Alpha-Spending Boundaries for Weighted Logrank Monitoring

Why alpha spending is needed

In a group sequential trial, a weighted logrank statistic may be examined at several interim analyses as well as at the final analysis. Repeatedly testing at the ordinary one-look significance level does not generally preserve the planned overall type I error: under \(H_0\), the chance of crossing at least one efficacy boundary can exceed \(\alpha\). An alpha-spending approach controls this repeated-testing error by allocating a cumulative error target across the planned information times.

The allocation is not itself a list of critical values. At each look, a boundary must be chosen in light of the null distribution of the statistic at that look and its dependence on statistics from previous looks. This dependence is especially important for weighted logrank monitoring: the chosen weights define the score process, while the information times describe how much of that process has accumulated.

Definition: Let \(t_k=I_k/I_K\) be the information fraction at look \(k\), where \(I_k\) is the information at that look and \(I_K\) is the planned final information. An alpha-spending function \(A(t)\) is a nondecreasing function on \([0,1]\) with \(A(0)=0\) and \(A(1)=\alpha\). For an upper-tailed efficacy test, \(A(t_k)\) is the planned probability, under \(H_0\), of having crossed an efficacy boundary by look \(k\). The amount newly spent at look \(k\) is \(A(t_k)-A(t_{k-1})\), with \(A(t_0)=0\).

The sequential weighted logrank statistics

For a prespecified weighted logrank score process, let \(U_k\) denote the accumulated score at look \(k\), and let \(I_k\) denote its null variance, or information. The standardized statistic is \(Z_k=U_k/\sqrt{I_k}\). The direction of the score is chosen so that larger values favor the treatment benefit being tested. The hypothesis is \(H_0\) of no treatment effect against the prespecified upper-tailed alternative \(H_a\).

The usual canonical approximation requires that the weight function and monitoring plan be prespecified, that the score increments have mean zero under \(H_0\), and that the standardized increments are approximately independent normal variables under the null. It also relies on adequate large-sample behavior for the weighted logrank scores and their variance estimates. With one common score process observed at successive looks, these assumptions give the joint null model

(Z_1,\ldots,Z_K)\ \text{approximately multivariate normal},\qquad E_0(Z_k)=0,\qquad \operatorname{Corr}_0(Z_j,Z_k)=\sqrt{\frac{I_j}{I_k}} =\sqrt{\frac{t_j}{t_k}},\quad j<k.

This correlation follows because \(U_j\) is contained in the accumulated score \(U_k\): under independent increments, \(\operatorname{Cov}_0(U_j,U_k)=I_j\). Dividing by \(\sqrt{I_jI_k}\) gives \(\sqrt{I_j/I_k}\). Thus the statistics are not independent. When different weights or score definitions are used at different looks, the canonical correlation formula need not hold; the joint null covariance must reflect the actual cross-covariances of those scores.

From spending targets to efficacy boundaries

For an upper-tailed design, let \(c_k\) be the efficacy boundary at look \(k\). The trial stops for efficacy at the first look for which \(Z_k\ge c_k\). The boundary at look \(k\) is selected to satisfy

P_0\left(Z_1<c_1,\ldots,Z_{k-1}<c_{k-1},\,Z_k\ge c_k\right) =A(t_k)-A(t_{k-1}).

Equivalently, the cumulative probability of crossing by look \(k\) is \(A(t_k)\). This distinction matters: the incremental spending at a look is the probability of crossing there without an earlier crossing, not simply the marginal probability \(P_0(Z_k\ge c_k)\). Because the statistics are correlated, the marginal tail area at look \(k\) generally exceeds the new error spent at that look.

A short justification makes the calibration principle explicit. The events of first crossing at looks \(1,\ldots,k\) are mutually exclusive. Therefore,

P_0(\text{cross by look }k) =\sum_{j=1}^{k}P_0(\text{first cross at look }j) =\sum_{j=1}^{k}\{A(t_j)-A(t_{j-1})\} =A(t_k).

The equality is obtained by choosing each boundary using the joint null distribution of the statistics observed up to that look. Consequently, the final cumulative crossing probability is \(A(1)=\alpha\), subject to the accuracy of the assumed null model and numerical calibration.

Worked Example: Calculating an O’Brien–Fleming Spending Target

Suppose a hypothetical trial uses an upper-tailed test with total \(\alpha=0.025\), and plans interim looks at information fractions \(t_1=0.50\) and \(t_2=1.00\). Use the one-sided O’Brien–Fleming spending function

A(t)=1-\Phi\left(\frac{z_{1-\alpha}}{\sqrt{t}}\right),\qquad t>0.

Here \(z_{1-\alpha}=z_{0.975}=1.960\). At the interim look,

A(0.50)=1-\Phi\left(\frac{1.960}{\sqrt{0.50}}\right) =1-\Phi(2.772)\approx 0.0028.

At the final look, \(A(1)=1-\Phi(1.960)=0.0250\). Thus the plan allows about 0.0028 cumulative type I error by the interim look and 0.0250 by the end. The additional amount targeted after the interim look is approximately \(0.0250-0.0028=0.0222\). These are probability targets, not yet critical values: the actual boundaries must account for the dependence between \(Z_1\) and \(Z_2\).

Worked Example: Calibrating Two Upper Boundaries

Consider another hypothetical two-look design with information fractions \(t_1=0.50\) and \(t_2=1.00\), total one-sided \(\alpha=0.025\), and spending targets \(A(t_1)=0.010\) and \(A(t_2)=0.025\). Assume the prespecified weighted logrank score process satisfies the canonical joint null model. Then \(Z_1\) and \(Z_2\) are standard normal under \(H_0\), with correlation \(\rho=\sqrt{0.50}=0.7071\).

