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Although I think I understand the (1,1) point on ROC, meaning a classifier that unconditionally issues positive (or negative) classification for all tests in a binary classification scenario, I have trouble understanding why most real-life classifiers in a given test will converge to (1,1). Any intuitive or easy-to-understand explanation why the classifiers converge ? Thanks

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You are trying to predict whether a patient has a disease based on a blood test. If the value measured is above done threshold $x_0$, you diagnose the disease. The ROC curve shows you what happens as you move $x_0$ around.

Once you put the threshold for diagnosis lower than the lowest value you can measure (or have measured on your test set), you automatically diagnose all patients and thus, correctly diagnose all patients that have the disease (=sensitivity 1.0), but you never predict correctly any no disease cases (=specificity 0, this 1-specificity = 1).

Models and algorithms for more than one predictor are just the same in this respect, you simply get a score, predicted outcome or predicted probability out of them, on which you define the cut off in a similar way.

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  • $\begingroup$ @thanks a lot, Bjorn. Great example. Threshold is very easy to understand. Is the concept of "threshold" the same as CDF? It seems that we can use "threshold" to plot the ROC curve, which would then cross both (0,0) and (1,1) of e.g. detecting a disease, WITHOUT invoking the concept of CDF. Are they (threshold and CDF) the same thing? $\endgroup$
    – B Chen
    Oct 8, 2018 at 11:36
  • $\begingroup$ The two things are closely related, because if you move a threshold on a risk score (or some similar decision metric) then the cumulative distribution function of this score is going to eventually go to 0 in one direction and to 1 in the other direction. $\endgroup$
    – Björn
    Oct 8, 2018 at 13:38
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The TPR is the cumulative proportion of positives with scores above some value, across all score values. In this sense, it is a CDF. All CDFs have a maximum at 1.

Likewise, the same is true for the FPR.

Because of these facts, all ROC curves have points at $(0,0)$ and $(1,1)$.

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  • $\begingroup$ thanks for the input. I think I can appreciate the idea of CDF but if possible, can you use a real-life example to illustrate the application of CDF. $\endgroup$
    – B Chen
    Oct 8, 2018 at 4:21

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