Question

1. A publisher plans to boost the sales of its most popular magazine by sending out promotional mails. We refer to a customer as a responder if he/she subscribes to the magazine for the next year after receiving a promotional mail. Otherwise the customer is referred to as a non-responder. Denote responder by C1 and non-responder by C2. The cost of promotional mail is $3 per customer. The cost of printing and delivering the magazine is $80 per year. With the promotion a customer pays $120 per year for the magazine. Compute the average misclassification costs using the confusion matrix below. Keep at least three digits after the decimal point.

		Actual Class
		C1	C2
Predicted Class	C1	20	230
	C2	3	1017

Cost of classifying a responder as a non-responder:
Cost of classifying a non-responder as a responder:
Average misclassification costs:

PLEASE SHOW ME THE CALCULATION STEPS

Answer 1

Confusion matrix given

		Actual Class
		C1	C2
Predicted Class	C1	20	230
	C2	3	1017

As per confusion matrix below,

Actual Class
Predicted Class		C1- responder	C2- non-responder
	C1- responder	TP	FP
	C2- non-responder	FN	TN

Where

• True Positive (TP) where the actual class is responder and the model accurately predicted a responder
• True Negative (TN) where the actual class is non-responder and the model accurately predicted non-responder
• False Positive (FP) which is a where actual class was non-responder and the model inaccurately predicted responder
• False Negative (FN) which is a where actual was responder but the model inaccurately predicted non-responder

We will now look at creating cost matrix using the details given in the problem.

Cost matrix is similar to the confusion matrix except the fact that we are calculating the cost of wrong prediction or right prediction.

Cost Matrix	Cost (i | j)	Actual Class
		C1	C2
Predicted Class	C1	Cost (C1 | C1)	Cost (C1 | C2)
	C2	Cost (C2 | C1)	Cost (C2 | C2)

where Cost (i | j) is cost of misclassifying class j example as class i

Cost (C2 | C1) is the cost of misclassifying responder as non-responder

Cost (C1 | C2) is the cost of misclassifying non-responder as responder

Cost of promotional mail = $3

Cost of printing and delivering - $80

Customers pays- $120

Profit from each magazine = $120 - $80 = $40

Cost (C1 | C1) = $40 - $3 = $37

Cost (C2 | C2) = 0

Cost (C1 | C2) - If we send promotional mailers thinking that these customers are respondents but in actual they do not respond, then the cost of this misclassification is cost of sending promotional mail i.e. $3.

Cost (C2 | C1) - In this scenario, customer are predicted as non-respondents but in actual they are respondents. There is an opportunity cost or loss of $40 (due to profit loss) of this misclassification.

Cost Matrix	Cost (i | j)	Actual Class
		C1	C2
Predicted Class	C1	$37	- $3
	C2	- $40	$0

Hence, now for calculating cost we will multiply values from confusion matrix and cost matrix

Cost of classifying a responder as a non-responder = $40* 3 = - $120

Cost of classifying a non-responder as a responder = $3 *230 = - $690

Cost of classifying responder as a responder = $37 * 20 = $740

Cost of classifying non-responder as a non-responder = $0 * 1017 = $0

Total Cost = - 120 - 690 + 740 + 0 = - $70

Average misclassification cost = Total Cost/ Total Sample = $70 / (20 + 230 +3 + 1017) = 0.055

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