Why Convergence Must Be Checked
A recursive calculation of a group sequential crossing probability replaces continuous integrals with numerical approximations. It therefore returns an estimate, not automatically the exact probability. A result can look plausible while depending materially on the chosen state-grid spacing, the quadrature order, or both. Refinement checks ask whether the reported digits persist as these numerical resolutions are increased.
The goal is not to find a grid so fine that every numerical issue disappears. The practical goal is to show that the probability of interest is stable to changes in the discretization that are large enough to reveal relevant numerical error. Stability is evidence of accuracy, not a proof of it: different approximations can share a bias, and other numerical choices must remain controlled during the comparison.
What Is Being Refined?
Consider a standardized Gaussian statistic \(Z_j\) at look \(j\), observed at information \(t_j\), with \(0<t_1<\cdots<t_K\). Under the canonical null model, the increments of the unstandardized process are independent Gaussian increments. Consequently, for \(j>1\),
Here the second argument of \(N(\mu,\sigma^2)\) is the variance. The model assumes the stated information times, the canonical Gaussian increment structure, and boundaries evaluated on the same standardized scale. At each look, the continuation region is the set of states that have not crossed a stopping boundary. A survivor density records the density of paths that have remained in that region through the current look.
The recursive update integrates the preceding survivor density against the Gaussian transition density, retaining only states in the next continuation region. For an upper-boundary illustration, if continuation at look \(j\) means \(y<b_j\), the update has the form
The probability of crossing by a specified look is one minus the survivor probability at that look, provided the process starts with total probability one and all stopping events of interest are included. Numerical implementations approximate the integral and represent its domain using state cells. Two main resolutions can be varied: the grid spacing \(h\), which controls how finely the state axis is divided, and the quadrature order \(m\), which controls how many integration nodes are used. Boundary interpolation, state-space truncation, and all other settings should be held fixed when isolating the effect of either resolution.
Why Refinement Should Change the Estimate
A state grid replaces a continuous region by a finite representation. In a rectangular-cell method, for example, the density or mass assigned to a cell stands in for behavior across the whole cell. Reducing \(h\) narrows that representation and can better resolve rapid density changes or the location of a boundary. The approximation error depends on the method and the smoothness of the integrand; it is not generally guaranteed to be proportional to \(h\), nor to decrease monotonically at every refinement.
Quadrature approximates an integral over a specified interval or collection of intervals. A larger order typically gives a more accurate approximation for a smooth integrand, but performance can deteriorate when the integrand changes sharply or when a stopping boundary creates a discontinuity inside an integration interval. Splitting an integral at a boundary, or using the boundary interpolation method consistently, can matter as much as adding nodes. Increasing \(m\) cannot repair a state grid that represents the continuation region poorly; reducing \(h\) cannot repair inaccurate integration on each cell.
This explains the basic convergence strategy: first vary one resolution at a time to identify sensitivity, then assess combinations of finer grids and higher quadrature orders. Keep the model, boundaries, information times, truncation limits, and boundary treatment unchanged during these comparisons. If truncation limits also change, the change in the estimate no longer isolates grid or quadrature error.
A Short Justification for the Two-Axis Check
Let \(P\) denote the target crossing probability, and write the computed value as \(\widehat P(h,m)\). As a useful conceptual decomposition, write
This is an error-accounting description, not a claim that the components are independent or exactly separable. The quadrature error can depend on the grid because changing the grid changes the intervals and integrands being integrated. Holding \(m\) fixed while reducing \(h\) mainly probes grid sensitivity; holding \(h\) fixed while increasing \(m\) mainly probes quadrature sensitivity. Comparing both axes helps expose interactions that a one-dimensional refinement sequence can conceal. For instance, a coarse grid and low quadrature order could produce errors of opposite signs that happen to cancel, making one estimate appear unusually accurate.
For a smooth integrand, a consistent quadrature rule often approaches a limiting integral as its order increases. Similarly, a consistent cell approximation should approach the continuous-domain calculation as \(h\) decreases. Those expectations justify refinement as a diagnostic. They do not guarantee a particular rate or monotone sequence, especially near boundaries or truncated tails. A stable sequence across meaningful changes is therefore stronger evidence than a single very fine calculation, but it remains a numerical check rather than a formal error bound.
Worked Examples
Worked Example: Isolating State-Grid Sensitivity
Suppose a hypothetical two-look upper-boundary calculation uses information times \(t_1=40\) and \(t_2=100\), with the same standardized boundaries and the same truncation limits in every run. At the second look, the transition from a first-look state \(z\) has conditional mean and variance
Thus the conditional transition is \(N(\sqrt{0.4}\,z,0.6)\). The variance is \(0.6\), not \(0.4\); confusing the information fraction with the complementary variance would change the model being integrated. Imagine that the same implementation, with quadrature order fixed at \(m=32\), gives the following hypothetical estimates of the probability of crossing by the second look.
| Grid spacing \(h\) | Estimated crossing probability |
|---|---|
| 0.200 | 0.04431 |
| 0.100 | 0.04379 |
| 0.050 | 0.04355 |
| 0.025 | 0.04348 |
The successive absolute changes are
The changes shrink as the grid is refined, and the final two estimates agree to about \(0.0001\). This is evidence that the reported probability is settling near \(0.0435\) under this particular quadrature setting. It does not establish that every digit is correct: quadrature error, truncation error, or a bias shared across grid resolutions could remain. The right conclusion is that grid sensitivity appears small at the finest two spacings tested, not that the exact probability has been proved to equal \(0.04348\).
