Why Spend Very Little Alpha Early?
A group sequential design tests a primary hypothesis at planned interim looks as well as at the final analysis. If each look used the full nominal significance level, the chance of at least one false positive under \(H_0\) would generally exceed the intended overall level. An alpha-spending function instead specifies how much of the overall type I error may have been used by each information time.
An O’Brien-Fleming-like curve has a particular pattern: it allows very little cumulative type I error at early looks, then permits more spending as information accumulates. This is useful when the design is intended to make an early efficacy conclusion difficult while retaining the possibility of stopping for strong evidence. The curve is a target for cumulative crossing probability, not a list of critical values.
Here, \(A_{\mathrm{OF}}(t)\) is the target probability, under \(H_0\), of having crossed an efficacy boundary by information fraction \(t\). The function starts near zero and reaches \(\alpha\) at \(t=1\). For example, in the two-sided case, \(A_{\mathrm{OF}}(1)=2[1-\Phi(z_{1-\alpha/2})]=\alpha\). At very small \(t\), the quantity \(z_{1-\alpha/2}/\sqrt{t}\) is large, so the normal tail probability is small.
From a Spending Curve to Boundaries
To turn cumulative targets into sequential boundaries, the design needs the joint null distribution of the test statistics across looks. Under the canonical group sequential model, the standardized statistics \(Z_1,\ldots,Z_K\) have a joint multivariate normal distribution, each with mean zero and variance one under \(H_0\). At information fractions \(t_1<\cdots<t_K\), their correlations are
For a symmetric two-sided design, the efficacy boundaries are \(\pm c_i\). The first boundary is chosen so that \(P(|Z_1|\geq c_1\mid H_0)=A_{\mathrm{OF}}(t_1)\). At a later look \(i\), \(c_i\) is chosen so that the cumulative probability of crossing at any look through \(i\) equals \(A_{\mathrm{OF}}(t_i)\), accounting for the dependence between the statistics and for crossings already possible at earlier looks.
This distinction matters. The marginal tail probability \(P(|Z_i|\geq c_i\mid H_0)\) is not generally the same as the additional type I error spent at look \(i\), nor is it generally equal to \(A_{\mathrm{OF}}(t_i)\). The boundary is found by solving a joint-probability equation. In practice, that equation is evaluated numerically.
The classical O’Brien-Fleming design is often described using critical values proportional to \(1/\sqrt{t_i}\). Its boundary is very high early and falls as information accumulates. The O’Brien-Fleming-like spending curve imports that early-conservative shape into a cumulative spending framework. With a finite set of looks, however, the spending-function boundaries are calculated to match the cumulative crossing targets; they need not equal the classical fixed-boundary values exactly.
Choose the overall \(\alpha\), one- or two-sided testing, and the planned information fractions.
Evaluate the appropriate O’Brien-Fleming-like function at each information fraction.
Match its null crossing probability to the first cumulative spending target.
Use the joint null distribution to make the probability of any crossing by that look equal the corresponding cumulative target.
Worked Examples
Worked Example: Spending at Early Information Fractions
Consider a hypothetical two-sided design with overall \(\alpha=0.05\). The standard normal quantile is \(z_{0.975}\approx1.960\). Calculate the cumulative spending at \(t=0.25\), \(0.50\), and \(0.80\).
At \(t=0.25\), the cutoff in the normal tail is $$ \frac{1.960}{\sqrt{0.25}}=\frac{1.960}{0.5}=3.920. $$ Thus $$ A_{\mathrm{OF}}(0.25)=2[1-\Phi(3.920)]\approx0.000089. $$
At \(t=0.50\), the cutoff is \(1.960/\sqrt{0.50}\approx2.772\), giving $$ A_{\mathrm{OF}}(0.50)=2[1-\Phi(2.772)]\approx0.00558. $$ At \(t=0.80\), the cutoff is \(1.960/\sqrt{0.80}\approx2.191\), so $$ A_{\mathrm{OF}}(0.80)=2[1-\Phi(2.191)]\approx0.0284. $$
These are cumulative null crossing targets, not three independent significance levels. By \(t=0.25\), only about \(0.000089\) of type I error has been spent; by \(t=0.80\), the target is about \(0.0284\), out of the total \(0.05\).
Worked Example: Solving a Two-Look Boundary
Suppose a hypothetical two-sided design has \(\alpha=0.05\) and two looks at \(t_1=0.50\) and \(t_2=1.00\). At the first look, the spending target is approximately \(0.00558\). The first boundary is therefore $$ c_1=\frac{z_{0.975}}{\sqrt{0.50}}\approx2.772. $$ The canonical correlation between \(Z_1\) and \(Z_2\) is \(\sqrt{0.50/1.00}\approx0.7071\).
