Why Clinical Priorities Change the Graph
When a clinical trial tests several endpoints at interim and final looks, the order and allocation of testing opportunities can affect which claims the design is able to support. The endpoint considered most clinically important may deserve the larger initial allocation, but that choice is not the whole design: if that endpoint is rejected, the transition path determines how much of its allocation can be used for another endpoint. A graph therefore encodes both where testing begins and where testing opportunity can go next.
The purpose here is to compare those choices in a concrete sequential setting. The numerical examples use hypothetical, already-calibrated sequential evidence requirements. They illustrate how to trace a graph; they are not substitutes for calibrating valid local boundaries for the actual look schedule, dependence structure, and endpoint tests.
For two endpoints, call the clinically higher-priority endpoint \(A\) and the other endpoint \(B\). If \(H_A\) is rejected, the update is \(a_B'=a_B+g_{AB}a_A\). If \(H_B\) is rejected, the update is \(a_A'=a_A+g_{BA}a_B\). These levels are cumulative local levels for the prespecified sequential tests, not fresh significance levels to spend independently at every look.
A sequential local test must control its type I error over all planned looks at its allocated level. Thus, the graph comparison presumes that the local procedures and the intersection tests required by the multiplicity procedure have been calibrated to the look schedule. The evidence requirements below summarize the observed sequential paths on that calibrated scale: an endpoint with requirement \(r_i\) is rejected when its current local level is at least \(r_i\). Smaller \(r_i\) represents stronger sequential evidence. A nominal p-value at one look is not automatically such a sequential requirement.
How to Read the Graph Arithmetic
Initial weights and transition weights answer different design questions. Initial weights express the priority assigned before any endpoint has been rejected. A transition weight expresses how much of a rejected endpoint's testing allocation can support a remaining endpoint. A large initial weight for \(A\) does not by itself guarantee that \(B\) will later receive a large level; that depends on \(g_{AB}\). Conversely, a graph can give \(B\) a modest initial level but provide substantial additional opportunity if \(A\) is rejected.
The update does not create alpha. With two nodes, after \(i\) is rejected, the total level remaining for \(j\) is \(a_j+g_{ij}a_i\), which is no greater than \(a_j+a_i\), since \(g_{ij}\leq 1\). Any untransferred fraction is unavailable to the remaining node. In larger graphs, the outgoing transition fractions likewise cannot exceed the rejected node's allocation.
This arithmetic is necessary but not, on its own, a proof of strong familywise error control. That control comes from the graphical procedure's closed-testing construction and valid local tests for the relevant intersection hypotheses. In a sequential design, those tests must remain valid across looks; merely checking that the displayed levels sum to at most \(\alpha\) is not enough.
Worked Example: A Graph That Prioritizes Endpoint A
Suppose a hypothetical trial has two endpoints, \(A\) and \(B\), and familywise level \(\alpha=0.05\). The design team assigns \(w_A=0.75\) and \(w_B=0.25\), with transitions \(g_{AB}=0.50\) and \(g_{BA}=0.20\). A calibrated sequential analysis of the hypothetical observed data gives \(r_A=0.026\), with \(A\)'s evidence sufficient by look 2, and \(r_B=0.029\), with \(B\)'s evidence sufficient by look 3. Trace the graph.
The initial local levels are \(a_A=0.05(0.75)=0.0375\) and \(a_B=0.05(0.25)=0.0125\). At look 2, \(A\)'s requirement is met because \(0.026\leq 0.0375\), so \(H_A\) is rejected. Initially, \(B\)'s level is below its requirement, since \(0.029>0.0125\).
On rejecting \(H_A\), the transfer to \(B\) is \(g_{AB}a_A=0.50(0.0375)=0.01875\). Therefore \(B\)'s updated level is \(a_B'=0.0125+0.01875=0.03125\). At look 3, \(B\) can be rejected because \(0.029\leq 0.03125\). Under this graph, both endpoints are rejected, with the rejection of \(A\) preceding the additional opportunity for \(B\).
The transfer is only half of \(A\)'s allocation: \(0.01875\) of \(0.0375\). The remaining half is not transferred by this graph. The calculation is about the local levels allowed by the prespecified sequential procedure; it does not mean that \(0.03125\) can be spent anew at each look.
Comparing a Different Priority Pattern
Now compare that graph with one that initially favors \(B\) and returns relatively little alpha to \(A\). The same evidence can produce a different endpoint conclusion even though the familywise level and observed endpoint paths have not changed. This is the practical meaning of a clinical-priority choice: it allocates power and rejection opportunity differently, rather than changing the evidence in the data.
Worked Example: Same Data, Different Initial Weights and Transitions
Keep \(\alpha=0.05\), \(r_A=0.026\) by look 2, and \(r_B=0.029\) by look 3. Consider a second hypothetical graph with \(w_A=0.40\), \(w_B=0.60\), \(g_{AB}=0.25\), and \(g_{BA}=0.10\). Compare its decisions with the graph in the previous example.
The new initial levels are \(a_A=0.05(0.40)=0.020\) and \(a_B=0.05(0.60)=0.030\). At look 2, \(A\) is not rejected because \(0.026>0.020\). At look 3, \(B\) is rejected because \(0.029\leq0.030\).
The \(B\)-to-\(A\) transition then gives \(A\) an additional \(g_{BA}a_B=0.10(0.030)=0.003\). Its updated level is \(a_A'=0.020+0.003=0.023\). Since \(0.026>0.023\), \(A\) is not rejected. Thus this graph rejects \(B\) but not \(A\), unlike the first graph, which rejects both.
