Updating Graph Weights After Endpoint Rejection

What Changes When an Endpoint Is Rejected?

In a weighted graph for endpoint multiplicity, each node represents a hypothesis and has a local significance level. Directed edges specify how that level is redistributed if the node’s hypothesis is rejected. The graph therefore has two different kinds of quantities: local alpha levels, which are used for testing, and edge weights, which govern later reallocations.

This tutorial focuses on the update after a formal rejection. “Formal” matters: the endpoint must have met its applicable local testing criterion, using a test calibrated at that local level. Evidence that merely looks persuasive, or a p-value below the familywise alpha but above the endpoint’s current local level, does not trigger a graph transition. The introductory material on weighted graphs for endpoint multiplicity gives the broader setup; here we work through the transition calculations and their conditions.

Definition: Let \(H_1,\ldots,H_m\) be the endpoint hypotheses, and let \(\alpha\) be the familywise significance level. The current local alpha for \(H_i\) is \(\alpha_i\). For distinct nodes \(i\) and \(j\), \(g_{ij}\) is the weight on the directed edge from \(i\) to \(j\): if \(H_i\) is rejected, the fraction \(g_{ij}\) of \(\alpha_i\) is transferred to \(H_j\). A usual graph has \(g_{ij}\geq 0\), no self-loops, and \(\sum_{j\ne i}g_{ij}=1\) for each node with other nodes available.

Initially, one common specification is \(\alpha_i=\alpha w_i\), where the nonnegative initial weights satisfy \(\sum_i w_i=1\). After a rejection, the remaining local levels need not retain this initial proportional form. The edge weights also change: they describe where future transfers go after the rejected node has been removed. These updates concern allocation across endpoints; they do not themselves specify how significance is spent across interim looks.

The Transition Rule

Suppose \(H_r\) is formally rejected. Its current local alpha is transferred to the remaining nodes according to the outgoing edges from \(r\). For each remaining node \(j\), the new local level is:

\alpha_j^{\,*}=\alpha_j+\alpha_r g_{rj}.

The remaining graph must also be updated. For distinct remaining nodes \(i\) and \(j\), the transition rule is:

g_{ij}^{\,*}= \frac{g_{ij}+g_{ir}g_{rj}}{1-g_{ir}g_{ri}}, \qquad i,j\ne r,\ i\ne j.

The numerator combines two routes from \(i\) toward \(j\): the direct edge \(i\to j\), and the route \(i\to r\to j\). The denominator adjusts for the repeated cycling between \(i\) and \(r\). This formula is defined only when \(1-g_{ir}g_{ri}>0\). Because both weights are between zero and one, the excluded case is \(g_{ir}=g_{ri}=1\): a closed two-node cycle.

Condition for a defined update: For every pair of remaining nodes \(i,j\) to which the transition formula is applied, require \(1-g_{ir}g_{ri}>0\). Nonnegative edge weights and outgoing sums of one alone are not sufficient. If \(g_{ir}=g_{ri}=1\), the formula has denominator zero and must not be used.

Why the Updated Outgoing Weights Still Sum to One

For a remaining node \(i\), assume its original outgoing weights sum to one and that the update denominator is positive. Since there are no self-loops, its outgoing edges went either to \(r\) or to another remaining node. Thus \(\sum_{j\ne i,r}g_{ij}=1-g_{ir}\). Also, the outgoing weights from \(r\) sum to one, so \(\sum_{j\ne i,r}g_{rj}=1-g_{ri}\). Summing the updated weights from \(i\) gives:

\begin{aligned} \sum_{j\ne i,r}g_{ij}^{\,*} &=\frac{\sum_{j\ne i,r}g_{ij} +g_{ir}\sum_{j\ne i,r}g_{rj}}{1-g_{ir}g_{ri}}\\ &=\frac{(1-g_{ir})+g_{ir}(1-g_{ri})}{1-g_{ir}g_{ri}}\\ &=\frac{1-g_{ir}g_{ri}}{1-g_{ir}g_{ri}}=1. \end{aligned}

The positivity condition is essential: this cancellation proves the row sum is one only when the denominator is nonzero. In a graph with just one remaining node, there are no outgoing edges to update; the process ends if that node is the only hypothesis left.

Worked Examples

Worked Example: Two Endpoints

Suppose the familywise level is \(\alpha=0.05\), with initial weights \(w_A=0.60\) and \(w_B=0.40\). The only edges are \(g_{AB}=1\) and \(g_{BA}=1\). Assume the test of \(H_A\), at its current local level, meets its formal rejection criterion.

Initially, \(\alpha_A=0.05(0.60)=0.030\) and \(\alpha_B=0.05(0.40)=0.020\). When \(H_A\) is rejected, all of its local level goes to \(H_B\):

\alpha_B^{\,*}=\alpha_B+\alpha_Ag_{AB} =0.020+(0.030)(1)=0.050.

Thus the remaining hypothesis \(H_B\) has local alpha \(0.050\). Since it is the only remaining node, there is no outgoing edge to calculate. The formula for updating edges is not needed in this terminal two-endpoint case.

