Why Calibrate the Two Sides Separately?
A sequential design may allocate different amounts of crossing probability to its upper and lower boundaries. For example, an analysis plan may require a small probability of crossing the upper boundary under the null while allowing a different probability of crossing the lower boundary. If the boundaries are forced to be symmetric, those separate targets will generally not be met. Numerical calibration instead solves for each boundary from the survivor density at that look.
The key is that the relevant density is not usually the unconditional distribution of the test statistic. It is the survivor density: the probability density among paths that have not crossed either boundary at an earlier look. At the current look, the probability of crossing below the lower boundary is the integral of that density below the lower boundary, and the probability of crossing above the upper boundary is its integral above the upper boundary. Only the mass between the two boundaries continues.
From Cumulative Targets to Tail Integrals
Suppose the design specifies cumulative upper-crossing targets \(A_k\) and cumulative lower-crossing targets \(B_k\) through look \(k\). These are probabilities of having crossed the corresponding boundary at or before that look, under \(H_0\). Set \(A_0=B_0=0\). The probability increments to be assigned at look \(k\) are \(\Delta A_k=A_k-A_{k-1}\) and \(\Delta B_k=B_k-B_{k-1}\). Calibration must match these increments, not the cumulative targets again: earlier crossings have already removed probability mass from \(g_k\).
The two equations can be solved separately because \(g_k\) is determined by the boundaries at earlier looks; the current boundaries do not change the density arriving at look \(k\). They do, however, need to be jointly feasible. Write \(S_k=\int_{-\infty}^{\infty}g_k(z)\,dz\) for the probability of reaching look \(k\). If the targets are positive, a nonempty continuing interval requires \(\Delta A_k+\Delta B_k<S_k\). When the sum is greater than \(S_k\), the requested crossing increments cannot be supplied by the available mass. Equality leaves no survivor mass between the boundaries, apart from possible density-zero regions.
For a positive, continuous survivor density, each tail integral is monotone in its boundary: the upper tail decreases as \(u_k\) increases, while the lower tail increases as \(\ell_k\) increases. This supports one-dimensional root finding for each side. With a density represented on a grid, the integrals are numerical approximations, so interpolation and refinement may be needed to locate a root between grid points.
Why the Interval Is the Only Part That Continues
Under the null canonical Gaussian process, one common transition representation is \(Z_{k+1}\mid Z_k=z\sim N(\rho_k z,1-\rho_k^2)\), where \(\rho_k=\sqrt{I_k/I_{k+1}}\) for information levels \(I_k<I_{k+1}\). Let \(K_k(y\mid z)\) denote the corresponding Gaussian transition density. A path that crosses at look \(k\) has stopped, so it cannot contribute to the next look’s survivor density. Consequently, the recursion integrates over the interval between the current boundaries only.
This formula also explains why an incorrect interval of integration can undermine every later calibration. Integrating over the full real line would let already-stopped paths continue; integrating over only one side would discard surviving paths. In computation, the quadrature approximation must represent both the survivor density and the interval endpoints accurately enough for the desired probability tolerances.
Worked Example: Asymmetric Targets at the First Look
Consider a hypothetical first look under \(H_0\) with \(Z_1\sim N(0,1)\). There are no earlier stops, so \(g_1(z)=\phi(z)\), the standard normal density, and \(S_1=1\). Suppose the upper-crossing target is \(0.0100\) and the lower-crossing target is \(0.0400\).
The standard normal tail equations give \(\int_{u_1}^{\infty}\phi(z)\,dz=0.0100\) and \(\int_{-\infty}^{\ell_1}\phi(z)\,dz=0.0400\). Thus \(u_1=\Phi^{-1}(0.9900)\approx2.326\) and \(\ell_1=\Phi^{-1}(0.0400)\approx-1.751\). The boundaries are not negatives of one another because the targets differ.
The probability that continues to the next look is \(1-0.0100-0.0400=0.9500\), which is positive. Equivalently, integrating the standard normal density between the boundaries gives \(\Phi(2.326)-\Phi(-1.751)\approx0.9500\). The calculation checks both tail targets and the remaining mass.
Numerical Root Finding and Feasibility Checks
In a later look, \(g_k\) may not be a standard normal density and may have total mass less than one. Define the upper-tail function \(U_k(c)=\int_c^\infty g_k(z)\,dz\) and the lower-tail function \(L_k(c)=\int_{-\infty}^c g_k(z)\,dz\). Calibration finds a root of \(U_k(u_k)-\Delta A_k=0\) and a root of \(L_k(\ell_k)-\Delta B_k=0\). Each function is monotone in its argument, so a bracketed method such as bisection is reliable when a bracket is available. A Newton-type method can be quicker, but requires a usable density value near the root and safeguards against steps outside the bracket.
Before solving, check that the cumulative targets do not decrease, that both increments are nonnegative, and that their sum is no greater than the arriving survivor mass. After solving, check the two tail integrals against their targets, confirm \(\ell_k<u_k\), and verify that the mass in the continuing interval agrees with \(S_k-\Delta A_k-\Delta B_k\), up to numerical error. These checks are particularly useful when the density has been truncated to a finite state range: omitted tail mass can make a seemingly converged root wrong.
