# CH7-4 - Outline 7.1: Model for Nominal Response; Chapter 7....

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Chapter 7. Logit Models for Multivariate Responses Deyuan Li School of Management, Fudan University Feb. 28, 2011 1/1 Outline 7.1: Model for Nominal Response; 7.2: Model for Ordinal Response; 7.3: other link functions; 7.4: alternative ordinal-response models; 7.5: test of conditional independence; 7.6: discrete-choice multinomial logit models. 2/1 7.1 Nominal Responses: Baseline-Category Logit Models 7.1.1 Baseline-Category Logits Let π j ( x )= P ( Y = j | x ), j =1 , 2 , ... J .Th en J j =1 π j ( x )=1 . Logit models pair each response category with a baseline category (often the last one or the most common one): log π j ( x ) π J ( x ) = α j + β ± j x , j , 2 , ..., J . (1) Of course, log π a ( x ) π b ( x ) =log π a ( x ) π J ( x ) log π b ( x ) π J ( x ) . With categorical predictors, X 2 and G 2 goodness-of-±t statistics provide a model check when data are not sparse. When an explanatory variable is continuous or the data are sparse, X 2 and G 2 are still valid for comparing nested models di²ering by few terms (see Haberman (1974), pp. 372-373). 3/1 7.1.2 Alligator Food Choice Example Table 7.1 is from a study of factors inﬂuencing the primary food choice of alligators. TABLE 7.1 Primary Food Choice of Alligators Primary Food Choice Size Ž. Lake Gender m Fish Invertebrate Reptile Bird Other Hancock Male F 2.3 7 1 0 0 5 ± 2.3 4 0 0 1 2 Female F 2.3 16 3 2 2 3 ± 2.3 3 0 1 2 3 Oklawaha Male F 2.3 2 2 0 0 1 ± 2.3 13 7 6 0 0 Female F 2.3 3 9 1 0 2 ± 2.3 0 1 0 1 0 Trafford Male F 2.3 3 7 1 0 1 ± 2.3 8 6 6 3 5 Female F 2.3 2 4 1 1 4 ± 2.3 0 1 0 0 0 George Male F 2.3 13 10 0 2 2 ± 2.3 9 0 0 1 2 Female F 2.3 3 9 1 0 1 ± 2.3 8 1 0 0 1 Source: Data courtesy of Clint Moore, from an unpublished manuscript by M. F. Delaney and C. T. Moore. 4/1

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Y =food choice (Fsh, invertebrate, reptile, bird, other) L =Lake (Hancock, Oklawaha, Tra±ord, George); G =Gender (male, female); S =Size ( 2 . 3, > 2 . 3); Data are sparse since 219 observations scattered among 80 cells. Thus, G 2 is more reliable for comparing models than testing Ft. Model ( ): no predictor; Model ( L + S ): considering lake ( L ) and size ( S ) e±ects; Model ( G + L + S ): considering gender ( G ), lake ( L ) and size ( S ) e±ects. 5/1 TABLE 7.2 Goodness of Fit of Baseline-Category Logit Models for Table 7.1 a 22 Model GX df Ž. 116.8 106.5 60 G 114.7 101.2 56 S 101.6 86.9 56 L 73.6 79.6 48 L q S 52.5 58.0 44 G q L q S 50.3 52.6 40 Collapsed over G 81.4 73.1 28 S 66.2 54.3 24 L 38.2 32.7 16 L q S 17.1 15.0 12 a G , gender; S , size; L , lake of capture. See the text for details. G 2 [() | ( G )] = 116 . 8 114 . 7=2 . 1and G 2 [( L + S ) | ( G + L + S )] = 52 . 5 50 . 3=2 . 2, each based on df=4, implies simplifying by collapsing the table over gender. Other analysis, not presented here, show that adding interaction terms including G do not improve the Ft signiFcantly. 6/1 Table 7.3 lists Ftted values for model (L+S) for the collapsed table. TABLE 7.3 Observed and Fitted Values for Study of Alligator’s Primary Food Choice Primary Food Choice Size of alligator Lake meters Fish Invertebrate Reptile Bird Other Hancock F 2.3 23 4 2 2 8 Ž . Ž. Ž.
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## This note was uploaded on 11/20/2011 for the course ST 3241 taught by Professor Deyuanli during the Spring '11 term at Adams State University.

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CH7-4 - Outline 7.1: Model for Nominal Response; Chapter 7....

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