[R] Simulate phi-coefficient (correlation between dichotomous vars)

Bliese, Paul D LTC USAMH paul.bliese at us.army.mil
Tue Sep 27 08:57:44 CEST 2005

Newsgroup members, 

I appreciate the help on this topic.

David Duffy provided a solution (below) that was quite helpful, and came
close to what I needed.  It did a great job creating two vectors of
dichotomous variables with a known correlation (what I referred to as a

My situation is a bit more complicated and I'm not sure it is easily
solved.  The problem is that I must assume one of the vectors is
constant and generate one or more vectors that covary with the constant

In a continuous example I could use the following code that I got from
the S-PLUS newsgroup year ago:

sample.cor<-function (x, rho) 
    y <- (rho * (x - mean(x)))/sqrt(var(x)) + sqrt(1 - rho^2) * 
    cat("Sample corr = ", cor(x, y), "\n")

X<-rnorm(100)  #a constant vector
Y1<-sample.cor(X,.30) # a new vector that correlates with X .30
Y2<-sample.cor(X,.45) # a second vector that correlates with X .45

I can, of course, have X be a vector of zeros and ones, and I can
dichotomize Y1 and Y2, but the program always returns a phi-coefficient
correlation lower than the continuous correlation.  Mathematically, I
guess this is expected because the phi-coefficient is partially a
function of the percentage of positive responses.  This, in turn,
explains Pearson's (1900) interest in the whole area of tetrachoric
correlations -- a tetrachoric correlation being the Pearson product
moment correlation that would have been observed had two dichotomously
scored variables been measured on a continuous scale (Pearson, 1900).

Appreciate any additional input or possible solutions.


-----Original Message-----
From: r-help-bounces at stat.math.ethz.ch
[mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of David Duffy
Sent: Monday, September 12, 2005 1:34 AM
To: r-help at stat.math.ethz.ch
Subject: [R] Simulate phi-coefficient

> From: "Bliese, Paul D LTC USAMH" <paul.bliese at us.army.mil>
> Given a sample of zeros and ones, for example:
> > VECTOR1<-rep(c(1,0),c(15,10))
> How would I create a new sample (VECTOR2) also containing zeros and
> ones, in which the phi-coefficient between the two sample vectors was
> drawn from a population with a known phi-coefficient value?
> I know there are ways to do this with normally distributed numbers
> example the mvrnorm function in MASS), but am stumped when dealing
> dichotomous variables.
> Paul

One way is to sample from the 2x2 table with the specified means and
correlation (phi):

for a fourfold table, a b
                      c d
with marginal proportions p1 and p2
cov <- phi * sqrt(p1*(1-p1)*p2*(1-p2))
a <- p1*p2 + cov
b <- p1*(1-p2) - cov
c <- (1-p1)*p2 - cov
d <- (1-p1)*(1-p2) + cov
expand.grid(0:1,0:1)[sample(1:4, size=25, replace=TRUE,


| David Duffy (MBBS PhD)                                         ,-_|\
| email: davidD at qimr.edu.au  ph: INT+61+7+3362-0217 fax: -0101  /     *
| Epidemiology Unit, Queensland Institute of Medical Research   \_,-._/
| 300 Herston Rd, Brisbane, Queensland 4029, Australia  GPG 4D0B994A v

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