Why Divide a Two-Sided Error Target?
A two-sided group sequential test can stop early for evidence in either direction. Under the null hypothesis, an early upper-boundary crossing is a false claim in one direction, while an early lower-boundary crossing is a false claim in the other. Both contribute to the overall type I error probability. A design therefore needs to specify how its overall error target is assigned to those two directional possibilities.
The allocation is easy to misread. A plan may assign nominal budgets of 0.025 to each tail for an overall target of 0.05, but those numbers do not automatically describe the probabilities that the combined sequential procedure will actually stop at its upper and lower boundaries. Those actual probabilities depend on the full stopping rule and the joint behavior of the statistics across looks.
Assume the test has a fixed sequence of planned analyses and uses a prespecified stopping rule. At each analysis, it either stops for an upper rejection, stops for a lower rejection, or continues. Under \(H_0\), the statistics across looks have a joint probability distribution specified by the design model; in a common canonical model, the standardized statistics have a multivariate normal distribution with correlations determined by information times. The type I error calculation must use that joint distribution and the actual stopping rule, not just the marginal distribution at each look.
Because the procedure stops at its first efficacy crossing, it cannot both first reject upward and first reject downward on the same path. Thus \(A_U\) and \(A_L\) are disjoint events. It follows that
This identity is the basic accounting rule. It does not say that either directional probability must equal a particular nominal allocation. Instead, it says that the actual directional probabilities, evaluated under the complete design, add to the actual overall type I error.
What a Tail Allocation Means
A nominal tail allocation assigns portions of the planned overall error target to upper- and lower-direction rejection. Write these targets as \(\alpha_U\) and \(\alpha_L\). A common starting point for a two-sided target \(\alpha\) is an equal allocation, \(\alpha_U=\alpha_L=\alpha/2\). An unequal allocation can be appropriate when the design gives different error budgets to the two directions.
The word nominal matters. A tail allocation states how the design intends to budget error; it does not, by itself, establish the actual directional crossing probabilities. The meaning depends on how the boundaries are calibrated. If each tail's boundaries are calibrated separately in a one-sided calculation, the resulting stand-alone crossing probabilities are not necessarily the actual upper and lower errors after both sets of boundaries are combined. If instead the two sets of boundaries are calibrated jointly to achieve specified actual directional probabilities, those probabilities can be the design targets, subject to the chosen model and calculation.
For stand-alone calibration, let \(E_U\) denote the event that an upper-only procedure would cross its upper boundaries under \(H_0\), and define \(E_L\) analogously for a lower-only procedure. When the corresponding upper and lower boundaries are combined, the actual first-upper event is contained in \(E_U\), and the actual first-lower event is contained in \(E_L\). Consequently,
The reason is that a path that would eventually cross an upper boundary in the upper-only procedure might first cross a lower boundary in the combined procedure and stop before reaching that upper boundary. The reverse can happen to a lower-only path. Therefore, stand-alone tail crossing probabilities can serve as conservative bounds on the corresponding actual directional errors, but they need not equal them. In contrast, a joint calibration can account directly for which boundary is crossed first.
Worked Examples
Worked Example: Checking an Equal Nominal Split
Consider a hypothetical two-sided design with overall target \(\alpha=0.05\). Its planning document assigns nominal budgets \(\alpha_U=0.03\) and \(\alpha_L=0.02\). A calculation under the design's null model and complete stopping rule gives actual first-crossing probabilities of 0.0284 in the upper direction and 0.0187 in the lower direction. Find the actual overall type I error and interpret the difference from the nominal budgets.
The first-crossing events are disjoint, so the actual overall error is their sum:
The combined procedure has an actual type I error of 0.0471, below the target of 0.05. Its actual upper error is 0.0284, which is 0.0016 below the upper nominal budget; its actual lower error is 0.0187, which is 0.0013 below the lower nominal budget. The remaining overall slack is \(0.05-0.0471=0.0029\). These results are consistent with budgets used as upper limits, but they would not meet a specification requiring the actual directional errors to equal 0.03 and 0.02 exactly.
Worked Example: Why Stand-Alone Rates Can Be Larger
Suppose a hypothetical path calculation for a two-look test gives stand-alone upper and lower crossing probabilities of 0.030 and 0.020 under \(H_0\). When both directions are included in the stopping rule, the calculated probabilities of first crossing are 0.027 upward and 0.018 downward. The values are illustrative outputs from the respective null-path calculations. Compare the stand-alone totals with the actual overall error.
The stand-alone total is \(0.030+0.020=0.050\). In the combined procedure, the actual directional probabilities sum to
The actual error is 0.005 lower than the stand-alone total. This can happen because some paths that would have crossed one tail's boundary in its stand-alone procedure first cross the opposing boundary in the combined procedure. They then stop in the other direction rather than later producing a first crossing in the original direction. The result illustrates why adding stand-alone rates can give a conservative total; it does not imply that every design will have this amount of difference, or even that its directional differences will be equal.
