Why Stopping Time Depends on Direction
Two-sided monitoring can end with evidence in either direction: the statistic may cross an upper efficacy limit or a lower efficacy limit. When those limits are unequal, the monitoring rule is not directionally balanced. Even under the same sequence of planned looks, a positive effect may make upper-boundary stopping more likely, while a negative effect may make lower-boundary stopping more likely. The differences affect not only which boundary is reached, but also when the trial stops and how much information is collected on average.
The quantity of interest here is the distribution of the stopping look under a specified effect. It is distinct from power alone: two designs can have similar probabilities of eventually crossing a boundary yet differ in how quickly they stop. Conversely, an early expected stopping time does not by itself show that the design has adequate power or acceptable error rates.
The sign convention is important: an upper crossing is evidence in the positive direction and a lower crossing is evidence in the negative direction. Under a positive effect, an upper crossing is usually the clinically relevant directional stop; under a negative effect, a lower crossing may be. The probability of stopping at each boundary and the probability of reaching the maximum look should therefore be reported separately when direction matters.
Model and Stopping Probabilities
For calculation, use the canonical group sequential model: information increases at prespecified looks \(I_1<\cdots<I_K\), the standardized statistic has unit variance at each look, and the joint distribution is multivariate normal. Under a standardized effect \(\theta\), assume
This model is appropriate when the test statistics have the usual canonical joint-normal approximation, the effect and information schedule are as specified, and the limits are fixed before monitoring. It captures the dependence across looks: later statistics reuse information from earlier observations. The assumptions are not that the statistics at successive looks are independent, nor that a one-look tail probability can be used afresh at every look.
Let \(S_k=P_\theta(T\geq k)\) be the probability of reaching look \(k\). For a trial still continuing at look \(k\), define \(q_{k,U}\) and \(q_{k,L}\) as its conditional probabilities of crossing the upper and lower limits there. For \(k<K\), let \(q_{k,C}=1-q_{k,U}-q_{k,L}\) be the conditional probability of continuing. These probabilities depend on the effect, the limits, and the joint distribution of the statistics—not only on the limits at look \(k\).
At the final look, the remaining probability is assigned to \(T=K\), whether or not a boundary is crossed. Thus \(P_\theta(T=K)=S_K\); the final-look upper and lower stops are \(S_Kq_{K,U}\) and \(S_Kq_{K,L}\), while \(S_Kq_{K,C}\) is the probability of reaching the maximum look without an efficacy crossing.
Why the Recursion Works
Define the continuation event before look \(k\) as \(A_{k-1}=\{l_j<Z_j<u_j\text{ for every }j<k\}\), with \(A_0\) the certain event. The unconditional probability of an upper stop at look \(k\) is the probability of surviving all previous looks and crossing the upper limit now:
The same argument gives the lower-stop probability. If a trial continues, it has survived the previous looks and also remains between the current limits. Therefore \(S_{k+1}=S_kq_{k,C}\). This recursion is a compact way to obtain stopping probabilities from the joint-normal distribution; in a design calculation, the conditional probabilities can be computed by multivariate-normal integration. It is not valid to replace that integration with independent per-look tail calculations unless independence is genuinely part of the model.
Once the stopping distribution is available, the expected look is
The second equality follows by writing the integer-valued \(T\) as \(T=\sum_{k=1}^{K}I(T\geq k)\), where \(I(\cdot)\) is an indicator, and taking expectations. If the planned sample sizes are \(n_1,\ldots,n_K\), the expected sample size is \(E_\theta(n_T)=\sum_{k=1}^{K}n_kP_\theta(T=k)\). It is not generally \(n_K\) multiplied by the probability of stopping early; the stopping distribution at every look is needed.
