Endpoint Allocation Versus Interim Alpha Spending

Two Different Decisions in a Multiplicity Plan

A study with several endpoints and interim analyses has two distinct ways to organize its type I error budget. First, it assigns error probability across endpoint hypotheses. Then, for each endpoint, it specifies when that endpoint’s assigned probability may be spent as interim data accumulate. These are separate decisions: changing an endpoint’s spending schedule does not, by itself, change the amount allocated to that endpoint.

This distinction matters because a procedure can allocate a reasonable total error budget across endpoints but use an inappropriate sequential boundary schedule, or it can use valid sequential boundaries while assigning too much error probability across the endpoint family. For clarity, the examples below use fixed endpoint allocations and one-sided superiority tests. The same separation applies when a local level is supplied by another multiplicity procedure, provided the sequential test is calibrated to that local level.

Definition: An endpoint allocation is the local type I error budget \(\alpha_j\) assigned to endpoint hypothesis \(H_j\). An alpha-spending function for endpoint \(j\) is a nondecreasing function \(A_j(t)\) of information fraction \(t\), with \(A_j(0)=0\) and \(A_j(1)=\alpha_j\). It specifies the cumulative probability, under the endpoint null hypothesis, of having crossed a rejection boundary by information fraction \(t\).

For \(K\) planned looks, let \(t_k\) be the information fraction at look \(k\), with \(0<t_1<\cdots<t_K=1\). The amount associated with look \(k\) is the increase in cumulative spending since the previous look:

\Delta_{jk}=A_j(t_k)-A_j(t_{k-1}), \qquad A_j(t_0)=0.

The increments are not additional endpoint allocations. They partition the endpoint’s cumulative budget over time. In a sequential design, a rejection boundary at look \(k\) is calibrated using the entire joint distribution of the interim test statistics so that the probability of a first crossing by look \(k\) equals the target \(A_j(t_k)\). It is generally not correct to treat \(\Delta_{jk}\) as a standalone marginal significance level and use a one-look critical value without accounting for earlier looks.

Assumptions and Error-Control Logic

A standard calibration uses a one-sided group sequential statistic \(Z_{jk}\) for endpoint \(j\), with larger values favoring the alternative. Under its null hypothesis, suppose the statistics follow the canonical joint model: each \(Z_{jk}\) is standard normal, and for looks \(r\) and \(s\), their correlation is \(\sqrt{t_r/t_s}\) when \(r<s\). This model reflects accumulating information with independent increments and correctly specified information fractions. The boundary calibration also requires a prespecified look schedule and a spending function that is nondecreasing and reaches its assigned local level at the final look.

For endpoint \(j\), boundaries \(c_{j1},\ldots,c_{jK}\) are chosen so that, under \(H_j\),

P\left(\bigcup_{r=1}^{k}\{Z_{jr}\ge c_{jr}\}\mid H_j\right)=A_j(t_k), \qquad k=1,\ldots,K.

Thus the cumulative crossing probability, not a sequence of independent single-look probabilities, is controlled. In practice, the boundaries are usually obtained numerically from the joint null distribution. The spending function sets the targets; it does not by itself provide the critical values.

For fixed allocations with \(\sum_{j=1}^{m}\alpha_j\le\alpha\), suppose each endpoint’s sequential test has type I error at most \(\alpha_j\) under its null. Let \(T\) be the set of true endpoint null hypotheses. The probability of at least one false rejection is bounded by

P(\text{at least one false rejection}) \le \sum_{j\in T}P(\text{reject }H_j) \le \sum_{j\in T}\alpha_j \le \sum_{j=1}^{m}\alpha_j \le \alpha.

This is the union-bound justification for strong familywise error control under a fixed Bonferroni-style allocation. It does not require the endpoint test statistics to be independent. The sequential calibration handles repeated looks within an endpoint; the allocation and union bound handle multiplicity across endpoints. More elaborate multiplicity procedures can supply changing local levels, but their rules must be followed explicitly rather than replaced with this fixed-allocation argument.

Worked Examples: Allocation and Spending Are Separate

Worked Example: Assigning Local Budgets Across Two Endpoints

Hypothetical design. A trial has two one-sided endpoint hypotheses and a familywise level \(\alpha=0.05\). The protocol assigns 60% of the total level to the primary endpoint and 40% to the secondary endpoint. Calculate the endpoint allocations and check the total.

The fixed allocations are

\alpha_1=0.60(0.05)=0.030,\qquad \alpha_2=0.40(0.05)=0.020.

Their sum is \(0.030+0.020=0.050\), so the allocations use the full familywise budget under the fixed Bonferroni-style plan. These numbers say how much local error probability each endpoint test may use over all its looks combined. They do not say that the primary endpoint should spend 0.030 at its first look, or that the secondary endpoint should spend 0.020 at its final look. Those timing choices require separate spending functions.

Worked Example: Calculating Look-Specific Spending Increments

Hypothetical design. The primary endpoint has local level \(\alpha_1=0.030\) and three looks at information fractions \(0.40\), \(0.70\), and \(1.00\). A prespecified spending schedule assigns 10%, 55%, and 100% of its local level cumulatively by those looks. Calculate the cumulative targets and the increments.

The cumulative targets are \(A_1(0.40)=0.10(0.030)=0.00300\), \(A_1(0.70)=0.55(0.030)=0.0165\), and \(A_1(1)=1.00(0.030)=0.0300\). Differencing successive cumulative targets gives

\Delta_{11}=0.00300,\qquad \Delta_{12}=0.0165-0.00300=0.0135,\qquad \Delta_{13}=0.0300-0.0165=0.0135.

