Final-Look Behavior of Spending Curves

Why the Final Look Needs Its Own Check

A spending function describes how much of a trial’s type I error budget may have been used by a given information time. Its value at the final information time matters for a direct reason: if the design is intended to have overall one-sided type I error \(\alpha\), or overall two-sided type I error \(\alpha\), the cumulative spending at the endpoint must equal that budget. A curve can look plausible at early information times and still fail this basic endpoint check.

The endpoint check is necessary but not sufficient. At the final analysis, the test must also respect rejections already possible at earlier looks. The final boundary is calibrated against the joint distribution of the sequence of statistics, so the error probability assigned to a final crossing is the budget still available after earlier crossings—not a new, independent test at level \(\alpha\).

In the canonical group-sequential model under \(H_0\), the standardized statistics \(Z_k\) have standard normal marginal distributions and correlation \(\operatorname{Corr}(Z_i,Z_j)=\sqrt{t_i/t_j}\) for \(i<j\). This structure follows from independent information increments. Boundary calibration chooses critical values so that the probability of crossing by look \(k\) equals \(A(t_k)\). The arguments below use this model and assume that the planned information fractions and spending function are defined and that the boundaries can be calibrated to the stated targets.

Endpoint Behavior of Common Spending Curves

Several familiar spending families are constructed to meet the same endpoint while allocating error differently before it. For the following formulas, \(\alpha\) denotes the total type I error budget. The O’Brien–Fleming-like formula shown is two-sided; the other expressions are written as cumulative spending amounts for their stated \(\alpha\).

The endpoint equality can be verified by direct substitution. For the power family, \(A(1)=\alpha\cdot1^\rho=\alpha\). For Hwang–Shih–DeCani spending with \(\gamma\ne0\), numerator and denominator become \(1-\exp(-\gamma)\), so their ratio is one; when \(\gamma=0\), the limiting linear form also gives \(A(1)=\alpha\). For the O’Brien–Fleming-like expression, \(\Phi(z_{1-\alpha/2})=1-\alpha/2\), yielding \(A(1)=\alpha\). For the Pocock-like curve, \(\log\{1+(e-1)\}=\log(e)=1\).

The endpoint is shared, but the curves need not approach it at the same rate. For differentiable curves, the local slope \(A'(1)\) describes the first-order change in cumulative spending as \(t\) approaches one. For example, the power curve has \(A'(1)=\alpha\rho\); the Hwang–Shih–DeCani curve with \(\gamma\ne0\) has \(A'(1)=\alpha\gamma e^{-\gamma}/\{1-e^{-\gamma}\}\); and the Pocock-like curve has \(A'(1)=\alpha(1-e^{-1})\). For the displayed two-sided O’Brien–Fleming-like curve, writing \(c=z_{1-\alpha/2}\), differentiation gives \(A'(1)=c\phi(c)\). These slopes describe local curve behavior, not the final boundary itself.

What “Remaining Error” Means at the Final Look

Let \(R_k\) denote the event that the trial first crosses an efficacy boundary at look \(k\). Because first-crossing events are disjoint, the probability of rejection by look \(k\) is the sum of their probabilities through that look. If calibration achieves the spending target at the preceding look, then the probability of having rejected by look \(K-1\) is \(A(t_{K-1})\). At the endpoint, the overall target is \(A(1)=\alpha\). Subtraction gives the probability assigned to first rejection at the final look.

This is an unconditional probability under \(H_0\), across repeated trials. Among paths that have not rejected by look \(K-1\), the corresponding conditional crossing probability is \(\{\alpha-A(t_{K-1})\}/\{1-A(t_{K-1})\}\), provided the preceding calibration is exact. The final statistic’s distribution among those surviving paths is generally not an ordinary untruncated standard normal distribution: survival depends on earlier correlated statistics. Therefore, the conditional probability ratio cannot simply be converted to a usual normal cutoff as if the final test were independent of the interim tests.

Calibrating the Terminal Boundary

For a two-sided design with symmetric efficacy boundaries, let \(b_k\) be the positive boundary at look \(k\), so a crossing occurs when \(|Z_k|\geq b_k\). The last boundary is chosen so that the probability of no earlier crossing and a final crossing equals the remaining budget. In a one-sided upper-boundary design, replace \(|Z_k|\geq b_k\) with \(Z_k\geq b_k\). These are joint-probability calculations, not calculations based only on the final marginal distribution.

The equality assumes that the earlier boundaries have already been calibrated to make cumulative rejection probability \(A(t_{K-1})\). In practice, a boundary-calibration procedure solves the displayed equation using the canonical joint normal distribution. The particular numerical boundary depends on all earlier information times and boundaries, as well as the final look. The spending increment supplies the target crossing probability, while the joint distribution determines which boundary achieves it.

Diagnostics, Comparisons, and Limitations

A reliable endpoint review checks both the spending curve and the calibrated boundary sequence. The curve check is algebraic: verify \(A(0)=0\), \(A(1)=\alpha\), and that evaluated values remain within the intended cumulative budget. The boundary check is probabilistic: under \(H_0\), verify that cumulative crossing probability at each planned look matches \(A(t_k)\), especially that the terminal cumulative probability is \(\alpha\). A curve satisfying the endpoint condition does not guarantee correctly calibrated boundaries if the joint probabilities were calculated incorrectly.

Several failure modes are worth checking. If a formula is coded with the wrong endpoint or parameterization, it may not reach \(\alpha\). Rounding intermediate spending values too aggressively can also leave a small but avoidable mismatch at the endpoint. More seriously, a final boundary calibrated as though \(Z_K\) were independent of earlier statistics fails to account for the selection induced by non-crossing. Finally, if the boundary solver reports a cumulative terminal crossing probability different from \(\alpha\), inspect the information fractions, one- versus two-sided convention, tail allocation, and numerical accuracy before interpreting the result.

The endpoint condition alone does not establish that a spending curve is appropriate for every design, nor does it ensure power against a particular alternative. It also presumes the stated information time and canonical joint model are suitable. Final-boundary calculations can be sensitive to numerical integration and root-finding tolerances; the computed crossing probabilities should be checked independently when accuracy matters. A spending function specifies error allocation over information time, but does not remove the need to model the sequential statistics correctly.

Common Mistake

Wrong reasoning: “The design has \(\alpha=0.05\), so the last analysis should use a 0.05 test; the earlier looks are separate.” This treats repeated opportunities to reject as unrelated tests and can spend more than the total error budget.

Correction: If the cumulative probability of rejection by the preceding look is \(A(t_{K-1})\), the final first-crossing probability is \(\alpha-A(t_{K-1})\). Calibrate the terminal boundary jointly with the earlier non-crossing events. Do not confuse this remaining probability with either a fresh level-\(\alpha\) test or a marginal normal cutoff.