Assignment 2
1. Odds are defined: Odds = p/(1 – p). Use this relationship to show that p = Odds/(1 + Odds) =
1/(1 + Odds-1). For your solution, do not assume any specific functional form p = f(X) (so no x’s
should appear in your solution). In particular, do not assume the form of the logistic function
1/(1 + e^-f(X)) as we do for logistic regression. (Hint: What does 1 + Odds equal?).
Given,
𝑂𝑑𝑑𝑠 = 𝑝
1 − 𝑝
To prove:
𝑝 = 𝑂𝑑𝑑𝑠
1 + 𝑂𝑑𝑑𝑠 = 1
(1 + 𝑂𝑑𝑑𝑠−1)
Consider R.H.S,
1
(1 + 𝑂𝑑𝑑𝑠−1) = 𝑂𝑑𝑑𝑠 ∗ 1
𝑂𝑑𝑑𝑠(1 + 𝑂𝑑𝑑𝑠−1)
[
 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 & 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 𝑏𝑦 𝑂𝑑𝑑𝑠]
= 𝑂𝑑𝑑𝑠
(𝑂𝑑𝑑𝑠 + 1) = 𝑂𝑑𝑑𝑠
(1 + 𝑂𝑑𝑑𝑠)
𝑂𝑑𝑑𝑠
1 + 𝑂𝑑𝑑𝑠 =
𝑝
1 − 𝑝
1 + 𝑝
1 − 𝑝
[𝐺𝑖𝑣𝑒𝑛 𝑂𝑑𝑑𝑠 = 𝑝
1 − 𝑝]
=
𝑝
1 − 𝑝
1 − 𝑝 + 𝑝
1 − 𝑝
= 𝑝 [
𝐻𝑒𝑛𝑐𝑒 𝑃𝑟𝑜𝑣𝑒𝑑]
2. Consider the Card Study data highlighted at the end of the Week 2. Use the file “card_study.jmp”
rather than the Excel file. Note that the jmp file has value labels, a target level, and correct
nominal encoding already (whereas importing from Excel would assume continuous variables
by default). Here is the scatter plot:
For this data, determine the two thresholds in terms of Purchases that separate the Upgraded
and Not Upgraded regions, for both Extra Cards = No (0) and Extra Cards = Yes (1). For the
thresholds, use the cutoff propensity 0.5. (Hint: See the annotated Week 2 slides.) Show your
calculations. (You may use Excel, and it may be easy to copy and paste the coef