Pocock-Like Spending Curves

Why Use a Pocock-Like Spending Curve?

A group sequential design must control the probability of a false efficacy conclusion across all planned looks, not merely at the final analysis. The shape of the spending curve determines how that total error budget is distributed over information time. A Pocock-like curve assigns a comparatively larger share early than an O’Brien-Fleming-like curve, while still allowing monitoring to follow the prespecified spending rule if the actual look times differ from the original plan.

The curve describes cumulative type I error spending; it is not itself a sequence of critical values. Boundaries must be calculated using the joint null distribution of the interim statistics so that the probability of crossing any boundary by a given look matches the cumulative spending target.

Definition: For a two-sided design with total type I error level $\alpha$, the Pocock-like cumulative spending function is $$ \alpha_P(t)=\alpha\log\{1+(e-1)t\},\qquad 0\le t\le 1, $$ where $t$ is the information fraction and $e$ is the base of the natural logarithm. The function gives the target cumulative probability, under $H_0$, of having crossed an efficacy boundary by information fraction $t$. In a symmetric two-sided design, the target is divided equally between the upper and lower tails.

This function starts at zero and ends at the full error level: $\alpha_P(0)=0$ and $\alpha_P(1)=\alpha$. It is increasing and concave, so cumulative spending rises quickly at first and then more slowly. The derivative makes this shape explicit:

\alpha_P'(t)=\frac{\alpha(e-1)}{1+(e-1)t}, \qquad \alpha_P''(t)=-\frac{\alpha(e-1)^2}{\{1+(e-1)t\}^2}<0.

The decreasing derivative means that, per unit increase in information fraction, the curve allocates more error early than late. This does not mean the design spends a fixed proportion of $\alpha$ at every look, nor does it imply that critical values are constant.

Assumptions and Boundary Construction

The usual construction uses the canonical joint null model for the sequential statistics. Let $Z_i$ be the standardized test statistic at look $i$, and let $t_i$ be that look's information fraction, with $0<t_1<\cdots<t_K=1$. Under $H_0$, assume each $Z_i$ is standard normal and, for $i<j$,

\operatorname{Corr}(Z_i,Z_j)=\sqrt{\frac{t_i}{t_j}}.

This model is appropriate when the statistics are formed from accumulating information in the canonical way. The look times must correspond to the information used in the analysis, and the planned spending function and sidedness must be specified. If the endpoint, statistic, or dependence structure does not support this joint null model, the usual boundary calculation may not provide the intended error control.

For symmetric two-sided efficacy monitoring, let $b_i$ be the positive boundary at look $i$: a crossing occurs when $|Z_i|\ge b_i$. The boundary is chosen so that, under $H_0$,

P_0\left(\bigcup_{j=1}^{i}\{|Z_j|\ge b_j\}\right)=\alpha_P(t_i).

The subscript $0$ indicates probability under $H_0$. At the first look, $Z_1$ is standard normal, so $b_1=\Phi^{-1}\{1-\alpha_P(t_1)/2\}$, where $\Phi$ is the standard normal cumulative distribution function. At each later look, the new boundary is found numerically using the joint distribution and the boundaries already set.

Key distinction: The increment $\alpha_P(t_i)-\alpha_P(t_{i-1})$ is the unconditional probability mass added to cumulative crossing probability at look $i$: it is the probability of a first crossing at that look under $H_0$. It is not the probability of crossing at look $i$ conditional on no earlier crossing. That conditional probability is the increment divided by the probability of no earlier crossing.

Worked Examples

Worked Example: Calculating Early Cumulative Spending

Consider a hypothetical two-sided group sequential design with total $\alpha=0.05$. Suppose its first planned look is at information fraction $t_1=0.25$. The Pocock-like cumulative target is

\alpha_P(0.25)=0.05\log\{1+(e-1)(0.25)\}.

Using $e-1\approx1.71828$, the expression inside the logarithm is $1+1.71828(0.25)=1.42957$. Thus

\alpha_P(0.25)=0.05\log(1.42957)\approx0.05(0.35737)=0.01787.

The target cumulative null crossing probability by the first look is about 0.01787, or 1.787%. For a symmetric two-sided design, each tail receives half of that target. The curve has therefore spent about 35.7% of the total 0.05 error budget by one quarter of the planned information.

Worked Example: Finding the First Two-Sided Boundary

Continue the hypothetical design above. At its first look, $Z_1\sim N(0,1)$ under $H_0$, and the probability of crossing a symmetric boundary $b_1$ is $2\{1-\Phi(b_1)\}$. Set this equal to the first-look spending target:

2\{1-\Phi(b_1)\}=0.01787, \qquad b_1=\Phi^{-1}(1-0.01787/2) =\Phi^{-1}(0.991065)\approx2.3684.

The first-look efficacy rule is to stop for efficacy if $|Z_1|\ge2.3684$, subject to using the unrounded boundary in an actual implementation. Under the stated standard-normal null model, the crossing probability is approximately $2\{1-\Phi(2.3684)\}=0.01787$. The probability refers to a false boundary crossing under $H_0$, not to the probability that $H_0$ is true.

Worked Example: Allocating Error Across Four Looks

Suppose the same hypothetical design plans looks at $t=0.25,0.50,0.75,1.00$. Applying the curve at each point gives the following approximate cumulative targets. The final target equals 0.05 because $t=1$.

