An insurance agent wants to identify which prospective customers are worth contacting to sell supplemental health insurance, such as coverage for critical illness, accidents, hospital insurance, etc.
Consider the following logistic regression model that uses hourly wage, an indicator for heart disease (coded 1 if the customer has heart disease and 0 if not), and the number of insurance policies that the customer already has, to explain the log-odds of purchasing supplemental health insurance:
Ln(odds of purchasing insurance) = 1.2752 + 0.0229*Hourly wage + 0.2321*Heart disease – 0.0275*number of policies
What is the estimated effect on the odds of purchasing the insurance policy if a customer’s hourly wage is 5 dollars lower than another customer with the same number of insurance policies given that both customers do not have heart disease?
To determine the estimated effect on the odds of purchasing the insurance policy with a $5 lower hourly wage, we need to substitute the values into the logistic regression model:
Let's assume the hourly wage of the first customer (without heart disease) is X, and the number of policies both customers have is Y.
For the first customer:
Ln(odds of purchasing insurance) = 1.2752 + 0.0229X - 0.0275Y
For the second customer with a $5 lower hourly wage:
Ln(odds of purchasing insurance) = 1.2752 + 0.0229(X-5) - 0.0275Y
To find the estimated effect on the odds of purchasing the insurance policy, we compare the two equations:
Ln(odds of purchasing insurance) - Ln(odds of purchasing insurance) = 1.2752 + 0.0229X - 0.0275Y - (1.2752 + 0.0229(X-5) - 0.0275Y)
Simplifying the equation:
Ln(odds of purchasing insurance) - Ln(odds of purchasing insurance) = 1.2752 + 0.0229X - 0.0275Y - 1.2752 - 0.0229X + 0.1145 - 0.0275Y
The 1.2752 and -1.2752 terms cancel out, and the 0.0229X and -0.0229X terms also cancel out. Thus, we are left with:
0.1145
Therefore, the estimated effect on the odds of purchasing the insurance policy for a customer with a $5 lower hourly wage, compared to another customer with the same number of insurance policies and no heart disease, is approximately 0.1145.