A large bank is interested in identifying the probability of fraudulent online purchases. The model below uses a variable, SimilarityScore, which considers similarity with past purchases, and an indicator variable for large transaction amounts. LargeTransaction (coded 1 if a transaction is greater than $1,000 and coded 0 if not) to explain the log-odds of fraudulent purchases (coded 1 if the transaction is fraudulent and coded 0 if not)

Question

A large bank is interested in identifying the probability of fraudulent online purchases. The model below uses a variable, SimilarityScore, which considers similarity with past purchases, and an indicator variable for large transaction amounts. LargeTransaction (coded 1 if a transaction is greater than $1,000 and coded 0 if not) to explain the log-odds of fraudulent purchases (coded 1 if the transaction is fraudulent and coded 0 if not)

Ln(odds of purchase fraud) = 12 - 0.018*Similarity + 0.4*LargeTransaction

Holding LargeTransaction constant, how will the predicted odds of purchase fraud change for a purchase with a SimilarityScore of 900 compared to 675, in percentage form?

Answer 1

To determine the percentage change in the predicted odds of purchase fraud, we need to calculate the difference in log-odds between the two similarity scores and convert it to a percentage.

First, we need to substitute the similarity scores into the given model:

For SimilarityScore = 900:
Ln(odds of purchase fraud) = 12 - 0.018 * 900 + 0.4 * LargeTransaction

For SimilarityScore = 675:
Ln(odds of purchase fraud) = 12 - 0.018 * 675 + 0.4 * LargeTransaction

To isolate the effect of similarity score, we subtract the equation for SimilarityScore = 675 from the equation for SimilarityScore = 900:

Ln(odds of purchase fraud) (SimilarityScore = 900) - Ln(odds of purchase fraud) (SimilarityScore = 675) = [12 - 0.018 * 900 + 0.4 * LargeTransaction] - [12 - 0.018 * 675 + 0.4 * LargeTransaction]

Simplifying the equation:

Ln(odds of purchase fraud) (SimilarityScore = 900) - Ln(odds of purchase fraud) (SimilarityScore = 675) = -0.018 * 900 + 0.018 * 675

Ln(odds of purchase fraud) (SimilarityScore = 900) - Ln(odds of purchase fraud) (SimilarityScore = 675) = -9

Now, we calculate the percentage change in the predicted odds by taking the exponential of the difference in log-odds and subtracting 1:

Percentage Change = (exp(-9) - 1) * 100

Percentage Change ≈ -99.055%

Therefore, the predicted odds of purchase fraud will decrease by around 99.055% for a purchase with a SimilarityScore of 900 compared to 675, holding LargeTransaction constant.