Signature
pi_E = P^c / (P^c + (1 - P)^c)
| Inputs | Definition | Unit |
|---|---|---|
P | Current posterior probability that the experimental arm has the higher success rate | probability |
c | Zero or above; 0 gives equal randomisation and 1 Thompson sampling | none |
pi_E | Allocation probability for the experimental arm under the Thall and Wathen rule | probability |
|---|
Function
Bayesian interim decision function for an adaptive trial
Maps a prior distribution for the treatment effect, the data observed so far and pre-specified thresholds to the quantities that interim rules consult: the posterior probability that the effect exceeds a minimum value, which summarises the evidence now, and the predictive probability that the final analysis will succeed, which averages over the results still to come. Response-adaptive rules use the same posterior to set the chance of allocation to each arm. The notation follows the Bayesian Adaptive Design article.
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Implementations
Excel
Thall and Wathen allocation probability from named cells
With the posterior probability in PostProbBetter and the tuning constant in Tuning, the formula returns the allocation probability, held in AllocProb.
=PostProbBetter^Tuning/(PostProbBetter^Tuning+(1-PostProbBetter)^Tuning)
Assumptions
Two arms with outcomes observed quickly
The rule allocates between an experimental and a control arm, and the posterior probability is updated from outcomes already observed, so the outcome must be available soon after treatment for the allocation to adapt.
Stable patient population and standard of care over recruitment
Robertson and colleagues describe time trends, from changes in standard of care or drift in the patients recruited, as a major barrier to response-adaptive randomisation, because they can distort standard analyses.
Worked examples
Article's allocation with P of 0.918 and c of one half
With a posterior probability of 0.918 and c of 0.5, the square roots are about 0.958 and 0.286, so the next patient joins the experimental arm with probability about 0.770, as in the article.
P = 0.918; c = 0.5; pi_E = 0.7699
Thompson sampling with c of 1
With c of 1 the allocation probability equals the posterior probability, 0.918 (computed here for illustration).
P = 0.918; c = 1; pi_E = 0.918
Tuning constant rising with enrolment
With c = i / (2n) after 50 of 200 planned patients, c is 0.125 and the allocation probability about 0.575, closer to equal allocation early in the trial (computed here for illustration).
P = 0.918; c = 0.125; pi_E = 0.5749
Common errors
Using Thompson sampling without regard to allocation variability
With c of 1 allocation follows every swing in the posterior. In the simulations reported by Robertson and colleagues, Thompson sampling with 200 patients had about a 14% chance of a sample size imbalance of more than 10% of the total in the wrong direction, and the Thall and Wathen rule reduced it.
Reading the allocation probability as the probability that the treatment is better
An allocation probability of 0.770 comes from a posterior probability of 0.918 shrunk towards one half by c. It is a design choice, not an estimate of benefit, and unequal arms leave fewer control patients for the costs and utilities an economic model needs.
Sources
Thall and Wathen allocation rule in a review of response-adaptive randomisation
Robertson DS, Lee KM, López-Kolkovska BC, Villar SS. Response-adaptive randomization in clinical trials: from myths to practical considerations. Statistical Science. 2023;38(2):185-208. Section on whether response-adaptive randomisation leads to a substantial chance of allocating more patients to an inferior treatment: the TW(c) rule randomises to the experimental arm with probability P^c / (P^c + (1 minus P)^c), where P is the posterior probability that it has the higher success rate; c of 0 gives equal randomisation and 1 Thompson sampling; c of 1/2 or i/(2n) is suggested; simulation results on imbalance; section 3.4 on time trends.
FDA guidance on response-adaptive randomisation
US Food and Drug Administration. Adaptive Designs for Clinical Trials of Drugs and Biologics: Guidance for Industry. Silver Spring, MD: FDA; November 2019. Section V.E: in response-adaptive randomisation the chance of assignment to each arm varies with accumulating outcome data; the statistical, ethical and pragmatic arguments for it are controversial.
Canonical Identity
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