Why Unequal Information Gains Matter
A group sequential design may have interim analyses at information fractions that are far apart, such as 0.10 and 0.90, rather than at evenly spaced fractions. The cumulative alpha spending function still specifies how much Type I error may have been spent by each look. But the amount spent at a look is not generally the marginal probability of exceeding that look’s cutoff. It is the probability of crossing there after not crossing at any earlier look.
That distinction matters when calibrating efficacy boundaries. The test statistics at different looks are correlated, and their correlations depend on the information fractions. A later cutoff must therefore be solved using both the spending increment and the probability of having remained below every earlier boundary. Large gaps between looks change those correlations and can make a marginal-tail shortcut especially misleading.
This model assumes the planned information fractions correctly describe the analysis times and that the null statistics follow the canonical joint normal distribution. The formula applies to the upper, one-sided efficacy boundaries considered here. The cumulative spending function \(A(t)\) is nondecreasing, satisfies \(A(0)=0\), and does not exceed the design’s one-sided Type I error level \(\alpha\). Earlier material on spending functions establishes how a spending curve specifies cumulative error; here the focus is how to translate its values into cutoffs when information increments differ.
From Cumulative Spending to First-Crossing Targets
Write \(A_i=A(t_i)\). The cumulative spending target at look \(i\) includes error spent at all previous looks. The target for a first crossing at look \(i\) is the increment \(\Delta_i=A_i-A_{i-1}\), where \(A_0=0\). For the first look, the cutoff is a standard normal upper quantile. At later looks, the incremental target is a joint probability that includes all earlier no-crossing conditions.
These equations calibrate the boundaries sequentially. At each look, the earlier cutoffs are already determined; the new cutoff is chosen to make the first-crossing probability equal to the new spending increment. The cutoff is not obtained by setting \(P(Z_i\ge b_i)=\Delta_i\), because that marginal tail includes paths that crossed earlier.
At each planned fraction \(t_i\), calculate \(A_i=A(t_i)\) and take differences to obtain \(\Delta_i=A_i-A_{i-1}\).
Set \(P(Z_1\ge b_1)=A_1\), so \(b_1=\Phi^{-1}(1-A_1)\), where \(\Phi\) is the standard normal cumulative distribution function.
Use the joint null distribution and all previous restrictions \(Z_j<b_j\) to solve for \(b_i\) such that the first-crossing probability equals \(\Delta_i\).
Sum the first-crossing probabilities through each look. The sum through look \(i\) should equal \(A_i\), up to numerical integration and rounding error.
Why the Boundary Equation Is Correct
Define the event \(C_i\) to be a first crossing at look \(i\): the statistic is below every earlier boundary and is at or above the current boundary. The events \(C_1,\ldots,C_i\) are disjoint, because a path can have only one first crossing. Their union is exactly the event of crossing at least one boundary by look \(i\). Thus \(P(\bigcup_{j=1}^i C_j)=\sum_{j=1}^iP(C_j)\).
If the boundaries are calibrated so that \(P(C_j)=A_j-A_{j-1}\), the sum telescopes: \(\sum_{j=1}^i(A_j-A_{j-1})=A_i-A_0=A_i\). This justifies the sequential equations: they allocate the cumulative spending into disjoint first-crossing probabilities. The argument also explains why the no-prior-crossing restrictions cannot be omitted.
Worked Example: A Large Gap Between the First Two Looks
Suppose a hypothetical one-sided design has looks at \(t_1=0.10\) and \(t_2=0.90\). The prescribed cumulative spending targets are \(A_1=0.0010\) and \(A_2=0.0200\). Find the second cutoff, given \(b_1=3.090\).
The first-look cutoff is consistent with the first target: \(P(Z_1\ge3.090)\approx0.00100\). The new target is not 0.0200, because 0.0010 was already spent at look 1. The incremental target is \(\Delta_2=0.0200-0.0010=0.0190\).
The canonical correlation is \(\operatorname{Corr}(Z_1,Z_2)=\sqrt{0.10/0.90}=1/3\). Consequently, the second cutoff must satisfy \(P(Z_1<3.090,Z_2\ge b_2)=0.0190\). Numerical evaluation of this bivariate normal probability and a one-dimensional cutoff solve give \(b_2\approx2.071\).
At this cutoff, the marginal upper tail \(P(Z_2\ge2.071)\) is about 0.0192, slightly larger than the required first-crossing probability. The difference is the small overlap with paths that had already crossed at look 1. The first-crossing probability is approximately 0.0190, so the cumulative error through look 2 is approximately \(0.0010+0.0190=0.0200\), as required.
Recalibrating a Final Look After Uneven Increments
A common calculation error is to calibrate a final cutoff using only the immediately preceding look, or to round a preceding cutoff and then reuse an old final cutoff without checking the full crossing probability. The correct final equation includes every earlier no-crossing condition. With three looks, that is a trivariate probability, even if conditional integration is used to evaluate it.
Worked Example: A Final-Look Increment of 0.0050
Continue with a hypothetical design at \(t_1=0.10\), \(t_2=0.90\), and \(t_3=1.00\). Let the cumulative targets be \(0.0010\), \(0.0200\), and \(0.0250\). Use \(b_1=3.090\) and the recalibrated \(b_2=2.071\) from the preceding example. The final increment is \(0.0250-0.0200=0.0050\).
The required final cutoff solves \(P(Z_1<3.090,Z_2<2.071,Z_3\ge b_3)=0.0050\). The correlations with the final statistic are \(\operatorname{Corr}(Z_1,Z_3)=\sqrt{0.10}\approx0.3162\) and \(\operatorname{Corr}(Z_2,Z_3)=\sqrt{0.90}\approx0.9487\). The final statistic is therefore strongly associated with the second-look statistic, and paths near or above the second boundary cannot be counted again as new crossings.
