Bayesian adaptive bandit-based designs using the Gittins index for multi-armed trials with normally distributed endpoints

Abstract

Adaptive designs for multi-armed clinical trials have become increasingly popular recently because of their potential to shorten development times and to increase patient response. However, developing response-adaptive designs that offer patient-benefit while ensuring the resulting trial provides a statistically rigorous and unbiased comparison of the different treatments included is highly challenging. In this paper, the theory of Multi-Armed Bandit Problems is used to define near optimal adaptive designs in the context of a clinical trial with a normally distributed endpoint with known variance. We report the operating characteristics (type I error, power, bias) and patient-benefit of these approaches and alternative designs using simulation studies based on an ongoing trial. These results are then compared to those recently published in the context of Bernoulli endpoints. Many limitations and advantages are similar in both cases but there are also important differences, specially with respect to type I error control. This paper proposes a simulation-based testing procedure to correct for the observed type I error inflation that bandit-based and adaptive rules can induce.

Keywords: Gittins index; Multi-armed bandit; normally distributed endpoint; patient allocation; response adaptive procedures; sequential sampling.

Figure 1.

Gittins Index values (normal reward process, known variance) for various discount factors d.

Figure 2.

The posterior mean ${\overline{x}}_{k,t}$ of each treatment arm's outcomes after each patient in a typical GI trial under $H_0$.

Figure 3.

Histograms of empirical distributions of the test statistic Z in GI trials, implemented under each hypothesis. Also marked is the standard normal distribution which Z should follow in the FR trial (red). The sample mean $\overline{Z}$, standard deviation $S_Z$ and an empirical $95\mathrm{th}$-percentile $C_{0.05}$ have been calculated under $H_0$. The empirical $95\mathrm{th}$-percentile under $H_0$ will correspond to the critical value for hypothesis testing, and is marked by a vertical dotted line on the histograms. (a) GI trial under $H_0$ (b) GI trial under $H_1$

Figure 4.

$\mathbb{E}({\overline{x}}_k^{\left(t\right)}-{\mu }_k)$, the mean (across the trial realisations) of the bias in the estimated outcome of each treatment after a total of t patients have been treated across both arms in the trial, under each scenario (two-arm trial simulations). (a) $H_0$, control arm k=0, (b) $H_0$, experimental arm k=1 (c) $H_1$, control arm k=0 and (d) $H_1$, experimental arm k=1.

Figure 5.

$\mathbb{E}({\overline{x}}_k^{\left(t\right)}-{\mu }_k)$, the mean (across the trial repeats) of the bias in the estimated treatment outcome of each drug under each scenario in the four-arm trial (large sample size). (a) $H_0$, control arm k=0, (b) $H_0$, experimental arm k=3, (c) $H_1$, control arm k=0, (d) $H_1$, experimental arm k=3.

Figure 6.

Empirical critical values $C_{0.05}$ for one-tailed testing to maintain 5% FWER in the four-arm trial design, against number T of patients in the trial.

Figure A.1.

Histograms of empirical distributions of the test statistic $Z_{0,1}$ in TS, RBI, RGI, UCB, KLU and CB two-arm trials, implemented under each hypothesis (as in Figure 3). Also marked is the standard normal distribution which $Z_{0,1}$ should follow in the FR trial (red). For each design, the sample mean ${\overline{Z}}_{0,1}$, standard deviation$S_{Z_{0,1}}$ and an empirical 95th-percentile $C_{0.05}$ have been calculated under $H_0$. 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. (a) TS trial under $H_0$, (b) TS trial under $H_1$, (c) RBI trial under $H_0$, (d) RBI trial under $H_1$, (e) RGI trial under $H_0$, (f) RGI trial under $H_1$, (g) UCB trial under $H_0$, (h) UCB trial under $H_1$, (i) KLU trial under$H_0$, (j) KLU trial under $H_1$, (k) CB trial under $H_0$, (l) CB trial under $H_1$.

Figure A.1.

Histograms of empirical distributions of the test statistic $Z_{0,1}$ in TS, RBI, RGI, UCB, KLU and CB two-arm trials, implemented under each hypothesis (as in Figure 3). Also marked is the standard normal distribution which $Z_{0,1}$ should follow in the FR trial (red). For each design, the sample mean ${\overline{Z}}_{0,1}$, standard deviation$S_{Z_{0,1}}$ and an empirical 95th-percentile $C_{0.05}$ have been calculated under $H_0$. 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. (a) TS trial under $H_0$, (b) TS trial under $H_1$, (c) RBI trial under $H_0$, (d) RBI trial under $H_1$, (e) RGI trial under $H_0$, (f) RGI trial under $H_1$, (g) UCB trial under $H_0$, (h) UCB trial under $H_1$, (i) KLU trial under$H_0$, (j) KLU trial under $H_1$, (k) CB trial under $H_0$, (l) CB trial under $H_1$.