A Three-Look Recursive Integration Example

Why a Three-Look Calculation Requires Two Updates

At a sequential analysis, a path that reaches the third look has already avoided stopping at the first two. Consequently, the stage-three crossing probability is not simply the probability that the third statistic exceeds its boundary. It is the probability of exceeding that boundary among paths that survived both earlier looks. The survivor-density recursion carries precisely that restriction forward.

Here we calculate stagewise probabilities for an upper-boundary design under the null hypothesis. The goal is to show how the two successive Gaussian transitions are used and how the resulting crossing probabilities can be checked. The earlier material on Gaussian transition kernels and survivor-density recursion supplies the general framework; the focus here is carrying it through three looks and interpreting the calculations.

Assumptions and notation: Let \(Z_k\) be the standardized statistic at look \(k\), at cumulative information \(t_k\), where \(0<t_1<t_2<t_3\). Under \(H_0\), assume the canonical Gaussian model: the underlying information process has independent Gaussian increments, and \(Z_k\) is that process divided by the square root of \(t_k\). Thus the statistics are jointly normal, have mean zero and variance one, and \( \operatorname{Corr}(Z_j,Z_k)=\sqrt{t_j/t_k} \) for \(j<k\). The upper stopping boundaries \(b_1,b_2,b_3\) are fixed. A path stops for efficacy at look \(k\) if it has not crossed an earlier boundary and \(Z_k\ge b_k\). Boundary equality has probability zero under this continuous model.

The Two Transition Steps

Under these assumptions, the conditional distribution of the next standardized statistic, given the current value \(x\), is normal. For adjacent looks \(k-1\) and \(k\), define \(a_k=\sqrt{t_{k-1}/t_k}\) and \(s_k=\sqrt{1-t_{k-1}/t_k}\). Then the transition density is \(K_k(y\mid x)=\phi((y-a_kx)/s_k)/s_k\), where \(\phi\) is the standard normal density. The mean \(a_kx\) and variance \(s_k^2\) follow from the Gaussian independent-increment model.

Let \(g_k(y)\) denote the unnormalized density of \(Z_k=y\) among paths that have survived through look \(k\). Its integral is the probability of surviving through that look, not one. For a one-sided upper boundary, the first update is

g_1(y)=\phi(y)\mathbf{1}(y<b_1).

At each later look, transition from the previous survivor density and retain only values below the current boundary:

g_k(y)=\mathbf{1}(y<b_k)\int_{-\infty}^{b_{k-1}}g_{k-1}(x)K_k(y\mid x)\,dx.

The probability of first crossing at look \(k\) is the mass that transitions to \(y\ge b_k\):

q_k=\int_{-\infty}^{b_{k-1}}g_{k-1}(x) \left[1-\Phi\left(\frac{b_k-a_kx}{s_k}\right)\right]dx,\qquad k=2,3.

Here \(\Phi\) is the standard normal cumulative distribution function. At the first look, \(q_1=1-\Phi(b_1)\). These \(q_k\) are stagewise first-crossing probabilities under \(H_0\), not conditional probabilities given survival to the look. A conditional probability among survivors would divide \(q_k\) by the probability of reaching that look without crossing.

Worked Example: Exact Three-Look Crossing Probabilities at Zero

Consider the hypothetical design \(t_1=1/3,\ t_2=2/3,\ t_3=1\), with \(b_1=b_2=b_3=0\). Calculate the probability of first crossing at each look under \(H_0\).

At look 1, \(Z_1\sim N(0,1)\), so \(q_1=P(Z_1\ge0)=1-\Phi(0)=0.5\).

For the transition from look 1 to look 2, \(a_2=1/\sqrt{2}\) and \(s_2=1/\sqrt{2}\). Given \(Z_1=x\), the probability of crossing zero at look 2 is \(1-\Phi((0-x/\sqrt{2})/(1/\sqrt{2}))=\Phi(x)\). Only paths with \(x<0\) reach that transition, so

q_2=\int_{-\infty}^{0}\phi(x)\Phi(x)\,dx =\left.\frac{\Phi(x)^2}{2}\right|_{-\infty}^{0} =\frac{(0.5)^2}{2}=0.125.