At the final look, choose \(c_2\) to satisfy the cumulative crossing equation $$ P\left(|Z_1|\geq c_1\ \text{or}\ |Z_2|\geq c_2\mid H_0\right)=0.0500, $$ where \((Z_1,Z_2)\) is bivariate standard normal with correlation \(0.7071\). Equivalently, the probability of remaining inside both boundaries must equal \(0.9500\): $$ P(-c_1<Z_1<c_1,\ -c_2<Z_2<c_2\mid H_0)=0.9500. $$
Numerically solving this bivariate normal probability equation gives \(c_2\approx1.9794\), or \(1.979\) to four significant figures. The final marginal probability \(P(|Z_2|\geq1.9794)\) is about \(0.0477\), but the cumulative crossing probability is \(0.0500\), because a small part of the final-look tail overlaps with paths that already crossed at the first look. The increase in the cumulative spending target from the first to final look is about \(0.0500-0.00558=0.0444\); it is not obtained by simply setting the final marginal tail probability to \(0.0444\).
Worked Example: Evaluating the Curve at an Actual Information Time
Suppose a hypothetical two-sided study with \(\alpha=0.05\) reaches an analysis at \(t=0.40\). The spending function is evaluated at the information fraction actually used for the analysis: $$ \frac{z_{0.975}}{\sqrt{0.40}}=\frac{1.960}{0.6325}\approx3.099, \qquad A_{\mathrm{OF}}(0.40)=2[1-\Phi(3.099)]\approx0.00194. $$
Thus, the target cumulative type I error by this analysis is about \(0.00194\). This value is not itself the boundary: the boundary must be obtained by matching the crossing probability to that target, taking account of any previous looks. If this is the first look, its two-sided boundary is about \(3.099\), because the marginal two-sided tail probability at that cutoff is the target. If it is a later look, its boundary requires the joint probability calculation illustrated above.
Why the Shape Is Useful, and How It Can Go Wrong
The O’Brien-Fleming-like shape is appropriate when early stopping for efficacy should require unusually strong evidence. It preserves most of the type I error for later analyses, when more information is available. Its advantages are therefore paired with a limitation: an early true effect may not cross the stringent early boundary, even if the study would ultimately have sufficient power at its final analysis.
Compared with fixed classical O’Brien-Fleming boundaries, a spending-function design expresses the type I error plan as cumulative targets and constructs boundaries for the planned looks. This is useful when actual analysis times are not perfectly fixed, provided the information fraction and canonical model are validly specified. Compared with the Power Spending Family, the defining emphasis here is exceptionally small early spending; the shape and resulting boundary schedule differ, even though both approaches allocate a total error budget across looks. Neither spending shape alone determines power: power also depends on the alternative effect, information schedule, and boundary construction.
Several checks help reveal an incorrect implementation. The computed spending must be nondecreasing with \(t\), be close to zero as \(t\) approaches zero, and equal \(\alpha\) at \(t=1\). The cumulative crossing probability at each look should match the target, within numerical tolerance. Also verify that the information fractions are in the right order and that the one-sided or two-sided quantile matches the planned test. A numerical boundary that decreases in an unexpected way, or a cumulative crossing probability that misses its target, calls for checking the model inputs and integration, not merely rounding the displayed cutoff.
Another limitation is that the formula depends on a meaningful information fraction and a correctly specified null distribution for the sequential statistics. If the statistics are poorly calibrated, the information measure is inappropriate, or the assumed dependence across looks fails, the nominal spending targets need not deliver the intended type I error control. The curve is a design rule under stated modeling conditions, not a remedy for invalid test statistics.
Check Your Understanding
Answer each question, then compare with the solutions below.
- For a two-sided design with \(\alpha=0.05\), what is the spending target at \(t=1\), and why?
- Why is \(A_{\mathrm{OF}}(0.25)\) much smaller than \(A_{\mathrm{OF}}(0.80)\)?
- In the two-look example, why is the final marginal tail probability not the same as the cumulative target of \(0.0500\)?
- What correlation does the canonical model imply for looks at \(t_1=0.36\) and \(t_2=0.81\)?
- Name two checks that can help identify an incorrectly implemented spending curve or boundary calculation.
Solutions. (1) It is \(0.05\), because \(A_{\mathrm{OF}}(1)=2[1-\Phi(z_{0.975})]=0.05\). (2) The normal-tail cutoff \(z_{0.975}/\sqrt{t}\) is much larger at the earlier fraction, so its tail probability is smaller. (3) The cumulative probability includes crossings at either look; the joint distribution means some paths in the final marginal tail have already crossed at look one. (4) The correlation is \(\sqrt{0.36/0.81}=\sqrt{4/9}=2/3\). (5) For example, check that spending is nondecreasing and reaches \(\alpha\) at \(t=1\), and verify that joint cumulative crossing probabilities match the targets at each look.