From Error-Budget Targets to a Spending Curve
A spending function lets a group sequential design specify how much of its type I error budget may have been used by each point in the study’s information timeline. A custom curve is useful when the design rationale is naturally expressed as targets: for example, no more than a small amount of error spent at an early look, a larger cumulative amount by a middle look, and the full budget by the final analysis. Turning those targets into a curve requires more than joining numbers: the curve must spend error in a valid order and meet the required endpoints.
Here, \(t\) denotes information fraction, scaled so that \(t=0\) is the start and \(t=1\) is the planned final analysis. The targets \(a_E\) and \(a_M\) are cumulative spending amounts at prespecified fractions \(t_E\) and \(t_M\), where \(0<t_E<t_M<1\). For a one-sided efficacy design, \(\alpha\) is the total type I error budget. For a two-sided design, the same construction can be applied to the cumulative total error budget across both tails.
The target values are probabilities, not boundary values and not nominal significance levels for separate tests. Their purpose is to describe cumulative null rejection probability. Once the curve is specified, its values at the planned analyses are used by the boundary-calibration procedure.
Assumptions and the Piecewise-Linear Construction
To connect a spending curve to efficacy boundaries, suppose the design uses the canonical group sequential statistics at planned information fractions \(0<t_1<\cdots<t_K=1\). Under \(H_0\), each \(Z_i\) has a standard normal distribution, and for \(i\leq j\) their canonical covariance structure is
This is the standard information-based model for the boundary construction discussed here. The analysis schedule and the statistical model must support this null distribution; changing the model or using a different testing rule can require a different calibration. For a one-sided design, boundaries \(b_i\) are calibrated so that the probability of crossing at or before look \(i\) under \(H_0\) equals \(A(t_i)\). Thus \(A(t_i)-A(t_{i-1})\) is the targeted increase in cumulative crossing probability between consecutive looks, with \(A(t_0)=0\).
For a custom curve, add the required endpoints to the user-specified targets. The resulting ordered knots are \((0,0)\), \((t_E,a_E)\), \((t_M,a_M)\), and \((1,\alpha)\). Between adjacent knots, use straight-line interpolation. If the ordered knot values are \((u_j,v_j)\) and \((u_{j+1},v_{j+1})\), then for \(u_j\leq t\leq u_{j+1}\),
This defines a continuous curve and preserves every target exactly. The piecewise-linear choice is one transparent construction, not the only possible curve through the targets. Other interpolations need their own checks: a smooth curve that overshoots between knots is not valid merely because its target values are ordered.
Why the Target-Ordering Check Works
On each segment, the slope is the difference in cumulative spending divided by the positive difference in information fraction. For example, the early-segment slope is \(a_E/t_E\), and the middle-segment slope is \((a_M-a_E)/(t_M-t_E)\). The final-segment slope is \((\alpha-a_M)/(1-t_M)\). Since every denominator is positive, all three slopes are nonnegative exactly when \(0\leq a_E\leq a_M\leq\alpha\).
A linear segment with nonnegative slope cannot decrease between its endpoints. The segments meet at their shared target values, so the whole curve is continuous and nondecreasing. At the first and final knots, the construction gives \(A(0)=0\) and \(A(1)=\alpha\). It also stays between 0 and \(\alpha\): on each segment, \(A(t)\) is a weighted average of its two endpoint values, with nonnegative weights summing to one. This proves the stated check, rather than relying only on a visual impression of the curve.
Worked Example: Constructing a One-Sided Curve
Consider a hypothetical one-sided design with \(\alpha=0.025\), an early target \(a_E=0.001\) at \(t_E=0.20\), and a middle target \(a_M=0.009\) at \(t_M=0.60\). The targets are ordered: \(0\leq0.001\leq0.009\leq0.025\). The piecewise-linear curve is therefore nondecreasing and has the required endpoints.
On the first segment, \(0\leq t\leq0.20\),
On the second segment, \(0.20\leq t\leq0.60\), the slope is \((0.009-0.001)/(0.60-0.20)=0.008/0.40=0.020\), giving
On the final segment, \(0.60\leq t\leq1\), the slope is \((0.025-0.009)/(1-0.60)=0.016/0.40=0.040\), so
For example, at \(t=0.40\), \(A(0.40)=0.001+0.020(0.40-0.20)=0.005\). At \(t=0.80\), \(A(0.80)=0.009+0.040(0.80-0.60)=0.017\). The curve reaches 0.001, 0.009, and 0.025 at the early, middle, and final targets, respectively. If planned analyses occur at those three fractions, the corresponding cumulative crossing-probability targets are 0.001, 0.009, and 0.025.
Reading the Curve as an Error Budget
Cumulative values and interval increments answer different questions. A target such as \(A(0.60)=0.009\) says that the cumulative null crossing probability by information fraction 0.60 is 0.009. It does not say that the analysis at 0.60 has a nominal level of 0.009. If an earlier analysis has cumulative target 0.001, then the increase in the cumulative target between those two fractions is \(0.009-0.001=0.008\). Under boundary calibration, that difference is the targeted additional crossing probability over that interval, accounting for the design’s earlier stopping rule.
The increments over all intervals add to the total budget because they telescope:
This identity is a useful audit. It checks that the increments implied by the curve reconcile with the final budget. The increments are not generally equal to separate one-look test levels that could be applied independently: group sequential statistics are correlated, and crossing probabilities at later looks are evaluated after accounting for earlier crossings.
Worked Example: Checking the Implied Interval Budgets
A hypothetical trial has \(\alpha=0.05\), targets \(A(0.25)=0.002\) and \(A(0.50)=0.015\), and the required final target \(A(1)=0.05\). The ordered target values satisfy \(0\leq0.002\leq0.015\leq0.05\), so they define a valid piecewise-linear curve.
