Adaptive Integration Near a Stopping Boundary

Why Boundary-Local Integration Matters

In a group-sequential calculation, a path contributes to later looks only if it has not crossed a stopping boundary. The survivor density is therefore integrated over a continuation region whose endpoints are the boundaries themselves. A small numerical error near an endpoint can change how much probability is assigned to stopping now versus surviving to a later look. Refining the whole state space uniformly can be wasteful, while leaving a boundary poorly resolved can distort the quantities of interest.

Adaptive integration addresses this imbalance: it places more quadrature work where the integrand changes rapidly or where an interval endpoint matters, and less where the integrand is smooth. The essential practice is not simply to request more precision from a routine. It is to represent the stopping boundary explicitly, split the integral there, and refine locally rather than allowing a quadrature rule to straddle a discontinuity.

Model and Survivor-Density Recursion

Use the canonical Gaussian group-sequential model. Let \(Z_i\) be the standardized statistic at look \(i\), and let \(0<t_1<\cdots<t_K\) be the information fractions. Under a drift parameter \(\theta\), assume the canonical joint normal model: \(E(Z_i)=\theta\sqrt{t_i}\) and \(\operatorname{Cov}(Z_i,Z_j)=\sqrt{t_i/t_j}\) for \(i\leq j\). Under the null, \(\theta=0\). The model assumes the specified information fractions and drift are appropriate and that the process has the stated Gaussian distribution.

These assumptions imply the Markov transition \(Z_i\mid Z_{i-1}=x\sim N\!\left(\sqrt{t_{i-1}/t_i}\,x+\theta(t_i-t_{i-1})/\sqrt{t_i},(t_i-t_{i-1})/t_i\right)\). For the first look, \(Z_1\sim N(\theta\sqrt{t_1},1)\). At look \(i\), let \(l_i\) and \(u_i\) be the lower and upper boundaries; a value in the continuation region \(C_i=(l_i,u_i)\) does not stop the trial at that look. An infinite endpoint represents the absence of a boundary on that side.

Define \(q_i(y)\) as the density at \(Z_i=y\) of paths that have survived every look through \(i\). It is an unconditional density: its integral over \(C_i\) is the probability of surviving through look \(i\), not one. The recursion integrates the previous survivor density against the conditional transition density and then retains only the current continuation region.

Here \(\phi\) is the standard normal density, \(f_i(y\mid x)\) is the normal transition density just specified, and \(\mathbf{1}_{C_i}\) is one inside \(C_i\) and zero outside. The limits of the integral are the previous look’s continuation boundaries; they are not arbitrary grid limits. Once \(q_i\) is available, the probability of crossing an upper boundary at look \(i\) is obtained by integrating the conditional upper-tail probability against \(q_{i-1}\). A lower crossing uses the corresponding conditional lower tail.

Where to Refine the Quadrature

There are two related integrations to distinguish. The survivor update computes \(q_i(y)\) by integrating over the previous survivor region. A stopping probability integrates over that same previous region, but its integrand includes a conditional tail probability. In both cases, the previous continuation endpoints should be explicit integration endpoints. Do not integrate across a region where the survivor density is abruptly set to zero and expect a smooth-integrand quadrature rule to behave as if no discontinuity were present.

A practical adaptive calculation proceeds as follows. First, express the integration interval using the exact boundary values. Next, evaluate the integrand on that interval and estimate the local quadrature error using an adaptive rule, such as an embedded Gauss–Kronrod rule. If the estimated error is too large, subdivide the interval; give particular attention to intervals adjacent to a boundary or to a region where the integrand has marked curvature. Finally, sum the accepted subinterval integrals. The embedded-rule difference is a useful refinement indicator, not an absolute guarantee of the true error.

Adaptive quadrature is especially useful when a boundary cuts through a coarse uniform grid, when survivor density is steep close to an endpoint, or when repeated recursive updates make grid spacing costly. It is not a substitute for a correct recurrence: if the transition model, boundary convention, or integration limits are wrong, a highly accurate integral will still give the wrong answer.

