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dataAnalysis_forestPlot.R
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## This r script is used to draw forest plot.
## Before drawing, two data frames are required to generate from the data.
## data frame one: point_est (columns are different coefficients, dim=(9,6) if there are five covariates.)
#### row names
## n=100 betahat
## n=100 gammabar
## n=100 beta_plugin
## n=100 betabar
## n=200 betahat
## n=200 gammabar
## n=200 beta_plugin
## n=200 betabar
## n=400 betahat
## n=400 gammabar
## n=400 beta_plugin
## n=400 betabar
## data frame two: CI (confidence interval) (dim = (9,12) if there are five covariates.)
#### column names beta0_lower_bound, beta0_upper_bound, beta1_lower_bound, beta1_upper_bound, etc.
#### row names
## n=100 betahat
## n=100 gammabar
## n=100 beta_plugin
## n=100 betabar
## n=200 betahat
## n=200 gammabar
## n=200 beta_plugin
## n=200 betabar
## n=400 betahat
## n=400 gammabar
## n=400 beta_plugin
## n=400 betabar
library(dplyr)
library(matlib)
library(dummies)
##---------------------------------------------------------------------------##
## Load and format BRAVA data
##---------------------------------------------------------------------------##
data <- read.csv("brava_penn_fake.csv")
# pre-process data
Y <- data[,"secBC"] # true disease status
S <- data[,"SBCE_spec"] # surrogate
X <- data[,c("dxyear", "age", "Stage", "ER_PR")] # predictors
# change the Stage 1, 2 to 0, 1
X$Stage <- X$Stage - 1 # change stage 2 to 1 and stage 1 to 0
# centerize the year
mean_year <- mean(X[,1])
X[,1] <- X[,1]-mean_year
# standardize the age
max <- max(X[,2])
X[,2] <- X[,2]/max
# change three-level categorical variables to two dummy variables
# outcome 1: dummy1 = 1, dummy2 = 1
# outcome 2: dummy1 = 0, dummy2 = 1
# outcome 3: dummy1 = 1, dummy2 = 0
x = X
x[,4:6] <- data.frame(dummy(x[,4]))
dummy1 <- rep(0,3152)
dummy2 <- rep(0,3152)
for (i in 1:3152){
if((x[i,5] == 0)&(x[i,6]==1)){
dummy2[i] = 1
}else if ((x[i,5] == 1)&(x[i,6]==0)){
dummy1[i] = 1
}else{
dummy1[i] = 0
dummy2[i] = 0
}
}
X_dummy <- X
X_dummy[,4] <- as.data.frame(dummy1)
X_dummy[,5] <- as.data.frame(dummy2)
colnames(X_dummy)[4] <- "dummy1"
colnames(X_dummy)[5] <- "dummy2"
dxyear <- X_dummy[,1] # predictors in the entire cohort
age <- X_dummy[,2]
stage <- X_dummy[,3]
D1 <- X_dummy[,4]
D2 <- X_dummy[,5]
# true coefficients
glm0 <- glm(Y ~ dxyear+age+factor(stage)+factor(D1)+factor(D2), family = "binomial")
betaTrue <- glm0$coefficients
var_beta = summary(glm0)$coefficients[,2]**2
betareal.ci <- c(t(betaTrue + qnorm(0.975)*sqrt(var_beta)%*%t(c(-1,1))))
##---------------------------------------------------------------------------##
## Function to carry out bias reduction
## Returns: naive estimate (gammabar), validation only estimate (betahat)
## adjusted estimate (betabar)
##---------------------------------------------------------------------------##
proposed_var <- function(x_mul,m,val_X,val_Y,val_S,val_Z1,val_Z2,X,Z3,val_Z3,n,betaHat,gammaHat,gammaBar){
