Why Intermediate Information Times Need Care
A trial may reach an interim analysis at an information fraction that was not one of the planned look times. The cumulative spending function still assigns a target amount of type I error at that fraction. But the boundary at the interim look is not generally obtained by interpolating the critical values planned for the neighboring looks: it must reflect both the spending target and the probability of having crossed an earlier boundary.
The first task is to evaluate cumulative spending at the actual information time. If the spending function is available in closed form, evaluate it there. If only a table of values is available, interpolation can provide an approximation, but the interpolation rule must be stated. The second task is to calibrate a boundary whose crossing probability, together with earlier crossings, matches the target cumulative spending.
Evaluating Spending Between Planned Looks
Suppose planned looks occur at \(t_a\) and \(t_b\), and an interim analysis occurs at \(t^*\), where \(t_a<t^*<t_b\). When the formula for \(A(t)\) is known, use \(A(t^*)\) directly. The fact that \(t^*\) was not planned does not change the definition of the spending function.
If only the endpoint values \(A(t_a)\) and \(A(t_b)\) are available, a common approximation is linear interpolation in information fraction:
This estimates cumulative spending, not a boundary. It is exact only when spending is linear over that interval. For a nonlinear spending curve, it may differ from the function’s actual value; the discrepancy can matter when the curve changes rapidly. Any interpolation rule used in a design should be specified in advance, and interpolated values should preserve the nondecreasing nature of cumulative spending.
Worked Example: Evaluating a Nonlinear Spending Curve
Consider a hypothetical two-sided design with \(\alpha=0.05\) and spending function \(A(t)=0.05(1-\exp(-2t))/(1-\exp(-2))\). Planned information fractions are 0.50 and 0.75, and the actual fraction is 0.60.
Evaluate the function at each information fraction:
Linear interpolation between the endpoint values gives \(\widetilde A(0.60)=0.03655293+(0.10/0.25)(0.044923-0.03655293)\approx0.039901\), using unrounded endpoint values. This is about \(0.000508\) below the function’s actual value. Thus, when the formula is available, evaluating it directly avoids interpolation error.
From a Spending Target to a Boundary
The cumulative target \(A(t^*)\) is not, by itself, the probability to allocate to the new look. If earlier looks have already used cumulative spending \(A(t_a)\), the new look is assigned the increment \(A(t^*)-A(t_a)\). Its boundary must be calibrated accounting for the fact that the test would already have stopped after crossing an earlier boundary.
For a standard canonical group sequential model under \(H_0\), the standardized test statistics satisfy \(Z(t)\sim N(0,1)\) and, for \(s\leq t\), \(\operatorname{Corr}\{Z(s),Z(t)\}=\sqrt{s/t}\). Assume this joint normal model applies, the information fractions are correctly specified, and the earlier boundaries have been calibrated to their cumulative spending targets. In a symmetric two-sided design, write the earlier boundaries as \(\pm b_i\). Define the no-crossing event through the last previous look by \(R_a=\{|Z(t_i)|<b_i\text{ for every previous look }i\}\).
This equation also explains why a marginal normal quantile is not generally enough at a later look. The event being calibrated includes survival past all earlier looks. Its probability depends on the joint distribution of the statistics and on the previous boundaries.
Why the Calibration Equation Works
The justification is a partition of the event of crossing by \(t^*\). Either a boundary was crossed at a previous look, or no earlier boundary was crossed and the new boundary is crossed at \(t^*\). These events are disjoint. By calibration at the previous look, the first event has probability \(A(t_a)\); the boundary equation assigns probability \(A(t^*)-A(t_a)\) to the second. Therefore:
For symmetric two-sided boundaries, increasing \(b^*\) decreases the probability of a crossing at the new look while no earlier crossing occurred. Numerical root-finding can therefore be used to solve the calibration equation. The joint null distribution, earlier boundaries, actual information time, and chosen spending target all enter that calculation.
Worked Example: Calibrating a First-Look Boundary
Suppose the information fraction \(t^*=0.50\) is the first look in a hypothetical two-sided design. Use the spending function and \(\alpha\) from the previous example. As there are no earlier looks, \(R_a\) is the whole sample space and \(A(t_a)=0\).
At the first look, \(Z(t^*)\sim N(0,1)\). Symmetry means the two tails share the cumulative target equally, so the boundary solves \(P(|Z(t^*)|\geq b^*)=0.03655293\). Thus each tail has probability \(0.03655293/2=0.01827647\), and
The calibrated rule is to reject \(H_0\) at this look if \(|Z(t^*)|\geq2.0907\). Under the stated null model, its total two-sided crossing probability at the first look is approximately 0.03655, the cumulative spending target. This marginal-quantile calculation applies because there is no earlier no-crossing condition to incorporate.
