Spending Alpha After A Graph Transition

Why a Graph Transition Changes the Spending Problem

A multiplicity graph reallocates local alpha when an endpoint hypothesis is rejected. In a group sequential design, however, each endpoint may already have been tested at one or more interim looks. A transition therefore affects a test that has a history: some looks have passed, some rejection boundaries have already been crossed or not crossed, and the future statistical information is conditional on what has been observed.

The key implementation rule is that a graph transition changes the future testing plan; it does not undo earlier looks or erase earlier boundary decisions. Nor does a new local alpha level by itself specify a valid continuation. The continuation must account for the observed history and preserve the error control required by the multiple-testing procedure.

Definition: A graph transition is the pre-specified update to endpoint weights and local alpha levels after an endpoint hypothesis is rejected. Alpha spending is the allocation of a local type I error allowance over planned interim looks, typically through cumulative spending targets at information fractions. A post-transition continuation is valid only if its future rejection rule, together with the decisions already made, satisfies the error-control requirements of the overall procedure.

This discussion assumes that the graph, transition rules, endpoint test statistics, analysis times or information fractions, and sequential monitoring plans are specified before examining the outcomes that trigger a transition. The usual group sequential model also needs to be appropriate for each endpoint: for example, under its null hypothesis, the standardized statistics at the planned looks have the joint null distribution used to calibrate the boundaries. If the endpoints have dependence or data structures that violate the calibration assumptions, the boundary calculation must address that issue rather than relying on an unverified marginal approximation.

Allocation Is Not the Same as Spending

A local alpha level is a budget for testing an endpoint hypothesis; a spending function describes how that budget is used over time. If endpoint \(B\) initially has local level \(\alpha_B\), its spending plan may assign cumulative targets \(A_B(t_1),\ldots,A_B(t_K)\) at information fractions \(t_1,\ldots,t_K\). These targets are used to calibrate sequential boundaries, accounting for the fact that statistics across looks are correlated.

When a graph transition increases endpoint \(B\)'s local level, it is incorrect to treat the new level as though it had been available at all prior looks. In particular, do not replace the old plan with a fresh spending function starting at the first look, or apply a newly calculated boundary to data from a look that has already occurred. The transition can only govern tests that remain to be conducted.

The arithmetic difference between a revised local level and an earlier nominal spending target can help maintain a planning ledger. It is not, by itself, a generally valid formula for the future boundary. Because the transition is triggered by observed rejections, the future distribution is conditional on the history; endpoint dependence and the graph's closed-testing requirements can also matter.

Worked Example: Recording a Graph Transition Without Rewriting History

Suppose two one-sided endpoint hypotheses, \(H_A\) and \(H_B\), begin with local levels \(0.030\) and \(0.020\), respectively. The pre-specified graph transfers 50% of \(H_A\)'s local level to \(H_B\) if \(H_A\) is rejected. At the first interim look, \(H_A\) is rejected and \(H_B\) is not.

The transfer to \(H_B\) is \(0.50(0.030)=0.015\), so the updated local level recorded for \(H_B\) is \(0.020+0.015=0.035\). This is the graph's updated allocation. It does not mean that \(H_B\) can be retested at the first look with level \(0.035\), nor does it determine the next boundary by itself.

The first-look decision for \(H_B\) remains the decision made under the original design. For future looks, the design must use a pre-specified valid continuation that incorporates the updated allocation and the observed history. For example, the continuation may be calibrated through conditional error for the relevant intersection tests. Simply starting a fresh \(0.035\) spending schedule after the transition would disregard the first-look history and is not justified by the ledger calculation.

Conditional Error for a Revised Continuation

Conditional error formalizes how much opportunity to reject remains after an observed history. For a specified null hypothesis or intersection hypothesis \(H_I\), let \(D_k\) denote all information available through look \(k\), including the statistics and decisions already observed. Under the null model for \(H_I\), the original design's conditional error is

e_I(D_k)=P(\text{the original design rejects }H_I\text{ at a future look}\mid D_k,H_I).

