How to generate bivariate categorical variables with defined correlation?

Question

Suppose I have two categorical variables A and B and both have three levels, 1, 2, 3 with prob 0.2 , 0.3 , and 0.5 for each level. How could I generate a list of random bivariate data of A and B with defined correlation 0.3? I know for a univariate A or B we can do

A=sample(1:3, 100, T, prob=c(0.2,0.3,0.5))
B=sample(1:3, 100, T, prob=c(0.2,0.3,0.5))

My question is how to sample cbind(A,B) with cor(A,B)=0.3 ?

Answer 1

Here's an example probability matrix. (Your own will depend on your chosen model.)

# describe probabilities over the space by a matrix
set.seed(1)
nA <- 3
nB <- 3
probmat <- matrix({r<-runif(nA*nB);r/sum(r)},ncol=nB)
#            [,1]       [,2]      [,3]
# [1,] 0.04868724 0.16654119 0.1732284
# [2,] 0.06823764 0.03698311 0.1211728
# [3,] 0.10504609 0.16474081 0.1153628

And here's one way to draw samples from it:

# rearrange
probs  <- c(probmat)
events <- as.matrix(expand.grid(A=1:nA,B=1:nB))

# draw samples
nSamp  <- 100
samp   <- events[sample.int(nA*nB,nSamp,prob=probs,replace=TRUE),]

Answer 2

Below is equivalent R code for this artical with a matlab example. You can use apply pnorm() and cut() to the columns the get correlated discrete random variables

# parameters
nrows  <-  10

# The desired correlation matrix
(M  <-    matrix(c(1.0,0.6,0.3,
                  0.6,1.0,0.5,
                  0.3,0.5,1.0),byrow=T,ncol=3))


#>      [,1] [,2] [,3]
#> [1,]  1.0  0.6  0.3
#> [2,]  0.6  1.0  0.5
#> [3,]  0.3  0.5  1.0

(U = chol(M))
#>      [,1] [,2]      [,3]
#> [1,]    1  0.6 0.3000000
#> [2,]    0  0.8 0.4000000
#> [3,]    0  0.0 0.8660254 

# generate a random matrix where columns have the desired correlatoin structure
matrix(rnorm(nrows*ncol(M)),ncol=ncol(M))%*%U

#>   -0.4326   -0.4089    0.0505
#>   -1.6656   -0.4187   -1.3665
#>    0.1253   -0.3955    0.4209
#>    0.2877    1.9192    2.3656
#>   -1.1465   -0.7970   -0.9976
#>    1.1909    0.8057    1.1459
#>    1.1892    1.5669    1.8695
#>   -0.0376    0.0248   -1.3678
#>    0.3273    0.1199   -1.1880
#>    0.1746   -0.5611    0.2141


# check that this works
cor(matrix(rnorm(1000000*ncol(M)),ncol=ncol(M))%*%U)

#>           [,1]      [,2]      [,3]
#> [1,] 1.0000000 0.5988445 0.2987633
#> [2,] 0.5988445 1.0000000 0.4992603
#> [3,] 0.2987633 0.4992603 1.0000000