Calibrating Sequential Boundaries to Cumulative Targets

Why Boundaries Must Be Calibrated to Cumulative Targets

At an interim analysis, a boundary controls whether the trial stops, but its error contribution cannot be assessed in isolation from earlier looks. A participant who crosses at an earlier look is no longer at risk of first crossing at a later one. Therefore, calibrating each look as if it were the only analysis generally fails to produce the intended overall type I error.

This tutorial considers a one-sided upper-boundary design under the null hypothesis. The task is to select each successive boundary so that the probability of crossing at least one boundary by that look equals a prespecified cumulative target. The earlier material on representing interim looks as numerical states and the survivor-density recursion supplies the computational framework; here the emphasis is the sequential calibration problem.

Definition: Let $Z_j$ be the standardized test statistic at look $j$, and let $c_j$ be its one-sided upper boundary. Under $H_0$, define the cumulative crossing probability through look $j$ by $$ A_j=P\left(\bigcup_{k=1}^{j}\{Z_k\ge c_k\}\mid H_0\right). $$ A cumulative target $a_j$ is the prespecified value that $A_j$ must attain. Thus $a_j$ includes crossings at all looks through $j$, not just crossings at look $j$.

Assume the null joint distribution of the standardized statistics is the canonical Gaussian model: each $Z_j$ has a standard normal marginal distribution, and for looks $i\le j$, their correlation is $\sqrt{I_i/I_j}$, where $I_j$ is the information at look $j$. Information increases across looks. The analysis plan, information times, and earlier boundaries are fixed while the current boundary is being calibrated. We consider only upper efficacy boundaries here; the calculations do not cover asymmetric two-sided stopping rules.

The Successive Boundary Equation

Let $S_{j-1}(z)$ denote the null survivor density at look $j-1$: it is the joint probability density for reaching that look with $Z_{j-1}=z$ without having crossed any earlier boundary. At the next look, the null transition distribution is Gaussian. Conditional on $Z_{j-1}=z$, its mean is $\rho_jz$ and its variance is $1-\rho_j^2$, where $\rho_j=\sqrt{I_{j-1}/I_j}$. The probability of surviving the new boundary is obtained by integrating this transition probability over the previous survivor density.

S_j(y)=\int_{-\infty}^{c_{j-1}} S_{j-1}(z) \frac{1}{\sqrt{1-\rho_j^2}} \phi\!\left(\frac{y-\rho_jz}{\sqrt{1-\rho_j^2}}\right)\,dz, \qquad y<c_j,

Here $\phi$ is the standard normal density. The survivor mass $M_j(c_j)=\int_{-\infty}^{c_j}S_j(y)\,dy$ is the probability of no crossing through look $j$. Consequently,

$$ A_j=1-M_j(c_j). $$

With the earlier boundaries held fixed, choosing $c_j$ means solving $1-M_j(c_j)=a_j$. Equivalently, the probability of surviving through look $j$ must equal $1-a_j$. This formulation avoids treating the look-specific crossing probability as though it were the cumulative target.

Calibration principle: At look $j$, compute the survivor distribution using the already calibrated boundaries $c_1,\ldots,c_{j-1}$, then solve $$ f_j(c)=A_j(c)-a_j=0. $$ The argument $c$ changes only the current boundary. Earlier boundaries remain fixed during this root solve.

For fixed earlier boundaries, $A_j(c)$ is nonincreasing in $c$: raising the current boundary can only remove current-look crossings, not create them. It is continuous under the Gaussian model. As $c$ tends to infinity, the chance of crossing at look $j$ tends to zero and $A_j(c)$ tends to $A_{j-1}$. As $c$ tends to negative infinity, nearly all paths that survived earlier looks cross at look $j$, so $A_j(c)$ tends to 1. Thus, for an ordinary finite boundary, the target must satisfy $A_{j-1}<a_j<1$. If $a_j\le A_{j-1}$, no current boundary can undo crossing probability already accumulated.

Worked Calibration Examples

Worked Example: Establishing the First Boundary

Suppose the null cumulative target at the first look is $a_1=0.0100$. Since there is no earlier look, $A_1=P(Z_1\ge c_1\mid H_0)$, with $Z_1\sim N(0,1)$. The equation is $$ 1-\Phi(c_1)=0.0100,\qquad c_1=\Phi^{-1}(0.9900)=2.3263. $$

The first boundary is therefore $c_1=2.3263$. Its null crossing probability is 0.0100, or 1.00%, at the first look. This is a cumulative target as well as a look-specific probability only because there has been no prior opportunity to cross.

Worked Example: Calibrating a Second Look to a 0.0250 Cumulative Target

Consider a hypothetical two-look design with information $I_1=0.40$ and $I_2=0.70$. The first boundary is $c_1=2.3263$, and the cumulative targets are $a_1=0.0100$ and $a_2=0.0250$. Under the stated canonical Gaussian model, $$ \rho=\sqrt{\frac{0.40}{0.70}}=0.7559,\qquad \sqrt{1-\rho^2}=0.6547. $$

For a candidate second boundary $c$, the cumulative crossing probability can be computed from the probability of no crossing at either look: $$ A_2(c)=1-P(Z_1<c_1,Z_2<c). $$ Using the conditional Gaussian distribution of $Z_2$ given $Z_1=z$, this becomes $$ A_2(c)=0.0100+\int_{-\infty}^{2.3263}\phi(z) \left[1-\Phi\!\left(\frac{c-0.7559z}{0.6547}\right)\right]dz. $$

Numerical integration gives $A_2(2.080)\approx0.0242$, below the target, and $A_2(2.000)\approx0.0277$, above it. These values bracket the target, so a bracketed root solver can be applied to $f_2(c)=A_2(c)-0.0250$. Solving gives approximately $c_2=2.059$, with $A_2(c_2)\approx0.0250$, subject to numerical integration precision.

