Simulate a continuous variable that correlates with an existing binary variable

Question

I am looking to mimic an age variable (limited range 18-35) that correlated 0.1 with an existing binary variable called use. Most of the examples I came across demonstrate how to simulate both variables at the same time.

# setup
  set.seed(493)
  n <- 134
  dat <- data.frame(partID=seq(1, n, 1),
                    trt=c(rep(0, n/2), 
                          rep(1, n/2)))

# set proportion
  a <- .8   
  b <- .2  
  dat$use <- c(rbinom(n/2, 1, b),
               rbinom(n/2, 1, a))

Answer 1

Not sure if this is the best way to get closer to this, but you can get closer using the answer from here: https://stats.stackexchange.com/questions/15011/generate-a-random-variable-with-a-defined-correlation- to-an-existing-variable

For example (using the code from the link):

x1    <- dat$use               # fixed given data

rho   <- 0.1                   # desired correlation = cos(angle)
theta <- acos(rho)             # corresponding angle
x2    <- rnorm(n, 2, 0.5)      # new random data
X     <- cbind(x1, x2)         # matrix
Xctr  <- scale(X, center=TRUE, scale=FALSE)   # centered columns (mean 0)

Id   <- diag(n)                               # identity matrix
Q    <- qr.Q(qr(Xctr[ , 1, drop=FALSE]))      # QR-decomposition, just matrix Q
P    <- tcrossprod(Q)          # = Q Q'       # projection onto space defined by x1
x2o  <- (Id-P) %*% Xctr[ , 2]                 # x2ctr made orthogonal to x1ctr
Xc2  <- cbind(Xctr[ , 1], x2o)                # bind to matrix
Y    <- Xc2 %*% diag(1/sqrt(colSums(Xc2^2)))  # scale columns to length 1

x <- Y[ , 2] + (1 / tan(theta)) * Y[ , 1]     # final new vector


dat$age <- (1 + x) * 25 

cor(dat$use, dat$age)
# 0.1

summary(dat$age)
#   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
# 20.17   23.53   25.00   25.00   26.59   30.50