Many contemporary applications are surveyed, and over new references are included. Over the past 40 years the Gittins index has helped theoreticians and practitioners to address a huge variety of problems within chemometrics, economics, engineering, numerical analysis, operational research, probability, statistics and website design.
This new edition will be an important resource for others wishing to use this approach. Multi-armed Bandit Allocation Indices. In the first edition of this book set out Gittins' pioneering index solution to the multi-armed bandit problem and his subsequent investigation of a wide of sequential resource allocation and stochastic scheduling problems. Also marked is the standard normal distribution which Z should follow in the FR trial red. The sample mean.
The empirical 95th-percentile under H 0 will correspond to the critical value for hypothesis testing, and is marked by a vertical dotted line on the histograms. The sample standard deviation of 1. Notice that because both left and right tails are heavier than the normal tails, the inflation of type I error rate when testing hypotheses at the normal cut-off value would be even greater in a two-tailed test. Instead, the empirical 95th-percentile of the distribution is.
The small left-hand peak has a weight of. Following the same procedure, 95th-percentiles of the test statistic distribution are estimated for the other adaptive trial designs. We now present results of 10 4 repetitions of each trial design using the estimated values described before as a priori critical values.
The s. The Upper Bound UB row displays a theoretical optimum for each measurement based on a design which assigns every patient to the best treatment i.
UB: theoretical UB from assigning all patients best treatment. All the adaptive rules achieve better patient welfare than the FR design under H 1. The TS trial is outperformed by the other adaptive designs in terms of patient welfare; this is explained by the tuning parameter c in the TS mechanism which stabilises the randomisation probabilities.
The standard deviation of 0. This indicates that most trials under CB and to some extent also GI and RBI are highly unbalanced, with one arm being dropped early on and most patients receiving the same treatment. Reduced power is also evident in the other trial designs c. The low standard deviation of 0.
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The results for the batched TS TSB illustrate the effects of a blocked implementation of the algorithm to deal with a moderate delay: a marginal increase in power and a considerable decrease of the patient welfare benefits. However, the patient-benefit advantages of TSB over FR are considerably large even if assuming a moderate delay in patient recruitment.
Trial simulations are then run using the computed quantiles as critical values; for each design the trial is run 10 4 times. As in the two-arm scenario, all the adaptive rules outperform the FR design under H 1 in terms of patient welfare, although TP only improves marginally over FR in this case. In particular, CB, which is essentially the simplest myopic approach, exhibits the worst performance in terms of power and variability. As in the two-arm trial, KLU achieves considerably greater power than UCB and the welfare benefit is only slightly reduced, offering a very good compromise between the two conflictive objectives.
Conversely, TPB results in a slightly reduced power than TP while the patient wlefare is practically identical. For the designs which were included in the two-arm simulation, the results here are similar. CG significantly lowers the bias in the estimates of control treatment outcomes, but it does not improve the issue of negatively biased estimates of unselected experimental treatment outcomes, where it performs almost identically to the original GI.
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Due to the greatly reduced sample sizes, all designs now achieve much lower power, a common situation in drug development for rare diseases. Both perform similarly well, achieving higher power than FR, and offering a marked improvement in patient welfare compared with FR. The results in the table for the batched approaches show that, as expected, as the delay in recruitment is more severe the advantages of TSB and TPB over FR are significantly reduced though both designs still offer important patient welfare advantages. Noticeably, the effect on power and patient welfare of a severe delay in the controlled version i.
The controlled version has its power levels reduced as the delay increases while the opposite happens to TSB. TP improves power over FR by matching the allocation of the control arm to that of the best performing arm, therefore increasing the allocation to these two arms over the other arms. With a larger delay TBP will allocate larger number of patients to all arms which therefore reduces its marginal power levels compared to TP.
For TSB the power improvement is explained because the design cannot skew allocation to the best arm as fast as with TS, thus allocating more patients to all arms when compared to TS. For most trial designs, there is little variation in C 0. However, for the UCB trial, C 0. Therefore, the 64 person trial conducted at the higher critical value of 2. As a result, the UCB mechanism may be unsuitable for trials where the total number of patients to be recruited is not known in advance.
This effect is less pronounced in the KLU variant making it more suitable in that case. Empirical critical values C 0. Since the trial size was much smaller than expected, there is a motivation to consider if using a smaller value for d would affect results, as a smaller discounting factor corresponds to putting less value on learning for the future. The simulation results provided by this paper illustrate how the index-based response-adaptive design derived from the MABP can lead to significant improvements in patient welfare also with a normally distributed endpoint.
In all situations, designs based on the Gittins index achieved the largest patient welfare gain over FR trials or myopic designs currently in use in drug development such as TP. However, there are a number of limitations to the effectiveness of the purely deterministic Gittins index design that still prevail. As in the binary case, the Gittins index rule exhibits considerably lower power than FR, and whilst the loss of power can be alleviated to some extent by the introduction of random perturbations to the indices RGI , in the two-arm trial the power achieved is still not sufficient for most clinical trials unless the exploration term is correctly calibrated.
In a multi-armed case, the patient welfare advantages of adaptive designs, and GI-based particularly, over FR are the largest. Moreover, there are adaptive designs that can offer more power than FR together with a patient-benefit advantage, making them suitable for drug development for common conditions. In the four-arm case based on a real trial we studied, a small deviation from optimality by protecting the allocation of the control treatment CG and CUC offers a power close to or even above FR's while still providing considerable patient-benefit.
In contexts where power is relatively less important if there are very few disease sufferers outside the trial , GI, RGI or UCB offer even better patient welfare at the expense of a power reduction. However, such power gains require a very large number of patients in the trial to be also accompanied by similar patient welfare advantages.
For example, KLU dominates over UCB under both criteria only in scenarios where trials had more than thousands of patients. For smaller and more realistic trial sizes, as the ones considered in this paper, UCB had better patient welfare and less power than KLU. Nevertheless, rules like KLU offer a good trade-off between the two objectives and can be suitable designs for common diseases. An important observation drawn from the simulations provided by this paper is that the type I error deflation of the GI observed for the Bernoulli case does not hold in the normally distributed case.
Actually, if no correction is introduced using a standard test will result in an important type I error inflation. In this work we have outlined a simulation-based procedure that can be used to prevent this inflation. Alongside the statistical community's faith in randomisation is a trust in frequentist inference, so this is generally used even in Bayesian trials to make the results as persuasive as possible. But, the inferential power and the potential patient-benefit from adaptive trials could be improved by applying Bayesian inference methods combined with the use of prior data.
Further research could seek an appropriate method of Bayesian inference based on index-based adaptive trials, for example, by considering which arm the adaptive design is favouring most at the end of the trial, or by incorporating information derived from historical data. Due to some mathematical properties of the problem, this is the same problem as choosing a patrol pattern, like the one in the picture above, that he can repeat over and over. This means that, even though he is choosing over and over for an infinite number of steps, he doesn't need an infinitely long plan, because he can just repeat his finite one.
Kaufmann : On Bayesian index policies for sequential resource allocation
To help solve this problem, we represent the locations using a graph, as shown in the picture above. The nodes circles represent the locations and a line between two locations means that the patroller can reach one from the other. In the picture, the green lines represent the patroller's path.
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- Kaufmann : On Bayesian index policies for sequential resource allocation.
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