# [R] How to simulate correlated data

(Ted Harding) Ted.Harding at nessie.mcc.ac.uk
Thu Dec 15 18:04:21 CET 2005

```On 15-Dec-05 Lisa Wang wrote:
> Hello there,
>
> I would like to simulate X --Normal (20, 5)
>                          Y-- Normal (40, 10)
>
> and the correlation between X and Y is 0.6. How do I do it in R?

... and, as well as using mvrnorm (MASS) or rmvnorm (mvtnorm),
as have been suggested, you could simply do it "by hand":

If U, V are independent and N(0,1), then

E(U + a*V)*(U - a*V) = 1 - a^2

E(U+a*V)^2 = E(U - a*V) = 1 + a*2

so the correlation between (U + a*V) and U - a*V) is

r = (1 - a^2)/(1 + a^2)

Hence, for -1 < r < 1, choose

a = sqrt((1 - r)/(1 + r))

which, for r = 0.6, gives a = sqrt(0.4/1.6) = sqrt(1/4) = 1/2
(how nice! ... ).

Then Var(U + a*V) = 1 + a^2 = 1 + 1/4 = 5/4 (I smell more smooth
numbers coming ... ).

Then, since the correlation between two variables is unchanged
if you add a constant to either, or multiply either by a constant,
you can give (U + a*V) variance 5 by multiplying it by 2, and
give (U - a*V) variance 10 by multiplying by 2*sqrt(2), both still
having expectation 0. So finally add 10 and 20:

X = 10 + 2*(U + V/2) ;  Y = 20 + 2*sqrt(2)*(U - V/2)

So you can get U and V by sampling from rnorm(), and then X and Y
as described.

(Which is how I used to do it before starting to use R, e.g. in
matlab/octave).

Best wishes,
Ted.

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Date: 15-Dec-05                                       Time: 17:04:18
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