Admissible Spending Function Conditions

Why a Spending Curve Needs Constraints

A cumulative spending function describes how much of a prespecified type I error budget has been assigned by each point in a trial’s monitoring progress. Its values are cumulative: spending at a later point includes what was assigned earlier. That makes its basic requirements more than convenient design preferences. If a candidate curve decreases, it asks the design to recover error probability already spent; if it exceeds the nominal level, it exceeds the budget.

Let \(t\) denote normalized monitoring progress, with \(t=0\) at the start and \(t=1\) at the maximum planned progress. The argument need not be calendar time. Let \(\alpha\) be the nominal type I error level for the specified test direction or error budget. The cumulative target \(A(t)\) is interpreted under \(H_0\): by progress \(t\), the design aims to have allocated cumulative probability \(A(t)\) to rejecting \(H_0\). The function specifies a target for boundary calibration; it is not itself a rejection boundary.

Definition: A cumulative spending function that fully uses a one-sided error budget \(\alpha\) on \(0\le t\le1\) satisfies \(A(0)=0\), \(A(1)=\alpha\), is nondecreasing, and has range \(0\le A(t)\le\alpha\). For type I error control without a requirement to use the entire budget, the final condition is instead \(A(1)\le\alpha\); the other conditions still apply. Such a function is often called an under-spending rule if \(A(1)<\alpha\).

The endpoint and range conditions make the interpretation explicit. There is no error budget spent at the start, and cumulative spending never exceeds the allowed amount. Monotonicity means \(A(t_2)\ge A(t_1)\) whenever \(t_2\ge t_1\). For a fully spending function, the endpoint values and monotonicity already imply the range condition; it is still useful to check the range directly when inspecting a proposed formula or a numerical implementation.

Why Monotonicity Is Necessary

Under \(H_0\), let \(E(t)\) be the event that the design has crossed its efficacy boundary at least once by progress \(t\). If \(t_1\le t_2\), then any crossing that has happened by \(t_1\) has also happened by \(t_2\). Thus \(E(t_1)\subseteq E(t_2)\), and probability cannot decrease when an event is enlarged:

P(E(t_1)\mid H_0)\le P(E(t_2)\mid H_0).

A target cumulative probability therefore cannot decrease with progress. At the beginning there has been no interim opportunity to reject, so the cumulative target starts at zero. At the end, the probability of rejecting \(H_0\) under the null must be no greater than \(\alpha\). These facts motivate the endpoint, monotonicity, and budget conditions. The null-path derivation of overall alpha gives the related accounting principle: error spent at earlier looks remains part of the overall type I error.

For planned looks at \(t_1<\cdots<t_K=1\), the target increment at look \(k\) is \(\Delta A_k=A(t_k)-A(t_{k-1})\), where \(A(t_0)=A(0)=0\). Nondecreasing \(A\) makes every increment nonnegative, and the increments add to the final target:

\Delta A_k\ge0,\qquad \sum_{k=1}^{K}\Delta A_k =\sum_{k=1}^{K}\{A(t_k)-A(t_{k-1})\} =A(1)-A(0).

For a fully spending function, that total is \(\alpha\). This is a short but important justification: negative increments would mean that a later look is assigned less cumulative error than an earlier one, which cannot represent cumulative crossing probability.

Worked Example: Checking a Power Curve

Suppose a hypothetical design has one-sided \(\alpha=0.05\) and proposes \(A(t)=0.05t^{1.5}\) for \(0\le t\le1\). Check the conditions and find the cumulative targets at \(t=0.20,\ 0.60,\) and \(1.00\).

The endpoints are \(A(0)=0.05(0^{1.5})=0\) and \(A(1)=0.05(1^{1.5})=0.05\). For \(t>0\), the derivative is \(A'(t)=0.05(1.5)t^{0.5}=0.075\sqrt{t}\), which is nonnegative. The curve is therefore nondecreasing on the full interval, and its endpoint values imply \(0\le A(t)\le0.05\).

Substitution gives \(A(0.20)=0.05(0.20^{1.5})=0.004472\), \(A(0.60)=0.05(0.60^{1.5})=0.02324\), and \(A(1)=0.05000\). The increments from the beginning through these points are \(0.004472\), \(0.02324-0.004472=0.01877\), and \(0.05000-0.02324=0.02676\), up to rounding. They are all nonnegative and sum to \(0.05000\). This curve satisfies the conditions for a fully spending function.

Checking Candidate Curves

For a formula proposed on the entire interval, check it on the entire interval, not only at the planned looks shown in a design table. A reliable check is to verify the endpoints, establish monotonicity (often by examining the derivative), and confirm the range. For a piecewise-defined or numerically supplied curve, check monotonicity and range across each piece and at each join. A finite set of reassuring values cannot rule out a dip or overshoot between them.

For a fully spending function with endpoints \(0\) and \(\alpha\), monotonicity alone guarantees the range. If the endpoint is not \(\alpha\), distinguish two cases: an endpoint below \(\alpha\) can be valid under-spending, while an endpoint above \(\alpha\) violates the error budget. If the curve is not monotone, an endpoint below \(\alpha\) does not rescue it; a decrease still conflicts with the cumulative interpretation.

Worked Example: Endpoints Do Not Prove Validity

A hypothetical candidate for \(\alpha=0.05\) is \(A(t)=0.05\{t+0.4\sin(\pi t)\}\). It has \(A(0)=0\) and \(A(1)=0.05\). Are those endpoint checks sufficient?

