# Conceptually, what happens when you take (A - B) and divide it by (1 - B)? What is it used for?

I've come across this transformation a lot in my advanced stats course and I'm curious as to what it accomplishes. The basic structure is:

A - B

---------

1 - B

Here's an example of its use when calculating a squared partial correlation:

And a near-example when F-testing RSquared:

If you plug in some numbers to the transformation, a few patterns show up:

(.5 - .25) / (1 - .25) = (.25 / .75) = .333.

(.5 - .75) / (1 - .75) = (-.25 / .25) = -1

(.5 - .5) / (1 - .5) = (0 / .5) = 0

(.5 - 1) / (1 - 1) = (-.5 / 0), which is impossible. So it seems that this transformation is only meant to be used on decimals?

(.5 - .1) / (1 - .1) = (.4 / .9) = .44444

(.5 - .01) / (1 - .01) = (.49 / .99) = .49494949, nearly .5. And obviously if B is 0 then it simplifies to A dividing by 1, or A. So the further B gets from 0, the more distant the output gets from A.

These patterns are interesting but I'm struggling to see what meaning they might have, particularly in the context of manipulating R Squared values.

• Hint: there exists a meaningful $C$ for which $1=A+C,$ rending the fraction in the form $$\frac{A-B}{1-B}=\frac{A-B}{C+(A-B)}=1-\frac{C}{C+(A-B)}.$$ In particular, this analysis shows that the underlying concept involves decompositions of sums of squares rather than some kind of transformation of numbers. – whuber May 11 at 12:06