Information fraction	Cumulative target	Increment from prior look
0.25	0.01787	0.01787
0.50	0.03101	0.01314
0.75	0.04140	0.01039
1.00	0.05000	0.00860

For example, the second increment is $0.03101-0.01787=0.01314$. This is the unconditional probability of a first crossing at the second look under $H_0$, once the boundaries are calibrated to the spending targets. If one instead asks for the conditional probability of crossing at the second look given no crossing at the first, the calculation is

\frac{0.01314}{1-0.01787}\approx0.01338.

The difference matters: the increment is measured relative to all null outcomes, while the conditional probability is measured only among outcomes that survived the first look.

How Later Boundaries Are Determined

At a later look, the boundary must account for outcomes that would already have crossed an earlier boundary. For example, at look $i$, the desired increment is

P_0\left(|Z_i|\ge b_i,\ |Z_j|<b_j\text{ for all }j<i\right) =\alpha_P(t_i)-\alpha_P(t_{i-1}).

The left side depends on $b_i$, the earlier boundaries, and the canonical correlation structure. A numerical root-finding procedure adjusts $b_i$ until this probability equals the spending increment. Equivalently, the procedure checks the cumulative union-crossing probability through look $i$ against $\alpha_P(t_i)$. Because raising $b_i$ cannot increase the probability of a new crossing at look $i$, this is a monotone numerical search.

This is the key reason that one cannot simply use the single-look normal quantile associated with each increment. The increments are not independent per-look test levels: the interim statistics are correlated, and the target is cumulative probability of any crossing. The Lan-DeMets construction uses the cumulative spending targets to update boundaries at each observed information time under the assumed joint null model.

Specify the plan.
Set the overall $\alpha$, sidedness, Pocock-like spending function, and information fractions.

Calculate cumulative targets.
Evaluate $\alpha_P(t_i)$ at each look. The target must increase from zero to $\alpha$.

Calibrate boundaries sequentially.
Use the standard normal distribution at the first look, then the canonical joint null distribution and prior boundaries at later looks.

Check the operating rule.
Confirm that the probability of any boundary crossing through each look matches its cumulative spending target, allowing only numerical tolerance and rounding.

Comparison, Diagnostics, and Limitations

The O’Brien-Fleming-like spending curve is comparatively conservative early: it spends little error at small information fractions and allocates more of the total near the final analysis. The Pocock-like curve spends more early and less in its later increments. This can make early stopping for efficacy less demanding than under an O’Brien-Fleming-like design, while potentially requiring a more stringent final boundary. Exact critical values depend on the information times, number of looks, sidedness, and joint calibration; the curve alone does not determine them.

Other spending families, such as Hwang-Shih-DeCani and power spending, provide alternative shapes. The choice is a design decision about how to distribute the error budget over information time. A Pocock-like shape is not automatically preferable: earlier spending may be useful when early evidence is especially valuable, but it also changes the trade-off between early and late opportunities to stop.

Several checks help reveal a faulty implementation:

Check endpoints and monotonicity. The cumulative target should start at zero, never decrease, and equal $\alpha$ at $t=1$. A value outside $[0,\alpha]$ signals an input or implementation error.
Check cumulative crossing probabilities. Simulated or numerically evaluated null crossing probabilities should agree with the targets at each look, within the method's numerical tolerance.
Do not treat increments as conditional levels. Dividing an increment by the probability of surviving prior looks is necessary if a conditional probability is requested.
Use information fractions, not merely calendar time. The curve is evaluated at the information available for analysis. Substituting elapsed time for information without justification can misallocate error.
Retain precision in boundaries. Rounded critical values can shift crossing probabilities, especially when several looks are involved. Use full-precision values for implementation and report rounded values for display.

The spending curve controls type I error under its assumed null model; it does not guarantee power, specify a futility rule, or remove the need to check that the trial's test statistic and information measure are appropriate. Its practical flexibility also does not mean arbitrary changes to the analysis are harmless: the boundary must be recalculated using the actual information time and valid joint model.

Common pitfall: It is incorrect to say that an increment such as $\alpha_P(t_i)-\alpha_P(t_{i-1})$ is the probability of crossing at look $i$, conditional on not crossing earlier. The increment is unconditional probability mass for a first crossing at that look. The corresponding conditional probability divides the increment by the probability of no earlier crossing.

Check Your Understanding

Answer each question, then compare with the solutions below.

For $\alpha=0.05$, explain what $\alpha_P(0.25)$ represents in a two-sided design.
Why does a Pocock-like curve allocate comparatively more error early than late?
If the cumulative targets at two successive looks are 0.020 and 0.034, what is the unconditional probability mass assigned to a first crossing at the second look?
Why is it generally wrong to set a later boundary using only the normal quantile for that look's spending increment?
Name two checks that can identify a mistaken implementation of the spending curve or boundaries.

Solutions. (1) It is the target probability, under $H_0$, of crossing either efficacy boundary by information fraction 0.25; it is not a probability about whether $H_0$ is true. (2) The curve is concave: its derivative is positive but decreases as $t$ increases, so cumulative spending rises faster at early information fractions. (3) The increment is $0.034-0.020=0.014$, or 1.4% unconditionally. (4) Earlier and current statistics are correlated, and the target is cumulative probability of any crossing; later boundaries must account for earlier crossings under their joint null distribution. (5) Check that the cumulative targets start at zero, increase, and finish at $\alpha$; also verify that the calibrated null crossing probability through each look matches its target and that information fractions, rather than unverified calendar-time fractions, were used.

What You'll Learn