Table 1. A Sample Randomization Procedure with Mass Weighted Urn Design.

Number of treatment arms = 3; Target Allocation = 1: 1: √2; Sample size n = 300; α = 4.

i	Before Assignment			During Assignment						After Assignment
	n i-1,1	n i-1,2	n i-1,3	p i,1	p i,2	p i,3	g i	x i	T i	d i	ni,2/ni,1	ni,3/ni,1
1	0	0	0	0.293	0.293	0.414	0	0.664	3	0.717
2	0	0	1	0.366	0.366	0.268	0.179	0.718	2	0.738
3	0	1	1	0.439	0.189	0.371	0.184	0.098	1	0.297	1	1
4	1	1	1	0.263	0.263	0.475	0.074	0.763	3	0.420	1	2
5	1	1	2	0.336	0.336	0.328	0.105	0.044	1	0.712	0.5	1
6	2	1	2	0.159	0.409	0.432	0.178	0.314	2	0.594	1	1
7	2	2	2	0.232	0.232	0.536	0.149	0.350	2	1.309	1.5	1
8	2	3	2	0.305	0.055	0.639	0.327	0.147	1	1.609	1	0.667
9	3	3	2	0.129	0.129	0.743	0.402	0.727	3	0.891	1	1
10	3	3	3	0.202	0.202	0.596	0.223	0.006	1	1.567	0.75	0.75
......
291	85	85	120	0.278	0.278	0.445	0.037	0.033	1	0.964	0.988	1.395
292	86	85	120	0.101	0.351	0.548	0.241	0.250	2	1.163	1	1.395
293	86	86	120	0.174	0.174	0.652	0.291	0.709	3	0.446	1	1.407
294	86	86	121	0.247	0.247	0.505	0.111	0.937	3	0.272	1	1.419
295	86	86	122	0.321	0.321	0.359	0.068	0.621	2	0.745	1.012	1.419
296	86	87	122	0.394	0.144	0.462	0.186	0.905	3	0.856	1.012	1.430
297	86	87	123	0.467	0.217	0.316	0.214	0.735	3	1.392	1.012	1.442
298	86	87	124	0.540	0.290	0.169	0.348	0.132	1	0.692	1	1.425
299	87	87	124	0.364	0.364	0.273	0.173	0.468	2	0.731	1.011	1.425
300	87	88	124	0.437	0.187	0.377	0.183	0.318	1	0.323	1	1.409

α: Parameter used by the mass weighted urn design.

i: Randomization sequence order.

n_i-1,j: Treatment distribution before subject i. (j=1,2,3).

p_i,j: Conditional allocation probability for subject i. (j=1,2,3)

g_i: Allocation predictability for subject i.

x_i: Random number

T_i: Treatment assignment for subject i.

d_i: Treatment imbalance for subject i.

d_i: Treatment imbalance after subject i.

n_i,j/n_i,1: Allocation ratio between arm j and 1, (j=2,3)