Why One-Sided Efficacy Spending Needs Upper Boundaries
In a one-sided group sequential test, evidence in the prespecified direction is assessed at more than one analysis. At each look, an upper boundary determines whether the evidence is strong enough to stop for efficacy. The central calibration question is not simply how much probability lies above each boundary separately. It is how much probability, under the null hypothesis, has crossed at least one boundary by that look.
Alpha spending expresses that cumulative target as a function of the information fraction. It allows the target to be specified over the course of the trial, while the upper boundaries are calculated using the dependence among the sequential test statistics. The introductory material on cumulative alpha spending defines the general target; here the focus is its one-sided efficacy-boundary construction.
Model and Boundary Construction
For the standard canonical calculation, suppose the planned information fractions satisfy \(0<t_1<\cdots<t_K=1\). Under the null hypothesis at its boundary, the sequential statistics have mean zero, variance one, and joint normal dependence $$ \operatorname{Corr}_0(Z_i,Z_j)=\sqrt{\frac{t_i}{t_j}}, \qquad i\leq j. $$ Equivalently, the statistics can be represented as standardized Brownian motion observed at increasing information times. This model assumes that the test statistic and information measure support the canonical joint distribution, that information does not decrease, and that the direction of efficacy was specified in advance. For a one-sided composite null, calibration at its boundary controls the rejection probability over the null when larger values are stochastically more favorable to efficacy.
The spending function is nondecreasing, starts at \(A(0)=0\), and has final target \(A(1)=\alpha\). At look \(k\), the boundary is chosen so that the probability of crossing by that look equals \(A(t_k)\), given the boundaries already chosen. At the first look, there is no earlier boundary to account for, so $$ P_0(Z_1\geq b_1)=A(t_1), \qquad b_1=\Phi^{-1}\{1-A(t_1)\}, $$ where \(\Phi\) is the standard normal cumulative distribution function.
At a later look, the increase \(A(t_k)-A(t_{k-1})\) is the probability of a new crossing at look \(k\): the statistic must be above its new boundary and the path must not have crossed an earlier boundary. Thus \(b_k\) solves $$ P_0\left(Z_1<b_1,\ldots,Z_{k-1}<b_{k-1},Z_k\geq b_k\right) =A(t_k)-A(t_{k-1}). $$ The left side depends on the joint distribution, not just the marginal distribution of \(Z_k\). This is why later boundaries cannot generally be obtained by applying a separate normal quantile to the incremental spending amount.
Worked Example: Calculating the First Upper Boundary
Suppose a hypothetical one-sided design has information fraction \(t_1=0.40\) at its first look, and the prespecified spending target is \(A(t_1)=0.0100\). The canonical null model gives \(Z_1\sim N(0,1)\), so the boundary satisfies \(P_0(Z_1\geq b_1)=0.0100\).
Using the upper 1% standard normal quantile, $$ b_1=\Phi^{-1}(1-0.0100)=\Phi^{-1}(0.9900)\approx 2.326. $$
The trial stops for efficacy at this look if \(Z_1\geq 2.326\). Under the null model, the probability of crossing at this first look is 0.0100. This is also the cumulative spending through the first look because no earlier efficacy boundary exists.
Why Later Boundaries Use the Surviving Paths
At a later look, some null paths have already crossed an earlier efficacy boundary. They are already included in cumulative spending and must not be counted again. A complete justification of the recursion follows directly by splitting the event of crossing by look \(k\) into two disjoint events: crossing by look \(k-1\), or avoiding every earlier boundary and crossing at look \(k\). Therefore, $$ \begin{aligned} P_0\left(\max_{1\leq j\leq k}Z_j\geq b_j\right) ={}&P_0\left(\max_{1\leq j\leq k-1}Z_j\geq b_j\right)\\ &+P_0\left(Z_1<b_1,\ldots,Z_{k-1}<b_{k-1},Z_k\geq b_k\right). \end{aligned} $$
Setting the first probability on the right to \(A(t_{k-1})\) and the left side to \(A(t_k)\) gives the later-look equation above. For two looks, the canonical correlation is \(\rho=\sqrt{t_1/t_2}\), and \(Z_2\mid Z_1=z\) is normal with mean \(\rho z\) and variance \(1-\rho^2\). The new-crossing probability can then be written as a one-dimensional integral: $$ \int_{-\infty}^{b_1}\phi(z) \left[1-\Phi\left(\frac{b_2-\rho z}{\sqrt{1-\rho^2}}\right)\right]\,dz, $$ where \(\phi\) is the standard normal density. Solving this equation for \(b_2\) is the two-look boundary update.
Worked Example: Updating the Boundary at the Final Look
Consider a hypothetical one-sided design with two looks at \(t_1=0.50\) and \(t_2=1.00\). Let the cumulative spending targets be \(A(t_1)=0.00500\) and \(A(t_2)=0.0250\). The first boundary is $$ b_1=\Phi^{-1}(0.99500)\approx 2.576. $$
The final look must add \(0.0250-0.00500=0.0200\) of null crossing probability among paths that did not cross at look 1. Here \(\rho=\sqrt{0.50/1.00}=0.7071\), so the final boundary solves $$ \int_{-\infty}^{2.576}\phi(z) \left[1-\Phi\left(\frac{b_2-0.7071z}{0.7071}\right)\right]\,dz=0.0200. $$
Numerical evaluation of this integral gives \(b_2\) approximately \(2.00\) (about \(2.00\) to three significant figures). The cumulative null crossing probability is consequently \(0.00500+0.0200=0.0250\), as targeted. The boundary is not the ordinary one-look 2.5% quantile, approximately \(1.960\): paths that crossed the first boundary must be excluded from the new-crossing probability at the final look.
