Why Two-Sided Spending Requires an Allocation
A two-sided group sequential test can stop for efficacy because the interim statistic is unusually high or unusually low. A total cumulative spending function specifies how much type I error may have been used by each information time, but it does not, by itself, say how that error is divided between the two directions. That division matters: it determines the separate upper and lower boundaries and the chance, under the null hypothesis, of crossing each one first.
The earlier material on one-sided efficacy spending describes how a cumulative target is translated into boundary-crossing probabilities over successive looks. For a two-sided test, the same principle applies, but the two boundaries must be calibrated together. In particular, assigning half of the total spending to each side is justified by symmetry, not merely by the fact that the test is called two-sided.
These are probabilities of a first efficacy crossing by that look under the null model. They are not probabilities of crossing a boundary at look \(j\) regardless of what happened earlier. The corresponding incremental targets at look \(j\) are \(A_U(t_j)-A_U(t_{j-1})\) and \(A_L(t_j)-A_L(t_{j-1})\), with both cumulative spending functions set to zero before the first look.
Equal Tail Allocation and Its Conditions
A common two-sided design allocates half of the cumulative spending to each direction:
This is called equal tail allocation. At each planned look, it targets equal cumulative null probabilities of first crossing in the upper and lower directions. It does not mean that each boundary separately may use the entire total spending \(A(t)\); doing so could use twice the allowed type I error.
Equal tail allocation is especially natural when the null-path distribution of the canonical statistics is symmetric about zero and the boundaries are symmetric: the upper boundary at each look is \(b_j\), and the lower boundary is \(-b_j\). Under the null, changing the sign of the entire statistic path leaves its probability distribution unchanged. This sign change maps every path that first crosses the upper boundary by look \(j\) to a path of equal probability that first crosses the lower boundary by look \(j\).
For example, under the canonical group sequential model, the null statistics have a joint centered normal distribution with \(P_{H_0}(Z_i,Z_j)\) determined by the information fractions. In the usual standardized setting, \(Z_j\sim N(0,1)\) under the null and \(\operatorname{Corr}(Z_i,Z_j)=\sqrt{t_i/t_j}\) for \(i<j\). This joint distribution is sign-symmetric. When equal tail allocation is selected, the upper boundary can be calibrated with the matching negative lower boundary, using the joint path probabilities rather than treating the looks as independent.
How the Allocation Determines Boundary Targets
Suppose the total cumulative spending function is \(A(t)\), and choose an upper allocation function \(A_U(t)\). The lower allocation must then satisfy \(A_L(t)=A(t)-A_U(t)\). At the planned looks, both allocations must be nondecreasing and begin at zero; their increments specify the target probabilities of first crossing in each direction at each look. For equal tail allocation, the upper and lower increments are each one half of the total increment.
At a look after the first, a boundary is not generally found by applying a single-look normal quantile to that look's incremental target. The probability of reaching the boundary depends on the joint distribution across looks and on the requirement that no earlier boundary was crossed. With symmetric boundaries, the two directional targets can be calibrated jointly; the upper and lower first-crossing probabilities then match by symmetry.
Worked Example: Equal Tails at the First Look
Consider a hypothetical two-sided design with total type I error \(\alpha=0.05\). At its first planned look, the cumulative spending function assigns \(A(t_1)=0.010\). Assume a standard normal null statistic at that look and choose symmetric efficacy boundaries.
Equal tail allocation gives \(A_U(t_1)=A_L(t_1)=0.010/2=0.005\). With no earlier look, the upper boundary \(b_1\) solves \(P_{H_0}(Z_1\geq b_1)=0.005\). Since \(Z_1\sim N(0,1)\), \(b_1=\Phi^{-1}(0.995)\approx2.576\). The lower boundary is \(-2.576\).
The total first-look crossing probability is \(P_{H_0}(Z_1\geq2.576)+P_{H_0}(Z_1\leq-2.576)\approx0.005+0.005=0.010\). Thus the boundaries use the specified first-look spending, and neither direction is assigned the full 0.010.
Worked Example: Cumulative Targets and Incremental Spending
For a hypothetical symmetric design, suppose the total cumulative spending at three planned looks is 0.008, 0.025, and 0.050. Calculate the cumulative and incremental targets for each direction under equal tail allocation.
At the three looks, the cumulative spending for each direction is respectively \(0.008/2=0.004\), \(0.025/2=0.0125\), and \(0.050/2=0.025\). Taking the difference between successive cumulative targets, the upper-direction increments are \(0.004\), \(0.0125-0.004=0.0085\), and \(0.025-0.0125=0.0125\). The lower-direction increments are the same.
The total increments are \(0.008\), \(0.025-0.008=0.017\), and \(0.050-0.025=0.025\). In each look, the two directional increments sum to the total increment: \(0.004+0.004=0.008\), \(0.0085+0.0085=0.017\), and \(0.0125+0.0125=0.025\). These are probability targets for first crossings at each look. The later-look boundaries still require calibration against the joint null distribution and prior noncrossing.