Numerical integration of the trivariate probability, followed by a one-dimensional root solve, gives a final cutoff of about \(b_3=2.10\) (approximately 2.101 at three decimals). Substituting the cutoff into the probability with both prior restrictions gives a first-crossing probability of approximately 0.0050. The cumulative crossing probability is therefore approximately \(0.0010+0.0190+0.0050=0.0250\).
This check is essential: solving with \(Z_2<2.071\) alone, or using only \(P(Z_3\ge b_3)\), does not impose the complete history of the trial. Even when the first boundary is high and its additional effect is small, it belongs in the stated probability equation.
Working With a Spending Curve and Irregular Looks
The spending increments can vary substantially even when the final \(\alpha\) is fixed. The following hypothetical calculation illustrates how to obtain targets from a spending curve before calibrating the later cutoffs. It uses the Hwang–Shih–DeCani form \(A(t)=\alpha\{1-\exp(-\gamma t)\}/\{1-\exp(-\gamma)\}\), with \(\gamma=-4\), \(\alpha=0.025\), and looks at \(t=0.10,0.60,1.00\). The curve is being used only to obtain cumulative targets; each efficacy cutoff still requires the joint-probability calculation.
Worked Example: Unequal Increments From a Spending Curve
At \(t=0.10\), the spending is \(A(0.10)=0.025\{e^{0.4}-1\}/\{e^4-1\}\approx0.000229\). At \(t=0.60\), \(A(0.60)=0.025\{e^{2.4}-1\}/\{e^4-1\}\approx0.004675150\). At the final look, \(A(1)=0.025\).
Subtract consecutive cumulative values to obtain the first-crossing targets: \(\Delta_1\approx0.000229\), \(\Delta_2\approx0.004446\), and \(\Delta_3\approx0.020325\). They sum to 0.025000, which checks the arithmetic. The first cutoff is \(b_1=\Phi^{-1}(1-0.000229)\approx3.50\).
For look 2, the correlation is \(\sqrt{0.10/0.60}\approx0.4082\), and \(b_2\) solves \(P(Z_1<b_1,Z_2\ge b_2)=0.004446\). For the final look, the relevant correlations are \(\sqrt{0.10}\approx0.3162\) and \(\sqrt{0.60}\approx0.7746\), and \(b_3\) solves \(P(Z_1<b_1,Z_2<b_2,Z_3\ge b_3)=0.020325\). The increments are therefore fully determined by the curve, while the later cutoffs are determined by those increments together with the joint null distribution. A marginal normal quantile for either later increment would not solve these equations.
Diagnostics, Alternatives, and Limitations
A useful diagnostic is to calculate the first-crossing probabilities after obtaining the boundaries and form their cumulative sums. At look \(i\), that sum should agree with \(A(t_i)\). Also check that the spending increments are nonnegative and that the solved boundaries behave consistently with their targets: for a fixed history of earlier restrictions, a larger incremental target requires a lower cutoff. Small numerical discrepancies can arise from integration tolerances and rounding, but a discrepancy larger than those tolerances calls for checking the probability equation, correlations, and inputs.
The main shortcuts and alternatives differ in what they control:
| Approach | What it uses | Consequence |
|---|---|---|
| Incremental first-crossing calibration | Spending differences and all earlier no-crossing restrictions | Matches the planned cumulative spending under the canonical null model |
| Marginal normal cutoffs | The tail probability at each look alone | Does not account for crossings at prior looks and generally misstates cumulative error |
| Fixed boundary design | A prespecified boundary pattern rather than the spending targets at each information time | Can be a valid alternative design, but it need not match a chosen spending curve at irregular looks |
Unequal information increments are not themselves a failure of spending-function calibration. They do, however, make it especially important to use the actual \(t_i\) values in the correlation structure. The method also depends on the assumed canonical model and on the planned analysis times. If the actual information differs from the plan or the analysis schedule changes, the original numerical boundaries may no longer deliver the intended spending at the realized looks. Spending-function calibration controls the specified null crossing probability; it does not ensure adequate power, correct implementation, or robustness to departures from the model.
Common Mistakes
- Using cumulative spending as the next-look probability. The correct target is \(A_i-A_{i-1}\), not \(A_i\).
- Equating the incremental target to a marginal tail. The target is a first-crossing probability and must exclude earlier crossings.
- Using equal-look correlations for an uneven schedule. Under the canonical model, the correlation is \(\sqrt{t_i/t_j}\), calculated from the actual fractions.
- Checking only the last cutoff. Verify the first-crossing probabilities and cumulative sums at every look, using all earlier restrictions.
- Rounding too early. Retain unrounded boundaries during calibration and verification; round only when reporting the final values.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Cumulative spending is 0.002 at look 1 and 0.018 at look 2. What first-crossing probability should the second cutoff target?
- At information fractions \(t_1=0.16\) and \(t_2=0.64\), what is the canonical null correlation between the two statistics?
- Why is \(P(Z_2\ge b_2)\) generally not the correct calibration equation for a second-look cutoff?
- Three cumulative spending targets are 0.001, 0.012, and 0.025. Calculate the three incremental targets and verify their sum.
Solutions. (1) \(0.018-0.002=0.016\). (2) \(\sqrt{0.16/0.64}=0.50\). (3) The marginal tail includes paths that crossed at look 1; the correct probability is \(P(Z_1<b_1,Z_2\ge b_2)\). (4) The increments are 0.001, \(0.012-0.001=0.011\), and \(0.025-0.012=0.013\); their sum is \(0.001+0.011+0.013=0.025\), the final cumulative target.