Unavailable Endpoint Results At An Interim Look

When an Endpoint Has No Interim Result

In a multi-endpoint trial, an interim analysis may be ready for one endpoint but not another. For example, the statistic for a clinical outcome may be calculable at the planned data cut, while an endpoint requiring additional assessment has too little usable information. The monitoring team then faces two distinct questions: which tests can actually be performed, and how should the planned type I error control be maintained?

The key distinction is between a test that was performed and did not reject its null hypothesis, and a test that could not be performed because no valid statistic was available. A non-rejection supplies an observed statistic and is part of the endpoint’s testing history. An unavailable result supplies neither. Treating the latter as a non-rejection, assigning it a made-up statistic, or transferring its alpha as though a rejection occurred can change the operating characteristics of the procedure.

Set the Monitoring Rule Before Using It

Suppose endpoint \(j\) has a null hypothesis \(H_{0j}\), and let \(Z_{jk}\) denote its standardized statistic at look \(k\). For the usual normal-theory group sequential calculation, assume that under \(H_{0j}\), each \(Z_{jk}\) has a standard normal marginal distribution. When looks use accumulating data with independent increments, the joint null distribution is also specified; for example, the correlation between statistics at information fractions \(t_r\) and \(t_s\), with \(t_r \leq t_s\), is \(\sqrt{t_r/t_s}\). A sequential boundary calculation must use the relevant joint distribution for the looks that were actually tested.

An alpha-spending function \(A_j(t)\) specifies the cumulative type I error that the endpoint’s sequential test is allowed to use by information fraction \(t\). It is nondecreasing, starts at zero, and is bounded by the local type I error available to that endpoint under the multiplicity procedure. With exact spending boundaries, the null probability of crossing at any performed look up to information fraction \(t\) is calibrated to the specified cumulative spending, subject to the procedure’s boundary conventions. The local amount available to an endpoint may be affected by a prespecified multiplicity graph; only a formal rejection can trigger the graph’s specified transfer.

If a look is skipped, there is no statistic to include in the crossing event. Recalibrate the later boundary using the spending target at the later performed look and the null distribution of the statistics at the performed looks. Do not count an unavailable look as a crossing, a non-rejection, or a source of alpha transfer. If the test has not been performed at any earlier look, the later test is a one-look test at that point: its boundary follows from its spending target and its marginal null distribution, without a joint distribution across unperformed looks.

A One-Look Boundary From the Spending Target

The simplest case makes clear why an unavailable earlier result is not itself an obstacle to a later valid test. Suppose an endpoint has no usable statistic at an earlier planned look and is first tested at information fraction \(0.90\). If its cumulative two-sided spending target there is \(A(0.90)=0.025\), and its null statistic is standard normal, the sole performed test must have null rejection probability \(0.025\). Thus the two-sided boundary is determined directly by the target:

Here \(\Phi^{-1}\) is the standard normal quantile function. No joint distribution across looks is needed because only one test has been performed. With multiple performed looks, by contrast, the later boundary must account for the probability of crossing at earlier tested looks as well as the later statistic.

How the Rule Fits a Multiplicity Graph

A multiplicity graph assigns local testing levels to endpoint hypotheses and specifies how those levels may change after rejections. Its transitions are triggered by the formal decision events in the procedure, not by the passage of time or by a missing result. Therefore, if endpoint \(A\) is unavailable, it has not rejected its null hypothesis and cannot pass its allocated alpha to endpoint \(B\). Endpoint \(B\) may still be tested using the level assigned to it by the current graph state, provided its own sequential test is valid.

This is different from a performed test of \(H_{0A}\) that does not reject. Both situations produce no rejection at that look, but only the latter has an observed statistic that belongs in endpoint \(A\)’s sequential testing history. Neither situation triggers a rejection-based graph transition. The distinction matters because later boundaries for \(A\) depend on its performed-look history, while the graph’s current allocations depend on its formal rejection history.

Justification for a Conservative Skipped-Look Rule

Exact group sequential boundaries use the joint null distribution of the performed statistics to meet a cumulative spending target. When a quick, transparent implementation is needed, an increment-wise Bonferroni rule provides a conservative alternative. Allocate a two-sided error increment \(\delta_r\) to each performed look \(r\), and test at that look with a marginal level no greater than \(\delta_r\). For any dependence among the statistics,

where \(\mathcal{P}\) is the set of performed looks. This follows because the probability of a union of events cannot exceed the sum of their probabilities. An unavailable look has no rejection event in this union. The rule can therefore accommodate a skipped test while preserving the bound, though it may spend less efficiently than exact joint-distribution calibration.

Diagnostics, Alternatives, and Limits

A monitoring record should make it possible to verify which tests were performed and why. For each endpoint and look, record whether the statistic was available, the information fraction, the test and boundary used, the decision, and any graph transition. For a later endpoint test, check that its boundary matches the prespecified spending rule applied to the actual performed-look schedule. If exact boundaries are recalculated, verify that the calculation uses the correct null joint distribution for those performed looks.

Common warning signs include assigning an unavailable endpoint a p-value of \(1\), inserting \(Z=0\), or copying its last observed statistic forward. These substitutions create a testing history that did not occur. Another error is to use the same nominal level at every performed look without a sequential adjustment; repeated opportunities to reject can exceed the intended error rate. Finally, an endpoint’s allocation should not be passed to another endpoint merely because its data are late. A graph transfer requires the graph’s specified rejection event.

Two defensible operational choices are to pause testing until all endpoint results are available, or to continue testing the available endpoints under a procedure that remains valid for the resulting schedule. Pausing simplifies synchronization but may delay decisions. Continuing can be more timely, but boundaries and any multiplicity-graph transitions must follow the prespecified rules. Exact spending-boundary recalibration can use the joint distribution of performed looks and may be less conservative than increment-wise Bonferroni testing. The latter is easier to justify under dependence but can sacrifice power.

These methods do not automatically solve outcome missingness, biased measurement, or adaptive timing. If the chance that a result is available depends on unobserved outcomes or on interim results, the availability pattern may itself affect the procedure’s operating characteristics. Data-dependent decisions about which endpoints to test, or when to test them, require a rule that accounts for that selection or timing mechanism. A boundary valid for a prespecified schedule should not be assumed valid after an unplanned change.