Worked Example: Isolating Quadrature-Order Sensitivity
Continue with the same hypothetical boundaries, information times, and truncation limits. Now fix the state-grid spacing at \(h=0.025\) and vary only the quadrature order. Suppose the estimates are as follows.
| Quadrature order \(m\) | Estimated crossing probability |
|---|---|
| 4 | 0.04402 |
| 8 | 0.04370 |
| 16 | 0.04351 |
| 32 | 0.04348 |
The changes from one order to the next are \(0.00032\), \(0.00019\), and \(0.00003\), respectively. In particular,
At this fixed grid, the estimate appears stable to the displayed precision at orders 16 and 32. That finding does not show that the grid itself is adequate; the first example was needed to assess that separate source of sensitivity. Also, an order increase is informative only if the integrals are being evaluated on appropriately defined intervals. If a boundary discontinuity lies inside a quadrature interval, high order can give a misleading appearance of precision without resolving the discontinuity.
Worked Example: Checking for an Interaction Between Resolutions
A final hypothetical check compares combinations of grid spacing and quadrature order for the same calculation. The estimates below illustrate why checking only one resolution can be insufficient.
| Grid spacing \(h\) | Order \(m=8\) | Order \(m=32\) |
|---|---|---|
| 0.100 | 0.04402 | 0.04379 |
| 0.025 | 0.04370 | 0.04348 |
| 0.0125 | 0.04365 | 0.04346 |
At \(h=0.100\), increasing the order changes the estimate by \(0.00023\). At \(h=0.025\), the same order increase changes it by \(0.00022\). At order 32, reducing \(h\) from \(0.025\) to \(0.0125\) changes the estimate by \(0.00002\). The finest-grid, higher-order result is \(0.04346\), close to the preceding finer-grid result \(0.04348\). Together, these comparisons support reporting approximately \(0.0435\), with the numerical precision chosen to match the observed stability.
The table does not show a dramatic interaction, but it still checks for one: it demonstrates that the apparent grid convergence is not merely an artifact of a low quadrature order. If estimates had varied substantially across the table, the calculation would need further investigation before a stable probability could be claimed.
Diagnostics, Alternatives, and Limitations
A useful diagnostic is a table of estimates, not just the final value. Record \(h\), \(m\), the truncation limits, and the resulting probability. Look at absolute changes between neighboring refinements and at whether the same digits persist across different combinations. A difference small relative to the precision needed for the design calculation may be practically acceptable; a difference large enough to affect the conclusion is not. There is no universal tolerance independent of the use of the probability.
Warning signs include estimates that continue to drift at the finest resolution, large changes when \(m\) is increased at fixed \(h\), unexpected oscillation, or results that depend strongly on whether a boundary falls near a cell edge. Nonmonotone estimates do not by themselves prove a failure: approximations can approach a limit from alternating directions. But persistent oscillation or large irregular changes call for checks of implementation, boundary handling, quadrature intervals, and truncation. A convergence check cannot distinguish every coding error from a numerical error; independent verification of the recurrence and probability accounting is still needed.
The main alternatives are not substitutes for a two-axis check. Using a finer grid alone is cheaper to interpret but can leave quadrature error hidden. Using a higher quadrature order alone can leave state-cell error hidden. Relying on one exceptionally fine calculation is difficult to assess because it offers no comparison and may simply be expensive rather than accurate. A coarse calculation may be useful for a quick exploratory result, but it should not be mistaken for a converged estimate. Numerical methods that adapt integration to difficult regions can be useful in suitable problems, but their own tolerances and boundary treatment must also be checked.
Refinement has limits. It can be computationally costly, and a stable sequence may still share bias from an incorrect transition variance, an inadequate state-space range, or a misapplied boundary. This check concerns discretization and quadrature sensitivity; it does not validate the statistical model, the stopping rule, or the assumed information times. Keep the scope of the conclusion precise: “stable across the tested grid and quadrature refinements” is justified by the evidence; “exact” is not.
A Common Mistake
Practical Refinement Sequence
Specify the probability, model, information times, boundaries, truncation limits, and boundary treatment. Record all settings that must stay fixed.
Run at several decreasing values of \(h\) with quadrature order \(m\) held fixed. Compare successive probability estimates.
At a sufficiently fine fixed grid, increase \(m\). Check whether the estimate is stable to the precision needed.
Compare more than one fine-grid, high-order combination. If results still drift materially, report the calculation as unresolved and investigate the cause.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- For information times \(t_{j-1}=25\) and \(t_j=100\), state the conditional mean and variance of \(Z_j\) given \(Z_{j-1}=z\).
- A calculation changes from \(0.0312\) to \(0.0311\) when quadrature order increases, but no grid refinement is performed. What has been checked, and what remains unknown?
- Why might a higher quadrature order fail to resolve an integral accurately when a stopping boundary lies inside its integration interval?
- Two successive grid refinements produce estimates that move in opposite directions, with smaller changes each time. Does this alone rule out convergence? Explain.
- Give one reason that a stable estimate across grid and quadrature refinements may still be inaccurate.
Solutions. 1. The information fraction is \(25/100=0.25\), so the conditional mean is \(\sqrt{0.25}\,z=0.5z\), and the variance is \(1-0.25=0.75\). 2. Quadrature sensitivity has been checked at the chosen grid; grid sensitivity remains unknown. 3. The boundary can create a discontinuity or sharp change within the interval, undermining the smooth-integrand behavior on which high-order quadrature relies. 4. No. Nonmonotone estimates can still approach a limit; the decreasing size of the changes is useful evidence, though more checks may be needed. 5. For example, all runs may use truncation limits that omit relevant probability mass, or they may share an incorrectly specified transition variance.