Why Coordinate Interim Schedules Across Endpoints?
A trial may monitor several endpoints at interim analyses. If every endpoint is reviewed at the same planned data cuts, the monitoring process is easier to coordinate: the team can evaluate the endpoints together, use a consistent look index, and apply its endpoint-wise monitoring rules on a shared schedule. But shared dates alone do not guarantee that the endpoints are at comparable stages of information accumulation. A schedule intended to align both dates and information fractions must be designed and checked on both scales.
The central design distinction is between when an analysis occurs and how much information has accumulated for each endpoint by then. Information is endpoint-specific: event counts, measurement variability, missing data, and the analysis model can differ. Consequently, equal calendar time or equal numbers of participants need not represent equal information fractions across endpoints.
The information unit must be appropriate to the endpoint’s analysis. For an event-driven endpoint, information may be represented by the planned number of relevant events; for a continuous endpoint, it may be based on the planned variance of the treatment-effect estimator. The schedule should state how \(I_{jk}\) is defined and how it will be estimated or projected. An information fraction is a ratio within an endpoint: requiring \(t_{jk}=t_k\) does not require the endpoints to have equal absolute information \(I_{jk}\).
Constructing a Shared Schedule
A practical design starts with candidate common analysis times, such as common data-cut dates or planned event milestones. For each endpoint, project the information available at each candidate time and the final information under the trial’s design assumptions. Then compute \(t_{jk}\) endpoint by endpoint. If the fractions differ materially, the schedule is date-aligned but not information-aligned. The design team can revise planned times, define endpoint-specific monitoring grids, or explicitly accept common dates with endpoint-specific fractions and calibrate monitoring accordingly.
The schedule is a design target, not a promise that realized information will equal its projection. The protocol should specify the planned information measure, the target fractions, the operational rule for triggering each analysis, and how deviations from planned information will be handled. Projection uncertainty in event timing is one reason that a scheduled analysis may occur at a fraction different from its target.
Specify shared data-cut dates or shared operational triggers, and state whether the final analysis is included in the schedule.
For every endpoint and analysis, calculate \(I_{jk}\) using the endpoint’s planned information measure and the design assumptions.
Calculate \(t_{jk}=I_{jk}/I_{jK}\). Compare the fractions across endpoints at each shared analysis.
If fractions align, use the common grid \(t_1,\ldots,t_K\) in endpoint-wise monitoring plans. If they do not, document the endpoint-specific fractions and calibrate to the actual planned grids rather than treating the fractions as common.
Worked Example: Matching Fractions With Different Event Totals
Consider a hypothetical trial with two event-driven endpoints. The planned common analysis dates are months 6, 10, 14, and 18. The final planned event totals are 240 for endpoint A and 160 for endpoint B. Suppose the projected cumulative events at those dates are 60, 120, 180, and 240 for A, and 40, 80, 120, and 160 for B. Check whether the schedule shares information fractions.
For endpoint A, the fractions are \(60/240=0.25\), \(120/240=0.50\), \(180/240=0.75\), and \(240/240=1.00\). For endpoint B, they are \(40/160=0.25\), \(80/160=0.50\), \(120/160=0.75\), and \(160/160=1.00\).
Thus the plan has both common dates and common fractions. The absolute information differs: at the second analysis, the planned event counts are 120 and 80. That difference does not conflict with alignment, because each fraction is measured relative to its own endpoint’s planned final information.
Why Common Fractions Simplify Endpoint-Wise Monitoring
Under the usual canonical group sequential model for one endpoint, standardized test statistics at successive analyses have a covariance structure determined by their information fractions. Specifically, assume the endpoint’s score process has independent increments, its information increases over analyses, and the standardized statistics are approximately jointly normal under the null hypothesis. If \(k\leq \ell\), the score accumulated by analysis \(k\) is part of the score accumulated by analysis \(\ell\). In the canonical model, the covariance of the unstandardized scores is \(I_{jk}\). Dividing each score by the square root of its information gives the correlation below.
The last equality holds because both information values use the same endpoint-specific final information \(I_{jK}\) as denominator. If every endpoint uses the same fractions \(t_1,\ldots,t_K\), this within-endpoint temporal correlation pattern is the same across endpoints under the canonical model. That makes it easier to organize endpoint-wise boundary calculations and to communicate which look corresponds to each planned fraction. It does not mean that test statistics for different endpoints are independent, have the same variance in original units, or must use the same boundary.
For a multiplicity procedure, each endpoint’s sequential monitoring still has to be calibrated to the local error level and spending rule assigned to it. The familywise error requirement concerns the endpoint family as well as repeated looks; shared timing does not itself provide that control. Endpoint allocations, graph transitions, and other multiplicity rules remain part of the design.
Worked Example: Checking a Date-Aligned Plan
In a second hypothetical trial, both endpoints are reviewed at the same three interim dates and at the final analysis. Endpoint A has projected information \(50,100,150,200\); endpoint B has \(30,70,110,160\). Calculate the fractions and decide whether this is a common information schedule.
For A, division by its planned final information of 200 gives \(0.25,0.50,0.75,1.00\). For B, division by 160 gives \(0.1875,0.4375,0.6875,1.00\). For example, at the second interim analysis, the fractions are \(100/200=0.50\) and \(70/160=0.4375\).