The comparison is not simply “more alpha for the important endpoint.” The first graph starts \(A\) at \(0.0375\), above \(A\)'s requirement. The second starts \(A\) at \(0.020\), and its modest \(B\)-to-\(A\) transition cannot make up the difference. If clinical policy instead requires that \(B\) be tested first, this second pattern may better represent that policy, even though it yields a different set of rejections for these hypothetical data.
Separating Initial Allocation from Transition Design
A useful graph review changes one feature at a time. The next example holds the initial weights fixed and changes only the transition from \(B\) to \(A\). This isolates the effect of a transition path: the initial opportunity to test either endpoint is identical, but rejection of \(B\) has a different consequence for \(A\).
Worked Example: Increasing the Return Path to Endpoint A
Use the second graph's initial weights, \(w_A=0.40\) and \(w_B=0.60\), so \(a_A=0.020\) and \(a_B=0.030\). Retain \(g_{AB}=0.25\), but compare \(g_{BA}=0.10\) with \(g_{BA}=0.60\). Suppose a different hypothetical evidence path has \(r_B=0.018\), sufficient by look 2, and \(r_A=0.030\), sufficient by look 3.
In both versions, \(B\) is rejected at look 2 because \(0.018\leq0.030\). With the smaller return path, \(g_{BA}=0.10\), the update is \(a_A'=0.020+0.10(0.030)=0.023\). Because \(0.030>0.023\), \(A\) is not rejected.
With the larger return path, \(g_{BA}=0.60\), the update is \(a_A'=0.020+0.60(0.030)=0.038\). Now \(0.030\leq0.038\), so \(A\) is rejected by look 3. The change in outcome is due to the transition fraction, not a change in initial allocation or endpoint evidence.
The transfer is \(0.60(0.030)=0.018\), leaving \(0.012\) of \(B\)'s former level untransferred. The remaining levels total \(0.038\), below the original \(0.050\); the larger return path does not increase the familywise level.
Assumptions, Diagnostics, and Alternatives
A defensible comparison depends on design assumptions being made explicit. At minimum, the endpoint hypotheses, clinical priority rationale, initial weights, transition weights, look schedule, and local sequential tests must be specified before interpreting the graph. The sequential tests must control their error over the planned looks, and the local intersection tests must satisfy the dependence assumptions used by the multiplicity procedure. If endpoint measurements mature at different times or a result is unavailable at a look, the planned graph rule must specify how that node is handled; silently treating an unavailable result as a nonrejection can change the operating characteristics.
Useful diagnostics include tracing each possible rejection order, checking every updated level, confirming that outgoing transition weights do not exceed 1, and verifying that the sequential local boundaries remain valid after a transfer. Also ask whether a clinically important endpoint can be blocked by the chosen initial weight or a weak return path. These checks reveal design consequences, but they do not replace a formal error-control assessment or a planned evaluation of power.
A fixed-sequence procedure is a different choice: it imposes an order and generally offers no opportunity to test a later endpoint unless the preceding hypothesis is rejected. A balanced allocation gives endpoints more similar initial levels, while a weighted graph can encode unequal priorities and recycle some opportunity after rejection. Neither strategy is universally best. Equal or weighted Bonferroni-style allocation may be easier to explain, but it can differ in how opportunity is recovered after a rejection. The graph is valuable when the intended clinical decision logic includes those paths and the team can communicate them clearly.
A common mistake is to look only at the initial weights and conclude that an endpoint with a small starting allocation has little chance of rejection. That ignores transitions. The reverse mistake is to assume that any rejected endpoint gives all its allocation to the remaining endpoint. The correction is to calculate the actual transfer, \(g_{ij}a_i\), and compare the updated local level with the endpoint's valid sequential evidence requirement. A graph's clinical rationale should be assessed alongside these numerical consequences.
Check Your Understanding
Answer each question using the graph rules and definitions above. The solutions are included for checking your reasoning.
- With \(\alpha=0.05\), weights \(w_A=0.60\), \(w_B=0.40\), and \(g_{AB}=0.25\), what is \(B\)'s level after \(A\) is rejected?
- If \(a_A=0.018\), \(a_B=0.032\), \(g_{AB}=0.50\), and \(A\) is rejected, does \(B\) meet a sequential evidence requirement \(r_B=0.039\)?
- Why is it invalid to compare a one-look nominal p-value directly with a graph's allocated level when the endpoint has been monitored at several looks?
- For a two-node graph with \(a_A=0.020\), \(a_B=0.030\), and \(g_{BA}=0.40\), calculate \(A\)'s level after \(B\) is rejected. If \(r_A=0.031\), is \(A\) rejected?
- What is the design difference between increasing \(w_A\) and increasing \(g_{BA}\)?
Solutions. (1) Initially \(a_A=0.05(0.60)=0.030\) and \(a_B=0.05(0.40)=0.020\). After rejecting \(A\), \(B\)'s level is \(0.020+0.25(0.030)=0.0275\). (2) \(B\)'s updated level is \(0.032+0.50(0.018)=0.041\). Since \(0.039\leq0.041\), \(B\) meets the stated requirement. (3) A nominal p-value at one look does not account for the sequential testing rule across all planned looks; use a valid sequential local test or its calibrated evidence requirement. (4) \(A\)'s updated level is \(0.020+0.40(0.030)=0.032\). Since \(0.031\leq0.032\), \(A\) is rejected. (5) Increasing \(w_A\) raises \(A\)'s initial allocation before any rejection; increasing \(g_{BA}\) raises the fraction of \(B\)'s allocation that can move to \(A\) only after \(B\) is rejected.