Worked Example: Calibrating From a Quadrature Density
Suppose a hypothetical numerical grid represents the survivor density at a look by a piecewise-constant approximation on \([-3,3]\). Its density is \(0.10\) on \([-3,0)\) and \(0.20\) on \([0,3]\), and is zero outside that range. This is a simple illustrative quadrature representation; its total survivor mass is \(0.10(3)+0.20(3)=0.900\). Let the requested lower and upper crossing increments be \(0.120\) and \(0.180\).
For a lower boundary \(\ell\) between \(-3\) and \(0\), the lower-tail integral is \(0.10(\ell+3)\). Setting it equal to \(0.120\) gives \(0.10(\ell+3)=0.120\), so \(\ell=-1.800\). For an upper boundary \(u\) between \(0\) and \(3\), the upper-tail integral is \(0.20(3-u)\). Setting it equal to \(0.180\) gives \(0.20(3-u)=0.180\), so \(u=2.100\).
The crossing increments sum to \(0.300\), less than the available \(0.900\), and the boundaries are correctly ordered. Integrating only between them gives \(0.10(1.800)+0.20(2.100)=0.180+0.420=0.600\). This equals \(0.900-0.120-0.180\), as required. A real calculation would use the density values and quadrature weights produced by the chosen numerical representation, rather than this deliberately simple piecewise-constant density.
Worked Example: Using Tabulated Tail Integrals to Find Roots
At another hypothetical look, suppose earlier cumulative targets are \(A_{k-1}=0.012\) and \(B_{k-1}=0.018\), while the new cumulative targets are \(A_k=0.035\) and \(B_k=0.050\). The required increments are therefore \(\Delta A_k=0.035-0.012=0.023\) and \(\Delta B_k=0.050-0.018=0.032\). Assume the arriving survivor mass is \(S_k=0.920\), so the combined increment \(0.055\) is feasible.
Suppose quadrature gives the following tail integrals near the roots:
| Boundary candidate | Upper-tail integral | Lower-tail integral |
|---|---|---|
| \(u=1.600\) | 0.030 | — |
| \(u=1.700\) | 0.020 | — |
| \(\ell=-1.600\) | — | 0.024 |
| \(\ell=-1.700\) | — | 0.036 |
Linear interpolation between the upper-tail values places the target \(0.023\) at \(u_k\approx1.600+[(0.030-0.023)/(0.030-0.020)](0.100)=1.670\). For the lower tail, the target \(0.032\) lies two-thirds of the way from \(0.024\) to \(0.036\), so \(\ell_k\approx-1.600-(2/3)(0.100)=-1.667\). These are interpolated numerical roots, not exact values; a more refined grid or a bracketed root solve can improve them.
The estimated continuing mass is \(0.920-0.023-0.032=0.865\). The upper and lower tail targets are distinct, and the two boundaries leave a nonempty interval. In a full sequential calculation, only the survivor density on approximately \([-1.667,1.670]\) is propagated through the next Gaussian transition.
Diagnostics, Alternatives, and Limitations
Separate calibration is useful when the design’s upper and lower crossing allocations differ. A symmetric-boundary approach imposes a structural constraint, often \(u_k=-\ell_k\) on the standardized scale, and therefore cannot generally satisfy two independently specified tail increments. Calibrating to cumulative targets, by contrast, is a way to state the design requirements across looks; the numerical task at each look still uses the increments left after earlier crossings. The two ideas work together rather than compete.
Several failures have recognizable signs. A root solver that returns a boundary outside its bracket may have been given a nonmonotone interpolation or a poorly chosen bracket. If the two computed boundaries are reversed, the requested increments may exceed the available survivor mass, or one or both tail integrals may have been evaluated with the wrong orientation. If the boundary residuals are small but the continuing mass does not match the expected remainder, check integration limits, truncation, and quadrature weights. Repeating the calculation on a refined grid helps distinguish numerical error from an infeasible target.
Calibration is only as accurate as the model and numerical integration. The Gaussian transition assumption must match the state process being analyzed; an incorrect information schedule or transition variance changes the survivor density and therefore the roots. Truncation can omit probability mass, and coarse grids can smooth or miss sharp changes in density near a boundary. Numerical convergence does not establish that the assumed statistical model is appropriate, and a calibrated boundary does not by itself guarantee desired operating characteristics under alternatives.
Check Your Understanding
Use the definitions and calculations above to answer these questions; concise solutions follow each question.
- At a look, the cumulative upper target rises from \(0.020\) to \(0.028\). What upper-crossing increment should be calibrated at this look?
- The arriving survivor mass is \(0.060\), and requested upper and lower increments are \(0.035\) and \(0.030\). Can both be met while leaving positive survivor mass? Explain.
- Why must the next-look density be integrated only over \([\ell_k,u_k]\)?
- A computation meets both tail targets, but its reported continuing mass is larger than \(S_k-\Delta A_k-\Delta B_k\). Name two numerical or implementation checks to make.
Solutions. (1) The increment is \(0.028-0.020=0.008\). (2) No: the increments sum to \(0.065\), which exceeds the available mass \(0.060\); the targets are infeasible. (3) Paths outside the interval have crossed a boundary and stopped, so only states inside it can continue. (4) Check that the interval endpoints and integration limits are correct, and inspect quadrature weights, truncation, and the accuracy of the tail integrals. The continuing mass should equal the arriving mass minus the two crossing increments, within numerical tolerance.