# Pocock and Simon's Method in the Two-Arms Case Allocates patients to one of two treatments using Pocock and Simon's method proposed by Pocock S J, Simon R (1975) . ```r PocSimMIN(data, weight = NULL, p = 0.85) ``` ## Arguments - `data`: a data frame. A row of the dataframe corresponds to the covariate profile of a patient. - `weight`: a vector of weights for within-covariate-margin imbalances. It is required that at least one element is larger than 0. If `weight = NULL` (default), the within-covariate-margin imbalances are weighted with an equal proportion, `1/cov_num`, for each covariate-margin. - `p`: the biased coin probability. `p` should be larger than `1/2` and less than `1`. The default is `0.85`. ## Details Consider $I$ covariates and $m_i$ levels for the $i$th covariate, $i=1,\ldots,I$. $T_j$ is the assignment of the $j$th patient and $Z_j = (k_1,\dots,k_I)$ indicates the covariate profile of this patient, $j=1,\ldots,n$. For convenience, $(k_1,\dots,k_I)$ and $(i;k_i)$ denote the stratum and margin, respectively. $D_j(.)$ is the difference between the numbers of patients assigned to treatment $1$ and treatment $2$ at the corresponding levels after $j$ patients have been assigned. The Pocock and Simon's minimization procedure is as follows: (1) The first patient is assigned to treatment $1$ with probability $1/2$; (2) Suppose that $j-1$ patients have been assigned ($1Imb_j^{(2)},$ $$ P(T_j=1|Z_j,T_1,\dots,T_{j-1})=p $$ for $Imb_j^{(1)}50"), 1000, TRUE), "jobs" = sample(c("stu.", "teac.", "others"), 1000, TRUE), stringsAsFactors = TRUE) weight <- c(1, 2, 1) Res <- PocSimMIN(data = df, weight) ## view the output Res ## view all patients' profile and assignments Res$Cov_Assig ## Simulated Data cov_num = 3 level_num = c(2, 3, 3) pr = c(0.4, 0.6, 0.3, 0.3, 0.4, 0.4, 0.3, 0.3) Res.sim <- PocSimMIN.sim(n = 1000, cov_num, level_num, pr) ## view the output Res.sim ## view the detials of difference Res.sim$Diff N <- 5 n <- 1000 cov_num <- 3 level_num <- c(2, 3, 5) # Set pr to follow two tips: # (1) length of pr should be sum(level_num); # (2)sum of probabilities for each margin should be 1. pr <- c(0.4, 0.6, 0.3, 0.4, 0.3, rep(0.2, times = 5)) omega <- c(0.2, 0.2, rep(0.6 / cov_num, times = cov_num)) weight <- c(2, rep(1, times = cov_num - 1)) ## generate a container to contain Diff DH <- matrix(NA, ncol = N, nrow = 1 + prod(level_num) + sum(level_num)) DP <- matrix(NA, ncol = N, nrow = 1 + prod(level_num) + sum(level_num)) for(i in 1 : N){ result <- HuHuCAR.sim(n, cov_num, level_num, pr, omega) resultP <- PocSimMIN.sim(n, cov_num, level_num, pr, weight) DH[ , i] <- result$Diff; DP[ , i] <- resultP$Diff } ## do some analysis require(dplyr) ## analyze the overall imbalance Ana_O <- matrix(NA, nrow = 2, ncol = 3) rownames(Ana_O) <- c("NEW", "PS") colnames(Ana_O) <- c("mean", "median", "95%quantile") temp <- DH[1, ] %>% abs tempP <- DP[1, ] %>% abs Ana_O[1, ] <- c((temp %>% mean), (temp %>% median), (temp %>% quantile(0.95))) Ana_O[2, ] <- c((tempP %>% mean), (tempP %>% median), (tempP %>% quantile(0.95))) ## analyze the within-stratum imbalances tempW <- DH[2 : (1 + prod(level_num)), ] %>% abs tempWP <- DP[2 : 1 + prod(level_num), ] %>% abs Ana_W <- matrix(NA, nrow = 2, ncol = 3) rownames(Ana_W) <- c("NEW", "PS") 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((tempWP %>% apply(1, mean) %>% mean), (tempWP %>% apply(1, median) %>% mean), (tempWP %>% 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 tempMP <- DP[(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", "PS") 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((tempMP %>% apply(1, mean) %>% mean), (tempMP %>% apply(1, median) %>% mean), (tempMP %>% apply(1, mean) %>% quantile(0.95))) AnaHP <- list(Ana_O, Ana_M, Ana_W) names(AnaHP) <- c("Overall", "Marginal", "Within-stratum") AnaHP ```

PocSimMIN function