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Unformatted text preview: For the twoway table, the loglinear model can be used to test indepen dence of the row and column variables or marginal homogeneity across rows. Both hypotheses are equivalent to the hypothesis H : XY ij = 0 for all i, j , based on the saturated model log( ij ) = + X i + Y j + XY ij H can be tested by comparing the deviance from this model with the deviance from the nested model log( ij ) = + X i + Y j ( * ) It can be shown that an equivalent test is to compare G 2 = 2 X i,j n ij log( n ij / ij ) with the 2 (( I 1)( J 1)) distribution, where ij s are the MLEs under independence (based on model (*)). It is important to realize that loglinear models make no distinction between response and explanatory variables. They are inherently multivariate. (We model the n ij s not the values of X or Y .) When such a distinction is appropriate, it may be more natural to use multinomial response models (logit, multinomial logit, others well talk about these later). For many multinomial response logit models, equivalent loglinear models exist. 201 Example 2 2 Table: In this case, Y = 1 Y = 2 X = 1 11 12 X = 2 12 22 represents the table of cell probabilities and Y = 1 Y = 2 X = 1 11 12 X = 2 12 22 represents the table of cell means. Logit model: log i 1 i 2 = , i = 1 , 2 Loglinear model: log( ij ) = + X i + Y j , i = 1 , 2 , j = 1 , 2 Both of these are independence models. The loglinear model parameters have interpretations in terms of , the common log odds of Y = 1 over the two rows: = logit( i 1 ) = log( i 1 ) log( i 2 ) = Y 1 Y 2 * = 2 Y 1 * under sumtozero constraints. 202 Another Example 2 2 2 Table: Suppose a 3way table is formed by the combinations of levels of X, Y, Z where each variable has 2levels. Let i index the levels of X , j index the levels of Y , and k index the levels of Z . Consider the model where Y is jointly independent of X and Z . This corresponds to the logit model log i 1 k i 2 k = , i = 1 , 2 , k = 1 , 2 When Y is jointly independent of X, Z , ijk = i k j ijk = n ijk log( ijk ) = log( n ) + log( i k ) + log( j ) = + X i + Z k + XZ ik + Y j is the corresponding loglinear model. The constant logodds that Y = 1 over the levels of X and Z is = log i 1 k i 2 k = log i 1 k i 2 k = + X i + Z k + XZ ik + Y 1 ( + X i + Z k + XZ ik + Y 2 ) = Y 1 Y 2 * = 2 Y 1 * under sumtozero constraints. For the setthelastleveltozero constraints, = Y 1 ....
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This note was uploaded on 11/13/2011 for the course STAT 8620 taught by Professor Hall during the Fall '11 term at University of Florida.
 Fall '11
 Hall

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