Title: Tau
Course: UGQuants (SSPS) > Section: Association for Nominal and Ordinal Variables

Tau

Tau, also referred to as Goodman and Kruskal’s Tau, is a PRE measure for nominal data.

Let us imagine we have a group of 278 people, of whom 133 are smokers and 145 are non smokers. If, without knowing any of their other characteristics, we had to predict whether any individual from the group of 278 people was a smoker or not, our best guess would have to be that they had a .478 probability (that is, 133 / 278) of being a smoker.

If we used that best guess to classify all of our 278 respondents, we would correctly classify .478 * 133 = 63.57 of the smokers as smokers and .522 * 145 = 75.69

of the non-smokers as non-smokers.

Meanwhile we would incorrectly classify .522 * 133 = 69.43 of the smokers as non-smokers, and .478 * 145 = 69.31 of the non-smokers as smokers.

Out of 278 guesses we would make 69.43 + 69.31 = 138.74 errors.

However, imagine we now obtain information about another characteristic of our group, their educational qualifications, and the relationship between this and smoking is as shown in the table below.

 | University degree | No University degree | All
Smoker | 51 | 82 | 133
Non-Smoker | 99 | 46 | 145
All | 150 | 128 | 278

Since those with different educational levels also have different probabilities of smoking, we can use this knowledge to make our guesses slightly more accurate.

Starting with those with a university degree, we can predict that 51/150 = 0.34 will be smokers.

We will correctly classify 0.34*51 = 17.34 smokers and 0.66 * 99 = 65.34 non-smokers.

We will incorrectly classify 33.66 smokers and 33.66 non-smokers.

For those without a university degree, we can predict that 82/128 = 0.64 will be smokers.

We'll correctly classify 0.64 * 82 = 52.53 smokers and 0.36 * 46 = 16.56 non-smokers, and incorrectly classify 29.47 smokers and 29.44 non smokers.

So now, out of 278 guesses we will have made 33.66 + 33.66 + 29.47 + 29.44 = 126.23

errors.

The proportional reduction in the errors we have made will therefore be 138.74 - 126.23) / 138.74 = 0.09