Spending Functions Versus Fixed Boundaries

Two Ways to Describe Interim Error Control

A group sequential design may be described by the cutoffs investigators will use at each planned look, or by how much cumulative Type I error the design is allowed to spend as information accumulates. These descriptions are related, but they are not interchangeable. A boundary is a decision threshold at a particular look; a spending function is a rule for allocating cumulative error across information time. Keeping that distinction clear matters especially when an interim analysis occurs earlier or later than planned.

Suppose \(Z(t)\) is the canonical standardized test statistic at information fraction \(t\), where \(0<t\leq 1\) and \(t=1\) denotes the planned final information. Under the null hypothesis and the canonical model, \(Z(t)\sim N(0,1)\), and statistics at two information fractions have correlation \(\sqrt{t_1/t_2}\) when \(t_1\leq t_2\). The model presumes the relevant information is correctly measured and that the joint null distribution is adequately represented by this canonical process. These assumptions are what make null-path error calculations meaningful.

The boundary sequence answers, “What cutoff applies at this look?” The spending function answers, “How much null rejection probability may have accumulated by this amount of information?” For a fixed schedule, the boundary sequence induces cumulative spending values: at look \(k\), the induced value is the null probability of crossing at or before that look. Conversely, given a spending rule and the actual look times, boundaries are calibrated so that the cumulative crossing probability at each look matches the rule.

How the Two Descriptions Are Connected

Write \(t_1<\cdots<t_K=1\). For a symmetric two-sided boundary sequence, define the event of crossing by look \(k\) as \(C_k=\{\text{there is some }j\leq k\text{ for which }|Z(t_j)|\geq c_j\}\). The spending constraint at look \(k\) is \(P_{H_0}(C_k)=A(t_k)\). This probability is cumulative: it includes a crossing at any earlier look, not just a crossing at look \(k\).

At the first look, there is no earlier crossing event to account for. Thus, for a two-sided first-look spend \(A(t_1)\), the boundary satisfies \(2\{1-\Phi(c_1)\}=A(t_1)\), where \(\Phi\) is the standard normal cumulative distribution function. At later looks, simply setting the marginal tail probability at that look equal to the increase in spending is generally wrong. The calculation must account for paths that already crossed a previous boundary.

This identity explains why a spending increment is not, in general, the marginal probability of crossing the current boundary. It is the probability of a new crossing at this look among paths that have not crossed before. The earlier material on cumulative alpha spending and boundary construction develops how these probabilities are used to determine cutoffs. Here the important point is that the function and the cutoffs represent different parts of that calculation.

What a Fixed Sequence Does—and Does Not—Specify

A fixed sequence is a complete operational rule only in conjunction with its planned looks and the statistical model used to calibrate it. At the planned times, its overall Type I error is the probability of at least one null crossing. For a two-sided design this is \(P_{H_0}(\bigcup_{k=1}^K\{|Z(t_k)|\geq c_k\})\). Because the statistics across looks are correlated, adding marginal tail probabilities does not usually give that overall probability.

The sequence alone does not say how its cutoffs should change if a look takes place at a different information fraction. Nor does it specify an interpolation rule between planned looks. If investigators apply the same numerical cutoff at a new time, they have made a choice about the boundary; they have not automatically preserved the original cumulative error allocation. In contrast, a spending rule supplies a target cumulative error at the actual information fraction, allowing the boundary calibration to be updated for that timing.

Induced Spending and a Two-Look Calculation

A fixed sequence can always be interpreted through the cumulative error it induces at its planned looks, provided its null joint distribution and look schedule are specified. That interpretation does not make it a spending rule for other schedules: the induced values are tied to the particular sequence of information times and cutoffs.

When Timing Changes, and How to Diagnose Trouble

The practical advantage of a spending rule is that its target is indexed by information time rather than by a fixed list of calendar dates. If an analysis is delayed, brought forward, or conducted at an unplanned information fraction, the rule can be evaluated at the observed information fraction and the boundary recalibrated to meet the cumulative target. This flexibility relies on using valid information fractions and on monitoring conditions that support the canonical null model. A rule indexed by information does not license choosing when to look after examining the test statistic in a way that invalidates the assumed error calculation.

A useful diagnostic is to compare, at every look, the cumulative null crossing probability implied by the implemented boundaries with the intended cumulative spending \(A(t_k)\). The comparison should be based on the joint path probability, not just the marginal tail probability at each look. Warning signs include using planned-time boundaries at different information times without recalibration, treating a spending increment as the current-look marginal tail probability, and reporting only the final-look tail area as the overall Type I error.

The main alternatives have different strengths. A fixed sequence is transparent and convenient when the number and timing of looks are genuinely fixed and the sequence has been calibrated for that schedule. A spending rule is more adaptable to changes in information timing, because it states the cumulative error target at each information fraction. Neither description removes the need to model the joint null distribution correctly. A spending rule also does not decide the scientific monitoring schedule, supply futility criteria, or protect against arbitrary outcome-dependent looks without suitable validity conditions.

Common Mistakes and Limits

• Calling a spending value a boundary. A value such as \(A(t)=0.0100\) is a cumulative probability, not a \(Z\)-statistic cutoff. A boundary is obtained by translating the probability target through the null distribution and accounting for earlier looks.
• Adding marginal error probabilities. Crossing events at different looks are dependent. Overall Type I error is a probability of a union of events, not generally the sum of their marginal probabilities.
• Using an increment as a marginal tail area. \(A(t_k)-A(t_{k-1})\) is the probability of a new crossing at look \(k\) among paths not previously rejected. It is not generally \(P_{H_0}(|Z(t_k)|\geq c_k)\).
• Assuming fixed cutoffs travel unchanged to new times. A cutoff sequence is calibrated for its specified look schedule. When timing changes, its induced cumulative spending can differ from the intended spending rule.

These approaches are not competing definitions of alpha. For a fixed schedule and a fully specified null model, a boundary sequence has an induced cumulative error profile. The spending-function formulation makes the desired cumulative profile explicit and provides a basis for constructing boundaries at the information times actually used. Its flexibility remains conditional on valid information measurement, appropriate monitoring conditions, and correct calibration.