Worked Example: A Positive Effect Favors Upper Stops
Consider a hypothetical three-look design with planned sample sizes \(n_1=40{,}000\), \(n_2=70{,}000\), and \(n_3=100{,}000\). Suppose a canonical-model calculation under a specified positive effect gives the following conditional probabilities among trials that have reached each look. At the final look, “no crossing” means the maximum sample size is reached without an efficacy stop.
| Look | Upper crossing | Lower crossing | Continue / no crossing |
|---|---|---|---|
| 1 | 0.2000 | 0.0100 | 0.7900 |
| 2 | 0.3500 | 0.0200 | 0.6300 |
| 3 | 0.6000 | 0.0500 | 0.3500 |
At look 1, \(S_1=1\), so the total probability of stopping there is \(0.2000+0.0100=0.2100\). The probability of reaching look 2 is \(S_2=1(0.7900)=0.7900\). The probability of stopping at look 2 is therefore \(0.7900(0.3500+0.0200)=0.2923\), and the probability of reaching look 3 is \(S_3=0.7900(0.6300)=0.4977\).
The look-3 upper and lower probabilities are \(0.4977(0.6000)=0.2986\) and \(0.4977(0.0500)=0.02489\). The probability of reaching the maximum look without crossing is \(0.4977(0.3500)=0.1742\). The overall upper-stop probability is \(0.2000+0.7900(0.3500)+0.4977(0.6000)=0.7751\); the overall lower-stop probability is \(0.0100+0.7900(0.0200)+0.4977(0.0500)=0.05069\).
Since the probabilities of stopping at looks 1, 2, and 3 are \(0.2100\), \(0.2923\), and \(0.4977\), respectively, \(E(T)=1(0.2100)+2(0.2923)+3(0.4977)=2.288\) looks. The expected sample size is \(40{,}000(0.2100)+70{,}000(0.2923)+100{,}000(0.4977)=78{,}631\) participants. The direction-specific probabilities show that upper stops dominate for this positive effect; the expected sample size summarizes timing but does not replace those directional results.
Worked Example: A Negative Effect Favors Lower Stops
For a second hypothetical effect and boundary configuration, suppose the conditional transition probabilities are as follows. The sample sizes are \(50{,}000\), \(75{,}000\), and \(100{,}000\) at the three looks.
| Look | Upper crossing | Lower crossing | Continue / no crossing |
|---|---|---|---|
| 1 | 0.0050 | 0.2500 | 0.7450 |
| 2 | 0.0100 | 0.4000 | 0.5900 |
| 3 | 0.0200 | 0.7000 | 0.2800 |
The probability of reaching look 2 is \(0.7450\), and the probability of reaching look 3 is \(0.7450(0.5900)=0.43955\). The total probability of stopping at look 2 is \(0.7450(0.0100+0.4000)=0.30545\). At look 3, the lower-stop probability is \(0.43955(0.7000)=0.3077\), and the upper-stop probability is \(0.43955(0.0200)=0.008791\).
Across all looks, the lower-stop probability is \(0.2500+0.7450(0.4000)+0.43955(0.7000)=0.8557\). The upper-stop probability is \(0.0050+0.7450(0.0100)+0.43955(0.0200)=0.02124\). The probabilities of stopping at the three looks are \(0.2550\), \(0.30545\), and \(0.43955\), so \(E(T)=0.2550+2(0.30545)+3(0.43955)=2.185\) looks.
The expected sample size is \(50{,}000(0.2550)+75{,}000(0.30545)+100{,}000(0.43955)=79{,}614\) participants, to the nearest participant. Here, the lower-stop probability is much larger than the upper-stop probability, as is consistent with a negative effect and the stated conditional crossing probabilities. This does not mean the lower limit is intrinsically more permissive in every design: the effect, information schedule, and limits jointly determine the transition probabilities.
Worked Example: Near-Null Effects Can Still Have Unequal Directional Stops
Suppose a third hypothetical calculation, under an effect close to zero, reports conditional probabilities \((q_{k,U},q_{k,L},q_{k,C})\) of \((0.0400,0.0150,0.9450)\), \((0.0600,0.0250,0.9150)\), and \((0.0800,0.0350,0.8850)\) at looks 1, 2, and 3. The last entry is the probability of no boundary crossing at the final look. Planned sample sizes are \(30{,}000\), \(60{,}000\), and \(90{,}000\).
