R help - Apr 2011 - Random Normal Variable Correlated to an Existing Binomial Variable

Hi, R-Helpers!

I have a dataframe that contains a binomial variable.  I need to add another
random variable drawn from a normal distribution with a specific mean and
standard deviation.  This variable also needs to be correlated with the existing
binomial variable with a specific correlation (say .75).  Any ideas?

Thanks!

Shane

On Sun, Apr 24, 2011 at 07:00:26PM -0400, Shane Phillips
wrote:
> Hi, R-Helpers!
> 
> I have a dataframe that contains a binomial variable.  I need to add
another random variable drawn from a normal distribution with a specific mean
and standard deviation.  This variable also needs to be correlated with the
existing binomial variable with a specific correlation (say .75).  Any ideas?
Hi.

If X, Y are dependent random variables and we want to generate y, so
that (x, y) is a pair from their joint distribution with known x,
then y should be generated from the conditional distribution P(Y|X=x).
If the probability P(X=x) is not too small, then this may be done by
rejection sampling: Generate pairs (X, Y) until the condition X=x is
satisfied and use the corresponding Y.

It remains to generate pairs (X, Y), where Y is a normal variable
and X a binomial one. The parameters of Y are known, the parameters
of X should be chosen somehow and the correlation of X and Y is
known. I suggest the following. Compute the distribution of X as a
vector of probabilities p_0, ..., p_n (see ?dbinom). Find a nondecreasing
function f() from reals to {0, .., n} such that f(Y) has distribution
p_0, ..., p_n. The function may be determined by a sequence of
cutpoints a_1, ..., a_n defining f(y) as follows

  y              f(y)  
  (-infty, a_1)  0
  [a_1, a_2)     1
  ...
  [a_n, infty)   n

For each i, the cutpoint a_i is the (p_0 + ... + p_{i-1})-quantile of Y
(see ?qnorm). See ?cut for computing f().

The pair (f(Y), Y) has the required marginal distributions and, in my
opinion, the maximal possible correlation. If this correlation is lower
than the requested one, then i think there is no solution.

If the correlation of (f(Y), Y) is at least the required one, then use
a mixture of the distribution (f(Y), Y) and (X, Y), where X has the 
required marginal distribution of X, but is generated independently
from Y. The mixture parameter may be determined as a solution of an
equation with one variable.

Hope this helps.

Petr Savicky.

Hi,

do you know the parameters of the binomial variate? then maybe you could use
something like the code below. as Petr pointed out, it is generally not
guaranteed that you can create variates with any linear correlation (ie,
depending on the parameters of the binomial)

n <- 100    # how many variates

# your binomial variate (example)
size <- 10; prob <- 0.2
vecB <- rbinom(n, size = size, prob = prob)

rho <- 0.75  # desired cor
m   <- 0.5   # mean and sd of Gaussian
sig <- 2 

rho <- 2*sin(rho*pi/6)  # a small correction
C <- matrix(rho, nrow = 2, ncol = 2)
diag(C) <- 1; C <- chol(C)

# (1) transform binomial to Gaussian
X1 <- qnorm(pbinom(vecB, size = size, prob = prob))
# (2) create another Gaussian
X2 <- rnorm(n)
X <- cbind(X1,X2)
# (3) induce correlation (does not change X1)
X <- X %*% C
# (4) make uniforms
U <- pnorm(X)
# (5) ... and put them into the inverses
vecB1 <- qbinom(U[,1],size,prob)
vecG <- qnorm(U[,2], mean = m, sd = sig)

# check
plot(vecB1,vecG)
cor(vecB1,vecG)
all.equal(vecB1,vecB)
sd(vecG)

(linear correlation is not affected by linear transformation, so you can
enforce exactly your desired mean and standard deviation for the Gaussian by
rescaling it in the end)


regards,
enrico
> -----Urspr?ngliche Nachricht-----
> Von: r-help-bounces at r-project.org 
> [mailto:r-help-bounces at r-project.org] Im Auftrag von Shane Phillips
> Gesendet: Montag, 25. April 2011 01:00
> An: R-help at r-project.org
> Betreff: [R] Random Normal Variable Correlated to an Existing 
> BinomialVariable
> 
> Hi, R-Helpers!
> 
> I have a dataframe that contains a binomial variable.  I need 
> to add another random variable drawn from a normal 
> distribution with a specific mean and standard deviation.  
> This variable also needs to be correlated with the existing 
> binomial variable with a specific correlation (say .75).  Any ideas?
> 
> Thanks!
> 
> Shane
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