First-Passage Events In Opposite Tails

Why First Passage Matters

In two-sided group sequential monitoring, a test statistic is checked at several planned looks. At each look, it may cross an upper efficacy boundary, cross a lower efficacy boundary, or remain in the continuation region. The key question for the stopping decision is not simply whether a path ever reaches a boundary: it is which boundary it reaches first. A path that would eventually reach both tails has only one first stopping decision under a rule that stops at the first crossing.

This distinction is especially important when describing paths that cross one tail early and the opposite tail later. After the first efficacy crossing, the trial stops, so the later statistic is not observed under that rule. The later crossing can still be described on a hypothetical extension of the statistic path, but it is not a second observed stopping outcome.

Definition: At look $k$, let $Z_k$ be the monitored test statistic, with lower boundary $l_k$ and upper boundary $u_k$, where $l_k < u_k$. A lower crossing occurs when $Z_k \leq l_k$; an upper crossing occurs when $Z_k \geq u_k$. A look is a continuation look when $l_k < Z_k < u_k$. Equality counts as crossing.

Assumptions and First-Passage Times

Suppose there are $K$ scheduled looks, indexed $k=1,\ldots,K$. The boundaries are specified before monitoring, and the statistic is evaluated at those looks. The stopping rule considered here stops efficacy monitoring at the first look with a boundary crossing. This is a discrete-look rule: a crossing between scheduled looks is not counted unless the design also monitors there.

The event definitions below apply to any realized sequence of statistics and do not require normality. For probability calculations in the usual canonical group sequential model under $H_0$, the vector of standardized statistics is modeled as jointly normal with mean zero and correlations $\operatorname{Corr}(Z_i,Z_j)=\sqrt{t_i/t_j}$ for $i<j$, where $t_k=I_k/I_K$ is the information fraction and $0<t_1<\cdots<t_K=1$. That model assumes the planned information times and analysis schedule are followed. This tutorial uses the model only as context: path classification itself requires the observed statistics and boundaries, not a calculation of crossing probabilities.

Definition: For a complete sequence $Z_1,\ldots,Z_K$, define the upper and lower first-passage times by $$ T_U=\min\{k:Z_k\geq u_k\},\qquad T_L=\min\{k:Z_k\leq l_k\}. $$ If the relevant set is empty, its minimum is defined to be $\infty$. The overall first-passage time is $T=\min(T_U,T_L)$.

Because $l_k<u_k$ at each look, $Z_k$ cannot cross both boundaries at the same look. Thus a finite first passage is either upper or lower, never both. If $T_U<T_L$, the upper boundary is reached first on the complete path; if $T_L<T_U$, the lower boundary is reached first. If both times are infinite, neither boundary is reached during the $K$ looks. The case $T_U=T_L<\infty$ is impossible under the stated boundary ordering.

Operational Stopping Events

For an actual trial, the first-passage event at look $k$ includes not only the crossing at look $k$, but also continuation at every earlier look. Define

A_{U,k}=\{l_j<Z_j<u_j\text{ for every }j<k,\ Z_k\geq u_k\},\qquad A_{L,k}=\{l_j<Z_j<u_j\text{ for every }j<k,\ Z_k\leq l_k\}.

Here $A_{U,k}$ is the event that the trial stops for an upper crossing at look $k$; $A_{L,k}$ is the event that it stops for a lower crossing at look $k$. For $k=1$, the condition on earlier looks is empty. The event of no efficacy crossing through the final look is

A_0=\{l_j<Z_j<u_j\text{ for every }j=1,\ldots,K\}.

These operational events are mutually exclusive: once a path crosses, it cannot later be in an operational stopping event at another look. Together, the $2K$ crossing events and $A_0$ cover all possible monitored paths, assuming a statistic value is available at each look reached while the trial continues.

Worked Example: Find the First Observed Crossing

Consider three scheduled looks with upper boundaries $u=(2.8,2.4,2.1)$ and lower boundaries $l=(-2.8,-2.4,-2.1)$. Suppose the hypothetical full statistic sequence is $Z=(1.3,2.5,-2.6)$. Classify the first passage and state what the monitoring rule observes.

At look 1, $-2.8<1.3<2.8$, so the trial continues. At look 2, $2.5\geq 2.4$, so the upper boundary is crossed. Therefore $T_U=2$, and $T_L$ is not yet determined from the observed trial.

The value $-2.6$ at look 3 is at or below the lower boundary $-2.1$. On the hypothetical full sequence, it would give $T_L=3$, so $T_U<T_L$: the upper tail is crossed first and the lower tail later. Under the stated stopping rule, however, the trial stops at look 2. The look-3 value is not observed as part of that trial's monitored path.

Complete Paths and Opposite-Tail Crossings

It is useful to distinguish two descriptions. An observed stopping path ends at the first crossing. A complete or extended path supplies values at all planned looks, even after a crossing, as a mathematical device for describing what would happen if monitoring continued. The times $T_U$ and $T_L$ can both be defined on that extended path. A later opposite-tail crossing is a statement about that extension, not a second decision made by the stopping rule.

For example, $T_U=i<T_L=j$ describes an extended path that first crosses the upper boundary at look $i$ and later crosses the lower boundary at look $j$. The actual stopping outcome remains an upper crossing at $i$. In the reverse ordering, $T_L=i<T_U=j$, the actual outcome is a lower crossing at $i$. If $T_U=T_L=\infty$, neither tail is crossed at any scheduled look.