When the Allocation Need Not Be Equal
A two-sided test need not use equal tail spending. For instance, investigators may choose a larger early allocation to the direction considered more important, or a test statistic and null model may not have the symmetry needed for mirrored boundaries. In such cases, specify \(A_U(t)\) and \(A_L(t)\) separately, and verify at every planned look that they are nondecreasing and sum to the chosen total spending.
Worked Example: An Unequal Early Allocation
Suppose a hypothetical design has total cumulative spending 0.006, 0.020, and 0.050 at three looks. Its chosen upper cumulative spending targets are 0.002, 0.008, and 0.025. Find the lower targets and check the incremental allocations.
Subtract the upper targets from the total targets. The lower cumulative targets are \(0.006-0.002=0.004\), \(0.020-0.008=0.012\), and \(0.050-0.025=0.025\). The upper increments are \(0.002\), \(0.008-0.002=0.006\), and \(0.025-0.008=0.017\). The lower increments are \(0.004\), \(0.012-0.004=0.008\), and \(0.025-0.012=0.013\).
All increments are nonnegative, and the cumulative directional targets sum to the totals at every look. The increments also add to the total increments: \(0.002+0.004=0.006\), \(0.006+0.008=0.014\), and \(0.017+0.013=0.030\). This allocation is not equal in the first two looks, so it does not justify symmetric boundaries. The final cumulative allocation is equal, but that alone does not make the earlier boundaries symmetric.
Justification and Design Checks
The equal-tail result follows directly from a pathwise symmetry argument. Let \(E_U(j)\) be the event that the first efficacy crossing occurs by look \(j\) at an upper boundary, and \(E_L(j)\) the corresponding event for a lower boundary. Under the assumptions of symmetric boundaries and a sign-symmetric null path, the transformation \((Z_1,\ldots,Z_j)\mapsto(-Z_1,\ldots,-Z_j)\) maps \(E_U(j)\) exactly onto \(E_L(j)\) and preserves probability. Hence \(P_{H_0}(E_U(j))=P_{H_0}(E_L(j))\). Since the events are disjoint and their union is the event of an efficacy crossing by look \(j\),
which gives \(A_U(t_j)=A_L(t_j)=A(t_j)/2\). This argument depends on the entire null path and the boundaries being symmetric; symmetry of the statistic at one isolated look is not enough to establish equal first-crossing probabilities for a sequential design.
Before using an allocation to construct boundaries, check that the total and directional cumulative functions are zero at the start, do not decrease across information times, and remain within the total type I error budget. Also check that \(A_U(t_j)+A_L(t_j)=A(t_j)\) at each look. For a symmetric design, confirm both that the null-path model is sign-symmetric and that the specified boundaries are mirror images. If either condition fails, equal spending numbers alone do not establish equal crossing probabilities.
A practical alternative to equal allocation is to specify two directional spending functions whose sum is the total function. This allows different directional priorities, but requires explicit targets for both sides and joint calibration of their boundaries. The central limitation is that an allocation is only a probability budget: it does not by itself give boundary values, and valid boundary values depend on the null joint distribution, planned information times, and previous noncrossing.
Common Mistakes
- Giving each side the full total spend. If the total cumulative spend is 0.02, assigning 0.02 to each tail would target 0.04 in total. Under equal allocation, each side receives 0.01.
- Confusing cumulative and incremental spending. A cumulative target of 0.012 at look 2 is not necessarily the probability to spend at look 2. Subtract the previous cumulative target to obtain the incremental target.
- Assuming “two-sided” automatically means equal tails. Equal allocation requires the design choice and, for symmetric boundaries, a symmetric null-path distribution. Otherwise the directions may have different targets or require nonsymmetric boundaries.
- Calibrating each look as if it were the first. Later boundaries must account for the correlated statistics and paths that did not cross earlier boundaries. A single-look quantile generally does not achieve the specified sequential spending.
Check Your Understanding
Answer each question, then compare your work with the solutions below.
- A symmetric design has cumulative total spending 0.014 at a planned look. What are the equal cumulative targets for the upper and lower directions?
- Total cumulative spending is 0.010 at look 1 and 0.028 at look 2. Under equal allocation, what is the upper-direction incremental target at look 2?
- At one look, total cumulative spending is 0.020, upper spending is 0.006, and lower spending is 0.014. Is this a valid equal-tail allocation? Is the total allocation consistent?
- Name the two symmetry conditions that justify equal first-crossing probabilities for upper and lower boundaries.
Solutions. (1) Each direction receives \(0.014/2=0.007\). (2) Upper cumulative targets are \(0.010/2=0.005\) at look 1 and \(0.028/2=0.014\) at look 2, so the upper incremental target at look 2 is \(0.014-0.005=0.009\). (3) The targets sum to \(0.006+0.014=0.020\), so they are consistent with the stated total, but they are not equal-tail targets; equal allocation would give 0.010 to each side. (4) The null joint distribution of the statistic path must be unchanged by reversing every sign, and the upper and lower efficacy boundaries must be mirror images at each look.