Table 1.

Comparison of performance measures for a two‐armed trial using different designs when the variance is assumed unknown (with the exception of FLGI‐known) and $T=72$, averaged over 50 000 trial replications

	${\mathit{\mu }}_0={\mathit{\mu }}_1=0.155$					${\mathit{\mu }}_0=0.155,{\mathit{\mu }}_1=0.529$
Design	${\mathit{t}}_{1-\mathit{\alpha }}$	$\mathit{\alpha }$	$\mathbb{E}\left({\mathit{p}}^{*}\right)$ (SD)	Rel $\mathit{E}\mathit{T}\mathit{O}$% (SD)	Bias (SD)	$1-\mathit{\beta }$	$\mathbb{E}\left({\mathit{p}}^{*}\right)$ (SD)	Rel $\mathit{E}\mathit{T}\mathit{O}$% (SD)	Bias (SD)
ER
b = 1	1.654	0.0518	0.49990 (0.06)	–0.19 (5.45)	–0.0005 (0.15)	0.7884	0.5005 (0.06)	0.20 (5.66)	–0.0013 (0.15)
FLGI‐known
$b=1$	1.991	0.0526	0.4997 (0.33)	–0.06 (5.42)	–0.0004 (0.49)	0.2290	0.8823 (0.16)	41.69 (7.01)	0.2167 (0.45)
$b=2$	1.969	0.0505	0.5002 (0.32)	0.34 (5.42)	–0.0003 (0.44)	0.2701	0.8777 (0.16)	41.27 (6.85)	0.1870 (0.41)
$b=6$	1.911	0.0481	0.4987 (0.29)	–0.03 (5.45)	0.0010 (0.34)	0.3599	0.8605 (0.14)	39.38 (6.59)	0.1147 (0.30)
$b=9$	1.864	0.0492	0.4983 (0.28)	–0.15 (5.43)	0.0008 (0.30)	0.4235	0.8483 (0.13)	38.14 (6.53)	0.0843 (0.26)
$b=18$	1.766	0.0513	0.5017 (0.24)	0.14 (5.44)	–0.0010 (0.23)	0.5653	0.8074 (0.12)	33.72 (6.30)	0.0389 (0.20)
$b=36$	1.682	0.0495	0.5000 (0.19)	0.26 (5.45)	0.0001 (0.17)	0.7124	0.7139 (0.09)	23.25 (5.98)	0.0087 (0.17)
FLGI
$b=1$	2.1820	0.0525	0.5013 (0.29)	–0.15 (5.43)	0.0013 (0.29)	0.3289	0.8712 (0.12)	40.62 (6.39)	0.0955 (0.27)
$b=2$	2.1590	0.0497	0.5016 (0.28)	0.21 (5.45)	0.0014 (0.28)	0.3432	0.8651 (0.12)	39.95 (6.41)	0.0902 (0.26)
$b=6$	2.1180	0.0477	0.4985 (0.27)	0.18 (5.42)	–0.0008 (0.26)	0.3790	0.8521 (0.12)	38.48 (6.36)	0.0801 (0.25)
$b=9$	2.0450	0.0514	0.5011 (0.26)	–0.01 (5.41)	0.0019 (0.25)	0.4236	0.8412 (0.12)	37.13 (6.36)	0.0698 (0.24)
$b=18$	1.8980	0.0517	0.5008 (0.24)	–0.04 (5.42)	0.0005 (0.22)	0.5277	0.8047 (0.12)	33.30 (6.23)	0.0356 (0.20)
$b=36$	1.7330	0.0505	0.4997 (0.18)	0.01 (5.43)	–0.0009 (0.18)	0.6973	0.7128 (0.09)	23.23 (6.00)	0.0097 (0.17)
FLGI‐HZ ($\gamma =2$)
$b=1$	1.658	0.0509	0.5000 (0.05)	0.12 (5.43)	0.0008 (0.15)	0.6510	0.7784 (0.04)	30.46 (5.51)	0.0001 (0.18)
$b=2$	1.660	0.0509	0.5003 (0.05)	–0.14 (5.44)	0.0004 (0.15)	0.6499	0.7786 (0.04)	30.61 (5.54)	–0.0007 (0.18)
