1. §7.2: The Proportional Odds Model for Ordinal Response

2. Warm Up Give an example of an ORDINAL variable Suppose p(Y=1)=0.5,
p(Y=3)=0.2. Find P(Y≤2).
What is logit(P(Y≤2))?
What is logit(P(Y≤3))?

3. Our Goal Y – ORDINAL response variable
X1, X2 , … XN predictor variables
Find the distribution of Y in terms of the X’s Y
1-Disagree
2-Neutral
3-Agree
Probability
P(Y=1|x’s)
P(Y=2|x’s)
P(Y=3|x’s)

4. Proportional Odds Model
Express “Cumulative Logit” as a linear combination of predictor variables.

5. You Try
For the ordinal variable you came up with earlier, what are two predictor variables?
Write out the model with your example variables Logit(P(Y≤j|x’s)=αj + β1x1+ β2x2
How many αj’s are in your model?

6. Proportional Odds vs. More Complicated Model
9 parameters Y 1 x1 x2
Y=1
a11
b11
b12
Y=2
a21
b21
b22
Y=3
a31
b31
b32
Y=4
n/a
Proportional Odds
5 parameters Y 1 x1 x2
Y=1
α11
β11
β 12
Y=2
α21
Y=3
α31
Y=4
n/a

7. Score Test Compares Proportional Odds Model to more complicated model
Null Hypothesis – Proportional Odds appropriate
Alternative Hypothesis – More complicated model is needed.