Signature
alpha_P = alpha * log(1 + (exp(1) - 1) * t)
| Inputs | Definition | Unit |
|---|---|---|
alpha | Overall one-sided significance level of the trial, for example 0.025 | probability |
t | Information fraction at which the spend is evaluated | proportion from 0 to 1 |
alpha_P | Cumulative one-sided type I error that may have been used by information fraction t | probability from 0 to alpha |
|---|
Function
Alpha spending function for group sequential trials
Maps the information fraction reached at an analysis of a group sequential trial to the cumulative one-sided type I error that may have been used by that point. The function rises from zero at the start of the trial to the overall significance level at the planned end. Each efficacy boundary is then set so that the probability, under the null hypothesis, of crossing any boundary up to and including analysis k equals the cumulative spend, with correlation sqrt(t_l / t_k) between the test statistics at analyses l and k. That step needs recursive numerical integration and is done in group sequential software, so the records here cover the closed-form parts: the information fraction, the two Lan-DeMets spending functions, the increment available at each look and the hazard ratio that crosses a given boundary.
Try this function
Implementations
Excel
Pocock type cumulative spend in one cell
Excel uses LN, the natural logarithm, and EXP(1) for e, on named cells holding the one-sided level and the information fraction.
=Alpha*LN(1+(EXP(1)-1)*InfoFraction)
Assumptions
Pocock type spend measured on the information scale
t is the information fraction, deaths observed over deaths planned for a survival outcome, not the share of calendar time or of patients recruited.
Pocock type boundaries solved from the joint normal distribution
The function fixes how much error is available, not the critical values. Apart from the first look, where the critical value is Phi^-1(1 minus alpha_P(t_1)), the boundaries are solved numerically from the joint distribution of the test statistics.
Worked examples
Pocock type spend at a quarter of the information
With a one-sided level of 0.025 and t of 0.25, the natural logarithm of 1.42957 is 0.35737, so the cumulative spend is 0.025 times 0.35737, about 0.008934. The first boundary is then 2.368.
alpha = 0.025; t = 0.25; alpha_P = 0.008934
Pocock type spend at half the information
At t of 0.5 the cumulative spend is about 0.015503, already more than half of the 0.025 budget.
alpha = 0.025; t = 0.5; alpha_P = 0.015503
Common errors
Using Excel LOG instead of LN for Pocock type spending
Excel LOG uses base 10 by default. At t of 0.25 it gives a spend of about 0.00388 instead of 0.008934, so the first boundary is set far too high.
Reading the cumulative spend as the p-value needed at a look
By the second of four equally spaced looks the Pocock type plan has spent about 0.0155, but the nominal one-sided p-value needed to stop there is about 0.009, because the first look has already used part of the budget.
Sources
Pocock type spending function formula in gsDesign
Anderson KM. gsDesign R package reference for sfLDOF and sfLDPocock (Lan-DeMets spending function approximations). Gives f(t; alpha) = alpha ln(1+(e minus 1)t) for the Pocock approximation.
Printed Pocock type spend and bounds for four looks
Anderson KM. gsDesign Technical Manual, chapter 8.1 (spending function definitions). Prints the spend 0.00893435, 0.01550286, 0.02069972 and 0.025 and the bounds 2.368328, 2.367524, 2.358168 and 2.350030 for alpha of 0.025 and four equally spaced looks.
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0