Suppose \(n\) units (participants) are to be assigned to \(K\)
treatment groups. For each unit \(i, i = 1, ..., n\) and
treatment \(j, j = 1, ..., K\), define the assignment
matrix \([T_{ij}]^{n*K}\), where
\(T_{ij}=1\) indicates unit \(i\) receives treatment \(j\).
Consider \(p\) continuous covariates, let \(x_i =
(x_{i1},...,x_{in})^T\).
The proposed method, namely the adaptive randomization
via Mahalanobis distance for multi-arm design (ARMM),
is outlined below. The implement of ARMM is similar to ARM.
First assume that \(n\) units are in a sequence
and then assign the first \(K\) units to \(K\) treatment
groups randomly
as the initialization. Then,
the following units are assigned in blocks of \(K\)
sequentially and
adaptively until all the units
are assigned. For \(K\) units are assigned to \(K\)
groups, there are in total \(K!\) possible allocations.
Calculate \(K!\) potential overall
covariate imbalance measurement
according to pairwise Mahalanobis
distance under the \(K!\) possible allocations.
Choose the allocation which corresponds to the smallest
Mahalanobis
distance with a probability of \(q\) across all potential allocations.
Repeat the process until all units are assigned.
For any pair of treatments \(s\) and \(t\) among the \(K\)
treatment groups, calculate the Mahalanobis distance by:
$$M_{s,t}(n) = 2n/K/K(\hat{x}_1 -\hat{x}_2)^Tcov(x)^{-1}(\hat{x}_1 -\hat{x}_2)$$
In total, there are \(C_K^2\) pairs of Mahalanobis
distances among \(K\) treatment groups.Finally, calculate
the mean, the median or the maximum to represent the total imbalance.
See the reference for more details.