A revised continuation has conditional rejection probability \(e_I^*(D_k)\). It preserves the original conditional error guarantee if, for every relevant history and every null hypothesis or intersection that must be protected,

e_I^*(D_k)\leq e_I(D_k).

To see why, average the conditional probability over histories under \(H_I\). By the law of total probability,

P(\text{future rejection under }H_I) =E_{H_I}\{e_I^*(D_k)\} \leq E_{H_I}\{e_I(D_k)\}.

Thus the revised continuation cannot have a larger unconditional chance of future rejection than the original continuation under that null. In a graph-based multiple-testing procedure, this check must be made for the relevant intersection tests, or the alternative branchwise calibration must otherwise establish strong familywise error control. Checking only a single endpoint's marginal test may not be sufficient.

For a canonical group sequential statistic at two looks, with information fractions \(t_1<t_2\), the standardized statistic satisfies

Z_2\mid Z_1=z_1\sim N\left(\sqrt{\frac{t_1}{t_2}}z_1,\ 1-\frac{t_1}{t_2}\right)

under the null. This follows because the canonical joint null distribution has \(E(Z_1)=E(Z_2)=0\), unit variances, and correlation \(\sqrt{t_1/t_2}\). The conditional mean is the correlation times \(z_1\), and the conditional variance is one minus the squared correlation. This distribution makes explicit why a future boundary should be assessed in light of the observed interim statistic.

Worked Example: Calculating a Conditional Boundary

Consider a one-sided canonical test with two looks at information fractions \(t_1=0.50\) and \(t_2=1.00\). Suppose no rejection occurred for this endpoint at look 1 and the observed statistic was \(z_1=0.40\). A pre-specified, intersection-valid recalibration assigns a conditional future rejection allowance of \(0.010\) on this branch. Find the final boundary \(c\) that uses this conditional allowance.

Under the null, the conditional distribution has mean and standard deviation

\mu_{2\mid1}=\sqrt{\frac{0.50}{1.00}}(0.40)=0.2828,\qquad \sigma_{2\mid1}=\sqrt{1-\frac{0.50}{1.00}}=0.7071.

The boundary must satisfy \(P(Z_2\geq c\mid Z_1=0.40)=0.010\). The upper-tail standard normal quantile for this probability is \(z_{0.99}\approx2.326\), so

c=0.2828+0.7071(2.326)=1.9275\approx1.93.

The recalibrated boundary is approximately \(1.93\) to three significant figures. This calculation uses the stated conditional allowance; it does not establish that \(0.010\) is available in an arbitrary graph. That allowance must come from a valid pre-specified allocation and must satisfy the required checks for the relevant intersections. The calculation shows how to turn an established conditional allowance into a future boundary.

Ways to Implement the Transition

There is no single boundary update that is automatically valid for every graph and spending plan. Common defensible approaches make the branch structure explicit and establish error control for the complete procedure.

Conditional-error recalibration. At the transition, calculate the original conditional error for each relevant endpoint or intersection test. Choose future boundaries so the revised conditional rejection probabilities meet the applicable error limits. The graph's reallocation can be used only as allowed by a calibration that preserves the overall multiplicity guarantee.
Pre-calibrated transition branches. Specify the possible graph states and transition paths in advance, then calibrate the future boundaries for each branch. The calibration must account for the rule that selects a branch, and must control familywise error across the configurations of true and false null hypotheses.
Keep the original local sequential test unchanged. A procedure may update graph bookkeeping for future hypotheses while leaving a surviving endpoint's already specified sequential test unchanged, if the complete graph procedure and its error control support that choice. This avoids an unverified boundary change but may not use all available reallocation.

These methods differ from a naive reset. A fresh spending curve applied after a transition may be convenient to calculate, but convenience does not show that its conditional or familywise error is controlled. Likewise, a newly updated endpoint alpha cannot be interpreted as permission to ignore intersection tests or the path that led to the update.

Worked Example: A Transition at a Later Look

Suppose a three-look design has planned information fractions \(0.40\), \(0.70\), and \(1.00\). Endpoint \(A\) is rejected at look 2, so the graph increases the allocation to endpoint \(B\). Endpoint \(B\) has not been rejected at either earlier look.

