Signature
HR_k = exp(-2 * c_k / sqrt(d_k))
| Inputs | Definition | Unit |
|---|---|---|
c_k | Critical value of the standardised test statistic at analysis k under the chosen spending plan | standard normal deviate |
d_k | Number of deaths observed in both arms combined at analysis k | events |
HR_k | Largest observed hazard ratio, experimental arm against control, that crosses the efficacy boundary at analysis k | ratio, dimensionless |
|---|
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
Stopping hazard ratio from a critical value in one cell
Excel returns the hazard ratio threshold from named cells holding the critical value and the number of deaths at the analysis.
=EXP(-2*CritValue/SQRT(Deaths))
Assumptions
Equal allocation and log-rank information of d/4
Patients are randomised 1:1, so the information is about one quarter of the deaths and the standard error of the log hazard ratio is about 2 divided by the square root of d_k. A lower hazard ratio favours the experimental arm.
Proportional hazards for the stopping hazard ratio
The conversion assumes proportional hazards. With a delayed treatment effect the hazard ratio thresholds, and the chance of stopping at early looks, can mislead, although type I error control is unaffected.
Worked examples
Second O'Brien-Fleming type look at 200 deaths
With a critical value of 2.963 at 200 deaths, 2 times 2.963 is 5.926, divided by the square root of 200, 14.142, gives 0.4190, and exp(-0.4190) is about 0.658. An observed hazard ratio of 0.658 or lower crosses the boundary.
c_k = 2.963; d_k = 200; HR_k = 0.658
First Pocock type look at 100 deaths
With a critical value of 2.368 at 100 deaths the trial stops only if the observed hazard ratio is about 0.623 or lower.
c_k = 2.368; d_k = 100; HR_k = 0.623
Final O'Brien-Fleming type analysis at 400 deaths
At the final analysis the critical value of 2.014 at 400 deaths corresponds to an observed hazard ratio of about 0.818.
c_k = 2.014; d_k = 400; HR_k = 0.818
Common errors
Feeding the stopping hazard ratio into the model as the true effect
Every early stop reports a hazard ratio at or below the threshold. If the true hazard ratio were 0.80, any Pocock type stop by the second look would report about 0.715 or lower, so a survival model built on the naive estimate overstates the gain. Estimates adjusted for the stopping rule are needed.
Sources
Log-rank statistic and information for a hazard ratio
Anderson KM. gsDesign Technical Manual, chapter 7 (other parameterisations), section on time-to-event outcomes: the log-rank statistic is approximately normal with mean equal to the log hazard ratio times V(d), and V(d) is approximately d/4 for equally sized groups under proportional hazards.
Bias of conventional estimates after group sequential stopping
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 conventional estimates tend to be biased towards greater effects when a group sequential design is used.
Canonical Identity
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