Odds Ratios and the Non-Constant Probability Effect
A credit risk model fits:
logit(p_default) = -3.0 + 0.05·utilization + 0.9·late_payment_history
where utilization is credit utilization in percent (0-100) and
late_payment_history is 1 if the applicant has any late payment on
record, 0 otherwise.
- Compute the odds ratio for
late_payment_historyand state it in plain English. - For an applicant with
utilization = 20and no late-payment history, computep_default. Now compute it again withutilization = 60. By how many probability points did it move? - A risk analyst says "the coefficient 0.05 means each percentage point of utilization adds 5% to default risk." Correct the statement.
1. Odds ratio for late payment history
e^{0.9} = 2.46. Holding utilization fixed, an applicant with a late payment on record has about 2.46 times the odds of default (a 146% increase in odds) compared to one without.
2. Probability at two utilization levels
At utilization = 20, no late history:
z = -3.0 + 0.05(20) = -2.0, p = 1/(1+e^{2.0}) = 0.119.
At utilization = 60, no late history:
z = -3.0 + 0.05(60) = 0.0, p = 1/(1+e^{0}) = 0.50.
The probability moved from 11.9% to 50.0% — a 38.1 point increase over 40 points of utilization, i.e. roughly 0.95 points of probability per point of utilization in this range, even though the coefficient is a constant 0.05 in log-odds.
3. Correcting the analyst
0.05 is the change in log-odds per percentage point of utilization, not a percentage-point change in probability. The odds multiply by e^{0.05} = 1.051 per point — a 5.1% increase in odds, not probability. And even that multiplier applies to odds, not to probability directly, so it does not translate into "5% more risk" in any everyday sense. As part 2 shows, the actual probability effect of one point of utilization is small near p=0.12 (roughly +0.6 pts) and much larger near p=0.50 (roughly +1.25 pts, since the sigmoid's slope peaks at p=0.5 where \partial p/\partial z = p(1-p) = 0.25, giving 0.25 \times 0.05 = 0.0125 per unit of utilization there). The correct statement: "each percentage point of utilization multiplies the odds of default by 1.051; the effect on the probability itself depends on where the applicant currently sits."
Share this question