By NEAS - 6/24/2018 3:05:39 PM
MS Module 15: Linear and logistic regression models – practice problems
(The attached PDF file has better formatting.)
Exercise 15.1: Logistic regression
A probability Y is related to the independent variable X by logistic regression:
Y = p(x) = exp(β0 + β1 x) / (1 + exp(β0 + β1 x) )
● When X = 7, the probability Y is 20%.
● When X = 8, the probability Y is 25%.
A. At X = 7, what is the odds ratio of Y?
B. At X = 8, what is the odds ratio of Y?
C. At X = 11, what is the odds ratio of Y?
D. At X = 11, what is the probability of Y?
Part A: The odds ratio of Y at x = 7 is 20% / (1 – 20%) = 0.2500.
Part B: The odds ratio of Y at x = 8 is 25% / (1 – 25%) = 0.3333.
Part C: The slope parameter β1 is the change in the log odds for a 1-unit increase in x, so the odds ratio itself changes by the multiplicative factor exp(β1) when x increases by 1 unit. This factor is
0.333333 / 0.25 = 1.33333
11 is 3 units more than 8, so the odds ratio of Y at x = 11 is 0.333333 × 1.3333333 = 0.79012
Part D: If Y = the probability and R = the odds ratio, R = Y / (1–Y) ➾ Y = R / (1+R).
The probability is the odds ratio / (1 + odds ratio), so the probability of Y at x = 11 is
0.79012 / 1.79012 = 44.14%
Exercise 15.2: Logistic regression
A statistician uses a logistic regression model:
● The independent variable X is a quantitative predictor.
● The dependent variable Y is 1 if the observation is a success and 0 otherwise.
The estimate of β1 is –0.20.
The odds of success at X = 1 are 50%.
A. What is the probability of success at X = 1?
B. What are the odds of success at X = 3?
C. What is the probability of success at X = 3?
D. What are the odds of success at X = 0?
E. What is the probability of success at X = 0?
F. What is β0?
Part A: If the probability of success is P, the odds of success are P/(1-P).
Given that P/(1-P) = 50%, P = ½ – ½P ➾ P = ⅓.
The formula is probability = odds / (1 + odds) = 50% / (1 + 50%) = 0.3333
Part B: For each one unit increase in X, the odds of success increase by a factor exp(1) = e–0.20 = 0.81873
3 is 2 units more than 1, so the odds of success at X = 3 are 50% × 0.818732 = 0.335159
Part C: P/(1-P) = 0.33516 ➾ 1.33516 P = 0.33516 ➾ P = 0.33516 / 1.33516 = 0.25103
Part D: 0 is 1 unit less than 1, so the odds of success at X = 0 are 50% / 0.81873 = 0.610702
Part E: The probability of success at X = 0 is 0.610702 / 1.610702 = 0.379153
Part F: For logistic regression, Y = exp(β0) / (1 + exp(β0 + β1 × X) ).
If Y = 0.379153 at X = 0, then exp(β0) / (1 + exp(β0) ) = 0.379153 ➾
(1 – 0.379153) × exp(β0) = 0.379153 ➾
exp(β0) = 0.379153 / (1 – 0.379153) = 0.610703
β0 = ln(0.610703) = –0.49314
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