MS Module 15 Logistic regression odds ratio practice exam questions

[Module 15 for the 2nd edition of the textbook and Module 21 for the 3rd edition of the textbook.]

(The attached PDF file has better formatting.)

A probability Y is related to the independent variable X by logistic regression:

    Y = probability(x) = exp(β0 + β1 x) / (1 + exp(β0 + β1 x) )

●    When X = 5.4, the probability Y is 31%
●    When X = 6.2, the probability Y is 39%


Question 15.1: Odds ratio

At X = 5.4, what is the odds ratio of Y?

Answer 15.1: 31% / (1 – 31%) = 0.449

(odds ratio = probability / (1 – probability)


Question 15.2: Odds ratio

At X = 6.2, what is the odds ratio of Y?

Answer 15.2: 39% / (1 – 39%) = 0.639

(odds ratio = probability / (1 – probability)


Question 15.3: Multiplicative change in the odds ratio

What is the multiplicative change in the odds ratio, or exp(β1), when x increases by 1 unit?

Answer 15.3: (0.639 / 0.449)(1/(6.2 – 5.4) ) = 1.554

(the multiplicative change in the odds ratio, or exp(β1), when x increases by 1 unit, =

(odds at point X2 / odds at point X1)(1 / (value of X2 – value of X1) )


Question 15.4: Odds ratio

At X = 6.9, what is the odds ratio of Y?

Answer 15.4: 0.639 × 1.554(6.9 – 6.2) = 0.870

(odds ratio at point X3 = odds ratio at point X2 × (multiplicative change in odds ratio)(value of point X3 – value of point X2)


Question 15.5: Probability of Y

At X = 6.9, what is the probability of Y?

Answer 15.5: 0.870 / (1 + 0.870) = 46.52%

(probability = odds ratio / (1 + odds ratio)