Thus 0.125, or 12.5%, of null paths cross for the first time at look 2. The survivor mass after look 2 is \(1-q_1-q_2=1-0.5-0.125=0.375\).

For look 3, the two transition parameters are \(a_2=1/\sqrt{2}\), \(s_2=1/\sqrt{2}\), followed by \(a_3=\sqrt{2/3}\), \(s_3=1/\sqrt{3}\). The recursion gives

q_3=\int_{-\infty}^{0}\phi(x) \int_{-\infty}^{0}K_2(y\mid x) \left[1-\Phi\left(\frac{0-\sqrt{2/3}\,y}{\sqrt{1/3}}\right)\right]dy\,dx.

This integral can be checked exactly using the joint Gaussian signs. For standard bivariate normal variables with correlation \(\rho\), the probability of different signs is \(\arccos(\rho)/\pi\), so the expected product of their signs is \(2\arcsin(\rho)/\pi\). Expanding the indicator of the sign pattern \(Z_1<0,Z_2<0,Z_3>0\), and using the zero expectations of individual signs and of products of three signs, gives

q_3=\frac18+\frac{1}{4\pi} \left[ \arcsin(\rho_{12})-\arcsin(\rho_{13})-\arcsin(\rho_{23}) \right].

The correlations are \(\rho_{12}=1/\sqrt{2}\), \(\rho_{13}=1/\sqrt{3}\), and \(\rho_{23}=\sqrt{2/3}\). Their inverse sines are approximately \(0.7854,\ 0.6155,\ 0.9553\), respectively. Therefore \(q_3=0.125+(0.7854-0.6155-0.9553)/(4\pi)=0.06250\). The total probability of crossing by look 3 is \(0.5+0.125+0.06250=0.6875\), leaving survivor mass \(0.3125\).

Reading the Transition Kernels

The kernel describes a conditional transition from a particular current state; it is not itself a stagewise crossing probability. Integrating that conditional crossing probability against the unnormalized density of survivors is what produces the stagewise probability. The distinction is especially useful when inspecting an implementation: at the same current boundary and information times, a more negative current statistic generally has a smaller chance of crossing at the next look.

Worked Example: Conditional Crossing Probabilities at Look 3

Use the hypothetical information times \(t_2=2/3,\ t_3=1\), and an upper boundary \(b_3=0\). Suppose a path has survived to look 2 with \(Z_2=x=-1\). The next-look conditional mean is \(\sqrt{2/3}(-1)=-0.8165\), and the conditional standard deviation is \(\sqrt{1/3}=0.5774\). Hence

P(Z_3\ge0\mid Z_2=-1) =1-\Phi\left(\frac{0-(-0.8165)}{0.5774}\right) =1-\Phi(1.4142) \approx0.07865.

At \(Z_2=0\), the same conditional crossing probability is \(0.5000\). At \(Z_2=-2\), the standardized threshold is \(2.8284\), giving a crossing probability of about \(0.00234\). These are conditional probabilities at specified states. None is \(q_3\): to obtain \(q_3\), average the conditional probability over the look-two survivor density \(g_2(x)\), as in the recursive integral.

A Numerical Integration Check

In practice, the nested integral is evaluated numerically. The first update forms the look-two survivor density from the look-one survivors. The second update uses that density, not the original standard normal density, to obtain the look-three crossing mass. A quadrature rule approximates each integral, while the indicator restrictions enforce survival at the earlier looks and non-crossing at the current look for the updated density.

Worked Example: A Deliberately Coarse Midpoint Approximation

For the zero-boundary design above, the exact look-three probability is \(0.06250\). To see what a nested numerical calculation is integrating, approximate only the part of its double integral over \([-2,0]\times[-2,0]\) with one midpoint cell in each dimension. Each cell has width 2 and midpoint \(-1\). This deliberately crude calculation omits the rest of the integration region.

