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
If a purchase has a SimilarityScore of 710 and a transaction amount of $400, what is the estimated probability that the purchase is fraudulent?
To calculate the estimated probability that the purchase is fraudulent, we need to convert the log-odds into actual odds and then into a probability using the logistic function.
The log-odds of the purchase being fraudulent is given by:
Ln(odds of purchase fraud) = 12 - 0.018 * Similarity + 0.4 * LargeTransaction
Let's substitute the given values:
SimilarityScore = 710
Transaction amount = $400
Ln(odds of purchase fraud) = 12 - 0.018 * 710 + 0.4 * 0
Simplifying the equation:
Ln(odds of purchase fraud) = 12 - 12.78 + 0 = -0.78
To convert the log-odds into odds, we use the exponential function:
odds of purchase fraud = exp(-0.78)
Now, to convert the odds into a probability, we use the logistic function:
P(purchase is fraudulent) = odds of purchase fraud / (1 + odds of purchase fraud)
P(purchase is fraudulent) = exp(-0.78) / (1 + exp(-0.78))
Using a calculator, we find:
P(purchase is fraudulent) ≈ 0.313
Therefore, the estimated probability that the purchase is fraudulent is approximately 0.313 or 31.3%.