Reading and Checking a One-Sided Design
A boundary is a rule for the sequential procedure, not an independent significance threshold divorced from earlier decisions. If the observed \(Z_k\) reaches or exceeds \(b_k\), the efficacy criterion is met at that look. If it is below \(b_k\), the procedure has not stopped for efficacy at that look; whether the trial continues depends on the prespecified design and any other stopping rules.
The spending target has a direct repeated-testing interpretation. At look \(k\), \(A(t_k)\) is the probability, under the null calibration model, that the procedure would have crossed an efficacy boundary at or before that look. It is not the marginal probability \(P_0(Z_k\geq b_k)\) in general, and it is not the probability that the null hypothesis is true. At the final look, the cumulative target is \(\alpha\), giving the design's one-sided overall type I error under the assumed null model.
Worked Example: Applying Boundaries to Hypothetical Results
Use the two-look design above, with \(b_1=2.576\) and \(b_2\approx2.00\). Suppose a hypothetical trial reports \(Z_1=2.40\) and, after continuing, \(Z_2=2.05\).
At look 1, \(2.40<2.576\), so the efficacy boundary is not crossed and the trial continues under this design. At look 2, \(2.05\geq2.00\), so the final efficacy boundary is crossed and the one-sided test rejects \(H_0\) at the prespecified sequential criterion.
The final statistic's ordinary one-sided normal tail probability is approximately \(1-\Phi(2.05)=0.0202\). That marginal tail probability does not replace the sequential calibration: the decision rule is the comparison with the look-specific boundary, and the design's cumulative null crossing target at the final look is \(0.0250\). The distinction matters because the trial had an earlier opportunity to stop.
Diagnostics, Alternatives, and Limitations
Several checks help detect a faulty construction. Confirm that \(A(t_k)\) never decreases and that the final target equals the intended one-sided \(\alpha\). Confirm that the first boundary has upper-tail probability \(A(t_1)\). For later looks, verify the cumulative crossing probability by integrating or simulating under the specified joint null model; checking only \(P_0(Z_k\geq b_k)\) is insufficient. Boundaries should also be compatible with the actual information fractions used in calibration. If the assumed canonical covariance, information ordering, test direction, or spending targets do not match the design, the calculated crossing probabilities need not be valid for the trial.
A fixed set of classical boundaries instead specifies cutoffs for a particular look schedule and calibration. Spending-based construction specifies cumulative error targets across information time and obtains boundaries to meet them. Incremental alpha allocation is a useful way to describe the added crossing probability at each look, but it is not generally a sequence of independent marginal tests: each increment is assigned among paths that survived earlier efficacy looks. These approaches differ in how the design is specified, not in the need to account for repeated looks.
One-sided efficacy spending also has limits. It controls the specified upper-tail crossing probability, not a lower-tail or two-sided error criterion. It relies on the assumed joint null distribution and on a prespecified direction of benefit. A boundary calculation alone does not establish adequate power, address the clinical value of an effect, or determine whether a trial should continue for futility. Those are distinct design questions.
Common Mistake
- Wrong reasoning: “The final spending increment is 0.020, so I can set the final boundary to the normal cutoff with upper-tail probability 0.020.” This treats the final statistic as if no earlier look occurred.
- Correction: The increment is \(P_0(\text{survive earlier efficacy looks and cross now})\). Calculate it under the joint distribution and solve for the boundary using only paths that have not previously crossed.
- Wrong reasoning: “The cumulative target \(A(t_k)\) is the tail probability beyond \(b_k\) at look \(k\).”
- Correction: It is the probability of crossing any efficacy boundary through look \(k\). At the first look these probabilities coincide; at later looks they generally do not.
Check Your Understanding
Answer each question using the one-sided null calibration described above.
- If \(A(t_1)=0.0200\), what equation determines the first boundary \(b_1\)?
- Why is the later-look increment \(A(t_k)-A(t_{k-1})\) not usually the marginal upper-tail probability at look \(k\)?
- In a two-look canonical design with \(t_1=0.25\) and \(t_2=1\), what is the null correlation between \(Z_1\) and \(Z_2\)?
- What does \(A(t_k)\) mean in words, and what does it not mean?
- Name one model or design feature that should be checked when diagnosing a computed boundary set.
Solutions: (1) \(P_0(Z_1\geq b_1)=0.0200\), so \(b_1=\Phi^{-1}(0.9800)\approx2.054\). (2) The increment counts only paths that did not cross earlier; the marginal tail at look \(k\) includes paths that may already have crossed. (3) \(\sqrt{0.25/1}=0.500\). (4) It is the null probability of crossing at least one efficacy boundary by look \(k\); it is not the probability that \(H_0\) is true or, generally, the marginal tail probability at look \(k\). (5) Examples include the assumed canonical joint distribution, information fractions and their ordering, the specified one-sided direction, and whether the final spending target equals \(\alpha\).