Recovering Crossing Probabilities From Survivor Mass

Why Track Probability Removed at Each Look?

A group sequential design can stop at several interim looks. At each look, some probability paths cross a stopping boundary; the remaining paths continue. The density used at a later look therefore describes only paths that have not already stopped. Keeping those two ideas separate—incoming probability mass and survivor probability mass—is essential for calculating crossing probabilities correctly.

This tutorial focuses on recovering those probabilities from the survivor-density recursion. The same accounting applies under a null or an alternative distribution, provided the densities and transition kernels correspond to the distribution being evaluated. A crossing probability is not automatically a type I error probability: it has that interpretation only when evaluated under the null and for the relevant rejection boundaries.

Definition: At look $j$, let $Z_j$ be the monitoring statistic, with lower and upper stopping boundaries $L_j$ and $U_j$. The continuation region is $C_j=(L_j,U_j)$. Let $h_j(y)$ be the incoming density at look $j$, before applying its boundaries, and let $g_j(y)$ be the survivor density, after restricting to paths that continue past look $j$. The lower- and upper-boundary crossing probabilities at look $j$ are $q_j^-$ and $q_j^+$, respectively.

The distinction between the two densities is operational. At the first look, the incoming density is the unrestricted density $f_1$ of $Z_1$. At a later look, the incoming density is formed by propagating the previous survivor density through the transition kernel. Only after calculating the mass outside $C_j$ do we restrict the incoming density to obtain $g_j$.

h_1(y)=f_1(y),\qquad q_1^-=\int_{-\infty}^{L_1} f_1(y)\,dy,\qquad q_1^+=\int_{U_1}^{\infty} f_1(y)\,dy, \qquad g_1(y)=f_1(y)\mathbf{1}\{y\in C_1\}.

For $j\geq 2$, let $K_j(y\mid x)$ be the transition density for $Z_j=y$ given $Z_{j-1}=x$. The recursion first propagates survivors to form $h_j$, then records the boundary-crossing mass, and finally retains only paths in the continuation region:

h_j(y)=\int_{C_{j-1}}g_{j-1}(x)K_j(y\mid x)\,dx,\qquad q_j^-=\int_{-\infty}^{L_j}h_j(y)\,dy,\qquad q_j^+=\int_{U_j}^{\infty}h_j(y)\,dy,\qquad g_j(y)=h_j(y)\mathbf{1}\{y\in C_j\}.

The densities $h_j$ and $g_j$ need not integrate to one. Their integrals are probability masses in the original experiment: $h_j$ is the probability of reaching look $j$, and $g_j$ is the probability of reaching that look and continuing past it. Normalizing either density to integrate to one would instead describe a conditional distribution among survivors; that distribution can be useful, but its crossing probabilities must then be multiplied by the probability of survival to the look.

Stage-Specific and Cumulative Probabilities

The quantities $q_j^-$ and $q_j^+$ are unconditional stage-specific probabilities: each is the probability, from the start of the monitoring process, of first crossing the corresponding boundary at look $j$. They are not conditional probabilities given that look $j$ is reached. Write $S_j$ for the probability of surviving past look $j$, and set $S_0=1$. The cumulative crossing probability through look $j$ is the sum of all stage-specific crossing probabilities through that look.

S_j=\int_{C_j}g_j(y)\,dy,\qquad Q_j^-=\sum_{k=1}^{j}q_k^-,\qquad Q_j^+=\sum_{k=1}^{j}q_k^+,\qquad Q_j=Q_j^-+Q_j^+=\sum_{k=1}^{j}(q_k^-+q_k^+).

For a continuous state distribution, the probability of landing exactly on a boundary is zero, so the open continuation interval and the tail integrals give a complete partition. If a model has point mass on a boundary, the design must specify how equality is classified, and that mass must be assigned consistently.

