I recently wrote about the Yule-Simon distribution. The same Yule, George Udny Yule, is also known for the statistics **Yule’s Y** and **Yule’s Q**. The former is also known as the **coefficient of colligation**, and the latter is also known as the **Yule coefficient of association**.

Both measure how things are related. Given a 2 × 2 contingency table

with non-negative entries, Yule’s Y is defined by

and Yule’s Q is defined by

Both essentially measure how much bigger *ad* is than *bc* but are weighted differently.

The algebraic properties of these two statistics may be more interesting than their statistical properties. For starters, both *Y* and *Q* produce values in the interval (-1, 1), and each is an inverse of the other:

There’s some simple but interesting algebra going on here. Define

This simple function comes up surprisingly often. It’s a Möbius transformation, and it comes up in diverse applications such as designing antennas and mental math shortcuts. More on that here.

If we define

and

then *W* generalizes both *Q* and *Y*: setting *p* = 1 gives us *Q* and setting *p* = 1/2 gives us *Y*.

Given the value of *W* with subscript *p*, we could easily solve for the value of W with another subscript *q*, analogous to solving *Q* for *Y* and *Y* for *Q* above.

If you’re expecting *f*^{-1} rather than *f* over the first arrow, you’re right, but *f* is its own inverse so we could just write *f* instead.

Perhaps of interest for you. Bonett and Price (2005) derived a similar conclusion. They used the odds ratio and the generic formula (OR^x – 1)/(OR^x + 1)

Using x = 1 you then obtain Yule Q, using x = pi/4 we get Edward’s Q (1957), using x = 3/4 we get Digby’s H (1983), and although not mentioned in the article using x = 0.5 we get Yule’s Y.

Edit: Bonett and Price should be ‘Becker and Clogg (1988)’