Early Information Behavior of Spending Curves

Why Early Spending Matters

A spending function allocates a cumulative Type I error budget across information time. Its behavior near zero is especially consequential when the first interim analysis occurs early: a curve that spends more alpha by that information fraction generally permits a lower first-look efficacy boundary, while a curve that spends almost none requires a higher boundary. Comparing curves only at the final information time can therefore conceal important differences in early stopping opportunities.

Here the focus is the first look for a one-sided efficacy test. Let \(t\) denote the information fraction, with \(0\leq t\leq1\), and let \(A(t)\) be cumulative one-sided alpha spent by information fraction \(t\). Assume \(A(0)=0\), \(A(1)=\alpha\), and that \(A(t)\) is nondecreasing. In the canonical group-sequential model under \(H_0\), the first standardized statistic satisfies \(Z_1\sim N(0,1)\). At the first look, there are no earlier looks to account for, so the boundary is determined directly by the marginal upper-tail probability.

This formula assumes the canonical null distribution is standard normal at the first look and that the design assigns cumulative one-sided spending \(A(t_1)\) through that look. If the test is two-sided and \(A(t_1)\) denotes total spending allocated equally to both tails, the corresponding positive boundary is \(\Phi^{-1}\{1-A(t_1)/2\}\). The examples below use one-sided spending throughout so the curves and boundaries are directly comparable.

How Curves Grow Near Zero

For a differentiable spending function with \(A(0)=0\), a finite initial slope \(A'(0)\) gives the local approximation \(A(t)=A'(0)t+o(t)\). The initial slope therefore describes the leading amount spent at very small information fractions. A power curve with exponent below one has no finite initial slope: it grows faster than a linear curve near zero. An O’Brien–Fleming-like curve has a different pattern, with spending that is smaller than every power of \(t\) as \(t\) approaches zero.

Several common families illustrate these patterns. For the power family, \(A(t)=\alpha t^\rho\), where \(\rho>0\). If \(\rho<1\), its spending is relatively front-loaded near zero; if \(\rho=1\), it is linear; and if \(\rho>1\), it is relatively back-loaded near zero. The Hwang–Shih–DeCani family is \(A(t)=\alpha\{1-\exp(-\gamma t)\}/\{1-\exp(-\gamma)\}\) when \(\gamma\neq0\), and \(A(t)=\alpha t\) when \(\gamma=0\). Its initial slope is \(\alpha\gamma/\{1-\exp(-\gamma)\}\). The Pocock-like curve considered here is \(A(t)=\alpha\log\{1+(e-1)t\}\), whose initial slope is \(\alpha(e-1)\).

For comparison, a standard one-sided O’Brien–Fleming-like spending curve is \(A(t)=1-\Phi\{z_{1-\alpha}/\sqrt{t}\}\) for \(t>0\), with \(A(0)=0\). Here \(z_{1-\alpha}=\Phi^{-1}(1-\alpha)\). As \(t\) becomes small, this curve evaluates a normal upper tail at a very large positive value. The normal-tail approximation gives \(A(t)\) approximately \(\sqrt{t}\exp\{-z_{1-\alpha}^2/(2t)\}/\{z_{1-\alpha}\sqrt{2\pi}\}\). This exponential term decays especially rapidly as \(t\) approaches zero, explaining the very high early boundary.

Worked Comparisons at an Early Look

The following examples use hypothetical design choices with one-sided \(\alpha=0.025\) and \(t_1=0.10\). The canonical null assumption is \(Z_1\sim N(0,1)\) in each case. These are design calculations, not results from an actual trial.

Why Spending Changes the Boundary

The link between spending and the boundary can be justified directly. Under the stated first-look null model, \(Z_1\sim N(0,1)\). A boundary \(b_1\) has upper-tail crossing probability \(P(Z_1\geq b_1)=1-\Phi(b_1)\). To spend exactly \(A(t_1)\) by that look, set \(1-\Phi(b_1)=A(t_1)\). Rearranging gives \(\Phi(b_1)=1-A(t_1)\), hence \(b_1=\Phi^{-1}\{1-A(t_1)\}\). Because \(\Phi^{-1}\) is increasing, \(b_1\) decreases as \(A(t_1)\) increases.

The numerical comparisons also show why slope alone is not enough for every practical comparison. For differentiable curves, comparing initial slopes ranks their spending only sufficiently close to zero when the leading terms differ. At a specific \(t_1\), higher-order terms may matter; calculate the actual \(A(t_1)\) before calculating the boundary. For a power curve with \(\rho\neq1\), a finite-slope comparison is not available at zero, so compare its actual spending or its order of growth instead.

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to tabulate \(A(t)\) at the planned first-look information fraction and verify that it lies between zero and the total \(\alpha\). Then check the boundary by substituting it back into the null tail: \(1-\Phi(b_1)\) should equal the reported spending, up to rounding. Large discrepancies often indicate a tail-direction error, a mistaken information fraction, or confusion between one-sided alpha and total two-sided alpha.

The alternatives differ in the early error budget they allocate. A power curve allows direct control through \(\rho\); Hwang–Shih–DeCani provides a shape parameter through its exponential form; the Pocock-like example has a finite, relatively large initial slope; and the O’Brien–Fleming-like form has extremely small early spending. Linear spending is a simple benchmark. A fixed boundary is a different design approach: repeatedly applying a constant threshold does not, in general, allocate the same cumulative Type I error as a spending-function design. The spending-function approach specifies cumulative error at information times and calibrates boundaries to that allocation.

These comparisons do not choose a universally best curve. A lower early boundary can make early stopping for efficacy easier, but the curve must still be appropriate for the design’s intended error allocation and operating characteristics. Small-information approximations describe behavior near zero, not the entire curve. In addition, the boundary formula here is specifically for the first look under the canonical null distribution. Later-look boundaries must account for the joint distribution and previous crossing regions; the first-look calculation cannot simply be reused unchanged.

A common mistake is to infer that a curve with a steep initial rise must have a low first boundary at every early look, or to calculate the boundary from the slope alone. The slope describes a local approximation, while the boundary at the planned look depends on the exact cumulative spending \(A(t_1)\). Another frequent error is to divide alpha by two without first deciding whether the specified spending is one-sided or total two-sided spending. State the tail convention, evaluate the spending function, and only then apply the appropriate normal quantile.