nr <- dim(x_mul)[2]
rho <- m/nrow(x_mul)
info_matrix1 <- matrix(0 , nrow = nr, ncol = nr) # information matrix
for (i in 1:nr){
for (j in 1:nr){
if (i == j){
info_matrix1[i,j] <- sum( val_X[,i]^2 * exp(val_Z1) / (1+exp(val_Z1))^2 ) /m
}else
info_matrix1[i,j] <- sum( val_X[,i] * val_X[,j] * exp(val_Z1) / (1+exp(val_Z1))^2) /m
info_matrix1[j,i] <- info_matrix1[i,j]
}
}
info_matrix2 <- matrix(0 , nrow = nr, ncol = nr) # information matrix
for (i in 1:nr){
for (j in 1:nr){
if (i == j){
info_matrix2[i,j] <- sum(X[,i]^2 * exp(Z3) / (1+exp(Z3))^2 ) /n
}else
info_matrix2[i,j] <- sum(X[,i] * X[,j] * exp(Z3) / (1+exp(Z3))^2) /n
info_matrix2[j,i] <- info_matrix2[i,j]
}
}
p1 <- matrix(0, nrow = nr, ncol =nr)
p2 <- matrix(0, nrow = nr, ncol =nr)
p3 <- matrix(0, nrow = nr, ncol =nr)
for (i in 1:m){
phi1 <- (val_Y[i] - (exp(val_Z1[i])/(1+exp(val_Z1[i])))) %*% as.numeric(val_X[i,])
phi1 <- inv(info_matrix1) %*% as.vector(phi1)
p1 <- p1 + phi1 %*% t(phi1)
phi2 <- (val_S[i] - (exp(val_Z3[i])/(1+exp(val_Z3[i])))) %*% as.numeric(val_X[i,])
phi2 <- inv(info_matrix2) %*% as.vector(phi2)
p2 <- p2 + phi2 %*% t(phi2)
p3 <- p3 + phi1 %*% t(phi2)
}
sigma <- p1 / m
sigma_star <- ((1-rho)) * p2 / m
omega <- (1-rho) * p3 / m
betaBar = t(betaHat - omega %*% inv(sigma_star) %*% (gammaHat - gammaBar))
var_betaBar = (sigma - omega %*% inv(sigma_star) %*% t(omega)) /m
var_betaBar = diag(var_betaBar)
return(list(betaBar = betaBar, var_betaBar = var_betaBar))
}
plugin <- function(val_X, val_S, val_Y, x_mul, S, betaHat){
d = dim(x_mul)[2]
nr <- d
# calculate plugin estimator and variance
##################################################################
##### Plugin estimator
t <- table(val_S,val_Y) # only validation set of y is obvered, so we can only use y in the validation set to do the estimation
est_pooled_a1 <- t[2,2]/sum(t[,2]) # estimate sensitivity
est_pooled_a2 <- t[1,1]/sum(t[,1]) # estimate specificity
## function for plugin estimator
nloglik=function(t)
{
beta = t[1:d];
alpha1 = t[length(t)-1]; alpha2 = t[length(t)]
tem= beta %*% t(x_mul)
p = (1-alpha2)+(alpha1+alpha2-1)*exp(tem)/(1+exp(tem))
likelihood= S*log(p)+(1-S)*log(1-p)
return(-sum(likelihood))
}
beta_plugin_temp <- nlminb(c(betaHat,est_pooled_a1,est_pooled_a2),
nloglik,lower=c(rep(-5,d),est_pooled_a1,est_pooled_a2),
upper=c(rep(5,d),est_pooled_a1,est_pooled_a2))
beta_plugin <-beta_plugin_temp$par[1:d]
#### variance
Zp <- beta_plugin %*% t(x_mul)
info_matrix <- matrix(0 , nrow = nr, ncol = nr) # information matrix
for (i in 1:nr){
for (j in 1:nr){
if (i == j){
info_matrix[i,j] <- sum( x_mul[,i]^2 * exp(Zp) / (1+exp(Zp))^2 ) /n
}else
info_matrix[i,j] <- sum( x_mul[,i] * x_mul[,j] * exp(Zp) / (1+exp(Zp))^2) /n
info_matrix[j,i] <- info_matrix[i,j]
}
}
var_beta_plugin_matrix <- inv(info_matrix)/n
var_beta_plugin <- diag(var_beta_plugin_matrix)
return(list(beta_plugin = beta_plugin, var_beta_plugin=var_beta_plugin))
}
par_est <- function(m){
n = dim(data)[1]
sample <- F
