1 Everyday is a new beginning in life. Every moment is a time for self vigilance.

2 Multicategory Logit Models Nominal responsesNominal responses Ordinal responsesOrdinal responses

3 Logit Models for Nominal Responses Response Y is a nominal variable with J categories { 1,  2, …,  J } denote the response probabilities All  i / j, i=j can be found once  i / J, i=1,…,J-1 are found

4 Baseline-Category Logit Model Baseline-category logits: log( i / J ), i=1,…,J-1 Baseline-category logit model: 

5 Interpretation for Baseline- category 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.

6 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

7 Example: Primary Food Parameter estimates (standard errors): Food choice for logit parameter(fish/other)(invert./other) Intercept Slope-0.110(.517)-2.465(.900)

More Baseline-Category Logit Model Categorical factors A and B For each i=1, 2, …, J: Using dummy variables for factor A only model: 8

9 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

10 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.

11 Cumulative Logit Models for Ordinal Responses Cumulative probabilities: Cumulative logits: Proportional odds model:

12 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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14 Example: Arthritis pain relief Koch and Edwards 1988 (a randomized study) Improvement SexTreatmentMarkedSomeNonetotal FemaleActive FemalePlacebo MaleActive52714 Maleplacebo101011

15 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)