Probability accounting identity: At every look $j$, the cumulative probability of crossing either boundary by that look plus the probability of surviving past that look equals one: $$ Q_j+S_j=1. $$ Equivalently, the incoming mass at look $j$ is partitioned into lower crossing, upper crossing, and continuation mass: $$ \int_{-\infty}^{\infty}h_j(y)\,dy=q_j^-+q_j^++S_j. $$

Why the Accounting Identity Holds

The identity follows from two facts: the transition kernel preserves probability, and the three regions at a look divide the state space. For the first look, $f_1$ integrates to one. At any later look, $K_j(\,\cdot\mid x)$ integrates to one for each $x$, so propagating the survivor density does not change its total mass. The boundary tails and continuation region then partition that incoming mass.

\int_{-\infty}^{\infty}h_j(y)\,dy = \int_{C_{j-1}}g_{j-1}(x) \left[\int_{-\infty}^{\infty}K_j(y\mid x)\,dy\right]dx = \int_{C_{j-1}}g_{j-1}(x)\,dx =S_{j-1}.

The partition at look $j$ gives $S_j=S_{j-1}-q_j^--q_j^+$. Starting from $S_0=1$, repeatedly substituting this relation yields $S_j=1-\sum_{k=1}^{j}(q_k^-+q_k^+)=1-Q_j$, which proves the identity. This argument assumes a proper initial density, probability-preserving transition kernels, and a complete partition into the two crossing regions and continuation region at each look.

S_j=S_{j-1}-(q_j^-+q_j^+) =1-\sum_{k=1}^{j}(q_k^-+q_k^+) =1-Q_j.

Worked Probability Accounting

The examples below are hypothetical. Their purpose is to show how to interpret and check probability mass, not to describe a real trial. Numerical incoming masses in the later-look examples are stipulated as results of integrating the appropriate incoming density; the emphasis here is the accounting once those masses are available.

Worked Example: One Look With Standard Normal Boundaries

Suppose there is one look and $Z_1\sim N(0,1)$. Use lower and upper boundaries $L_1=-1.96$ and $U_1=1.96$. The model assumption is that the unrestricted statistic has the stated standard normal distribution under the distribution being evaluated; there are no earlier looks or survivor restrictions.

The incoming density is $f_1$, the standard normal density. By symmetry, each tail probability is approximately $0.0250$:

q_1^-=\int_{-\infty}^{-1.96}f_1(y)\,dy =\Phi(-1.96)\approx0.0250,\qquad q_1^+=\int_{1.96}^{\infty}f_1(y)\,dy =1-\Phi(1.96)\approx0.0250.

The continuation probability is therefore $S_1=1-0.0250-0.0250=0.9500$, to the displayed precision. The check is $Q_1+S_1=0.0500+0.9500=1.0000$. These are probabilities, not percentages; expressed as percentages, about 5.00% crosses either boundary at this look and about 95.0% continues.

Worked Example: Two Looks and Cumulative Tail Probabilities

Consider two looks with continuous statistic distributions and valid transition densities. At the first look, suppose the boundaries are $-2.00$ and $2.00$, and the incoming distribution is standard normal. At the second look, suppose integration of the incoming density $h_2$, obtained by propagating the first-look survivors, gives lower crossing mass $0.0120$, upper crossing mass $0.0180$, and continuation mass $0.9245$. These stipulated masses are internally checked below.

At look 1, standard normal tail probabilities are approximately $0.02275$ each. Thus $S_1=1-0.02275-0.02275=0.9545$ to four significant figures. The total incoming mass at look 2 must equal this survivor mass. The stated second-look components satisfy that requirement:

$$ 0.0120+0.0180+0.9245=0.9545. $$

The cumulative lower crossing probability through look 2 is the lower crossing probability at look 1 plus the lower crossing probability at look 2:

$$ Q_2^-=0.02275+0.0120=0.03475. $$

Likewise, $Q_2^+=0.02275+0.0180=0.04075$. The total cumulative crossing probability is $Q_2=0.03475+0.04075=0.07550$. Including the second-look survivor mass gives $Q_2+S_2=0.07550+0.9245=1.0000$. In context, 7.55% of paths cross a boundary by the second look under the specified model, and 92.45% survive past that look.

Worked Example: Conditional Crossing Rate Versus Stage Probability

Suppose the survivor mass before a particular look is $S_{j-1}=0.8000$. Integrating the incoming density at that look gives lower crossing mass $q_j^-=0.0100$, upper crossing mass $q_j^+=0.0300$, and continuation mass $S_j=0.7600$. The total mass checks because $0.0100+0.0300+0.7600=0.8000$.

