ARMM {CARM}

Adaptive Randomization via Mahalanobis distance for Multi-arm design

Description

Randomize patients into treatment groups for multi-arm trials using ARMM proposed by Haoyu Yang, Yichen Qin, Yang Li, Fan Wang, and Feifang Hu.(2022)

Usage

ARMM(covariate, assignment, K, q = 0.75, method)

Arguments

covariate

a data frame. A row of the dataframe corresponds to the covariate profile of a patient.

assignment

a vector. If partial patients had been allocated , please input their allocation. IF all the patients are not be allocated, please input 'assignment = NA' directly.

K

an integer; number of arms of the trial.

q

the biased coin probability. q should be larger than 1/2 and less than 1, default = 0.75

method

Methods for calculating Mahalanobis distance, input one of these texts: 'mean', 'max' or 'median'.

Details

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.

Value

An object of class "ARMM" is a list containing the following components:

assignment

Allocation of patients.

sample_size

The number of patients from treatment 1 to treatment K respectively.

Mahalanobis_Distance

Mahalanobis distance among treatment groups .

References

Yang H, Qin Y, Wang F, et al. Balancing covariates in multi-arm trials via adaptive randomization. Computational Statistics & Data Analysis, 2023, 179: 107642. https://doi.org/10.1016/j.csda.2022.107642

Examples

library(MASS)
#simulate covariates of patients
p <- 6; n <- 30
sigma <- diag(p); mean <- c(rep(0,p))
data <- mvrnorm(n, mean, sigma)
covariate <- as.data.frame(data)
#IF all the patients are not be allocated
ARMM(covariate = covariate, assignment = NA, K = 3, q = 0.75, method = 'mean')
#IF you had allocated partial patients
ARMM(covariate = covariate, assignment = c(1,2), K=4, q=0.75, method = 'max')