$b=6$	1.688	0.0487	0.5001 (0.05)	0.16 (5.42)	0.0000 (0.15)	0.6417	0.7781 (0.04)	30.40 (5.54)	–0.0001 (0.18)
$b=9$	1.661	0.0529	0.4994 (0.05)	0.43 (5.43)	0.0017 (0.15)	0.6501	0.7777 (0.04)	30.27 (5.52)	–0.0004 (0.18)
$b=18$	1.684	0.0490	0.4999 (0.05)	0.11 (5.41)	0.0003 (0.15)	0.6570	0.7644 (0.04)	28.99 (5.54)	–0.0011 (0.18)
$b=36$	1.665	0.0516	0.5000 (0.05)	0.10 (5.41)	0.0017 (0.15)	0.7779	0.5865 (0.05)	9.57 (5.64)	0.0003 (0.15)
TS
$b=1$	1.751	0.0496	0.4999 (0.11)	–0.1108 (5.44)	–0.0001 (0.17)	0.7425	0.6961 (0.11)	21.35 (6.14)	0.0302 (0.19)
$b=2$	1.739	0.0497	0.4997 (0.11)	–0.0431 (5.41)	–0.0016 (0.17)	0.7479	0.6934 (0.11)	21.27 (6.11)	0.0290 (0.19)
$b=6$	1.741	0.0513	0.4994 (0.11)	0.3098 (5.44)	–0.0001 (0.17)	0.7489	0.6825 (0.10)	19.88 (6.14)	0.0257 (0.18)
$b=9$	1.729	0.0499	0.5000 (0.10)	0.1311 (5.42)	0.0001 (0.17)	0.7547	0.6747 (0.10)	18.95 (6.13)	0.0229 (0.18)
$b=18$	1.722	0.0494	0.5008 (0.10)	0.4446 (5.42)	0.0013 (0.16)	0.7602	0.6509 (0.10)	16.40 (6.11)	0.0184 (0.17)
$b=36$	1.697	0.0507	0.4999 (0.08)	0.4332 (5.41)	0.0013 (0.16)	0.7726	0.6040 (0.10)	11.45 (6.07)	0.0095 (0.16)
CGI
(c = 2)	1.887	0.0496	0.4871 (0.28)	0.19 (5.42)	0.0004 (0.24)	0.4298	0.8294 (0.11)	37.59 (6.19)	0.0340 (0.21)
MZR
$n^{\text{ER}}=2$	1.794	0.0516	0.5005 (0.19)	0.2794 (5.41)	0.0002 (0.19)	0.7471	0.6569 (0.12)	17.15 (5.76)	0.0229 (0.17)
$n^{\text{ER}}=6$	1.780	0.0507	0.4998 (0.17)	0.3487 (5.43)	0.0001 (0.18)	0.7632	0.6414 (0.10)	15.47 (5.49)	0.0202 (0.16)
$n^{\text{ER}}=11$	1.751	0.0508	0.5001 (0.14)	–0.2534 (5.41)	0.0007 (0.17)	0.7755	0.6173 (0.08)	12.86 (5.29)	0.0155 (0.16)
Gwise
$n^{\text{ER}}=2$	1.877	0.0495	0.4997 (0.13)	–0.08 (5.43)	0.0012 (0.18)	0.7193	0.4999 (0.13)	–0.11 (6.47)	–0.0013 (0.18)
$n^{\text{ER}}=6$	1.697	0.0482	0.5003 (0.06)	–0.08 (5.45)	–0.0000 (0.15)	0.7833	0.5005 (0.06)	0.02 (5.67)	–0.0013 (0.15)
$n^{\text{ER}}=11$	1.705	0.0492	0.4999 (0.06)	0.07 (5.41)	0.0007 (0.15)	0.7837	0.5000 (0.06)	–0.20 (5.66)	0.0000 (0.15)

Note: The true variance of the response is ${\sigma }_k^2={0.64}^2$ for $k\in \{0,1\}$.

Abbreviations: ER, Equal Randomization; ETO, Expected Total Outcome; FLGI, Forward‐Looking Gittins Index; MZR, Modified Zhang and Rosenberger; TS, Thompson Sampling; CGI, Constrained Gittins Index; FLGI‐HZ, Forward‐Looking Gittins Index Implemented Using Hu and Zhang.