Phi coefficient and odds ratios

Question

I have something strange going on. My phi coefficient of two binary variables is .07 while my odds ratio of the same two binary variables is 1.80. How is this possible?

Answer 1

Note that an odds ratio of 1.80 is in the same direction as a phi of 0.07 - they both represent positive association. But they're not monotonically associated.

Binary phi may be either positive or negative:

$\phi = \frac{ad-bc}{\sqrt{(a+b)(c+d)(a+c)(b+d)}}$

It will be in the same direction as log(OR) (log(OR) is symmetric) -- at least when all your counts are >0. This is easy to see, because when $bc>0$:

$\phi = (OR-1)\frac{bc}{\sqrt{(a+b)(c+d)(a+c)(b+d)}} = k(OR-1)$

log(OR) and OR-1 are monotonically related and if $bc>0$ then $k>0$.

There's no particular basis on which to be surprised at the combination of an odds-ratio of 1.8 and a phi of 0.07; it's perfectly easy to get a result like that. Indeed it's trivial to construct cases where phi is lower and the odds ratio is higher or vice-versa.

---

You can hold OR roughly constant and yet manipulate $\phi$, pushing it up or down. Consider:

1) a=12,b=156,c=4,d=94: \begin{array}{cc} 12 &156\\4 &94 \end{array}

2) a=80,b=60,c=59,d=80: \begin{array}{cc} 80 &60\\59 &80 \end{array}

both have odds ratios close to 1.81, but their phis are quite different.

We can play with $\phi$ while holding ad/bc close to constant by manipulating $k$ above, which we can do fairly readily by changing (at least) two of the numbers a,b,c,d at a time.