StrBCD: Shao's Method in the Two-Arms Case

Description

Allocates patients to one of the two treatments using Shao's method proposed by Shao J, Yu X, Zhong B (2010) <Doi:10.1093/biomet/asq014>.

Usage

# S3 method for carandom
StrBCD(data, p = 0.85)

Arguments

data

a dataframe. A row of the dataframe contains the covariate profile of a patient.

the probability of assigning one patient to treatment 1. p should be larger than 1/2 to obtain balance. The default is 0.85.

Value

It returns an object of class "carandom".

The function print is used to obtain results. The generic accessor functions Cov_Assig, Diff, data, All strata and others extract various useful features of the value returned by StrBCD.

An object of class "carandom" is a list containing at least the following components:

cov_num

the number of covariates.

the number of patients.

Cov_Assign

a (cov_num + 1) * n matrix containing covariate profiles for all patients and corresponding assignments. The $i$th column represents the $i$th patient. The first cov_num rows include patients' covariate profiles, and the last row contains the assignment.

All strata

a matrix containing all the strata involved.

Diff

a matrix with only one column. There are final differences at the overall, within-stratum, and marginal levels.

Data Type

the data type. Real or Simulated.

Details

Consider $I$ covaraites and $m_i$ levels for the $i$th covariate. $T_j$ is the assignment of the $j$th patient and $Z_j = (k_1,\dots,k_I)$ indicates the covariate profile of this patient. For convenience, $(k_1,\dots,k_I)$ and $(i;k_i)$ denote the stratum and margin respectively. $D_n(.)$ is the difference between the numbers of assigned patients in treatment $1$ and treatment $2$ at the corresponding level after $n$ patinets have been assigned.Then Shao's procedure is as follows:

(1) The first patient is assigned to treatment $1$ with probability $1/2$;

(2) Suppose $n-1$ patients have each been assigned to a treatment ($n>1$) and the $n$th patinent falls within $(k_1^*,\dots,k_I^*)$;

(3)If the $n$th patient was assigned to treatment $1$, then the potential within-stratum difference between the two groups is $$D_n^{(1)}(k_1^*,\dots,k_I^*)=D_n(k_1^*,\dots,k_I^*)+1.$$ Similarly, the potential differences would be obtained in the same way if the $n$th patinent was assigned to treatment $2$.

(4) An imbalance measure is defined by $$Imb_n^{(l)}=[D_n^{(1)}(k_1^*,\dots,k_I^*)]^2,l=1,2;$$

(5)Conditional on the assignments of the first ($n-1$) patients as well as the covariates'profiles of the first $n$ patients, assign the nth patient to treatment $1$ with probability $$P(T_n=1|Z_n,T_1,\dots,T_{n-1})=q,$$ for $Imb_n^{(1)}>Imb_n^{(2)},$ $$P(T_n=1|Z_n,T_1,\dots,T_{n-1})=p,$$ for $Imb_n^{(1)}<Imb_n^{(2)}$, and $$P(T_n=1|Z_n,T_1,\dots,T_{n-1})=0.5,$$ for $Imb_n^{(1)}=Imb_n^{(2)}.$

References

Shao J, Yu X, Zhong B. A theory for testing hypotheses under covariate-adaptive randomization[J]. Biometrika, 2010, 97(2): 347-360.

