ROC Curve in Multivariate Logistic Regression

So far you have learned about some evaluation metrics and saw why they're important to evaluate a logistic regression model. Now, recall that the sensitivity that you got (~53.768%) was quite low and clearly needs to be dealt with. But what was the cause of such a low sensitivity in the first place?

If you remember, when you assigned 0s and 1s to the customers after building the model, you arbitrarily chose a cut-off of 0.5 wherein if the probability of churning for a customer is greater than 0.5, you classified it as a 'Churn' and if the probability of churning for a customer is less than 0.5, you classified it as a 'Non-churn'. 

Now, this cut-off was chosen at random and there was no particular logic behind it. So it might not be the ideal cut-off point for classification which is why we might be getting such a low sensitivity. So how do you find the ideal cutoff point? Let's start by watching the following video. For a more intuitive understanding, this part has been demonstrated in Excel. You can download the excel file from below and follow along with the lecture.

So you saw that the predicted labels depend entirely on the cutoff or the threshold that you have chosen. For low values of threshold, you'd have a higher number of customers predicted as a 1 (Churn). This is because if the threshold is low, it basically means that everything above that threshold would be one and everything below that threshold would be zero. So naturally, a lower cutoff would mean a higher number of customers being identified as 'Churn'. Similarly, for high values of threshold, you'd have a higher number of customer predicted as a 0 (Not-Churn) and a lower number of customers predicted as a 1 (Churn).

Now, let's move forward with our discussion on how to choose an optimal threshold point. For that, you'd first need a few basic terminologies (some of which you have seen in earlier sections.). So let's hear what these terminologies are.

So you learned about the following two terminologies -

True Positive Rate (TPR)

This value gives you the number of positives correctly predicted divided by the total number of positives. Its formula as shown in the video is:

$TruePositiveRate\left(TPR\right)=\frac{TruePositives}{TotalNumberofActualPositives}$

Now, recall the labels in the confusion matrix,

As you can see, the highlighted portion shows the row containing the total number of actual positives. Therefore, the denominator term, i.e. in the formula for TPR is nothing but -

$TotalNumberofActualPositives=TruePositives+FalseNegatives$

So, the formula for True Positive Rate (TPR) becomes -

$TruePositiveRate\left(TPR\right)=\frac{TruePositives}{TruePositives+FalseNegatives}=\frac{TP}{TP+FN}$

As you might remember, the above formula is nothing but the formula for sensitivity. Hence, the term True Positive Rate that you just learnt about is nothing but sensitivity.

The second term which you saw was -

False Positive Rate (FPR)

This term gives you the number of false positives (0s predicted as 1s) divided by the total number of negatives. The formula was -

$FalsePositiveRate\left(FPR\right)=\frac{FalsePositives}{TotalNumberofActualNegatives}$

Again, using the confusion matrix, you can easily see that the denominator here is nothing but the first row. Hence, it can be written as -

$TotalNumberofActualNegatives=TrueNegatives+FalsePositives$

Therefore, the formula now becomes -

$FalsePositiveRate\left(FPR\right)=\frac{FalsePositives}{TrueNegatives+FalsePositives}=\frac{FP}{TN+FP}$

Again, if you recall the formula for specificity, it is given by - 

$Specificity=\frac{TN}{TN+FP}$

Hence, you can see that the formula for False Positive Rate (FPR) is nothing but (1 - Specificity). You can easily verify it yourself.

So, now that you have understood what these terms are, you'll now learn about ROC Curves which show the tradeoff between the True Positive Rate (TPR) and the False Positive Rate (FPR). And as was established from the formulas above, TPR and FPR are nothing but sensitivity and (1 - specificity), so it can also be looked at as a tradeoff between sensitivity and specificity. 

So you can clearly see that there is a tradeoff between the True Positive Rate and the False Positive Rate, or simply, a tradeoff between sensitivity and specificity. When you plot the true positive rate against the false positive rate, you get a graph which shows the trade-off between them and this curve is known as the ROC curve. The following image shows the ROC curve that you plotted in Excel.

As you can see, for higher values of TPR, you will also have higher values of FPR, which might not be good. So it's all about finding a balance between these two metrics and that's what the ROC curve helps you find. You also learnt that a good ROC curve is the one which touches the upper-left corner of the graph; so higher the area under the curve of an ROC curve, the better is your model.

You'll learn more on ROC curves in the coming segments but first, try out some questions.