No. At \(t=0.80\), \(\sin(0.8\pi)\approx0.5878\), so \(A(0.80)=0.05\{0.80+0.4(0.5878)\}=0.05176\), which is greater than \(0.05\). Also, \(A'(t)=0.05\{1+0.4\pi\cos(\pi t)\}\). At \(t=1\), the derivative is \(0.05(1-0.4\pi)\approx-0.01283\), so the curve is decreasing near the endpoint. The proposed function has the right endpoint values but violates both the range and monotonicity requirements. It is not an admissible cumulative spending function.

The error is not repaired by the fact that the curve returns to \(0.05\) at \(t=1\). It already exceeded the budget before the endpoint, and a cumulative error target cannot later decrease to undo that excess.

Worked Example: Turning a Valid Curve into Look Increments

For another hypothetical design with \(\alpha=0.05\), consider \(A(t)=0.05(2t-t^2)\). The planned looks occur at \(t=0.20,\ 0.60,\) and \(1.00\). Check the curve and calculate each target increment.

The endpoints are \(A(0)=0\) and \(A(1)=0.05\). Its derivative is \(A'(t)=0.10(1-t)\), which is nonnegative on \(0\le t\le1\). Thus the curve is nondecreasing and remains between its endpoint values. The cumulative targets are \(A(0.20)=0.05(0.40-0.04)=0.018\), \(A(0.60)=0.05(1.20-0.36)=0.042\), and \(A(1)=0.050\).

The first target increment is \(0.018-0=0.018\). The second is \(0.042-0.018=0.024\). The third is \(0.050-0.042=0.008\). These increments are nonnegative and total \(0.018+0.024+0.008=0.050\). They describe the cumulative targets assigned at the three looks; they are not, by themselves, standardized test-statistic cutoffs.

Regularity, Implementation, and Alternatives

The endpoint, monotonicity, and range requirements express the basic probability-budget logic. In flexible-timing applications, spending functions are commonly taken to be continuous as well. Continuity avoids a sudden jump in the target as monitoring progress changes by an arbitrarily small amount. It is an additional regularity choice, not a substitute for the budget conditions. A nondecreasing step function can make sense for a design with only specified discrete looks, but its jumps do not provide a smoothly varying target between looks.

Most importantly, \(A(t)\) is a calibration target, not an automatic guarantee that a boundary has been correctly constructed. At each planned look, the boundary must be calibrated so the design’s cumulative null crossing probability matches its target, accounting for crossings at earlier looks. If a boundary is not recalculated for a changed schedule, the intended spending targets need not be achieved. The earlier material on Lan-DeMets boundary construction addresses this calibration task; the conditions here are checks on the spending rule supplied to that procedure.

A spending curve is useful when the timing of looks may change: it specifies cumulative targets as a function of progress rather than listing only increments for one fixed schedule. A fixed allocation across planned looks can be simpler when the schedule is certain, but it does not by itself define what target to use at a newly timed look. Neither approach removes the need to control the overall error or calibrate the boundaries. Spending functions also have limitations: the basic conditions do not choose a desirable early-versus-late spending pattern, establish power, or guarantee that a particular practical design is efficient.

Common Mistakes and Diagnostics

Checking only the endpoints. Matching \(A(0)=0\) and \(A(1)=\alpha\) does not rule out an interior decrease or an overshoot. Check monotonicity and range across the full domain.
Treating incremental allocations as cumulative values. \(A(t_k)\) is the total target through look \(k\); the increment is \(A(t_k)-A(t_{k-1})\). Confusing them can make a valid curve appear to exceed the budget.
Assuming the final target must always equal \(\alpha\). Equality describes full use of the nominal budget. A nondecreasing curve ending below \(\alpha\) can still control type I error, though it leaves part of the budget unused.
Assuming valid targets prove valid boundaries. The curve must be paired with boundary calibration under the null. If the schedule changes but the boundaries are left unchanged, the realized cumulative crossing probabilities may no longer match the targets.
Ignoring numerical artifacts. A computed curve that rises at most points but falls slightly at one point may reflect rounding or interpolation, or a genuinely invalid implementation. Inspect unrounded values and the method used to evaluate the curve.

Key takeaway: A fully spending cumulative function starts at zero, ends at \(\alpha\), never decreases, and stays in \([0,\alpha]\). An error-controlling function may end below \(\alpha\), but it must still be nondecreasing and never exceed the budget.

Check Your Understanding

For each question, state the reasoning as well as the conclusion.

A candidate is nondecreasing, begins at zero, and ends at \(0.045\) when \(\alpha=0.05\). Is it valid for error control? Does it fully spend the budget?
A candidate has \(A(0)=0\) and \(A(1)=0.05\), but falls from \(0.031\) at \(t=0.6\) to \(0.029\) at \(t=0.7\). Is it a valid cumulative spending function?
At three planned looks, the cumulative targets are \(0.012,\ 0.030,\) and \(0.050\). Find the incremental targets and verify their total.
Why does checking a curve at only the planned looks fail to establish validity between those looks?

Solutions. (1) Yes, if it remains within range on the full interval: its endpoint is below \(\alpha\), so it controls the budget by under-spending. It does not fully spend the budget because \(A(1)=0.045<0.05\). (2) No. The decrease contradicts monotonicity, even though the endpoints are correct; cumulative spending cannot be taken back. (3) The increments are \(0.012\), \(0.030-0.012=0.018\), and \(0.050-0.030=0.020\). Their sum is \(0.012+0.018+0.020=0.050\), the final cumulative target. (4) A curve may decrease or exceed \(\alpha\) between checked points and return to acceptable values at the looks. Validity requires the conditions throughout the domain, not just at a finite set of points.