The probabilities of reaching looks 2 and 3 are \(S_2=0.9450\) and \(S_3=0.9450(0.9150)=0.864675\). Thus, the probability of stopping at look 2 is \(0.9450(0.0600+0.0250)=0.080325\). At look 3, the upper-stop probability is \(0.864675(0.0800)=0.06917\), and the lower-stop probability is \(0.864675(0.0350)=0.03026\). The probability of reaching the final look without crossing is \(0.864675(0.8850)=0.7652\).
The overall upper- and lower-stop probabilities are \(0.0400+0.9450(0.0600)+0.864675(0.0800)=0.1659\) and \(0.0150+0.9450(0.0250)+0.864675(0.0350)=0.06889\). The expected look is \(E(T)=1(0.0550)+2(0.080325)+3(0.864675)=2.810\). The expected sample size is \(30{,}000(0.0550)+60{,}000(0.080325)+90{,}000(0.864675)=84{,}290\) participants.
A near-null effect does not guarantee equal probabilities of stopping in the two directions when the boundaries are asymmetric. The exact result depends on the boundary configuration and cross-look dependence. These illustrative transition probabilities are not, by themselves, a claim that a particular design has a specified type I error rate; error-rate calibration is a separate design property.
Interpretation, Comparisons, and Diagnostics
Expected stopping is useful for comparing the operational burden of designs under different plausible effects. It complements the probability of upper or lower stopping, the probability of reaching the maximum look, and the design’s error and power characteristics. For example, a design might stop earlier on average under a positive effect because upper crossings occur frequently at early looks, while still having a substantial chance of continuing to the final look.
A symmetric-boundary design under a symmetric null model has mirror-related directional behavior: reversing the sign of the effect interchanges the upper and lower outcomes. Unequal limits break that balance. A fixed-sample analysis has no early stopping distribution at all: every participant contributes through the planned final sample size. Comparing expected sample sizes can quantify a potential efficiency advantage of sequential monitoring, but it does not make sequential and fixed-sample operating characteristics interchangeable.
- Check probability accounting. At each look, the conditional upper, lower, and continue probabilities must sum to one. The unconditional probabilities of upper stops, lower stops, and reaching the maximum without crossing must also sum to one.
- Keep the effect direction explicit. Pooling upper and lower stops can hide an important directional imbalance, particularly when limits are unequal.
- Respect dependence across looks. Reusing a one-look crossing probability at every look, or multiplying unconditional look probabilities as if looks were independent, generally gives the wrong stopping distribution.
- Report the endpoint convention. If \(T=K\) includes trials that reach the final look without crossing, distinguish those trials from final-look efficacy stops when interpreting results.
The calculations rely on the specified model and prespecified monitoring rule. Misspecified information, noncanonical statistic behavior, changes to the limits during monitoring, or unplanned looks can make calculated stopping probabilities unreliable. Expected stopping is also a summary over repeated hypothetical trials; it does not predict the exact stopping time of an individual trial or guarantee a particular reduction in sample size.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- If \(S_k=0.60\), \(q_{k,U}=0.20\), and \(q_{k,L}=0.10\), what are the unconditional upper- and lower-stop probabilities at look \(k\), and what is the probability of continuing?
- Why can a near-zero effect still produce unequal overall upper- and lower-stop probabilities?
- For a three-look design, the probabilities of stopping at looks 1, 2, and 3 are \(0.20\), \(0.30\), and \(0.50\). Find \(E(T)\).
- What error is made by treating the statistics at successive looks as independent when calculating the stopping distribution?
Solutions. (1) The unconditional upper-stop probability is \(0.60(0.20)=0.12\), the lower-stop probability is \(0.60(0.10)=0.06\), and the continuation probability is \(0.60(1-0.20-0.10)=0.42\). (2) Unequal limits can yield unequal directional crossing probabilities, and the joint distribution across looks affects how many trials remain at risk for later crossings. (3) \(E(T)=1(0.20)+2(0.30)+3(0.50)=2.30\) looks. (4) It ignores the correlation created by accumulating information across looks and generally miscalculates both the surviving probability and the first-crossing probabilities.