The dates are common, but the interim information fractions are not. Calling this a common-fraction schedule would therefore be inaccurate. The design could retain those dates and calibrate the endpoint-wise monitoring plans using the displayed endpoint-specific fractions, or it could revise its planned timing if common fractions are an operational or analytical goal. The final fraction is 1 for both endpoints by definition when each endpoint reaches its planned final information.
Illustrating the Error-Budget Bookkeeping
A shared fraction grid also gives a compact way to display interim spending for endpoint-wise sequential tests. For illustration, suppose two endpoints are each assigned a one-sided local level of \(0.025\), so that testing each endpoint across all its looks at that level gives a Bonferroni familywise bound of \(0.050\). Suppose each endpoint uses the linear cumulative spending function \(a(t)=0.025t\) on the common fractions \(0.25,0.50,0.75,1.00\). This example illustrates bookkeeping; it does not supply numerical critical boundaries.
At those fractions, the cumulative planned spending values for either endpoint are \(0.00625,0.01250,0.01875,0.02500\). The increments are obtained by subtracting the previous cumulative value, starting with zero. Under a group sequential design, the spending function specifies cumulative type I error to be spent by each look; critical boundaries must be calibrated under the endpoint’s sequential null distribution to achieve that spending. The spending increments are not, by themselves, critical values.
| Look | Common fraction | Cumulative spending per endpoint | Increment at look |
|---|---|---|---|
| 1 | 0.25 | 0.00625 | 0.00625 |
| 2 | 0.50 | 0.01250 | 0.00625 |
| 3 | 0.75 | 0.01875 | 0.00625 |
| 4 | 1.00 | 0.02500 | 0.00625 |
Because both endpoints share the fraction grid and spending rule in this illustration, the table can be presented once for both endpoint-wise plans. If endpoints have different local levels or spending functions, the shared timing still helps align the rows, but their spending values and boundaries can differ. The stated Bonferroni bound depends on valid endpoint-wise tests at their assigned levels; synchronizing analyses does not replace that error-control argument.
Worked Example: Calculating Spending on a Common Grid
Using the illustrative linear spending function \(a(t)=0.025t\), find the cumulative spending and the new increment at the second common look, \(t_2=0.50\). At the first look, \(t_1=0.25\).
The cumulative amount by look 2 is \(a(0.50)=0.025(0.50)=0.01250\). The amount already spent by look 1 is \(a(0.25)=0.025(0.25)=0.00625\). Therefore the increment associated with look 2 is \(0.01250-0.00625=0.00625\). This is the cumulative error-spending bookkeeping for one endpoint, not a boundary or a p-value. Boundary calibration must also account for the joint distribution of that endpoint’s statistics across looks.
Diagnostics, Alternatives, and Limitations
A schedule should be checked both at design time and when actual analysis information is available. Useful diagnostics include comparing endpoint-specific projected and realized fractions at every data cut, confirming that the final information denominator is still appropriate, and checking whether changes in event accrual, measurement variability, or missingness have materially shifted the intended grid. For endpoints with different information definitions, the design should make clear that each ratio is on that endpoint’s own scale.
Common calendar dates are often operationally attractive, especially when endpoints can be extracted from the same data cut. If information accrues differently, the main alternative is not to pretend that fractions match: retain shared dates and use endpoint-specific fractions in the endpoint-wise monitoring plan. Designs may also use endpoint-specific analysis occasions when that is necessary for their information targets. These approaches involve different operational and calibration choices; common dates are not universally preferable.
Common timing has important limits. It does not make endpoint statistics independent, resolve dependence among endpoints, guarantee that each endpoint reaches its planned information on schedule, or determine how multiplicity should be controlled. Nor does identical timing require identical hypotheses, effect scales, spending functions, local levels, or boundaries. The correlation structure across endpoints and the treatment of endpoint multiplicity require their own design specifications.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Endpoint A has planned information 45 at an interim analysis and 180 at the final analysis. Endpoint B has planned information 30 and 120, respectively. Are their information fractions equal at that interim analysis?
- Two endpoints have the same interim dates, but their information fractions at the second analysis are 0.40 and 0.55. What is shared, and what is not shared?
- Under the canonical model, what is the correlation between an endpoint’s standardized statistics at fractions \(t_k=0.36\) and \(t_\ell=0.81\), where \(k<\ell\)?
- For \(a(t)=0.025t\), calculate the cumulative spending at \(t=0.60\) and the increment from a preceding look at \(t=0.40\).
- Does a common fraction grid establish familywise error control across endpoints? Explain briefly.
Solutions. (1) Yes. The fractions are \(45/180=0.25\) and \(30/120=0.25\). (2) The calendar analysis dates are shared, but the information fractions are endpoint-specific and are not aligned at the second analysis. (3) The correlation is \(\sqrt{0.36/0.81}=\sqrt{4/9}=2/3\), approximately 0.667. (4) Cumulative spending at 0.60 is \(0.025(0.60)=0.0150\); at 0.40 it is \(0.0100\), so the increment is \(0.0150-0.0100=0.00500\). (5) No. Shared fractions coordinate the timing grid; familywise control still depends on a valid multiplicity procedure and valid endpoint-wise sequential tests.