#################################################################################################################################
### Figure 1: R Code to reproduce simulation results (tables and figures)
#################################################################################################################################

library(ggplot2)
library(xtable2)

parameter = seq(from=0.01, to=0.10,by=0.01)
result = data.frame(min = rep(0,length(parameter)), sigma1Star2 = 0, sigma2Star2 = 0, pStar = 0, N = 0, N12=0, kappa = 0)

shape1 <- c(3,3,3)
shape2 <- c(3,2,1)

output = data.frame(min = 0, N = 0, N12 = 0, pStar = 0, kappa = 0)

sim <- 10^6

set.seed(0)
for(k in 1:length(shape1)) {
  for(kk in 1:length(parameter)){
    cat(kk,"/",length(parameter),"\n")

    x1 = rbeta(sim, 5,5)
    x2 = rbeta(sim, shape1[k],shape2[k])
    alpha=parameter[kk]
    beta=0.8
    m1 <- length(x1)
    m2 <- length(x2)
    # ranks among union of samples:
    R <- rank(c(x1,x2), ties.method="average")
    R1 <- R[1:m1]
    R2 <- R[m1+(1:m2)]
    # ranks within samples:
    R11 <- rank(x1, ties.method="average")
    R22 <- rank(x2, ties.method="average")
    # placements:
    P1 <- R1 - R11
    P2 <- R2 - R22
    # effect size:
    pStar <- (mean(R2)-mean(R1)) / (m1+m2) + 0.5
    # variances:
    sigmaStar <- sqrt(sum((R11-((m1+1)/2))^2) / m1^3)
    sigma1Star <- sqrt(sum((P1-mean(P1))^2) / (m1*m2^2))
    sigma2Star <- sqrt(sum((P2-mean(P2))^2) / (m1^2*m2))

    ss = function(t){
      return((sigmaStar*qnorm(1-alpha/2) + qnorm(beta)*sqrt(t*sigma2Star^2 + (1-t)*sigma1Star^2))^2 / (t*(1-t)*(pStar-0.5)^2))
    }

    kappa=sigma2Star/sigma1Star


    result[kk,1]=optimize(ss,interval=c(0.3,0.7), maximum=FALSE,tol = .Machine$double.eps)$minimum
    result[kk,2]=sigma1Star^2
    result[kk,3]=sigma2Star^2
    result[kk,4]=pStar
    result[kk,5]=optimize(ss,interval=c(0.3,0.7), maximum=FALSE,tol = .Machine$double.eps)$objective
    result[kk,6] = ss(1/2)
    result[kk,7] = kappa
  }

  output = rbind(output, result[,c("min","N","N12","pStar","kappa")])
}


output <- output[-1,]
output$parameter = parameter
output$Effect = factor(gl(length(shape1),10), labels = c("small", "medium", "large"))

library(ggplot2)

p1 <- ggplot(data=output, aes(x=parameter, y= min, col = Effect,group=Effect)) +
  geom_line()+
  geom_point() +
  xlab(expression(paste("Type-I error rate ", alpha))) +
  ylab(expression(paste("Optimal allocation rate t"))) +
  theme(legend.position = "top") +
  theme(axis.text=element_text(size=12), axis.title=element_text(size=12,face="bold")) +
  scale_colour_discrete(name = "Relative Effect") +
  labs(size = expression(paste("Ratio Variances"))) +
  theme(plot.margin = unit(c(2,2,2,2), "cm")) +
  theme(axis.title.y = element_text(margin = margin(t = 0, r = 10, b = 0, l = 0))) +
  theme(axis.title.x = element_text(margin = margin(t = 10, r = 0, b = 0, l = 0))) +
  annotate("text", label = "p = 0.50", x = 0.05, y= 0.48, color = "red") +
  annotate("text", label = "p = 0.65", x = 0.05, y= 0.4725, color = "darkgreen") +
  annotate("text", label = "p = 0.84", x = 0.05, y= 0.4675, color = "blue")


ggsave("Simulation_1.pdf", plot = p1)

print(xtable(output, digits = 3), include.rownames = FALSE)




