How does AUC of ROC equal concordance probability? [duplicate]

Question

If this answer on Quora is correct then I think I understand what concordance probability is. However, I also find that this answer on StackExchange provides the formula of concordance probability that is a little bit different from the one on Quora: It include the counts of tied pairs with weight $ 0.5 $. Which one should I trust?

I also understand AUC stands for Area Under a Curve and how to compute it... visually.

What I do not understand is how AUC equal concordance probability?

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I have just found my question is the same as this answered question on StackExchange.

Answer 1

I finally have just figured it out. I'm not sure if my reasoning is correct so I will leave it here for anyone to point out the errors.

The discrete formula of AUC should be:

$$ \sum_{i} \frac{TP_{i}}{P}\frac{FP_{i+1}-FP_{i}}{N} $$

(borrowed from http://mlwiki.org/index.php/ROC_Analysis)

That is, I also happen to calculate the (sometimes maybe just approximate) number of concordance pairs in the numerator ($ TP_{i}(FP_{i+1}-FP_{i}) $) and the number of total pairs in the denominator ($ PN $).

For an illustration:

• When $ i = 2 $, $ TP_{2} = 2 $ and $ FP_{3} - FP_{2} = 1$, that is we have found $ 2 $ possible concordance pairs at $ i = 2 $ (i.e. 1-3, 2-3).

• When $ i = 6 $, $ TP_{6} = 5 $ and $ FP_{7} - FP_{6} = 1$, that is we have found more $ 6 $ possible concordance pairs at $ i = 6 $ (i.e. 1-7, 2-7, 4-7, 5-7, 6-7). The iterative process keeps going on.

• ...

However, this formula of AUC does not count tied pairs with 0.5 weight according to this answer on StackExchange.

Answer 2

Please forgive the shoddy pictures.

I use basic integration to prove that Area Under ROC and % of Concordant pairs are the same. I am adding a description below each illustration to make the understanding clear.

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Figure 1 contains the distribution of 0's and 1's , $x(c), y(c)$ respectively. The cutoff $c$ is represented on the X-axis.

Figure 2 contains the distribution of ROC curve. The sensitivity is $Y$ and the (1-specificity) is $X$

In the above picture, we begin by counting the total number of concordant pairs. The number of concordant pairs, for a given $x(c)dc$ number of 1's, is given by $[x(c)dc]*[\int_0^c y(c)]$.

So, the total number of concordant pairs is found by integrating the above expression from 0 to 1, i.e. $\int_0^1[x(c)*[\int_0^c y(c)]]dc$

Proportion of concordant pairs is found by ${(\int_0^1[x(c)*[\int_0^c y(c)]]dc) \over ((\int_0^1 x(c)dc)*(\int_0^1 y(c)dc))} \tag{1}\label{1} $

Now we turn to the ROC plot. For a given cutoff $c$, we obtain the specificity ($1-X$) by ${\int_0^c y(c)dc \over \int_0^1 y(c)dc} $.

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We obtain $dX = {y(c)dc \over \int_0^1 y(c)dc} $. As $X$ varies from 0 to 1, $c$ varies from 1 to 0.

Similarly, we obtain the expression for Sensitivity. And we go on to get the area under ROC = $\frac{[\int_0^1[\int_0^1 x dc] y dc]}{(\int_0^1 x(c)dc)(\int_0^1 y(c)dc)} \tag{2}\label{2}$

Now we have to show that the numerators of (1) and (2) are the same. This can be easily shown by using integration by parts.

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