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We could now choose a classification threshold of 45, for example. In this case, of the 5 people with the disease, we would correctly classify 4 as "diseased" and 1 incorrectly as "healthy." Thus, we correctly classified 4 out of 5 as "diseased."

We simply select the two variables Diseased and Blood Value and specify what we consider a positive event, in our case the answer yes. Now we get the ROC curve. In the table below the ROC curve we find the respective threshold value for each point of the curve.

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On the other hand, out of the 5 healthy individuals, we have 2 misclassified as "diseased" and 3 correctly classified as "healthy". So, we misclassified 2 out of 5 as "diseased". This value is called False Positive Rate (FPR).

Very often, 50% is simply taken as the threshold to classify whether a person is "diseased" or not. But of course this does not have to be the case! Any threshold can be used.

The False Positive Rate is equal to the false positives divided by the false positives plus the true negatives. The false positives are the healthy individuals misclassified as "diseased" and the true negatives are the individuals correctly classified as "healthy".

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We can do this for all other thresholds, finishing the ROC curve. At the marked point below, for example, 80% of the diseased people were correctly classified as "diseased" and 20% of the healthy people were incorrectly classified as "diseased".

Cite DATAtab: DATAtab Team (2024). DATAtab: Online Statistics Calculator. DATAtab e.U. Graz, Austria. URL https://datatab.net

Of course, we can easily output a ROC curve online with DATAtab. To do this, we simply copy our data into this table and click on ROC Calculator. Alternatively, you can also create an ROC curve in the Regression Calculator in Logistic Regression.

In the same way, however, we also misclassify all 5 healthy persons as "diseased". Our False Positive Rate is therefore 5 out of 5, i.e. 1.

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Using the ROC curve, we can now also compare different classification methods. A classification model is better the higher the curve is. Therefore, the larger the area under the curve, the better the classifier. Exactly this area, the Area under the Curve, is reflected by the AUC value.

Now we can increase the threshold. Here we still classify all 5 diseased people correctly as "diseased". So we still have a True Positive Rate 5 / 5. However, of the 5 healthy individuals, we now only misclassify 4 out of 5 as "diseased". So we have 4 out of 5, or 0.8.

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At the next threshold, we still have a True Positive Rate of 1. All 5 diseased are correctly classified. The False Positive Rate takes the value of 3/5, so 0.6.

If we choose the threshold value to be very small, i.e. pushing it all the way to the left, we correctly classify all 5 diseased individuals. Our True Positive Rate is thus 5 out of 5 i.e. 1.

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But what about the ROC curve and the logistic regression? We could, for example, create a new classifier using logistic regression. Here we could use, in addition to the blood value, the age and gender of the person.

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This classification is done with the help of a certain blood value, where high values indicate cancer. The question now is which value we choose as the classification threshold. So from which value do we predict a disease?

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A ROC curve is a graphical representation of the performance of a binary classification model for all classification thresholds. Here, ROC stands for Receiver Operating Characteristic.

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The True Positive Rate is equal to the true positives divided by the true positives plus the false negatives. The true positives are those correctly classified as "diseased" and the false negatives are those incorrectly classified as "healthy".

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At the next threshold, for the first time, a diseased person is misclassified as "healthy". We therefore obtain a True Positive Rate of 4/5, i.e. 0.8, and a False Positive Rate of 3/5, i.e. 0.6.

We can now calculate for each threshold what the True Positive Rate and the False Positive Rate are. These two values are plotted on the ROC curve. The True Positive Rate is plotted on the y-axis and the False Positive Rate on the x-axis.