Why Adding a Look Requires Recalibration
An interim analysis creates an additional opportunity to reject the null hypothesis. Even if the treatment data and the planned final analysis remain unchanged, inserting a look can increase the probability of at least one false efficacy claim. The new boundary must therefore be chosen with the earlier looks in view: it must deliver the cumulative type I error that the spending function has allocated by the added look’s information time.
The target is cumulative alpha, not a fresh allowance to use at the added analysis. Some null paths that cross the new boundary have already crossed an earlier boundary; those paths cannot be counted a second time. Conversely, a boundary calibrated only from a marginal tail probability ignores those earlier crossings and can spend too much error.
Model and Calibration Equation
For the standard canonical group sequential model under \(H_0\), the standardized test statistics have a joint multivariate normal distribution with mean zero, unit variances, and correlation \(\operatorname{Corr}_0\{Z(s),Z(t)\}=\sqrt{s/t}\) for \(s\leq t\). This is the usual information-time representation of a test with independent increments. It assumes the information fractions are correctly specified, the null model and standardization are valid, and the one-sided upper-boundary design is the one being calibrated. If the statistic or its null distribution differs, the probability calculation must use that design’s appropriate joint null model.
Write \(A(t_*)\) for the spending function’s cumulative target at the added look. The existing boundaries determine how much of this target has already been used. In particular, if the earlier looks have cumulative crossing probability \(A(t_{k-1})\), the probability of a first crossing at the added look should be \(A(t_*)-A(t_{k-1})\). This difference is the incremental spending; it is not generally equal to \(P_0\{Z(t_*)\geq b_*\}\).
Equivalently, the probability of staying below every boundary through the added look must be \(1-A(t_*)\). For two looks, this is a bivariate normal probability. With several earlier looks, it is a joint probability over all those looks, with their correlations determined by the information fractions. The calculation can be made by numerical integration or an appropriate multivariate-normal probability routine.
Worked Example: Calibrating an Added Look After One Earlier Look
Suppose a hypothetical one-sided trial has an existing look at \(t_1=0.25\) with boundary \(b_1=2.326\), giving a null upper-tail probability of approximately \(0.0100\). An interim look is added at \(t_*=0.50\). The chosen spending curve has cumulative spending \(A(0.50)=0.0125\). Find the new boundary.
Under the canonical model, the correlation between the two statistics is \(\rho=\sqrt{0.25/0.50}=0.7071\). Let \(\Phi_2(x,y;\rho)\) denote the bivariate standard normal cumulative distribution function with correlation \(\rho\). The probability of no crossing at either look is \(\Phi_2(2.326,b_*;0.7071)\). Therefore the calibration equation is:
Numerically solving this equation gives \(b_*\) approximately \(2.65\) (the unrounded numerical root is used to satisfy the target). At that root, the probability of crossing at the new look but not at the first is approximately \(0.0025\). Adding the earlier crossing probability gives \(0.0100+0.0025=0.0125\), as required. The marginal tail at the added look is about \(P_0\{Z(0.50)\geq 2.65\}=0.0040\); it is larger than the incremental spending because some of that tail overlaps with paths that already crossed at the first look.
This calculation uses a valid canonical joint null model and retains the earlier boundary. The information fractions are ordered, and the target at the new look exceeds the earlier cumulative spending, so the added look has positive incremental alpha available.
Why the Calibration Equation Works
The added boundary partitions paths that have not crossed earlier into those that cross at the new analysis and those that do not. For fixed earlier boundaries, increasing \(b_*\) can only reduce the probability of a first crossing at the added look. This monotonicity makes one-dimensional root-finding appropriate when the target is feasible.
The earlier crossing event is determined by the existing \(b_j\). Do not count those paths again as new error at \(t_*\).
Compute \(A(t_*)\) from the selected spending function at the added information fraction, not at a calendar-time fraction unless those fractions coincide.
Find \(b_*\) such that the probability of crossing at least one boundary through \(t_*\) equals \(A(t_*)\).
Re-evaluate the joint null crossing probability using the unrounded numerical boundary and check that it agrees with \(A(t_*)\) within the calculation tolerance.
The target must be feasible. If earlier boundaries already produce cumulative crossing probability \(A(t_{k-1})\), then a nondecreasing spending function requires \(A(t_*)\geq A(t_{k-1})\). When the inequality is strict, the added analysis has positive incremental spending. If the target equals the earlier cumulative probability, the new look must add no crossing probability among paths that survived earlier looks; the required boundary may be extremely high. A target below the probability already spent cannot be attained by choosing a new boundary while leaving the earlier boundaries unchanged.
A useful formal view is to define \(G(c)\) as the probability of crossing any earlier boundary or crossing \(Z(t_*)\geq c\). The function \(G(c)\) is nonincreasing and continuous under the joint normal model. As \(c\) increases without bound, \(G(c)\) approaches the earlier cumulative crossing probability; as \(c\) decreases without bound, it approaches one. Thus a root for \(G(c)=A(t_*)\) exists when the target lies between these limits. In practice the useful root is found over a finite numerical range, and the final probability is checked directly.