Examples

Run this code

# NOT RUN {
# a simple use
## Real Data
## creat a dataframe
df <- data.frame("gender" = sample(c("female", "male"), 1000, TRUE, c(1 / 3, 2 / 3)), 
                 "age" = sample(c("0-30", "30-50", ">50"), 1000, TRUE), 
                 "jobs" = sample(c("stu.", "teac.", "others"), 1000, TRUE), 
                 stringsAsFactors = TRUE)
Res <- StrBCD(data = df)
## view the output
Res
# }
# NOT RUN {
## view all patients' profile and assignments
Res$Cov_Assig
# }
# NOT RUN {
## Simulated Data
cov_num = 3
level_num = c(2, 3, 3)
pr = c(0.4, 0.6, 0.3, 0.4, 0.3, 0.4, 0.3, 0.3)
Res.sim <- StrBCD.sim(n = 1000, cov_num, level_num, pr)
## view the output
Res.sim
# }
# NOT RUN {
## view the detials of difference
Res.sim$Diff
# }
# NOT RUN {
# }
# NOT RUN {
N <- 5
n <- 1000
cov_num <- 3
level_num <- c(2, 3, 5) 
# Set pr to follow two tips:
# (1) length of pr should be sum(level_num);
# (2)sum of probabilities for each margin should be 1
pr <- c(0.4, 0.6, 0.3, 0.4, 0.3, rep(0.2, times = 5))
omega <- c(0.2, 0.2, rep(0.6 / cov_num, times = cov_num))

## generate a container to contain Diff
DH <- matrix(NA, ncol = N, nrow = 1 + prod(level_num) + sum(level_num))
DS <- matrix(NA, ncol = N, nrow = 1 + prod(level_num) + sum(level_num))
for(i in 1 : N){
  result <- HuHuCAR.sim(n, cov_num, level_num, pr, omega)
  resultS <- StrBCD.sim(n, cov_num, level_num, pr)
  DH[ , i] <- result$Diff; DS[ , i] <- resultS$Diff
}

## do some analysis
require(dplyr)

## analyze the overall imbalance
Ana_O <- matrix(NA, nrow = 2, ncol = 3)
rownames(Ana_O) <- c("NEW", "Shao")
colnames(Ana_O) <- c("mean", "median", "95%quantile")
temp <- DH[1, ] %>% abs
tempS <- DS[1, ] %>% abs
Ana_O[1, ] <- c((temp %>% mean), (temp %>% median),
                (temp %>% quantile(0.95)))
Ana_O[2, ] <- c((tempS %>% mean), (tempS %>% median),
                (tempS %>% quantile(0.95)))

## analyze the within-stratum imbalances
tempW <- DH[2 : (1 + prod(level_num)), ] %>% abs
tempWS <- DS[2 : 1 + prod(level_num), ] %>% abs
Ana_W <- matrix(NA, nrow = 2, ncol = 3)
rownames(Ana_W) <- c("NEW", "Shao")
colnames(Ana_W) <- c("mean", "median", "95%quantile")
Ana_W[1, ] = c((tempW %>% apply(1, mean) %>% mean),
               (tempW %>% apply(1, median) %>% mean),
               (tempW %>% apply(1, mean) %>% quantile(0.95)))
Ana_W[2, ] = c((tempWS %>% apply(1, mean) %>% mean),
               (tempWS %>% apply(1, median) %>% mean),
               (tempWS %>% apply(1, mean) %>% quantile(0.95)))

## analyze the marginal imbalance
tempM <- DH[(1 + prod(level_num) + 1) :
              (1 + prod(level_num) + sum(level_num)), ] %>% abs
tempMS <- DS[(1 + prod(level_num) + 1) :
               (1 + prod(level_num) + sum(level_num)), ] %>% abs
Ana_M <- matrix(NA, nrow = 2, ncol = 3)
rownames(Ana_M) <- c("NEW", "Shao")
colnames(Ana_M) <- c("mean", "median", "95%quantile")
Ana_M[1, ] = c((tempM %>% apply(1, mean) %>% mean),
               (tempM %>% apply(1, median) %>% mean),
               (tempM %>% apply(1, mean) %>% quantile(0.95)))
Ana_M[2, ] = c((tempMS %>% apply(1, mean) %>% mean),
               (tempMS %>% apply(1, median) %>% mean),
               (tempMS %>% apply(1, mean) %>% quantile(0.95)))

AnaHP <- list(Ana_O, Ana_M, Ana_W)
names(AnaHP) <- c("Overall", "Marginal", "Within-stratum")

AnaHP
# }

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