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# 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 . Then 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 = 1 , 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-fit 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 differing 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Male F 2.3 13 10 0 2 2 2.3 9 0 0 1 2 Female F 2.3 3 9 1 0 1 2.3 8 1 0 0 1 Source: Data courtesy of Clint Moore, from an unpublished manuscript by M. F. Delaney and C. T. Moore. 4 / 1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 (fish, invertebrate, reptile, bird, other) L =Lake (Hancock, Oklawaha, Trafford, 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 fit. Model ( ): no predictor; Model ( L + S ): considering lake ( L ) and size ( S ) effects; Model ( G + L + S ): considering gender ( G ), lake ( L ) and size ( S ) effects. 5 / 1 TABLE 7.2 Goodness of Fit of Baseline-Category Logit Models for Table 7.1 a 2 2 Model G X df Ž . 116.8 106.5 60 Ž . G 114.7 101.2 56 Ž . S 101.6 86.9 56 Ž . L 73.6 79.6 48 Ž . L q S 52.5 58.0 44 Ž . G q L q S 50.3 52.6 40 Collapsed over G Ž . 81.4 73.1 28 Ž . S 66.2 54.3 24 Ž . L 38.2 32.7 16 Ž . 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 . 1 and 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 fit significantly. 6 / 1 Table 7.3 lists fitted 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 Ž . Lake meters Fish Invertebrate Reptile Bird Other Hancock F 2.3 23 4 2 2 8 Ž . Ž . Ž . Ž . Ž .
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