# RE: [R] Generating a binomial random variable correlated with a

From: Ashraf Chaudhary <mchaudha_at_jhsph.edu>
Date: Sun 17 Apr 2005 - 06:46:07 EST

Method I: Generate two correlated normals (using Cholesky decomposition method) and dichotomize one (0/1) to get the binomial. Method II: Generate two correlated variables, one binomial and one normal using the Cholesky decomposition methods.

Here is how I did:

```X <- rnorm(100)
Y <- rnorm(100)
r<- 0.7
Y1 <- X*r+Y*sqrt(1-r**2)
cor(X,Y1)	    # Correlated normals using Cholesky decomposition
```
cor(X>0.84,Y1) # Method I

##
X1 <- rbinom(100,1,0.5)
Y2 <- X1*r+Y*sqrt(1-r**2)

cor(X1,Y2); # Method II

"Are you computing the correlation between the continuous variable and the dichotomized variable with the formula for the biserial correlation? If not, that is probably the root of your problem."

I looked at the biserial correlation which is a special case of Pearson correlation between a continuous and binomial random variable. I don't know how I can use it to generate the data. Any idea?

Regards,
Ashraf

-----Original Message-----
From: Ted Harding [mailto:Ted.Harding@nessie.mcc.ac.uk] Sent: Saturday, April 16, 2005 3:22 AM
To: Ashraf Chaudhary
Cc: r-help@stat.math.ethz.ch
Subject: RE: [R] Generating a binomial random variable correlated with a

On 15-Apr-05 Ashraf Chaudhary wrote:
> Hi,
> I am posting this problem again (with some additional detail)
> as I am stuck and could not get it resolved as yet. I tried to
> look up in alternative sources but with no success. Here it is:
>
> I need to generate a binomial (binary 0/1) random variable linearly
> correlated with a normal random variable with a specified correlation.
> Off course, the correlation coefficient would not be same at each run
> because of randomness.
>
> If I generate two correlated normals with specified correlation and
> dichotomize one, the correlation of a normal and the binomial random
> variable would not be the same as specified.
>
> I greatly appreciate your help.
> Ashraf

Hello Ashraf,

I do not know what you mean by "a binomial random variable linearly correlated with a normal random variable." You can certainly (and indeed your dichotomy method is one way) generate a binomial and a normal which are correlated. But apparently this gives a result which is "not the same as specified": however, I cannot see in your description a specification which would violated by the result of doing so.

You cannot expect a binomial variable to be such that, for instance, its expectation conditional on the value of a normal variable would be a linear function of the normal variable, since this would allow a situation where the expectation was greater than 1 or less than 0. But I wonder what else you could possibly mean by "linearly correlated".

Trying to help,
Ted.

E-Mail: (Ted Harding) <Ted.Harding@nessie.mcc.ac.uk> Fax-to-email: +44 (0)870 094 0861
```Date: 16-Apr-05                                       Time: 08:21:42
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