This distinction also explains why the event “ever crosses the upper boundary” is not the same as “stops at the upper boundary.” A complete path may cross upper after it has already crossed lower; in that case it ever crosses upper, but the operational stopping decision is lower. First passage gives the unambiguous decision category.

Worked Example: Lower First, Upper Later

Use the same three-look boundaries, $u=(2.8,2.4,2.1)$ and $l=(-2.8,-2.4,-2.1)$. Consider the extended sequence $Z=(-1.1,-2.5,2.3)$.

At look 1, $-2.8<-1.1<2.8$, so the process continues. At look 2, $-2.5\leq-2.4$, so it crosses the lower boundary and $T_L=2$. At look 3, $2.3\geq2.1$, so the extended path crosses the upper boundary and $T_U=3$.

Thus $T_L<T_U$: the lower tail is crossed first and the upper tail later on the extension. The operational decision is a lower crossing at look 2. Calling this an upper stopping event because the final extended value exceeds $u_3$ would ignore the earlier stop.

Worked Example: Neither Tail Is Crossed

With the same boundaries, consider $Z=(1.0,-1.8,2.0)$. At look 1, $-2.8<1.0<2.8$; at look 2, $-2.4<-1.8<2.4$; and at look 3, $-2.1<2.0<2.1$. Each value lies strictly inside its continuation interval.

Consequently $T_U=T_L=\infty$ through the planned looks, and the path belongs to $A_0$. The design has no efficacy boundary crossing by the final analysis. This classification alone does not say whether another prespecified action, such as a non-efficacy stopping rule, applies.

Why the Event Classification Is Complete

The operational categories can be justified directly. For any path, scan the looks in order. If a boundary is crossed, let $k$ be the first such look. At each earlier look $j<k$, neither boundary was crossed, which is exactly $l_j<Z_j<u_j$. At look $k$, the statistic crosses either the upper or lower boundary, so the path belongs to exactly one of $A_{U,k}$ or $A_{L,k}$. If no crossing occurs, it belongs to $A_0$. The ordering $l_k<u_k$ prevents both crossing types at the same look.

This argument also shows why the events are disjoint. If a path belongs to a crossing event at look $k$, it has crossed at $k$; it cannot meet the required continuation condition at that look for a later first-passage event. Upper and lower events at the same look are disjoint because a value cannot simultaneously satisfy $Z_k\geq u_k$ and $Z_k\leq l_k$ when $l_k<u_k$.

\{A_0\}\ \cup\ \bigcup_{k=1}^{K}\left(A_{U,k}\cup A_{L,k}\right)

is therefore a partition of the monitored outcomes. This is an event-classification result, not a calculation of the probability assigned to each event.

Diagnostics, Alternatives, and Limitations

A classification is going wrong if it labels a later crossing as the observed stopping decision, reports both tails as the first decision, or omits a look where the path remained inside the boundaries. To audit a path, list each look in order, compare $Z_k$ with both boundaries, and stop the operational scan at the first crossing. For a complete-path description, continue the comparisons but label any post-stop crossing as hypothetical.

First passage is different from several tempting alternatives:

Terminal-only classification uses only $Z_K$. It can miss an earlier crossing and assign a later terminal sign that is not the trial's stopping decision.
Ever-crossing classification records whether a full path crosses each tail at any time. It can assign both tails to one extended path, so it does not by itself identify the actual stop.
First-passage classification scans in time order and assigns the unique first stopping tail and look, or no crossing. It matches a rule that stops at the first efficacy boundary.

These definitions depend on the scheduled looks and their boundaries. A path can be classified differently if the look schedule or boundary values change. Discrete monitoring also means that values between looks are not assessed by this rule. Finally, first-passage labels do not by themselves quantify error rates, power, or the probability of a particular path category; those require a probability model for the joint statistic sequence and are separate calculations.

Common pitfall: “The path eventually crosses both tails, so the trial has both an upper and a lower stopping outcome.” This confuses the extended path with the observed stopping rule. The correction is to report the first crossing as the operational outcome and describe any later opposite-tail crossing only as a hypothetical feature of the extended path.

Check Your Understanding

Answer each question by applying the first-passage definitions. Solutions are included below.

With $u=(2.5,2.2)$, $l=(-2.5,-2.2)$, and $Z=(2.0,-2.4)$, what is the operational stopping event?
For a complete path with $T_L=1$ and $T_U=3$, which tail is the observed first stopping outcome? What does the later upper crossing mean?
If every $Z_k$ satisfies $l_k<Z_k<u_k$, what are $T_U$, $T_L$, and the path category?
Why can a path not cross both boundaries at the same look under the stated assumptions?

Solutions. (1) At look 1, $2.0<2.5$, so monitoring continues. At look 2, $-2.4\leq-2.2$, so the operational outcome is a lower crossing at look 2. (2) The lower boundary is the first stopping outcome at look 1. The upper crossing at look 3 is only a later crossing on the complete extension; the trial would have stopped at look 1. (3) Both first-passage times are $\infty$, and the path belongs to $A_0$, the no-crossing category. (4) At each look $l_k<u_k$. A single value cannot satisfy both $Z_k\leq l_k$ and $Z_k\geq u_k$, so simultaneous crossings at that look are impossible.