Conditional Crossing Integrals and Local Refinement

For an upper crossing at look \(i\), the conditional probability given \(Z_{i-1}=x\) is a normal tail. Write \(m_i(x)\) and \(v_i\) for the conditional mean and variance under the canonical transition. The stagewise upper-crossing probability is \(\int_{l_{i-1}}^{u_{i-1}}q_{i-1}(x)\{1-\Phi((u_i-m_i(x))/\sqrt{v_i})\}\,dx\). The lower-crossing probability replaces the braced term by \(\Phi((l_i-m_i(x))/\sqrt{v_i})\). These integrands are generally smoother than a density multiplied by a discontinuous indicator, but they still inherit any sharp variation in \(q_{i-1}\). Boundary-aligned integration and local refinement remain important.

Adaptive Integration Versus a Uniform Grid

A uniform grid is not inherently inaccurate. It can be reliable when its spacing is sufficiently fine, the boundaries are represented accurately, and results are stable under refinement. Adaptive integration is an alternative allocation of computational effort, not an automatic guarantee of superiority. For many recursive calculations, a useful comparison is to compute the same probability using a boundary-aligned adaptive rule and a successively refined uniform grid, then examine whether the two stabilize at a common value.

Derivation: Why Splitting at the Boundary Helps

Suppose a calculation is written over an interval containing a stopping boundary \(b\), with an integrand that equals a smooth function \(g(x)\) on the survivor side and zero on the stopped side. In a simple one-sided case, that integrand is \(g(x)\mathbf{1}(x<b)\). It has a jump at \(b\) whenever \(g(b)\ne0\). A quadrature rule that treats the whole interval as smooth may sample on both sides of the jump without locating it, so its usual smooth-function accuracy can be lost.

Instead, separate the integral at the known boundary: \(\int_a^c g(x)\mathbf{1}(x<b)\,dx=\int_a^b g(x)\,dx\), for \(a<b<c\). The integral on \([a,b]\) now has the smooth integrand \(g\), and no panel crosses the jump. The same logic applies to a continuation interval with two finite endpoints: integrate only from \(l_i\) to \(u_i\), and refine panels within that interval. This does not remove error from approximating a difficult smooth integrand, but it removes avoidable error from hiding a known discontinuity inside a panel.

There is also a simple sensitivity interpretation. If an endpoint is accidentally displaced by a small amount \(\varepsilon\), the resulting omitted or added mass is approximately \(g(b)\varepsilon\) when \(g\) is continuous at \(b\). Thus even an accurate interior quadrature cannot compensate for a misplaced boundary when the density near that boundary is appreciable. Boundary values should come from the analysis specification, not from rounding them to the nearest grid node.

Diagnostics and Limitations

Signs that boundary resolution may be inadequate include a reported stopping probability that changes materially when a grid is shifted, a large change under modest grid refinement, or a discrepancy between adaptive and uniform-grid calculations concentrated in the boundary-adjacent region. Also inspect whether the survivor mass responds plausibly when a boundary is moved slightly: moving an upper continuation limit inward should not increase the mass retained at that same look, all else equal. Such checks diagnose numerical behavior; they do not by themselves identify the source of an error.

• Do not smooth across a stopping rule. A discontinuity in the survivor indicator is a feature of the design. Split at its known location rather than approximating it with a smooth curve.
• Do not interpret quadrature tolerance as total numerical accuracy. The quadrature routine assesses its own integration approximation, not errors from truncating the state space, interpolating a density, or using rounded boundaries.
• Do not assume local refinement always saves time. If the integrand is smooth over a small, regular domain, a uniform grid may be simpler and equally effective.
• Do not confuse numerical stability with model validity. Stable values can still be wrong if the canonical model, drift, information fractions, or stopping boundaries do not represent the intended calculation.

Adaptive quadrature also has limits when the integrand is sharply peaked, has a narrow feature the initial evaluations miss, or is itself obtained from an inaccurate interpolation. In those cases, an error estimator can be falsely reassuring. Supply boundary endpoints explicitly, choose a reasonable initial partition, and compare with a method whose discretization differs. This tutorial’s comparison is about locating integration effort; a comprehensive accounting of all numerical error sources is a separate task.