Suppose that \(n\) patients are to be assigned to two treatment groups.
Consider \(p\) continuous covariates for each patient.
\(T_i\) is the assignment of the \(i\)th patient.
The proposed procedure to assign units to treatment groups, namely adaptive
randomization via Mahalanobis distance (ARM), is outlined below.
(1) Arrange all \(n\) units randomly into a sequence
\(x_1,...,x_n\).
(2) Assign the first two units with \(T_1=1\) and \(T_2=2\).
(3) Suppose that \(2i\) units have been assigned to
treatment groups,
for the \(2i+1\)-th and \(2i+2\)-th units:
(3a) If the \(2i+1\)-th unit is assigned to treatment 1 and
the \(2i+2\)-th
unit to treatment 2, then calculate the potential
Mahalanobis distance, between the updated treatment groups.
with \(2i+2\) units, \(M_1(2i + 2)\).
(3b) Similarly, if the \(2i+1\)-th unit is
assigned to treatment 2 and
the \(2i+2\)-th unit to treatment 1, then calculate the
other potential Mahalanobis distance, \(M_2(2i + 2)\).
(4) Assign the \(2i+1\)-th unit to treatment groups
according to the
following probabilities:
if \( M_1(2i + 2) < M_2(2i + 2)\), \(P(T_{2i+1} = 1)= q\);
if \( M_1(2i + 2) > M_2(2i + 2)\), \(P(T_{2i+1} = 1)= 1-q\);
if \( M_1(2i + 2) = M_2(2i + 2)\), \(P(T_{2i+1} = 1)= 0.5\).
(5) Repeat the last two steps until all units are assigned. If n is odd,
assign the last unit to two treatments with equal probabilities.
Mahalanobis distance \(M(n)\) between the sample means across
different treatment groups is:
$$M(n)= np(1-p)(\hat{x_1} - \hat{x_2})^Tcov(x)^{-1}(\hat{x_1} - \hat{x_2}$$
See the reference for more details.