#################################################################################################################################
### Figure 2: R Code to reproduce simulation results
#################################################################################################################################

parameter = seq(from=0.5, to=0.95,by=0.05)
result = data.frame(min = rep(0,length(parameter)), sigma1Star2 = 0, sigma2Star2 = 0, pStar = 0, N = 0, N12=0, kappa = 0)

shape1 <- c(3,3,3)
shape2 <- c(3,2,1)

output = data.frame(min = 0, N = 0, N12 = 0, pStar = 0, kappa = 0)

sim <- 10^6

set.seed(0)
for(k in 1:length(shape1)) {
  for(kk in 1:length(parameter)){
    cat(kk,"/",length(parameter),"\n")

    x1 = rbeta(sim, 5,5)
    x2 = rbeta(sim, shape1[k],shape2[k])
    alpha=0.05
    beta=parameter[kk]
    m1 <- length(x1)
    m2 <- length(x2)
    # ranks among union of samples:
    R <- rank(c(x1,x2), ties.method="average")
    R1 <- R[1:m1]
    R2 <- R[m1+(1:m2)]
    # ranks within samples:
    R11 <- rank(x1, ties.method="average")
    R22 <- rank(x2, ties.method="average")
    # placements:
    P1 <- R1 - R11
    P2 <- R2 - R22
    # effect size:
    pStar <- (mean(R2)-mean(R1)) / (m1+m2) + 0.5
    # variances:
    sigmaStar <- sqrt(sum((R11-((m1+1)/2))^2) / m1^3)
    sigma1Star <- sqrt(sum((P1-mean(P1))^2) / (m1*m2^2))
    sigma2Star <- sqrt(sum((P2-mean(P2))^2) / (m1^2*m2))

    ss = function(t){
      return((sigmaStar*qnorm(1-alpha/2) + qnorm(beta)*sqrt(t*sigma2Star^2 + (1-t)*sigma1Star^2))^2 / (t*(1-t)*(pStar-0.5)^2))
    }

    kappa=sigma2Star/sigma1Star


    result[kk,1]=optimize(ss,interval=c(0.3,0.7), maximum=FALSE,tol = .Machine$double.eps)$minimum
    result[kk,2]=sigma1Star^2
    result[kk,3]=sigma2Star^2
    result[kk,4]=pStar
    result[kk,5]=optimize(ss,interval=c(0.3,0.7), maximum=FALSE,tol = .Machine$double.eps)$objective
    result[kk,6] = ss(1/2)
    result[kk,7] = kappa
  }

  output = rbind(output, result[,c("min","N","N12","pStar","kappa")])
}


output <- output[-1,]
output$parameter = parameter
output$Effect = factor(gl(length(shape1),10), labels = c("small", "medium", "large"))

library(ggplot2)

p2 <- ggplot(data=output, aes(x=parameter, y= min, col = Effect,group=Effect)) +
  geom_line()+
  geom_point() +
  xlab(expression(paste("Power 1-", beta))) +
  ylab(expression(paste("Optimal allocation rate ", t))) +
  theme(legend.position = "top") +
  theme(axis.text=element_text(size=12), axis.title=element_text(size=12,face="bold")) +
  scale_colour_discrete(name = "Relative Effect") +
  labs(size = expression(paste("Ratio Variances"))) +
  theme(plot.margin = unit(c(2,2,2,2), "cm")) +
  theme(axis.title.y = element_text(margin = margin(t = 0, r = 10, b = 0, l = 0))) +
  theme(axis.title.x = element_text(margin = margin(t = 10, r = 0, b = 0, l = 0))) +
  annotate("text", label = "p = 0.50", x = 0.75, y= 0.487, color = "red") +
  annotate("text", label = "p = 0.65", x = 0.9, y= 0.467, color = "darkgreen") +
  annotate("text", label = "p = 0.84", x = 0.75, y= 0.465, color = "blue")

ggsave("Simulation_2.pdf", plot = p2)

print(xtable(output, digits = 3), include.rownames = FALSE)