Worked Example: Adding a Look With No Earlier Analysis
Suppose a hypothetical trial adds its first interim analysis at information fraction \(t_*=0.40\). It uses a one-sided Hwang-Shih-DeCani spending curve with \(\gamma=0\) and total \(\alpha=0.05\). For this limiting case, \(A(t)=\alpha t\). Find the interim boundary.
There are no earlier looks, so the cumulative crossing probability at this first analysis is simply the marginal upper-tail probability. The spending target is \(A(0.40)=0.05(0.40)=0.0200\). Thus the boundary satisfies \(P_0\{Z(0.40)\geq b_*\}=0.0200\), or:
The standard normal null model is appropriate here because this is the first analysis and \(Z(t_*)\) has unit variance under \(H_0\). The result means that the probability of crossing this upper efficacy boundary by this look, if \(H_0\) is true, is 0.0200. Once an earlier look exists, the marginal-quantile shortcut in this example no longer suffices: the joint crossing probability must be calibrated.
Check the Joint Probability, Not Just the Marginal Tail
A boundary that looks plausible as a single-look \(z\)-cutoff can still be wrong for a sequential design. The relevant new contribution is the probability of crossing at the new look without an earlier crossing. The following calculation shows how a seemingly reasonable cutoff can materially overspend.
Worked Example: Diagnosing an Undersized Boundary
Use the hypothetical two-look design from the first worked example: \(t_1=0.25\), \(b_1=2.326\), \(t_*=0.50\), and target \(A(t_*)=0.0125\). Suppose someone proposes \(b_*=2.287\). Under the same bivariate null model, the marginal upper-tail probability at the added look is about \(0.0111\), and the probability of crossing both boundaries is about \(0.0030\).
The probability of crossing at the added look without having crossed earlier is the marginal tail less the overlap: \(0.0111-0.0030=0.0081\). The cumulative crossing probability is therefore about \(0.0100+0.0081=0.0181\), not \(0.0125\). This cutoff overspends the target by about \(0.0056\). It is much too low; the calibrated boundary is around \(2.65\), as found by solving the joint probability equation.
The model assumptions are unchanged and the correlation remains \(0.7071\). The failure is not a different information fraction or a change in the spending curve: it is the incorrect boundary. In particular, the overlap is approximately \(0.0030\), not a value inferred from the marginal tail alone.
What Changes After the Look Is Added?
A new boundary that achieves \(A(t_*)\) preserves the cumulative null crossing probability only through the added look. If the trial has later planned analyses, their boundaries generally need recalibration as well. Each later target \(A(t_j)\) is cumulative across all analyses up to \(t_j\), including the new one. Simply retaining every later boundary from the old schedule does not, in general, preserve the original spending curve.
The main alternatives are less reliable for this purpose. Copying the boundary from a nearby planned look ignores the changed information time and the probability structure of the intervening analyses. Using \(1-\Phi(b_*)\) as the target treats the added look as if it were the only analysis. Dividing total alpha equally among looks is a different design rule and does not, by itself, reproduce a specified spending curve. A valid recalibration instead uses the target \(A(t_*)\) and the joint null distribution for all looks through \(t_*\).
Diagnostics should focus on both the inputs and the result. Confirm that \(t_*\) is computed from information, lies between the surrounding information fractions, and is the value used to evaluate \(A(t_*)\). Check that correlations follow the assumed information-time model and that the earlier boundaries correspond to the design being continued. Finally, calculate cumulative crossing probability using the full-precision root. A discrepancy may indicate a numerical integration problem, an incorrect correlation, an inconsistent boundary, or a confusion between cumulative and incremental alpha.
This method has limits. It controls the specified null crossing probability under the model used for calibration; it does not correct a misspecified null distribution, inaccurate information estimates, or unplanned changes to the analysis statistic. A one-sided upper-boundary calculation also does not automatically apply to a two-sided design or to futility boundaries. Those require their own event definitions and probability calculations.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A design has cumulative null crossing probability \(0.012\) before an added look, and the spending function gives \(A(t_*)=0.015\). What is the target incremental crossing probability at the added look?
- Why is \(1-\Phi(b_*)\) generally not the incremental alpha for a look that follows an earlier analysis?
- For an added first look at \(t=0.30\), a spending function gives \(A(0.30)=0.01\). What equation determines its one-sided boundary under a standard normal null statistic?
- If the earlier cumulative crossing probability is \(0.02\) but the proposed target at the added look is \(0.018\), can a new boundary alone attain the target while earlier boundaries stay fixed? Explain.
- In the two-look example, what is the cumulative crossing probability for the proposed cutoff \(2.287\), and why?
Solutions. (1) \(0.015-0.012=0.003\); this is the probability of a first crossing at the new look. (2) The marginal tail includes paths that crossed earlier, so subtracting the overlap—or, equivalently, calculating the joint cumulative crossing probability—is necessary. (3) Solve \(P_0\{Z\geq b_*\}=0.01\), giving \(b_*=\Phi^{-1}(0.99)\approx2.326\). (4) No. The earlier looks have already used probability \(0.02\), so a cumulative target of \(0.018\) is below the attainable limit with those boundaries fixed. (5) It is about \(0.0181\): approximately \(0.0100\) crossed at the first look, and \(0.0111-0.0030=0.0081\) crossed at the second without an earlier crossing.