Everyday is a new beginning in life.
Every moment is a time for self vigilance.
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Multicategory
Logit Models
•Nominal responses
•Ordinal responses
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Logit Models for Nominal
Responses
• Response Y is a nominal variable
with J categories
• {p1, p2, … , pJ} denote the response
probabilities
• All pi/pj, i=j can be found once pi/pJ,
i=1,…,J-1 are found
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Baseline-Category Logit Model
• Baseline-category logits:
log(pi/pJ), i=1,…,J-1
• Baseline-category logit model:
log(

pi 
pi
pJ
)   i   i x , i  1, 2 ,..., J
exp(  i   i x )
 exp( 
j
  j x)
, i  1,..., J  1
j
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Interpretation for Baselinecategory Logit Model
• Similar to logistic regression for a binary
response, the “response” for each
baseline-category logit regression is either
Y=j or J
• How to interpret , ?
• logit(odds) for any 2 categories of Y can
be found once all baseline-category logits
are found.
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Example: Primary Food
• Alligator size (meters) vs. primary
food choice
• F= fish, I=invertebrates, O=other
• Data (alligator.sas; lake of George only)
1.24 I, 1.30 I, 1.30 I, 1.32 F, 1.32 F,
…, 3.89 F
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Example: Primary Food
• Parameter estimates (standard errors):
Food choice for logit
parameter
Intercept
Slope
(fish/other)
(invert./other)
1.618
5.697
-0.110(.517)
-2.465(.900)
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More Baseline-Category Logit
Model
• Categorical factors A and B
• For each i=1, 2, …, J:
log(
pi
pJ
)   i  A k ( i )  B l ( i )  AB kl ( i )
• Using dummy variables for factor A
only model:
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Example: Primary Food
(Table 7.1)
• Primary food: fish, invertebrates,
reptile, bird, other
• Factors:
– size (big/small)
– lake (Hancock, Oklawaha, Trafford,
George)
• Alligator2.sas and table 7.2
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Connection with Loglinear
Models
• The loglinear model which
corresponds to a logit model is the
one with the most general interaction
among explanatory variables from
the logit model. It has the same
association and interaction structure
relating the explanatory variables to
the response.
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Cumulative Logit Models for
Ordinal Responses
• Cumulative probabilities:
P (Y  j )  p 1  p 2  ...  p j , j  1, 2 ,... J
• Cumulative logits:
 p 1  ...  p j
log it [ P (Y  j )]  log 
p
 j  1  ...  p J

 , j  1,..., J  1


• Proportional odds model:
log it [ P (Y  j | x )]   j   x , j  1,..., J  1
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Interpretation for
Proportional Odds Model
• Similar to logit regression for a
binary response, the “response” for
each cumulative logit regression is
either Y < j or not. But, there are NO
corresponding loglinear models
• How to interpret , ?
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Example: Breast Self-exam
Table of age by exam
age
Frequency
Total
exam
45-59
Monthly
150
Never
155
Occasion
200
Total
505
60+
109
172
198
479
<45
91
51
90
232
350
378
488
1216
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Example: Arthritis pain relief
Koch and Edwards 1988 (a randomized study)
Improvement
Sex
Treatment
Female
Active
Female
Marked
Some
None
total
16
5
6
27
Placebo
6
7
19
32
Male
Active
5
2
7
14
Male
placebo
1
0
10
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• Proportional odds model makes an
assumption: the effect of x is the same
for all cumulative odds; should check this
assumption before doing inference (SAS
can do the test)
• When this assumption fails, we can fit
baseline-category logit model
• (Optional) When this assumption fails, i.e.
the proportional odds model does not fit,
continuation-ratio model is a good
alternative (see Sec 7.4)
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