Signature
Delta_alpha_k = alpha_k - alpha_(k-1)
| Inputs | Definition | Unit |
|---|---|---|
alpha_k | Cumulative spend alpha(t_k) at analysis k from the chosen spending function | probability |
alpha_(k-1) | Cumulative spend at the previous analysis, equal to 0 at the first look | probability |
Delta_alpha_k | Additional type I error that may be used at analysis k | probability |
|---|
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
Alpha spending increment from two cumulative spends
With the cumulative spend at this look and at the previous look in named cells, Excel returns the increment. At the first look the previous spend cell holds 0.
=SpendThisLook-SpendPrevLook
Assumptions
Spending function chosen before the first look
Only the spending function has to be fixed in advance; the number and timing of looks may vary. Choosing analysis times in response to accumulating results can still inflate the type I error.
Increment is a probability of first crossing
The increment is the probability, under the null hypothesis, that the trial first crosses a boundary at analysis k. The critical value that achieves it depends on the earlier boundaries and the correlation between looks, so it is not the normal quantile of the increment.
Worked examples
Second look increment under O'Brien-Fleming type spending
With cumulative spends of about 0.001525 at t of 0.5 and 0.0000074 at t of 0.25, the O'Brien-Fleming type plan may use about 0.001518 at the second look.
alpha_k = 0.001525; alpha_(k-1) = 0.0000074; Delta_alpha_k = 0.001518
Second look increment under Pocock type spending
With cumulative spends of 0.015503 at t of 0.5 and 0.008934 at t of 0.25, the Pocock type plan may use 0.006569 at the second look, less than at the first.
alpha_k = 0.015503; alpha_(k-1) = 0.008934; Delta_alpha_k = 0.006569
Common errors
Testing every look at the full significance level
Testing each of four equally spaced looks at the one-sided 0.025 level, a critical value of 1.96, gives an overall false positive probability of about 0.063 rather than 0.025. Only the increment, not the whole budget, is available at each look.
Sources
Increment of the alpha spending function at each analysis
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 describes the increment alpha(t_k) minus alpha(t_(k-1)) as the additional type I error that can be used at the kth analysis, and notes that the number and timing of analyses need not be set in advance.
Analysis timing driven by accumulating results
US Food and Drug Administration. Adaptive designs for clinical trials of drugs and biologics: guidance for industry. November 2019. Section V.A, which states that choosing interim analysis times based on accumulating comparative results can inflate the type I error probability, and asks for a targeted number of analyses and an approximate schedule.
Canonical Identity
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