Signature
t_k = d_k / D
| Inputs | Definition | Unit |
|---|---|---|
d_k | Number of deaths (events) observed in both arms combined at analysis k | events |
D | Number of deaths expected by the planned final analysis | events |
t_k | Share of the planned final information that is available at analysis k | proportion from 0 to 1 |
|---|
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.
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Implementations
Excel
Information fraction from death counts in one cell
Excel divides the deaths observed at the analysis by the planned number of deaths, held in named cells.
=Deaths/PlannedDeaths
Assumptions
Deaths measure information for a survival outcome
With equal allocation the information in the log-rank statistic is approximately one quarter of the number of deaths, so a ratio of deaths approximates the ratio of information. Treating events as information assumes proportional hazards between the arms.
Final number of deaths estimated while the trial runs
D is a planning figure that has to be estimated while the trial is in progress. If the estimate of D is revised, the information fractions of later analyses, and so the error they may spend, change with it.
Worked examples
First interim look at 100 of 400 planned deaths
A survival trial planned to end at 400 deaths holds its first interim analysis at 100 deaths, so the information fraction is 0.25.
d_k = 100; D = 400; t_k = 0.25
Third interim look at 300 of 400 planned deaths
At the third analysis of the same trial 300 deaths have been observed, giving an information fraction of 0.75.
d_k = 300; D = 400; t_k = 0.75
Common errors
Measuring the information fraction in calendar time
Using elapsed calendar time, for example two years of a planned four, gives t of 0.5 even when only 100 of 400 deaths have occurred. Under the Pocock type function the trial would then spend about 0.0155 instead of about 0.0089 at that look, using error that belongs to later analyses.
Sources
Information fraction as deaths over expected deaths
DeMets DL, Lan KKG. Interim analysis: the alpha spending function approach. Statistics in Medicine. 1994;13:1341-1352. Section on the alpha spending function, which defines the information fraction as n/N for a comparison of means and approximates it by d/D, observed over expected deaths, for survival analyses.
Log-rank information approximately one quarter of the deaths
Anderson KM. gsDesign Technical Manual, chapter 7 (other parameterisations), section on the log-rank statistic, which states that for equally sized groups the information is approximately d/4 under proportional hazards.
Canonical Identity
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