Worked Example: Updating a Three-Endpoint Graph

Let \(\alpha=0.05\), with initial weights \(w_A=0.50\), \(w_B=0.30\), and \(w_C=0.20\). Set the outgoing edges to \(g_{AB}=0.60\), \(g_{AC}=0.40\), \(g_{BA}=0.30\), \(g_{BC}=0.70\), \(g_{CA}=0.50\), and \(g_{CB}=0.50\). Each node’s outgoing weights sum to one. Suppose \(H_A\) is formally rejected at its current local level.

The initial local levels are \(\alpha_A=0.025\), \(\alpha_B=0.015\), and \(\alpha_C=0.010\). Transfer \(H_A\)’s level along its outgoing edges:

\alpha_B^{\,*}=0.015+(0.025)(0.60)=0.030, \qquad \alpha_C^{\,*}=0.010+(0.025)(0.40)=0.020.

Now update the edge from \(B\) to \(C\). Its denominator is \(1-g_{BA}g_{AB}=1-(0.30)(0.60)=0.82\), which is positive, so the update is defined:

g_{BC}^{\,*} =\frac{g_{BC}+g_{BA}g_{AC}}{1-g_{BA}g_{AB}} =\frac{0.70+(0.30)(0.40)}{0.82} =\frac{0.82}{0.82}=1.

Similarly, the denominator for the edge from \(C\) to \(B\) is \(1-g_{CA}g_{AC}=1-(0.50)(0.40)=0.80\), and \(g_{CB}^{\,*}=[0.50+(0.50)(0.60)]/0.80=0.80/0.80=1\). With only \(B\) and \(C\) remaining, each must direct all its outgoing weight to the other. The revised local levels still sum to \(0.030+0.020=0.050\).

Worked Example: A Second Rejection After the Transition

Continue with the revised three-endpoint graph above. Suppose \(H_B\), now tested at local alpha \(0.030\), meets its formal rejection criterion. The updated edge from \(B\) to \(C\) is \(g_{BC}^{\,*}=1\), so \(C\) receives all of \(B\)’s current local alpha:

\alpha_C^{\,**}=\alpha_C^{\,*}+\alpha_B^{\,*}g_{BC}^{\,*} =0.020+(0.030)(1)=0.050.

Now \(C\) is the sole remaining hypothesis, with local alpha \(0.050\). There is no remaining edge row to update. The calculation uses the current local level of \(B\), not its initial level \(0.015\), and the current edge weight, not the original \(g_{BA}\) or \(g_{BC}\).

Checks, Alternatives, and Failure Modes

After each transition, check both the local levels and the edge rows. If the graph began with total level \(\alpha\), transferring the rejected node’s entire current level should preserve the total across remaining nodes: \(\sum_{j\ne r}\alpha_j^{\,*}=\sum_{j\ne r}\alpha_j+\alpha_r\sum_{j\ne r}g_{rj}=\sum_j\alpha_j\). Also verify that every updated edge is nonnegative and every remaining outgoing row sums to one, provided the update is defined. These checks can catch indexing errors, omitted transfer paths, or use of stale graph weights.

A common alternative is to keep fixed weighted Bonferroni allocations: each endpoint retains its original local level, with no recycling after rejection. That approach is simpler to implement but does not redistribute alpha to surviving endpoints. A graph allows such redistribution according to specified priorities. In either approach, the allocation must be paired with valid endpoint tests, and the graph transition itself does not establish familywise error control; that justification is a separate result.

Do not confuse an edge weight with an alpha level. An edge weight is a fraction between nodes; the transferred amount is the edge weight multiplied by the rejected node’s current local alpha.
Do not recycle on non-rejection. A transition is triggered by formal rejection, not by a favorable trend or a p-value that fails the current local test.
Do not use the original graph after a transition. Both the local levels and the edges may have changed.
Check denominators. If \(1-g_{ir}g_{ri}=0\), the edge formula is undefined. For example, \(g_{ir}=g_{ri}=1\) makes both numerator and denominator zero for transitions from \(i\) to other nodes. This is not an updated weight of zero or one; the graph requires separate handling or redesign.

The transition rule is limited to the endpoint-allocation graph. It does not determine interim boundary values, account for a changing information schedule, or resolve correlated endpoint statistics. Those features must be addressed by the corresponding testing design. In particular, a correct graph update cannot repair an endpoint test that was not calibrated at its assigned local level.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

With \(\alpha_A=0.012\), \(\alpha_B=0.018\), and \(g_{AB}=0.25\), what is \(B\)’s local alpha after \(A\) is rejected?
For a transition with \(g_{ir}=0.40\) and \(g_{ri}=0.50\), is the denominator positive, and what is its value?
Why is the graph transition formula undefined when \(g_{ir}=g_{ri}=1\), even if all outgoing rows sum to one?
After a rejection, should a later transfer use the rejected node’s original local alpha or its current local alpha?
What should be checked about each updated outgoing row when its transition denominator is positive?

Solutions. (1) \(0.018+(0.012)(0.25)=0.021\). (2) Yes: \(1-(0.40)(0.50)=0.80>0\). (3) The denominator is zero, so the displayed ratio is undefined; an outgoing-sum condition does not prevent this closed two-node cycle. (4) Use the current local alpha, which includes any previous transfers. (5) Its updated weights should be nonnegative and sum to one; the sum follows from the transition rule and the positive-denominator condition.