Worked Example: Interpreting an Unequal Allocation
A hypothetical design has overall target \(\alpha=0.05\), with nominal allocations \(\alpha_U=0.04\) and \(\alpha_L=0.01\). The study defines positive \(Z\)-statistics as evidence in the upper direction and negative \(Z\)-statistics as evidence in the lower direction. A joint null calculation reports actual first-crossing probabilities of 0.038 and 0.009. Check the budget and state the directional interpretation.
First, the nominal budgets add to the overall target:
The actual overall error is
Under \(H_0\), this design has a 0.038 probability of first rejecting at an upper boundary and a 0.009 probability of first rejecting at a lower boundary. The actual total is below 0.05, and each actual directional probability is below its nominal budget. The unequal allocation gives more nominal error to upper-direction rejection; it does not mean the upper-direction null probability must be four times the lower-direction probability. The reported actual probabilities, rather than the budget ratio alone, describe the combined procedure's directional error rates.
Justifying the Error Accounting
The identity for actual overall error follows directly from how the stopping rule defines rejection. Let \(R\) be the event of any rejection under \(H_0\). If rejection can occur only by first crossing an upper or lower efficacy boundary, then \(R=A_U\cup A_L\). A single path cannot have both types of first rejection, so \(A_U\cap A_L\) is empty. The addition rule for disjoint events therefore gives
This also clarifies the role of the nominal budgets. Suppose stand-alone boundary calculations are designed so that \(P_{H_0}(E_U)\leq\alpha_U\) and \(P_{H_0}(E_L)\leq\alpha_L\). Since \(A_U\subseteq E_U\) and \(A_L\subseteq E_L\), the combined procedure satisfies
Thus \(\alpha_U+\alpha_L\leq\alpha\) provides a sufficient error-control condition under this stand-alone calibration approach. It can be conservative because the combined procedure may use less than the sum of the two stand-alone crossing probabilities. A joint calibration may instead target \(P_{H_0}(A_U)=\alpha_U\) and \(P_{H_0}(A_L)=\alpha_L\) directly. In that case, if the targets are achieved and sum to \(\alpha\), the actual overall error is \(\alpha\). The calculation must still respect the planned look schedule, the null distribution of the statistics, and every stopping rule that affects whether a rejection can occur.
Diagnostics, Alternatives, and Limitations
A useful diagnostic is to report the overall error and both directional errors separately. A single statement that a design “controls two-sided alpha at 0.05” does not reveal whether the overall rate is close to its target, whether one direction consumes more of the error budget, or whether the nominal allocation was achieved. Numerical integration, recursive probability calculation, or null simulation can be used to evaluate these quantities, provided the calculation represents the planned analysis times and stopping rules.
- Check the event definition. Count first rejections, not every boundary a path might have crossed if monitoring had continued after stopping.
- Check the null model. Use the joint distribution across looks assumed by the design. Marginal tail probabilities at individual looks do not by themselves determine a sequential crossing probability.
- Check the reported quantities. Label stand-alone crossing rates, nominal budgets, and actual first-crossing rates distinctly. They answer different questions.
- Check the total. Confirm that the two actual directional errors sum to the reported overall type I error, apart from stated rounding.
Equal allocation, such as \(\alpha/2\) to each direction, is a familiar baseline. Unequal allocation changes the intended error budget by direction. A one-sided design is a different choice: it permits rejection in only one prespecified direction and is not made two-sided merely by reporting an overall error target. The appropriate choice depends on the scientific question and the consequences of false conclusions in each direction.
Allocation alone does not specify a complete group sequential design. It does not uniquely determine the shape of the boundaries across looks, and the same nominal budgets can accompany different stopping rules. Nor does a calculation guarantee error control if the actual analysis schedule, statistical model, or stopping rule differs from the one used to calibrate the boundaries. If there are additional rules that stop monitoring without rejection, the actual first-crossing probabilities must be calculated for the full planned procedure.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A two-sided sequential test has actual first-crossing probabilities 0.021 upward and 0.024 downward under \(H_0\). What is its actual overall type I error?
- Why can the actual upper first-crossing probability be smaller than the stand-alone upper crossing probability?
- A design assigns nominal budgets of 0.035 upward and 0.015 downward. Do these budgets sum to an overall target of 0.05? Does that establish the actual directional errors?
- What should be checked if a report gives the overall type I error but omits the two directional probabilities?
Solutions. (1) The first-crossing events are disjoint, so the overall error is \(0.021+0.024=0.045\). (2) In the combined design, a path that might later cross an upper boundary in an upper-only procedure can first cross a lower boundary and stop; it is then not an upper first rejection. (3) Yes, \(0.035+0.015=0.050\). The sum specifies nominal budgets, not necessarily actual directional probabilities; those require calculation under the full combined rule. (4) Check that the overall rate is computed under the planned null model and stopping rule, and request the actual upper and lower first-crossing probabilities to see how that total is divided.