The graph update applies to the continuation after look 2. The look-1 and look-2 statistics for \(B\), and the fact that \(B\) crossed neither earlier boundary, remain part of the observed history \(D_2\). A valid implementation first determines the future error allowance for the applicable tests conditional on \(D_2\), or uses boundaries calibrated in advance for this transition branch.

For example, if the branch-specific calibration gives \(B\) a conditional future allowance of \(0.005\) at the final look, the final boundary must be chosen so that the conditional null rejection probability is no greater than \(0.005\), while the intersection tests also meet their required limits. The number \(0.005\) is a hypothetical branch allowance, not something obtained merely by subtracting previous spending from the updated graph alpha. No new test is performed at looks 1 or 2, and no past boundary is revised.

Diagnostics, Alternatives, and Limitations

Before implementing a transition, check the analysis record and the calibration inputs. The information fractions must reflect the endpoint's planned information scale; the statistics and covariance assumptions must match the design; and the transition must be triggered exactly as specified. If an endpoint uses an event-driven look, event projection uncertainty can affect when information is available, but it does not authorize a post hoc change to the rejection rule.

Warning signs include a new boundary that was chosen after inspecting which result would be favorable, a spending function that appears to restart at the transition, a graph update applied to an endpoint that already rejected, or an error calculation that checks only endpoint-wise nulls while omitting relevant intersections. Another diagnostic is a simulation or numerical calibration that evaluates only one transition path. Strong control requires the procedure to be assessed under configurations in which any subset of hypotheses may be true, including paths where false nulls trigger transitions while true nulls remain active.

A fixed endpoint allocation with unchanged sequential boundaries is often easier to audit and can be preferable when transitions are rare or branch-specific calibration is impractical. Conditional-error recalibration can use information from the observed history and accommodate changes, but requires accurate conditional distributions and careful checks across intersections. Pre-calibrated branches make operational rules explicit, though the number of possible histories and graph states can make calibration demanding. None of these approaches removes the need to state the statistical model or verify the error-control claim.

Common pitfall: “The graph raised endpoint \(B\)'s local alpha, so I can restart its spending function at that larger level.” This reasoning treats a mid-study allocation as if it had existed from the beginning. The correction is to preserve all past decisions and calibrate only the future continuation, using conditional error or a pre-calibrated branch procedure that establishes the required familywise error control.

A Practical Implementation Sequence

Record the trigger and graph state.
Identify the rejection that caused the transition, the updated weights, and the local allocations under the pre-specified graph.

Freeze completed looks.
Retain all prior statistics, decisions, boundaries, and spending history; do not apply the updated allocation retroactively.

Determine the valid future allowance.
Use conditional-error calculations or the calibrated rule for the observed branch, including the relevant intersection tests.

Set and document future boundaries.
State the conditional model, information fractions, numerical boundary calculations, and error-control checks before conducting the next test.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Endpoint \(C\)'s local alpha increases after a rejection at look 2. May the new level be used to re-evaluate \(C\)'s look-1 statistic?
In a two-look canonical design, \(t_1=0.25\), \(t_2=1.00\), and \(Z_1=0.60\). Find the conditional mean and standard deviation of \(Z_2\) under the null.
Why is the difference between an updated local alpha and an earlier nominal spending target not automatically a valid future boundary allowance?
What must be checked when validating a graph transition that changes a sequential test?

Solutions. (1) No. The look-1 decision and statistic are part of the completed history; the transition governs only future testing. (2) The conditional mean is \(\sqrt{0.25/1.00}(0.60)=0.300\), and the conditional standard deviation is \(\sqrt{1-0.25/1.00}=\sqrt{0.75}\approx0.8660\). Thus \(Z_2\mid Z_1=0.60\sim N(0.300,0.75)\), where \(0.75\) is the variance. (3) The transition is data-dependent, and the future test is conditional on prior statistics and decisions; spending targets alone do not establish the conditional or intersection-wise error bound. (4) Check the conditional rejection probabilities or the pre-calibrated branch, under the specified null model, for all relevant intersection hypotheses and configurations needed for familywise error control.