At the first look the boundary is the upper normal quantile with tail probability 0.010:

c_1=\Phi^{-1}(0.990)=2.326.

At the second look, \(c_2\) must satisfy the cumulative spending target. Since the first crossing probability is 0.010, the probability of a second-look crossing without a first-look crossing must be 0.015:

P_0(Z_1<2.326,\ Z_2\ge c_2)=0.025-0.010=0.015.

For standard normal variables with correlation 0.7071, this probability can be evaluated by conditioning on \(Z_2=z\). The conditional distribution is \(Z_1\mid Z_2=z\sim N(0.7071z,\,0.50)\), so the equation is

\int_{c_2}^{\infty}\phi(z)\, \Phi\left(\frac{2.326-0.7071z}{\sqrt{0.50}}\right)\,dz=0.015.

Numerical evaluation gives \(c_2\approx 2.07\) (rounded). The second-look marginal tail probability at this boundary is about \(1-\Phi(2.07)=0.0192\), which is larger than the new 0.015 spending target. The difference is explained by paths that already crossed at the first look: those paths contribute to the cumulative 0.025 but cannot be counted as new second-look error. The calibrated boundaries are therefore approximately 2.326 and 2.07, not the independent-look quantiles obtained by assigning 0.010 and 0.015 as separate marginal tail areas.

Worked Example: Checking the Canonical Dependence at Three Looks

A hypothetical weighted logrank monitoring plan has cumulative information \(I_1=30\), \(I_2=70\), and \(I_3=100\) units. Assume the same fixed weight function defines the score process at every look and the independent-increment approximation is reasonable. The information fractions are \(t_1=0.30\), \(t_2=0.70\), and \(t_3=1.00\). The canonical correlations are

\rho_{12}=\sqrt{\frac{30}{70}}=0.6547,\qquad \rho_{13}=\sqrt{\frac{30}{100}}=0.5477,\qquad \rho_{23}=\sqrt{\frac{70}{100}}=0.8367.

Thus, the null correlation matrix supplied to a boundary calculation is approximately

\begin{pmatrix} 1 & 0.6547 & 0.5477\\ 0.6547 & 1 & 0.8367\\ 0.5477 & 0.8367 & 1 \end{pmatrix}.

These values show why the information fractions alone do not produce boundaries: they specify the canonical dependence structure only under the stated common-score assumptions, and the spending function still has to be translated into sequential crossing probabilities. If the weights were changed in a way that produced different score processes, these correlations could not be assumed from the fractions alone.

Diagnostics, alternatives, and limitations

Before using an alpha-spending boundary, check that the analysis schedule and the spending function are prespecified, the information fractions are calculated consistently with the weighted score, and the score direction matches the stated efficacy alternative. Confirm that the null covariance used for calibration matches the statistics actually monitored. In particular, a canonical correlation matrix is not justified merely because analyses occur at increasing sample sizes: it depends on the score-increment assumptions and on how the weights are used.

A useful numerical check is that \(A(t_k)\) is nondecreasing, \(A(1)=\alpha\), and the calibrated cumulative crossing probability at each planned look matches its spending target. A boundary sequence should also be assessed for sensible behavior under the chosen design. A failed probability calculation, an unexpected correlation, or a mismatch between simulated and target crossing probabilities can indicate a coding error, inaccurate information fractions, or an unsuitable joint null model. Simulation can help diagnose such issues, but it should use the design’s actual score construction and null data-generating assumptions.

Spending functions offer flexibility compared with a fixed critical value repeated at each look: they allow the timing of planned error expenditure to reflect when interim information is expected. They also avoid requiring a particular equally spaced look schedule, provided the spending function is evaluated at the actual information fractions. Other group sequential approaches may specify a named boundary shape or stagewise error allocation directly. Whatever the approach, the overall type I error depends on the joint distribution of the repeated statistics; a spending function is not a way to ignore that dependence.

The method controls type I error only to the extent that the null model and information calculations are appropriate. The normal approximation may be less accurate with limited numbers of events or sparse risk sets. Misspecified weights, unplanned changes to the analysis, or a covariance structure that differs from the assumed one can also affect error control. Finally, a boundary design does not by itself establish power, optimal timing, or the clinical meaning of a weighted logrank effect; those are distinct design and interpretation questions.

Common mistake

Pitfall: Treating the incremental spending \(A(t_k)-A(t_{k-1})\) as the marginal probability \(P_0(Z_k\ge c_k)\). That reasoning ignores the probability of crossing at earlier looks and generally produces boundaries with the wrong overall type I error. Instead, choose \(c_k\) so the probability of a first crossing at look \(k\), calculated under the joint null distribution, equals the incremental spending. Then verify the cumulative crossing probability at every look.

Check Your Understanding

Answer each question, then compare with the solutions below.

For a spending function with \(A(t_1)=0.006\) and \(A(t_2)=0.025\), what is the incremental spending at the second look?
Under the canonical model, if \(t_1=0.36\) and \(t_2=1\), what is \(\operatorname{Corr}_0(Z_1,Z_2)\)?
Why is the second-look boundary generally not obtained by setting its marginal upper-tail probability equal to the incremental spending?
Name one condition under which the canonical correlation formula should not be assumed without further calculation.

Solutions. (1) \(0.025-0.006=0.019\). (2) \(\sqrt{0.36/1}=0.600\). (3) Because some paths have already crossed at the first look; calibration must assign the new error to paths with no earlier crossing, using the joint null distribution. (4) For example, when different weights or score definitions are used at different looks, the cross-covariances must be derived for those statistics rather than inferred from information fractions alone.