The increments sum to \(0.00300+0.0135+0.0135=0.0300\), exactly the primary endpoint’s allocation. Under a calibrated sequential test, they correspond to the successive increases in the cumulative null probability of crossing by each look. They are not three extra allocations to add on top of \(\alpha_1\).

Worked Example: Same Allocation, Different Timing

Hypothetical design. A secondary endpoint has local level \(\alpha_2=0.020\) and looks at information fractions \(0.25\), \(0.60\), and \(1.00\). Compare two permissible, prespecified cumulative schedules: Schedule A spends 5%, 40%, and 100% of the local level by those looks; Schedule B spends 25%, 60%, and 100%.

For Schedule A, the cumulative targets are \(0.05(0.020)=0.00100\), \(0.40(0.020)=0.00800\), and \(0.0200\). Its increments are \(0.00100\), \(0.00800-0.00100=0.00700\), and \(0.0200-0.00800=0.0120\).

For Schedule B, the cumulative targets are \(0.25(0.020)=0.00500\), \(0.60(0.020)=0.0120\), and \(0.0200\). Its increments are \(0.00500\), \(0.0120-0.00500=0.00700\), and \(0.0200-0.0120=0.00800\).

Both schedules end at the same local level, \(0.020\), so they do not change the endpoint allocation. Schedule B permits more cumulative type I error early, while Schedule A reserves more for the final look. Their boundary values will therefore differ, even though each is calibrated to the same endpoint budget. A protocol must choose the schedule for its operating characteristics and decision context before examining unblinded interim results.

Why the Cumulative Targets Matter

The relation between cumulative targets and boundaries can be seen directly at the first look. Under the one-sided canonical null model, \(Z_{j1}\sim N(0,1)\). To spend \(A_j(t_1)\) by that look, choose \(c_{j1}\) such that \(P(Z_{j1}\ge c_{j1})=A_j(t_1)\). At the second look, however, \(Z_{j2}\) is correlated with \(Z_{j1}\), and the target is the probability of crossing at either look. The calibration condition is

P(Z_{j1}\ge c_{j1}\ \text{or}\ Z_{j2}\ge c_{j2}\mid H_j)=A_j(t_2).

Because the first-look crossing event is already included, the increase in this probability from look 1 to look 2 is the second spending increment. This explains why simply using a critical value whose single-look tail probability equals \(\Delta_{j2}\) need not yield the desired cumulative spending. The joint probability must account for both looks and their correlation.

Allocate across endpoints.
Specify each endpoint’s local level \(\alpha_j\), ensuring that the multiplicity procedure’s error-control requirements are met.

Choose within-endpoint timing.
For each endpoint, specify information fractions and a nondecreasing cumulative spending function ending at \(\alpha_j\).

Calibrate sequential boundaries.
Use the joint null distribution of the repeated statistics so cumulative crossing probabilities match the spending targets.

Check familywise control.
Verify that endpoint-wise type I error bounds combine according to the chosen multiplicity procedure, such as the fixed allocation and union-bound argument.

Diagnostics, Alternatives, and Limitations

Several checks help reveal whether allocation and spending have been confused or implemented inconsistently:

Endpoint totals: verify that fixed local allocations sum to no more than the familywise level when using a Bonferroni-style plan.
Monotonicity: cumulative spending targets must not decrease as information accumulates; every increment should be nonnegative.
Final target: check that \(A_j(1)=\alpha_j\). A smaller final target is conservative but does not use the full local allocation; a larger one exceeds it.
Boundary calibration: confirm that the probability of crossing by each look equals the cumulative target under the assumed joint null model, not merely that marginal tail areas look plausible.
Information fractions: use the information values corresponding to the analysis, rather than substituting calendar-time fractions without justification.

A simple alternative is to conduct a single analysis at the end using the endpoint’s local level. This avoids repeated interim testing but gives up interim efficacy decisions. Another alternative is to use fixed per-look critical values; unless their joint crossing probability has been calibrated to the endpoint budget, repeated testing can inflate its type I error. Spending functions offer flexibility in how error probability is distributed over time while retaining an overall local level, but they do not automatically address multiplicity across endpoints.

The canonical model is an assumption, not a guarantee. Incorrect information fractions, changes to the analysis statistic, unplanned looks, or a mismatch between the assumed and actual null distribution can make nominal spending targets inaccurate. Spending also does not determine whether interim stopping is desirable, nor does it repair a flawed endpoint allocation. The allocation, schedule, and calibration assumptions should be documented separately.

Common pitfall: Treating each look’s spending increment as a new alpha allocation, or adding the increments to the endpoint’s local level, double-counts error. The correction is to define cumulative targets first, calculate increments by differencing them, and calibrate the sequential boundaries so the cumulative probability of crossing by each look matches those targets.

Check Your Understanding

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

Two endpoints receive fixed local levels 0.025 and 0.015. What is their total allocation, and how much is available for a familywise level of 0.05 under a fixed Bonferroni-style plan?
An endpoint with local level 0.020 has cumulative spending targets 0.002, 0.009, and 0.020 at three looks. Find the three increments.
Why is a second-look critical value generally not obtained by using a one-look tail probability equal to the second spending increment?
Does changing a spending schedule from early spending to late spending change the endpoint’s local allocation if both schedules end at the same value? Explain.

Solutions. (1) The total allocation is \(0.025+0.015=0.040\), leaving \(0.050-0.040=0.010\) unused under that plan. (2) The increments are \(0.002\), \(0.009-0.002=0.007\), and \(0.020-0.009=0.011\); they sum to \(0.020\). (3) The repeated statistics are correlated, so the second boundary must make the probability of crossing at either look equal the second cumulative target; a marginal tail probability alone ignores first-look crossings. (4) No. The timing of cumulative spending changes, but the endpoint allocation remains the common final target, provided both schedules end at that local level.