CH8-4 - Outline 8.1: Loglinear Model for Two-way...

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Chapter 8. Loglinear Models for Contingency Tables Deyuan Li School of Management, Fudan University Feb. 28, 2011 1/1 Outline 8.1: Loglinear Model for Two-way contingency Tables; 8.2: Loglinear Model for Independence and Interaction in Three-way Contingency Tables; 8.3: Inference for Loglinear Models; 8.5: Loglinear-Logit Model Connection; 8.6: Loglinear Models Fitting: Likelihood Equations and Asymptotic Distributions. 2/1 8.1 Loglinear Models for Two-way Tables I × J contingency table, the numbers in each cell are assumed to be independent and have Poisson distributions with mean { μ ij } . The observed cell counts by { n ij } . 8.1.1 Independence model The loglinear independence model is { Insert . ..... } (i.e. μ ij = μα i β j ), with constraints λ X I = λ Y J =0 . The ML estimates are { ˆ μ ij = n i + n + j / n } . Goodness-of-±t can be checked by using X 2 or G 2 . 3/1 8.1.2 Interpretation of Parameters We illustrate with independence model for I × 2tab les .Inrow i , the logit equals logit [ P ( Y =1 | X = i )] = log P ( Y | X = i ) P ( Y =2 | X = i ) =log μ i 1 μ i 2 μ i 1 log μ i 2 =( λ + λ X i + λ Y 1 ) ( λ + λ X i + λ Y 2 )= λ Y 1 λ Y 2 . The ±nal term doses not depend on i . So, independence implies a model of form logit [ P ( Y | X = i )] = α . In each row, the odds of response in column equal exp( λ Y 1 λ Y 2 ). 4/1
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8.1.3 Saturated model The saturated model is log μ ij = λ + λ X i + λ Y j + λ XY ij , { λ XY ij } are association terms. All λ XY ij =0 independence. The constraints are λ X I = λ Y J =0and λ XY Ij = λ XY iJ =0fora l l i , j . So, the number of parameters in saturated model is 1+( I 1) + ( J 1) + ( I 1)( J 1) = IJ , the number of cells. There exist relationships between log odds and { λ XY ij } .Fo r example, for 2 × 2tab les , log θ =log μ 11 μ 22 μ 12 μ 21 μ 11 +log μ 22 log μ 12 log μ 21 =( λ + λ X 1 + λ Y 1 + λ XY 11 )+( λ + λ X 2 + λ Y 2 + λ XY 22 ) ( λ + λ X 1 + λ Y 2 + λ XY 12 ) ( λ + λ X 2 + λ Y 1 + λ XY 21 ) = λ XY 11 + λ XY 22 λ XY 12 λ XY 21 . Thus, { λ XY ij } determine the association. 5/1 8.1.5 Multinomial Models for cell probabilities Conditional on the sum n of the cell counts, Poisson loglinear models for { μ ij } become multinomial models for the cell probabilities { π ij = μ ij / ∑∑ μ ab } . For saturated models, it is π ij = exp( λ + λ X i + λ Y j + λ XY ij ) a b exp( λ + λ X a + λ Y b + λ XY ab ) . 6/1 8.2 Logistic Models for Independence and Interaction in Three-Way Tables In Section 2.3 we introduced three-way contingency tables and related structure such as conditional independence and homogeneous association. Loglinear models for three-way tables describe their independence and association patterns. 8.2.1 Types of independence I × J × K -table; X , Y , Z three response variables; n ijk : the number of observations in cell ( i , j , k ), independently from Poisson sampling with mean μ ijk .
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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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CH8-4 - Outline 8.1: Loglinear Model for Two-way...

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