while (!sample){
id <- sample(n, m)
val_Y <- Y[id] # validation set of Y
val_S <- S[id] # validation set of S
val_X <- X_dummy[id,] # validation set of X
v_dxyear <- dxyear[id] # predictors in the validation set
v_age <- age[id]
v_stage <- stage[id]
v_D1 <- D1[id]
v_D2 <- D2[id]
#check for empty cells
if (min(c(c(table(val_Y,v_D1)),c(table(val_Y,v_D2)),c(table(val_S,v_D2)),c(table(val_S,v_D1))))>0) sample <- T
}
# feed the glm
###############################################################
# with only validation Y and X
glm1 <- glm(val_Y~v_dxyear+v_age+factor(v_stage)+factor(v_D1)+factor(v_D2), family = "binomial")
betaHat <- glm1$coefficients
var_beta_hat = summary(glm1)$coefficients[,2]**2
# validation S and X
glm2 <- glm(val_S~v_dxyear+v_age+factor(v_stage)+factor(v_D1)+factor(v_D2), family = "binomial")
gammaHat <- glm2$coefficients
var_gamma_hat = summary(glm2)$coefficients[,2]**2
# all S and X
glm3 <- glm(S ~ dxyear+age+factor(stage)+factor(D1)+factor(D2), family = "binomial")
gammaBar <- glm3$coefficients
var_gamma_bar = summary(glm3)$coefficients[,2]**2
# calcualte Z and get validation set of Z
##################################################################
x_mul <- cbind(1,X_dummy)
Z1 <- betaHat %*% t(x_mul)
Z2 <- gammaHat %*% t(x_mul)
Z3 <- gammaBar %*% t(x_mul)
val_Z1 <- Z1[id]
val_Z2 <- Z2[id]
val_Z3 <- Z3[id]
val_X <- x_mul[id,]
# calculate proposed variance
##################################################################
adj <- proposed_var(x_mul,m,val_X,val_Y,val_S,val_Z1,val_Z2,x_mul,Z3,val_Z3,n,betaHat,gammaHat,gammaBar)
var_betaBar <- adj$var_betaBar
betaBar <- adj$betaBar
plugin <- plugin(val_X, val_S, val_Y, x_mul, S, betaHat)
beta_plugin <- plugin$beta_plugin
var_beta_plugin <- plugin$var_beta_plugin
est <- rbind(betaHat, gammaBar, beta_plugin, betaBar)
betahat.ci <- c(t(betaHat + qnorm(0.975)*sqrt(var_beta_hat)%*%t(c(-1,1))))
gammabar.ci <- c(t(gammaBar + qnorm(0.975)*sqrt(var_gamma_bar)%*%t(c(-1,1))))
beta_plugin.ci <- c(t(beta_plugin + qnorm(0.975)*sqrt(var_beta_plugin)%*%t(c(-1,1))))
betabar.ci <- c(t(c(betaBar) + qnorm(0.975)*sqrt(var_betaBar)%*%t(c(-1,1))))
ci <- rbind(betahat.ci, gammabar.ci, beta_plugin.ci, betabar.ci)
var <- rbind(var_beta_hat, var_gamma_bar, var_beta_plugin, var_betaBar)
return(list(est = est, CI = ci, VAR=var))
}
##---------------------------------------------------------------------------##
## Apply bias reduction method to get point estimates and standard errors
## for validation sample sizes: 200, 400, 800
##---------------------------------------------------------------------------##
set.seed(4321)
est1 <- par_est(200)
set.seed(4321)
est2 <- par_est(400)
set.seed(4321)
est4 <- par_est(800)
point_est <- rbind(est1$est, est2$est, est4$est)
CI <- rbind(est1$CI, est2$CI, est4$CI)
### forest plot
forest.plot <- function(i){
## xlim depends on the value of CI and point_est
plot(0,0,type="n", xlab="", ylab="", yaxt="n", xaxt="n", xaxs="i", yaxs="i",ylim=c(1,7),xlim=c(min(CI[,(2*i-1)])-0.5,max(CI[,(2*i)])+0.5), main="")