Worked Example: Finding the Increment at an Intermediate Look
Continue the hypothetical design, now with a previous look at \(t_a=0.50\) and an intermediate look at \(t^*=0.60\). From the earlier calculations, \(A(0.50)\approx0.03655293\) and \(A(0.60)\approx0.04040906\). The additional spending target is
If the previous two-sided boundary is \(\pm b_1\), the new boundary \(b^*\) must satisfy \(P_0(|Z(0.50)|<b_1,\ |Z(0.60)|\geq b^*)=0.00385613\). Under the canonical model the correlation is \(\sqrt{0.50/0.60}\approx0.9129\). For example, the probability on the left can be evaluated from
where \(\phi\) and \(\Phi\) are the standard normal density and cumulative distribution function. Given \(b_1\), numerical integration and root-finding determine \(b^*\). The increment alone does not determine a marginal quantile: the previous boundary and correlation are also needed. This is a complete calibration target, not a claim that the incremental target is itself the probability of crossing without conditioning on the earlier look.
Spending Interpolation Is Not Boundary Interpolation
A tempting shortcut is to interpolate between the neighboring planned critical values. That does not, in general, solve the boundary calibration equation. Critical values control tail probabilities, and at later looks the relevant crossing probability is additionally restricted to paths that did not cross earlier. Neither the cumulative spending curve nor the boundary sequence is generally linear.
The main alternatives are therefore distinct:
| Approach | What it does | When it is appropriate |
|---|---|---|
| Evaluate the specified spending function | Computes \(A(t^*)\) directly | Preferred when the function and actual information fraction are available |
| Interpolate cumulative spending | Approximates \(A(t^*)\) from neighboring tabulated values | Useful when only a table is available and the interpolation rule is specified |
| Calibrate a boundary using the joint null model | Matches \(P_0(R_a,\text{ crossing at the new look})\) to the spending increment; equivalently, the conditional crossing probability given \(R_a\) is the increment divided by \(P_0(R_a)\) | Required to translate the target into a valid boundary at a later look |
| Interpolate critical values | Estimates a boundary directly from neighboring boundaries | Not a substitute for calibration; generally does not guarantee the spending target |
Diagnostics, Limitations, and Common Mistakes
Before using an intermediate-look boundary, check that the actual information fraction is on the same scale as the spending function and that the previous boundaries correspond to the same design. Confirm that cumulative spending has not decreased and that the new target is at least the spending already assigned. A large difference between direct function evaluation and interpolation is a warning that the tabulated grid may be too coarse for the chosen approximation.
- Wrong: “The interim look uses the linearly interpolated critical value, so its alpha spending is automatically correct.” Correction: Interpolation of a boundary does not establish the required joint crossing probability. Calibrate the boundary against the cumulative spending increment.
- Wrong: “The increment \(A(t^*)-A(t_a)\) is the marginal tail probability at the new look.” Correction: It is the probability assigned to a crossing at the new look among paths with no earlier crossing.
- Wrong: “Interpolating between endpoint spending values is exact.” Correction: It is exact only if spending is linear across that interval; otherwise it is an approximation to the specified function.
- Warning sign: The computed boundary is based on planned information rather than the information actually available. Since the joint null correlation depends on information times, that mismatch can invalidate the calibration.
Intermediate-time calibration depends on the assumed information measure, the canonical joint normal model, the spending function, and the previous boundaries. If information is mismeasured, the planned joint distribution may not describe the observed test statistics. Interpolation also cannot recover the true curve between sparse tabulated points without additional assumptions. These are reasons to document the function, information-time calculation, and interpolation rule, not reasons to treat an interpolated boundary as automatically valid.
Check Your Understanding
Answer each question, then compare with the solutions below.
- If \(A(0.40)=0.020\) and \(A(0.70)=0.041\), what is the linear interpolation estimate of \(A(0.55)\)?
- In a two-sided first-look design, why can the boundary be found from a standard normal marginal quantile?
- At a later look, what probability does the boundary calibration equation assign to the new crossing event?
- Why is interpolating critical values not generally equivalent to interpolating cumulative spending?
Solutions. (1) The interpolation fraction is \((0.55-0.40)/(0.70-0.40)=0.5\), so the estimate is \(0.020+0.5(0.041-0.020)=0.0305\). (2) There are no earlier looks, so there is no earlier-survival event to include; under the null, the first-look statistic is standard normal. (3) It assigns \(A(t^*)-A(t_a)\) to crossing the new boundary among paths that did not cross any earlier boundary. (4) A boundary is a critical value, and its crossing probability depends on the null distribution and, at later looks, on the joint dependence with earlier statistics; linear interpolation of critical values does not enforce the cumulative spending target.