At \(x=-1\), the look-one density is \(\phi(-1)=0.2420\). The look-one-to-look-two transition density at \(y=-1\) is

K_2(-1\mid-1) =\frac{1}{\sqrt{2\pi}\sqrt{1/2}} \exp\left[-\frac{(-1+1/\sqrt{2})^2}{2(1/2)}\right] \approx0.5178.

The conditional probability of crossing at look 3 from \(y=-1\) is \(0.07865\), as calculated above. The one-cell midpoint approximation to the restricted double integral is therefore \(2\cdot2\cdot0.2420\cdot0.5178\cdot0.07865\approx0.0394\). This is not a reliable estimate of the full \(q_3\): it uses only one cell and omits values below \(-2\). Its purpose is to make the factors in the recursive integral explicit. A production calculation needs suitable quadrature over the full survivor region and numerical checks.

Diagnostics, Alternatives, and Limitations

Several checks help distinguish an integration error from an interpretation error. First, every \(q_k\) must lie between zero and one. Second, cumulative crossing probabilities cannot decrease: \(q_1\le q_1+q_2\le q_1+q_2+q_3\). Third, after the third look, the remaining survivor mass plus all stagewise crossing masses must equal one, up to numerical error:

q_1+q_2+q_3+\int_{-\infty}^{b_3}g_3(y)\,dy=1.

For the exact example, the terms are \(0.5000+0.1250+0.06250+0.3125=1.0000\). A mismatch can indicate that a boundary restriction was omitted, that crossing mass was counted more than once, or that the numerical integration is too coarse.

A direct multivariate normal probability calculation can evaluate the same joint event, and simulation can estimate it by generating many Gaussian paths and recording the first crossing look. These provide useful checks or alternatives, but the recursive integration makes the survivor restriction explicit and returns survivor densities that can be reused in later calculations. Simulation has sampling error; direct multivariate integration can be less transparent about stagewise survivor states. The recursion also has limitations: its result depends on the assumed Gaussian transition model, specified information times, and fixed boundaries. Approximation error can be substantial if the integration region or density is represented too coarsely.

A common mistake is to use \(P(Z_3\ge b_3)\) as the look-three first-crossing probability. That probability includes paths that may already have crossed at looks 1 or 2. The correction is to integrate the look-three conditional crossing probability against \(g_2(x)\), which contains only paths that survived both earlier looks. Another error is to renormalize \(g_2\) to integrate to one and then report a conditional crossing probability as though it were \(q_3\). If a normalized survivor distribution is used, its conditional crossing probability must be multiplied by the probability of reaching look 3 without a prior crossing.

Key takeaway: A three-look first-crossing calculation requires two successive survivor-density updates. Keep the survivor density unnormalized, integrate each new crossing mass only over paths that reached that look, and check that crossing and final-survivor masses sum to one.

Check Your Understanding

Use the definitions and calculations above. Answers are included so that each result can be checked.

For upper boundaries, why does \(g_2(y)\) include an indicator restricting \(y<b_2\)?
Solution: \(g_2\) represents paths that survive through look 2. Values \(y\ge b_2\) have crossed at look 2 and belong in \(q_2\), not in the look-two survivor density.
In the zero-boundary example, what is the survivor mass after look 1?
Solution: \(1-q_1=1-0.5000=0.5000\).
For \(t_2=2/3,\ t_3=1,\ b_3=0\), what is the conditional probability of crossing at look 3 from \(Z_2=0\)?
Solution: The conditional mean is zero and the conditional variance is \(1/3\). Thus \(P(Z_3\ge0\mid Z_2=0)=1-\Phi(0)=0.5000\).
In the exact zero-boundary example, what is the cumulative probability of crossing by look 2, and what is the survivor mass after look 2?
Solution: The cumulative probability is \(q_1+q_2=0.5000+0.1250=0.6250\). The survivor mass is \(1-0.6250=0.3750\).
A calculation reports \(q_3=0.40\) and final survivor mass \(0.20\) for the zero-boundary example. What accounting check fails?
Solution: The reported terms sum to \(q_1+q_2+q_3+\text{survivor}=0.5000+0.1250+0.40+0.20=1.225\), not 1. The stage-three crossing mass or survivor mass, or both, is inconsistent with probability conservation.