $q_{j-1}$ to $x<b_{j-1}$, previously stopped paths leak into later calculations and inflate cumulative error.
Inconsistent targets: cumulative spending should be nondecreasing, and each increment must be attainable from the surviving probability. A negative increment or an increment exceeding available mass signals a design or input problem.
Model mismatch: the canonical correlation structure depends on the test-statistic and information-time assumptions. Unequal information increments are allowed when correctly represented by the $t_j$, but incorrect information times or a different covariance structure invalidate these transitions.

Direct multivariate-normal integration can calculate the joint probability of no crossing, but its dimension grows with the number of looks. Recursive integration uses the Markov transitions and a sequence of lower-dimensional calculations. Simulation is a useful independent check, especially for complicated models, but finite simulation introduces random error and may be imprecise for small spending increments. It should not silently replace a sufficiently accurate calibration calculation.

The recursion is specific to the stated efficacy rule and null distribution. It does not by itself select a spending function, establish power under alternatives, handle binding futility rules, or correct a misspecified analysis plan. Those are separate design questions. The earlier material on spending functions provides the target $A(t_j)$; this calibration maps those targets to cutoffs under the chosen model.

Common pitfall: “The marginal tail at the second look is 0.02275, so a 0.0100 first-look spend plus a 0.02275 second-look spend gives 0.03275.” This counts some paths twice. The correction is to integrate the second-look tail only over first-look survivors; with $b_1\approx2.3263$ and $b_2=2.000$, that increment is about $0.01840$, giving cumulative crossing probability about $0.02840$.

Check Your Understanding

Use the definitions and recursive calculation above. Answers follow each question so the arithmetic and interpretation can be checked.

A design has $A_1=0.008$ and $A_2=0.020$. What second-look crossing increment must be calibrated?
Why is $1-\Phi(b_2)$ generally not the second-look increment?
If the surviving probability at look $j-1$ is $0.60$, can a requested new crossing increment of $0.65$ be attained? Explain.
For two looks, $t_1=0.25$ and $t_2=1$. What is the conditional mean and variance of $Z_2$ given $Z_1=x$ under the canonical model?
If a numerical calculation reports cumulative crossing probabilities $0.010, 0.008,$ and $0.025$ at successive looks, identify a diagnostic concern.

Solutions. (1) The increment is $A_2-A_1=0.020-0.008=0.012$. (2) It includes paths that may already have crossed the first boundary; integrate only over first-look survivors. (3) No: the probability of crossing at that look cannot exceed the surviving mass $0.60$, and $0.65>0.60$. (4) $\rho=\sqrt{0.25}=0.5$, so the conditional mean is $0.5x$ and variance is $1-0.25=0.75$. (5) Cumulative crossing probability cannot decrease as looks are added; the reported sequence is inconsistent and suggests an integration, truncation, or cumulative-summation error.

Recursive Calibration of Efficacy Boundaries

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