Odds ratios

Topic: Logistic regression · more questions on this topic
‹ Derivative of the logistic function Linear discriminant analysis implies a logistic model ›

Using the heart-failure output, find the odds ratio for a 10-year increase in age and its 95\% interval, from the profile-likelihood interval for the age coefficient. By how much does the predicted probability of death change for a 60-year-old without high blood pressure compared with a 70-year-old? Compute both probabilities.

Worked answer (click to show)

For ten years the log odds rise by 10\times 0.04618=0.4618, and the odds ratio is e^{0.4618}=1.587, reported as 1.59. Scaling the profile interval endpoints by 10 gives (0.2484,0.6851), and

e^{0.2484}=1.282,\qquad e^{0.6851}=1.984.

The odds ratio for ten years is 1.59 with 95\% interval (1.28,1.98). For a 60-year-old with high blood pressure equal to 0,

\eta=-3.70568+0.04618(60)=-0.9349, \qquad \hat p=\frac{1}{1+e^{0.9349}}=0.282.

For age 70, \eta=-0.4731 and \hat p=1/(1+e^{0.4731})=0.384. Under this reported fitted association, the predicted probability of recorded death during follow-up changes from about 28\% to 38\%. The ratio 0.384/0.282=1.36 is smaller than the odds ratio 1.59, because the probabilities are not small.