Numerical Integration of Crossing Probabilities

Why Integrate Crossing Probabilities?

A group sequential design can stop when a test statistic crosses an efficacy boundary at any planned analysis. Under the null hypothesis \(H_0\), the probability of crossing at least one such boundary is the design’s cumulative Type I error. A boundary’s marginal tail probability at one look is not, by itself, this cumulative probability: the statistics at different looks are dependent. Numerical integration evaluates the joint probability needed to account for that dependence.

This tutorial concerns evaluating crossing probabilities for fixed boundaries and information times. It does not choose or calibrate the boundaries. The key computational task is to integrate the canonical joint normal distribution over the region where one or more boundaries are crossed.

Definition: For \(K\) planned analyses, let \(Z_i\) be the standardized test statistic at information time \(t_i\), where \(0<t_1<\cdots<t_K\). Under the canonical model and \(H_0\), the vector \((Z_1,\ldots,Z_K)\) is multivariate normal with mean zero, variances 1, and correlations \(\operatorname{Corr}(Z_i,Z_j)=\sqrt{t_i/t_j}\) for \(i<j\). For one-sided efficacy boundaries \(b_i\), the crossing event is \(\bigcup_{i=1}^K\{Z_i\ge b_i\}\).

Model and Assumptions

The canonical model is appropriate when the standardized statistics arise from a score or estimator process with independent increments across information. Under \(H_0\), the increments have mean zero, and standardizing the accumulated statistic at each information time gives unit variance. The analyses must use the stated information times, and the correlation structure must be a reasonable representation of the design. For example, the model may not describe the data adequately if information is mismeasured or the test statistics do not have the assumed joint normal distribution.

For \(i<j\), the later statistic contains the earlier information increment as part of its total. Under the canonical model, that shared information implies correlation \(\sqrt{t_i/t_j}\). In particular, for two looks at \(t_1\) and \(t_2\), write \(\rho=\sqrt{t_1/t_2}\). Then \((Z_1,Z_2)\) is bivariate standard normal with correlation \(\rho\), where \(0<\rho<1\). Let \(\phi\) and \(\Phi\) denote the standard normal density and cumulative distribution function.

Reducing a Two-Look Probability to an Integral

The probability of crossing both two-look boundaries is a joint upper-tail probability. A conditional-normal calculation makes it a one-dimensional integral that can be evaluated by numerical quadrature.

Condition on the first statistic.
Under the bivariate canonical model, \(Z_2\mid Z_1=z\sim N(\rho z,1-\rho^2)\).

Write the conditional crossing probability.
\(P(Z_2\ge b_2\mid Z_1=z)=\Phi((\rho z-b_2)/\sqrt{1-\rho^2})\).

Integrate over first-look crossing values.
Multiply by the density of \(Z_1\) and integrate from \(b_1\) to infinity.

P(Z_1\ge b_1,\;Z_2\ge b_2) = \int_{b_1}^{\infty} \phi(z)\, \Phi\!\left(\frac{\rho z-b_2}{\sqrt{1-\rho^2}}\right)\,dz.

The event of crossing at least one boundary is the union of the two crossing events. Inclusion-exclusion gives

P(Z_1\ge b_1\ \text{or}\ Z_2\ge b_2) = \{1-\Phi(b_1)\}+\{1-\Phi(b_2)\} - P(Z_1\ge b_1,\;Z_2\ge b_2).

Subtracting the joint probability is essential because observations that cross both boundaries are counted in each marginal tail. For \(K\) looks, the same principle applies to the multivariate normal vector with correlation matrix \(R\), where \(R_{ij}=\sqrt{t_i/t_j}\) for \(i<j\). The crossing probability is the integral of its joint density over the union of the boundary-crossing regions. In more than two dimensions, numerical evaluation may require multivariate normal integration or conditional integration over several variables.

Worked Example: Two Looks With Equal Boundaries

Suppose a hypothetical design has planned information times \(t_1=0.25\) and \(t_2=1\). Under \(H_0\), it uses one-sided efficacy boundaries \(b_1=b_2=2\). The canonical correlation is \(\rho=\sqrt{0.25/1}=0.5\). The assumptions are that the standardized statistics follow the canonical joint normal model and that the specified information times and boundaries apply.

The joint crossing probability is

P(Z_1\ge2,\;Z_2\ge2) = \int_2^\infty \phi(z)\, \Phi\!\left(\frac{0.5z-2}{\sqrt{0.75}}\right)\,dz.

For a numerical calculation, first integrate over \([2,4]\) with composite Simpson quadrature and step \(h=0.25\). For an even number \(n\) of subintervals, Simpson’s formula is \(S=(h/3)[f(z_0)+f(z_n)+4\sum f(z_i)+2\sum f(z_i)]\), where the first sum is over odd interior indices and the second is over even interior indices. Here \(n=8\). The integrand values at \(z=2,2.25,\ldots,4\), rounded, are \(0.00670, 0.00496, 0.00339, 0.00214, 0.00125, 0.000675, 0.000337, 0.000156,\) and \(0.0000669\). Substitution in Simpson’s formula gives approximately \(0.004036\) on \([2,4]\). Numerical integration of the remaining tail from 4 to infinity contributes about \(0.000018\), so the joint crossing probability is approximately \(0.00405\).

Each marginal crossing probability is \(1-\Phi(2)=0.022750\). Therefore,

P(\text{cross at either look}) \approx 0.022750+0.022750-0.00405 =0.04145.