The first interval, from 0 to 0.25, has target increment \(0.002-0=0.002\). The next interval, from 0.25 to 0.50, has increment \(0.015-0.002=0.013\). The final interval, from 0.50 to 1, has increment \(0.05-0.015=0.035\). Adding them gives \(0.002+0.013+0.035=0.050\), the full type I error budget.
The curve’s value at \(t=0.75\) lies halfway through the final segment: \(A(0.75)=0.015+\{(0.75-0.50)/(1-0.50)\}(0.05-0.015)=0.015+0.5(0.035)=0.0325\). Thus the specified curve allows cumulative null crossing probability of 0.0325 by information fraction 0.75. This is a cumulative target, not a p-value threshold for a single analysis.
Infeasible Targets and How to Revise Them
A target specification can fail before any boundary calculation is attempted. For example, an early cumulative target cannot exceed a later cumulative target; otherwise the design would be required to “unspend” error. Similarly, the final target cannot exceed the overall budget, and a negative target is not a probability. A curve-fitting routine cannot make contradictory targets valid without changing at least one design choice.
Worked Example: Diagnosing an Out-of-Order Specification
Suppose a hypothetical one-sided design has \(\alpha=0.025\), \(t_E=0.20\), \(a_E=0.004\), \(t_M=0.60\), and \(a_M=0.003\). Although both targets lie between 0 and 0.025, their order is invalid because \(0.004>0.003\). The middle target would require cumulative spending to fall by \(0.001\).
The corresponding middle-segment slope would be \((0.003-0.004)/(0.60-0.20)=-0.001/0.40=-0.0025\), which is negative. Therefore no nondecreasing curve can pass through all the specified knots. If the design rationale requires keeping the early target at 0.004, the middle target must be at least 0.004; for example, revising it to 0.007 gives ordered targets \(0.004\leq0.007\leq0.025\). Alternatively, if the middle target 0.003 is essential, the early target must be no greater than 0.003. These are different design decisions, not mathematical repairs that can be made without consultation.
Calibration, Alternatives, and Practical Limits
Once the curve passes its checks, evaluate it at the planned information fractions. The resulting cumulative targets provide the inputs for the Lan-DeMets boundary construction: under the stated canonical null model, the boundaries are chosen to match the cumulative crossing probabilities. The curve alone does not give numerical boundary statistics. Those also depend on the planned looks, the joint distribution of the statistics, and whether the rule is one-sided or two-sided. For a two-sided design, specify clearly whether \(A(t)\) represents the combined error budget across both tails or a per-tail budget; do not treat the same number as both.
A custom piecewise-linear curve is especially interpretable because every knot has a stated cumulative target and every interval increment is visible. Its limitations are also worth recognizing. The slope changes abruptly at the knots, although the curve itself remains continuous. The targets reflect design priorities, so valid arithmetic does not establish that they provide acceptable power or a scientifically appropriate balance between early and late evidence. The final curve and its calibrated boundaries should be evaluated against the study’s operating goals.
Standard spending families, such as Hwang-Shih-DeCani or power spending, offer compact parameterized forms. They are convenient when a familiar shape is appropriate and the shape parameter can be calibrated to design preferences. A custom curve is an alternative when the desired cumulative budgets are easier to state directly than to express through a family parameter. Fixed boundaries are another approach, but they do not in general adapt their cumulative error spending to the information timeline in the same way. The choice is between design representations; each still requires correct calibration under the planned model.
Common diagnostics include checking that information fractions are strictly ordered and that all targets use the same cumulative-error convention and units. Verify \(A(0)=0\), \(A(1)=\alpha\), target ordering, and the interval-increment sum. If any segment has a negative slope, the specification is not a valid cumulative spending curve. If targets are ordered but the calibrated boundary results do not match the intended cumulative crossing probabilities, check the look schedule, null covariance model, sidedness, and implementation of the boundary calculation.
- Common mistake: Treating an early cumulative target of 0.002 and a middle cumulative target of 0.015 as independent per-look significance levels, then adding them to obtain the total error.
- Correction: They are cumulative targets. The increment between those looks is \(0.015-0.002=0.013\), and the boundary calibration accounts for dependence and earlier crossings. Check increments across the full schedule; they sum to \(\alpha\).
Check Your Understanding
Use the cumulative-target interpretation throughout. Solutions follow the questions.
- A one-sided design has \(\alpha=0.02\), with targets \(A(0.30)=0.003\) and \(A(0.70)=0.012\). Is the piecewise-linear construction valid? What are its three interval increments?
- For the targets in Question 1, find \(A(0.50)\).
- A design specifies \(A(0.25)=0.006\) and \(A(0.65)=0.004\), with \(\alpha=0.02\). What condition fails, and what is the slope between the targets?
- Explain why a target \(A(0.70)=0.012\) is not, by itself, the nominal significance level of the analysis at \(t=0.70\).
Solutions. 1. Yes: \(0\leq0.003\leq0.012\leq0.02\). The increments are \(0.003\), \(0.012-0.003=0.009\), and \(0.02-0.012=0.008\); they sum to 0.020. 2. The fraction 0.50 is halfway between 0.30 and 0.70, so \(A(0.50)=0.003+0.5(0.012-0.003)=0.0075\). 3. Monotonicity fails because the later cumulative target is smaller. The slope is \((0.004-0.006)/(0.65-0.25)=-0.002/0.40=-0.005\), so the segment decreases. 4. The value 0.012 is the cumulative null crossing-probability target by that information fraction. A boundary’s calibration accounts for the sequence of analyses and earlier crossings; 0.012 is not automatically a separate one-look test level.