Why a Classical Efficacy Design Has an Endpoint
A group sequential design allows efficacy to be assessed at several planned analyses rather than only once at the end. That flexibility does not mean the trial can continue indefinitely. The design specifies a maximum information, and the planned final analysis occurs at that limit. The maximum information therefore sets the endpoint of the efficacy boundary: it determines the last scheduled analysis and the final opportunity to reject \(H_0\) under the design.
The distinction between an interim stop and the endpoint matters. At an interim analysis, crossing the efficacy boundary may stop the trial early. If no interim boundary is crossed, the trial proceeds to the planned final analysis. That final analysis is not an early stop; it is the terminal decision at the information cap. A result below the final efficacy boundary does not authorize an additional look under the same prespecified design.
Here, information measures precision about the treatment effect; it is not necessarily the number of participants. Under a particular statistical model, information can be related to sample size, but that relationship depends on the outcome, variance, allocation, and other design assumptions. A fixed maximum sample size and a fixed maximum information are related design limits, not interchangeable quantities in every setting.
How Information Defines the Boundary Endpoint
Write \(Z_k\) for the standardized efficacy statistic at look \(k\), and \(b_k\) for the prespecified upper efficacy boundary at that look. For a one-sided efficacy design, the stopping rule is to stop and reject \(H_0\) at the first planned look \(k\) for which \(Z_k\ge b_k\). If this has not happened at an interim look, the trial continues to the next planned analysis. At \(K\), the statistic is compared with \(b_K\); the sequence ends there because \(I_K=I_{\max}\).
The boundary values are constructed for the planned sequence of information times and analyses. In the usual canonical group sequential model, the standardized statistics have marginal standard normal distributions under the null, and their dependence is determined by the information times: for \(j<k\), \(\operatorname{Corr}(Z_j,Z_k)=\sqrt{I_j/I_k}\). This model assumes the prespecified test statistic and information calculation are appropriate, information accumulates as planned, and the null distribution used to calibrate the efficacy rule is valid. The boundary is calibrated as a sequence, not by treating each look as an unrelated fixed-sample test.
The maximum information sets the final point at which that sequence is defined. If the planned fractions are \(t_1,\ldots,t_K\), then the final fraction must equal one. A value \(t_k>1\) would mean that a supposed interim look lies beyond the stated maximum, which is inconsistent with the schedule. Conversely, if a proposed final look has \(t_K<1\), then either the maximum information has not been reached or the information schedule has been specified inconsistently.
Worked Examples
Worked Example: Locating the Endpoint in a Four-Look Design
Suppose a hypothetical trial has planned information values \(I_1=25\), \(I_2=50\), \(I_3=75\), and \(I_4=100\) in units of inverse effect-squared. The design specifies upper efficacy cutoffs \(b_1=3.30\), \(b_2=2.70\), \(b_3=2.30\), and \(b_4=2.00\). Treat these cutoffs as supplied by the design; this example is about applying the endpoint and stopping rule, not calibrating the boundary.
The maximum information is \(I_{\max}=I_4=100\). The information fractions are \(\ t_1=25/100=0.25,\ t_2=50/100=0.50,\ t_3=75/100=0.75,\ t_4=100/100=1.00.\) Thus the fourth analysis is the terminal look.
Imagine the observed statistics are \(Z_1=2.40,\ Z_2=2.20,\ Z_3=2.35,\ Z_4=2.05\). At the first look, \(2.40<3.30\), so the efficacy boundary is not crossed. At the second, \(2.20<2.70\), and at the third, \(2.35>2.30\). The trial therefore stops for efficacy at look 3, before reaching the maximum information. Look 4 is not conducted under this stopping rule.
For a different hypothetical path, suppose the first three statistics remain below their cutoffs and the final statistic is \(Z_4=1.85\). Since \(1.85<2.00\), the terminal efficacy boundary is not crossed. The design has reached its endpoint, so this rule does not call for a fifth analysis.
Worked Example: Translating a Participant Cap Into Maximum Information
Consider a hypothetical two-arm trial with a continuous outcome. Assume independent observations in the two arms, equal allocation, a common known variance of \(400\) outcome-units squared per participant, and a treatment-effect estimate equal to the difference in sample means. Suppose the maximum total sample size is 400 participants, or 200 per arm.
For independent arm means, the variance of the difference is the sum of their variances: \[ \operatorname{Var}(\bar X_T-\bar X_C)=\frac{400}{n_T}+\frac{400}{n_C}. \] At the maximum sample size, \(n_T=n_C=200\), so \[ \operatorname{Var}(\bar X_T-\bar X_C) =\frac{400}{200}+\frac{400}{200} =2+2=4\ \text{outcome-units}^2. \] The information for the treatment effect is the reciprocal of this variance: \[ I_{\max}=\frac{1}{4}=0.2500\ \text{outcome-units}^{-2}. \]
Now consider an interim analysis after 150 participants per arm, 300 total. The assumptions remain the same, so \[ \operatorname{Var}(\bar X_T-\bar X_C) =\frac{400}{150}+\frac{400}{150} =2.6667+2.6667 =5.3333\ \text{outcome-units}^2. \] Therefore \(I_{\mathrm{interim}}=1/5.3333=0.1875\ \text{outcome-units}^{-2}\), and its information fraction is \[ t_{\mathrm{interim}}=\frac{0.1875}{0.2500}=0.7500. \] The interim look is at 75% of maximum information. Its 300 participants are within the 400-participant cap. This calculation checks both arms in the variance and compares like with like: interim information divided by final information.
