Notes 9 - ’ & ST3241 Categorical Data...

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Unformatted text preview: ’ & ST3241 Categorical Data Analysis I Multicategory Logit Models Logit Models For Nominal Responses 1 ’ & 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 ’ & 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 /π 3 ) and log( π 2 /π 3 ) . 3 ’ & 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 /π J π b /π 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 ’ & 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 ’ & 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 ’ & 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 F1....
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Notes 9 - ’ & ST3241 Categorical Data...

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