axis(1)
mtext("95% Confidence Interval", outer=FALSE, line=0.4,cex=0.9)
# segments for betahat
segments(CI[c(1,5,9),2*i-1], c(2.2,4.2,6.2),CI[c(1,5,9),2*i], c(2.2,4.2,6.2), lwd=2)
segments(point_est[c(1,5,9),i], c(2.2,4.2,6.2)-0.1,point_est[c(1,5,9),i], c(2.2,4.2,6.2)+0.1, lwd=2)
segments(CI[c(1,5,9),2*i-1], c(2.2,4.2,6.2)-0.05,CI[c(1,5,9),2*i-1], c(2.2,4.2,6.2)+0.05, lwd=2)
segments(CI[c(1,5,9),2*i], c(2.2,4.2,6.2)-0.05,CI[c(1,5,9),2*i], c(2.2,4.2,6.2)+0.05, lwd=2)
# segments for gammabar
segments(CI[c(2,6,10),2*i-1], c(1.9,3.9,5.9),CI[c(2,6,10),2*i], c(1.9,3.9,5.9), lwd=1, lty = 1)
segments(point_est[c(2,6,10),i], c(1.9,3.9,5.9)-0.1,point_est[c(2,6,10),i], c(1.9,3.9,5.9)+0.1, lwd=1)
segments(CI[c(2,6,10),2*i-1], c(1.9,3.9,5.9)-0.05,CI[c(2,6,10),2*i-1], c(1.9,3.9,5.9)+0.05, lwd=1)
segments(CI[c(2,6,10),2*i], c(1.9,3.9,5.9)-0.05,CI[c(2,6,10),2*i], c(1.9,3.9,5.9)+0.05, lwd=1)
# segments for beta_plugin
segments(CI[c(3,7,11),2*i-1], c(1.6,3.6,5.6),CI[c(3,7,11),2*i], c(1.6,3.6,5.6), lwd=1, lty = 2)
segments(point_est[c(3,7,11),i], c(1.6,3.6,5.6)-0.1,point_est[c(3,7,11),i], c(1.6,3.6,5.6)+0.1, lwd=1)
segments(CI[c(3,7,11),2*i-1], c(1.6,3.6,5.6)-0.05,CI[c(3,7,11),2*i-1], c(1.6,3.6,5.6)+0.05, lwd=1)
segments(CI[c(3,7,11),2*i], c(1.6,3.6,5.6)-0.05,CI[c(3,7,11),2*i], c(1.6,3.6,5.6)+0.05, lwd=1)
# segments for betabar
segments(CI[c(4,8,12),2*i-1], c(1.3,3.3,5.3),CI[c(4,8,12),2*i], c(1.3,3.3,5.3), lwd=1,col='red', lty = 3)
segments(point_est[c(4,8,12),i], c(1.3,3.3,5.3)-0.1,point_est[c(4,8,12),i], c(1.3,3.3,5.3)+0.1, lwd=1,col="red")
segments(CI[c(4,8,12),2*i-1], c(1.3,3.3,5.3)-0.05,CI[c(4,8,12),2*i-1], c(1.3,3.3,5.3)+0.05, lwd=1,col="red")
segments(CI[c(4,8,12),2*i], c(1.3,3.3,5.3)-0.05,CI[c(4,8,12),2*i], c(1.3,3.3,5.3)+0.05, lwd=1,col="red")
# real value of the ith data (confidence interval)
abline(v=betaTrue[i],lty=2,col="blue")
abline(v=betareal.ci[2*i-1],lty=2,col="green")
abline(v=betareal.ci[2*i],lty=2,col="green")
if (i != 2){
u = par("usr")
segments(u[1], c(2,4,6), u[1]-0.18, c(2,4,6), xpd=TRUE)
text(u[1]-0.18,2,'n=200',xpd=T,adj=1,cex=1)
text(u[1]-0.18,4,'n=400',xpd=T,adj=1,cex=1)
text(u[1]-0.18,6,'n=800',xpd=T,adj=1,cex=1)
legend("topleft",legend=c( expression(paste(beta[paste("V")])),
expression(paste(gamma[paste("F")])),
expression(paste(beta[paste("P")])),
expression(paste(beta[paste("A")]))),
lty=c(1,1,2,3),lwd=c(2,1,1,1),cex=0.8,col = c(1,1,1,2),bty="n")
}else{
u = par("usr")
segments(u[1], c(2,4,6), u[1]-0.03, c(2,4,6), xpd=TRUE)
text(u[1]-0.03,2,'n=200',xpd=T,adj=1,cex=1)
text(u[1]-0.03,4,'n=400',xpd=T,adj=1,cex=1)
text(u[1]-0.03,6,'n=800',xpd=T,adj=1,cex=1)
legend("topleft",legend=c( expression(paste(beta[paste("V")])),
expression(paste(gamma[paste("F")])),
expression(paste(beta[paste("P")])),
expression(paste(beta[paste("A")]))),
lty=c(1,1,2,3),lwd=c(2,1,1,1),cex=0.8,col = c(1,1,1,2),bty="n")
}
}
par(mfrow=c(3,2))
forest.plot(1)
forest.plot(2)
forest.plot(3)
forest.plot(4)
forest.plot(5)
forest.plot(6)
# dev.off()