Thus the null probability of crossing at least one of the two boundaries is about 0.04145, not the sum of the two marginal tail probabilities, 0.04550. The difference is the probability of crossing both boundaries.

Unequal Boundaries and First-Crossing Interpretation

The same integral applies when the boundaries differ. The joint probability is still subtracted from the sum of the marginal tails; only the integration limits and conditional tail change. The probability of a crossing at the second look after no crossing at the first can also be obtained by subtracting the joint crossing probability from the second-look marginal tail. This partition is useful for checking that the total crossing probability has been assembled consistently.

Worked Example: Unequal Boundaries

Consider a hypothetical two-look design with \(t_1=0.25\), \(t_2=1\), so \(\rho=0.5\), but with boundaries \(b_1=2.5\) and \(b_2=2\). Assume the canonical null model as in the previous calculation. Numerical integration gives a joint crossing probability of about \(0.00157\); for example, Simpson quadrature on \([2.5,4]\) with step \(0.25\), followed by numerical tail integration, gives this value to about three significant figures.

The marginal tails are \(1-\Phi(2.5)\approx0.006210\) and \(1-\Phi(2)\approx0.022750\). Hence the cumulative crossing probability is

0.006210+0.022750-0.00157\approx0.02739.

The probability of crossing the second boundary but not the first is approximately \(0.022750-0.00157=0.02118\). Adding that to the first-look crossing probability gives \(0.006210+0.02118=0.02739\), an independent arithmetic check on the inclusion-exclusion calculation. The first-look boundary is higher, so its marginal crossing probability is smaller; both contributions are still required to describe the chance of crossing at either analysis.

Quadrature Choices and Numerical Checks

Composite Simpson quadrature is convenient for a smooth, one-dimensional integrand when the interval is finite and divided into an even number of equal subintervals. Adaptive quadrature is often preferable when the integrand changes rapidly in one region or when a reliable error estimate is needed. Neither method removes the need to handle the infinite upper limit: integrate to a sufficiently large cutoff and bound or numerically evaluate the remaining tail.

For the equal-boundary example, Simpson quadrature with \(h=0.5\) on \([2,4]\) gives about \(0.00403\), while \(h=0.25\) gives about \(0.004036\). The coarser estimate is close to the finer one; it is not substantially larger. This agreement is a useful stability check, but agreement between two grids is not a formal guarantee of accuracy. Check the tail contribution separately and, where possible, compare against an independent method.

Worked Example: Checking the Tail and Grid

Use the equal-boundary example, whose joint crossing integral starts at \(z=2\). Integrating only to \(z=4\) gives approximately \(0.004036\). If the tail from 4 to infinity were simply discarded, the joint probability would be understated by about \(0.000018\), roughly 0.4% of the joint probability. Including that tail gives \(0.004036+0.000018=0.004054\), rounded to \(0.00405\).

As a separate check, use the \(h=0.5\) Simpson estimate, about \(0.00403\), on \([2,4]\), then add the same tail estimate. This gives about \(0.00405\) overall, consistent with the finer-grid result. Using this checked joint probability, the cumulative crossing risk remains \(0.045500-0.004054\approx0.04145\). Reporting more decimal places than the grid and tail checks justify would imply unwarranted numerical precision.

Diagnostics, Alternatives, and Limitations

A numerical result should be checked against the probability structure, not accepted merely because software returned a value. Useful diagnostics include:

Check probability bounds. The probability of crossing at least one boundary must be at least as large as the largest marginal crossing probability and no larger than the sum of the marginal probabilities.
Check the correlation matrix. It must be positive definite for a nondegenerate joint normal density. The canonical correlations should match the planned information times.
Check numerical stability. Refine the grid or use adaptive quadrature, vary the integration cutoff, and verify that the estimated probability changes by less than the precision you plan to report.
Check against simulation when useful. Simulated canonical normal paths provide an independent diagnostic, but Monte Carlo estimates have sampling error and may be imprecise for small crossing probabilities unless many paths are generated.

For two looks, conditional one-dimensional quadrature is transparent and usually efficient. A multivariate normal probability routine can be more convenient, especially for several looks, but its numerical tolerances and event definition still need checking. Simulation is flexible and can accommodate more complicated event regions, but it is generally less efficient for accurately estimating small tail probabilities. High-dimensional integration can become computationally demanding, and no numerical method repairs a misspecified information-time or joint normal model.

Common mistake: Adding the marginal probabilities of crossing at each look and calling the sum the cumulative Type I error. This counts paths that cross more than once multiple times. For two looks, subtract the joint crossing probability; for several looks, evaluate the probability of the union under the joint distribution rather than summing marginal tails.

Key takeaway: Numerical integration estimates the joint crossing probability under the canonical null model. Correct cumulative crossing risk depends on both the marginal tails and their dependence, and credible results require checks on quadrature, tail truncation, and the assumed correlation structure.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

For two canonical looks at \(t_1=0.36\) and \(t_2=1\), what is the null correlation between their standardized statistics?
Why does the two-look union probability subtract the joint crossing probability?
For equal boundaries of 2 in the first worked example, what is the approximate probability of crossing at the second look but not the first?
Name two checks that help assess whether a numerical crossing probability is reliable.

Solutions. (1) \(\rho=\sqrt{0.36/1}=0.6\). (2) Paths crossing both boundaries are included in each marginal tail, so subtracting the intersection counts them once. (3) The second-look marginal tail minus the joint probability is approximately \(0.022750-0.00405=0.01870\). (4) Examples include refining the quadrature grid, checking the omitted tail, varying the cutoff, and comparing with a multivariate normal calculation or simulation.