The stage-specific probability of crossing either boundary at this look is $0.0100+0.0300=0.0400$, an unconditional probability measured from the start of the process. Among paths that actually reach this look, the conditional probability of crossing is instead:

P(\text{cross at look }j\mid\text{reach look }j) =\frac{q_j^-+q_j^+}{S_{j-1}} =\frac{0.0400}{0.8000}=0.0500.

Thus 4.00% of all paths cross at this look, while 5.00% of paths that reach it cross. If cumulative crossing probability before this look was $0.2000$, then it becomes $0.2000+0.0400=0.2400$; the remaining mass is $1-0.2400=0.7600$. The distinction matters when reporting stagewise risks: a conditional rate must not be added directly to previous unconditional probabilities.

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to check mass at every look, not only at the end. The incoming mass at look $j$ should equal $S_{j-1}$; the lower crossing, upper crossing, and survivor masses should sum to that incoming mass; and cumulative crossing mass plus survivor mass should equal one. Small discrepancies can arise from rounding, but large or systematic discrepancies indicate a calculation or implementation error.

Using the survivor density to calculate the current crossing tails. The survivor density is zero outside the continuation region, so integrating it beyond the current boundaries gives zero rather than the crossing mass. Use $f_1$ at the first look and the incoming density $h_j$ at later looks for crossing integrals; apply the continuation restriction only to obtain $g_j$.
Adding conditional rates as if they were unconditional probabilities. A conditional crossing rate among paths reaching a look must be multiplied by the probability of reaching that look to obtain its unconditional stage-specific probability.
Counting paths more than once. Later incoming distributions must be formed from survivors only. Propagating the unrestricted distribution would include paths that already crossed and stopped.
Forgetting a side or a remaining mass. In a two-sided design, account for both lower and upper crossing probabilities. If the final continuation region is not a stopping decision, its mass remains part of survivor probability rather than being silently assigned to a crossing.

Survivor-mass accounting is one way to obtain crossing probabilities. An equivalent conceptual route is to calculate each first-crossing event directly as a joint probability: the statistic lies in continuation regions at all earlier looks and crosses the relevant boundary at the current look. The survivor recursion organizes those same events sequentially and makes the remaining mass explicit. Either route requires the correct joint or transition model; stagewise marginal distributions alone do not generally determine sequential crossing probabilities.

The identity checks probability conservation, not the correctness of the model or numerical integration. A wrongly specified transition kernel can still conserve mass perfectly. The accounting also does not by itself establish that a design meets a target error rate, nor does a total crossing probability identify how crossings divide between lower and upper boundaries. Those interpretations require the relevant model, hypotheses, and boundary meanings to be stated.

Key takeaway: Calculate crossing mass from the unrestricted incoming density at each look, then restrict that density to form the survivor density. Stage-specific probabilities add to cumulative crossing probabilities, and cumulative crossings plus remaining survivor mass must equal one.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

At look 1, a continuous statistic has density $f_1$ and continuation region $(L_1,U_1)$. Which density is used to calculate the lower crossing probability, and what is its integral expression?
At a look, the incoming mass is $0.70$, the lower crossing mass is $0.02$, and the upper crossing mass is $0.03$. Find the survivor mass and verify the local accounting identity.
A look has unconditional crossing probability $0.06$, and $0.75$ of all paths reach it. Find the conditional crossing probability among paths reaching the look.
Why is it incorrect to calculate current-look crossing mass by integrating the current survivor density outside the continuation region?

Solutions. (1) Use the unrestricted density $f_1$: $q_1^-=\int_{-\infty}^{L_1}f_1(y)\,dy$. (2) The survivor mass is $0.70-0.02-0.03=0.65$; the components sum to $0.02+0.03+0.65=0.70$. (3) The conditional probability is $0.06/0.75=0.0800$, or 8.00%. (4) The survivor density has already been set to zero outside the continuation region; the crossing mass must be calculated from the incoming density before that restriction.