The result is a cumulative probability of 2.50% of crossing by the second look under $H_0$, not a 2.50% probability of crossing specifically at the second look. The first-look crossing probability remains 1.00%.

Worked Example: A Lower Cumulative Target at the Second Look

Keep the same information times and first boundary, but suppose the desired second-look cumulative target is $a_2=0.0226$. The correlation and transition standard deviation remain $\rho=0.7559$ and $0.6547$. At candidate boundary $c=2.120$, the same integral gives $A_2(2.120)\approx0.0226$. Thus a calibrated boundary is approximately $c_2=2.120$.

This target is feasible because it exceeds the probability already accumulated at the first look: $0.0226>0.0100$. It also exceeds the first-look probability by $0.0126$, so the probability of first crossing at the second look is approximately 1.26 percentage points. The boundary is higher than the one calibrated to the 0.0250 target, as expected: increasing an upper boundary decreases the cumulative crossing probability.

Why the Root Solve Works

The key property is monotonicity. If $c^{(L)}<c^{(U)}$, every path that crosses at the higher boundary $c^{(U)}$ at the current look also crosses at the lower boundary $c^{(L)}$, while the converse need not hold. Earlier crossings are unchanged. Therefore $A_j(c^{(L)})\ge A_j(c^{(U)})$. This nesting of crossing events makes a bracketed method, such as bisection, reliable when the target is feasible.

Fix the design inputs.
Specify the null model, information times, cumulative target at each look, and any already calibrated boundaries.

Check feasibility.
Confirm that the new target is greater than the cumulative probability already attained and less than 1.

Evaluate the current-look function.
For candidate $c$, compute the survivor mass using the transition model and set $f_j(c)=1-M_j(c)-a_j$.

Bracket and solve.
Find $c_L,c_U$ for which $f_j(c_L)$ and $f_j(c_U)$ have opposite signs, then use bisection or another safeguarded root solver.

Verify and continue.
Recompute the cumulative probability at the returned boundary, record the residual, then use this boundary when calibrating the next look.

A short justification for the sequential procedure follows directly from this construction. At look 1, the one-look tail equation sets $A_1=a_1$. Suppose the calibrated boundaries through look $j-1$ give $A_{j-1}=a_{j-1}$. The next root solve holds those boundaries fixed and selects $c_j$ so $A_j(c_j)=a_j$. Hence, by induction, each completed step meets its own cumulative target, provided the numerical evaluations and root solves are sufficiently accurate.

Numerical Checks, Alternatives, and Limitations

A reliable implementation should verify more than a solver's success flag. Confirm that the residual $|A_j(c_j)-a_j|$ is small relative to the desired numerical tolerance, that the final candidate remains inside its root bracket, and that tightening integration tolerances or refining the state grid does not materially change the boundary. Check the monotonic direction as well: for an upper boundary, increasing $c_j$ must not increase $A_j$. A violation can indicate a sign error, inconsistent truncation, or an inaccurate integration.

The root finder and the probability calculator solve different problems. Bisection, Brent's method, or another safeguarded solver searches for a boundary; the survivor recursion, quadrature, or another valid probability method evaluates $A_j(c)$. A fast solver cannot correct a biased probability calculation. Bracketing is generally safer than an unguarded Newton step because it preserves a known interval containing the root even when the derivative is poorly estimated. A direct multivariate normal probability calculation can serve as an independent check for a small number of looks, but it does not change the definition of the cumulative target.

Target treated as look-specific. Setting the second-look tail probability alone to $a_2$ ignores paths that already crossed. Use the probability of crossing at any look through $j$.
Infeasible target. If $a_j\le A_{j-1}$, the boundary at look $j$ cannot reduce the crossing probability that has already occurred.
Wrong monotonic direction. For an upper boundary, a larger $c_j$ gives a smaller or equal cumulative crossing probability. Check this direction in both code and output.
False precision. Reporting many boundary digits is not justified if integration error or grid sensitivity is larger than the implied change in probability.

The procedure assumes the specified Gaussian null distribution and information structure are appropriate and that numerical survivor mass is computed accurately. A finite truncation of the state space, coarse grid, or inaccurate boundary interpolation can distort $A_j(c)$; apparent root convergence does not rule out those errors. Calibration also does not establish operating characteristics under an alternative, which require a separate distributional calculation.

Check Your Understanding

Answer each question, then compare with the solutions below.

At look $j$, what probability does a cumulative target $a_j$ specify?
If the cumulative crossing probability through look 2 is 0.018 and the target at look 3 is 0.015, can a finite upper boundary at look 3 attain the target? Why?
For a fixed set of earlier boundaries, what happens to the cumulative crossing probability when the current upper boundary is raised?
In the two-look example, why is the integral for $A_2(c)$ added to 0.0100 rather than used as the entire cumulative probability?
What numerical checks should be made after a root solver returns a boundary?

Solutions. (1) It specifies the probability under $H_0$ of crossing at least one boundary from the first look through look $j$. (2) No. Crossing through look 2 has already occurred with probability 0.018, and a later boundary cannot undo those events; the cumulative probability cannot be reduced to 0.015. (3) It decreases or stays unchanged, because raising the boundary removes current-look crossings while leaving earlier crossings unaffected. (4) The integral counts paths that survive look 1 and cross at look 2; 0.0100 accounts for paths that already crossed at look 1. (5) Re-evaluate the cumulative probability and residual, check that the root lies within a sign-changing bracket, confirm monotonicity, and assess sensitivity to integration tolerances or grid refinement.