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I'm trying to conceptually understand 'why' the horizontal gray line in ROC curve below would be considered random.

I know the basics of the ROC in that the farther the curved black line is from the horizontal the better, and that the screenshot below (the random point highlighted) is stating that a prediction threshold of .473 will result in .533 sensitivity and .785 specificity. With sensitivity being the true positive and specificity being the true negative.

That being said, I'm missing something basic and can't grasp why .6 sensitivity with .4 specificity would be the 'random' result. Hoping someone can help me grasp it in simple, layman's terms. Thank's in advance!

Bonus question.. sometimes I see Specificity along the x axis and sometimes it's the false positive rate instead. Not sure why you would use one vs the other or the meaning in that.

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  • $\begingroup$ You have two classes: 0,1. What is the specificity/sensitivity of a coin flip where 1 comes up with probably $p$ and 0 with $1-p$? Now trace out these pairs for all $p\in[0,1]$. $\endgroup$
    – Alex R.
    Commented Oct 9, 2019 at 1:59

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See the diagram here for how it is made: Understanding ROC curve.

A random classifier won't rank a positive higher than a negative on the whole so the diagnonal line represents pure chance.

It goes up when a positive is called correctly and right when it is called incorrectly, and the parameter varied is the probability threshold for calling a positive. All points are ranked according to highest positive probability so that is why it works.

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    $\begingroup$ Thank you that was helpful. If anyone else comes across this - found this video to be very helpful in conceptually understanding how the ROC works: youtube.com/watch?v=21Igj5Pr6u4 $\endgroup$
    – JD1
    Commented Oct 10, 2019 at 18:01

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