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Notes 9 - $ ST3241 Categorical Data Analysis I...

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& $ % ST3241 Categorical Data Analysis I Multicategory Logit Models Logit Models For Nominal Responses 1
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& $ % Models For Nominal Responses Y is nominal with J categories. Let { π 1 , · · · , π J } denote the response probabilities with π 1 + · · · + π J = 1 . If we have n independent observations based on these probabilities, the probability distribution for the no. of outcomes that occur for each J types is called multinomial. Multicategory (or polychotomous ) logit models simultaneously refer to all pairs of categories. They describe the odds of response in one category rather than another. Once the model specifies logits for a certain ( J - 1) pairs of categories, the rest are redundant. 2
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& $ % Baseline Category Logits Logit models for nominal responses pair each response category with a baseline category . The choice of baseline category is arbitrary. If the last category ( J ) is the baseline, the baseline category logits are: log( π j π J ) , j = 1 , · · · , J Given that the response falls in category j or J , this is the log odds that the response is j . For J = 3 , for instance, the logit model uses log( π 1 3 ) and log( π 2 3 ) . 3
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& $ % Baseline Category Logit Models The logit models using the baseline-category logits with a predictor x has form log( π j π J ) = α j + β j x, j = 1 , · · · , J Parameters in the ( J - 1) equations determine parameters for logits using all other pairs of response categories. For instance, for an arbitrary pair of categories a and b log( π a π b ) = log( π a J π b J ) = log( π a π J ) - log( π b π J ) = ( α a + β a x ) - ( α b + β b x ) = ( α a - α b ) + ( β a - β b ) x 4
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& $ % Notes The logit equation for categories a and b has intercept parameter ( α a - α b ) and slope parameter ( β a - β b ) . For optimal efficiency, one should fit J - 1 logit equations simultaneously. Estimates of the model parameters will then have smaller standard error than the estimates obtained by fitting the equations separately. For simultaneous fitting, the same parameter estimates occur for a pair of categories no matter which category is baseline. 5
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& $ % Alligator Food Choice Example The data is taken from a study by the Florida Game and Fresh Water Fish Commission of factors influencing the primary food choice of alligators. For 59 alligators sampled in Lake George, Florida, it shows the alligator length (in meters) and the primary food type, in volume, found in the alligator’s stomach. Primary food type has three categories: Fish, Invertebrate, and Other. 6
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& $ % Reading The Data data gator; input length choice $ @@; datalines; 1.24 I 1.30 I 1.30 I 1.32 F 1.32 F 1.40 F 1.42 I 1.42 F 1.45 I 1.45 O 1.47 I 1.47 F 1.50 I 1.52 I 1.55 I 1.60 I 1.63 I 1.65 O 1.65 I 1.65 F 1.65 F 1.68 F 1.70 I 1.73 O 1.78 I 1.78 I 1.78 O 1.80 I 1.80 F 1.85 F 1.88 I 1.93 I 1.98 I 2.03 F 2.03 F 2.16 F 2.26 F 2.31 F 2.31 F 2.36 F 2.36 F 2.39 F 2.41 F 2.44 F 2.46 F 2.56 O 2.67 F 2.72 I 2.79 F 2.84 F 3.25 O 3.28 O 3.33 F 3.56 F 3.58 F 3.66 F 3.68 O 3.71 F 3.89 F ; run; 7
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& $ % Fitting A Baseline-Category Logit Model proc logistic data=gator descending ; model choice (REFERENCE="O") = length / link=glogit scale=none aggregate; output out = prob PREDPROBS=I; run; 8
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& $ % Partial Output
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