Worked Example: A Terminal Analysis Is Not Permission to Extend
Suppose a hypothetical efficacy design has a maximum information of \(0.2500\ \text{units}^{-2}\), reached at 400 participants, and a terminal cutoff of \(b_K=1.98\). The interim looks have not crossed their efficacy cutoffs. At the planned final analysis, the observed statistic is \(Z_K=1.91\).
The terminal comparison is \(1.91<1.98\), so the prespecified efficacy boundary is not crossed. The correct design-based conclusion is that the efficacy stopping rule does not reject \(H_0\) at the final analysis. It is not correct to collect more participants and keep applying the same terminal cutoff as if the additional analysis had been planned. That would add information and a new opportunity to reject, changing the procedure’s operating characteristics.
For this example, the endpoint is defined by both the maximum information and the terminal analysis rule. Reaching the participant cap without reaching the intended information, or reaching the information cap at a different sample size because the variance differs, would need to be handled according to the design’s stated information and analysis conventions—not by silently redefining the endpoint after seeing the data.
Why the Endpoint Must Be Prespecified
A group sequential efficacy rule controls the chance of a false positive across its planned looks by accounting for their joint behavior. Adding an unplanned analysis after the final look introduces another chance to cross a threshold. Reusing a cutoff that was specified for the original sequence does not generally preserve the original error control. The endpoint is therefore part of the inferential rule, not merely an administrative target.
The information fraction is useful for checking that the stated schedule is coherent. For example, if \(I_{\max}=0.2500\ \text{units}^{-2}\) and a proposed interim value is \(0.1875\ \text{units}^{-2}\), then \(t=0.75\), which is below one. If an interim calculation instead produced information greater than \(0.2500\), investigate the variance calculation, the arm-specific sample sizes, the definition of the cap, and the units before proceeding. In a two-arm difference in means, omitting one arm’s contribution to the variance can lead to an incorrect information value.
The information-to-sample-size mapping can also be imperfect in practice. The simple calculation above relies on a known common variance, independent observations, and the stated difference-in-means estimator. With other endpoints or models, information may depend on event counts, nuisance-parameter estimates, clustering, missing data, or the observed allocation. Thus a planned maximum sample size may be a practical cap while the realized information at that cap differs from its target. The protocol and analysis plan need to specify how the information and terminal analysis are determined.
This tutorial concerns the endpoint of a fixed classical efficacy design. It does not determine which boundary family is best, calculate expected sample size, or prescribe how to redesign a study after a change in the look schedule. Those questions require additional design decisions. The essential point here is narrower: once the maximum information and final analysis are fixed, the boundary’s planned sequence stops there.
Common Mistake and Practical Checks
Wrong reasoning: “The final statistic is just short of the cutoff, so one more analysis at the same cutoff is harmless.” This treats a sequential rule as if each analysis were independent of the others and ignores that the original boundary was defined for a finite planned sequence.
Correction: Compare the statistic with the cutoff at the planned terminal analysis. If the boundary is not crossed, report that the prespecified efficacy criterion was not met. Do not add another look under the original rule. Before analysis, verify that the final information fraction is exactly one, that interim fractions are below one, and that the information calculation includes all terms in the variance of the treatment-effect estimate.
- Confirm that the stated maximum information corresponds to the planned final analysis.
- Check that information increases across looks and that \(I_k\le I_{\max}\) for every planned look.
- Confirm that the final information fraction is \(t_K=1\).
- For a sample-size-based information calculation, state the outcome model, variance assumptions, and arm-specific sample sizes.
- Keep the terminal decision distinct from an interim efficacy stop and from any later redesign.
Check Your Understanding
Answer each question using the endpoint and information definitions above. Solutions follow the questions.
- A design has maximum information \(I_{\max}=80\) and planned looks at information 20, 40, 60, and 80. What are the information fractions, and which look is terminal?
- With \(I_{\max}=0.30\ \text{units}^{-2}\), an interim look is at \(0.24\ \text{units}^{-2}\). Calculate its information fraction and say whether it exceeds the cap.
- In an equal-allocation two-arm trial, each arm has variance \(225\ \text{units}^2\) per participant and there are 100 participants per arm at the cap. What is the variance of the difference in means and the corresponding maximum information?
- At the planned final analysis, \(Z_K\) is below \(b_K\). Does the fixed design permit another analysis using the same cutoff? Explain.
- Give two assumptions that must be checked before using the inverse-variance calculation for a difference in means to translate sample size into information.
Solutions. (1) The fractions are \(20/80=0.25\), \(40/80=0.50\), \(60/80=0.75\), and \(80/80=1.00\); look 4 is terminal. (2) \(t=0.24/0.30=0.80\), so it is below the cap. (3) The variance is \(225/100+225/100=4.50\ \text{units}^2\), giving maximum information \(1/4.50=0.2222\ \text{units}^{-2}\). (4) No. The planned sequence ends at \(I_{\max}\); adding a look changes the testing procedure and requires a separately justified design. (5) Examples include independent observations between arms, the stated variance being appropriate and known or suitably handled, and the treatment-effect estimator being the difference in sample means. Any two, correctly explained, suffice.