lec3 - For the two-way table, the log-linear model can be...

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Unformatted text preview: For the two-way table, the log-linear 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 log-linear 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 log-linear 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 Log-linear model: log( ij ) = + X i + Y j , i = 1 , 2 , j = 1 , 2 Both of these are independence models. The log-linear 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 sum-to-zero constraints. 202 Another Example 2 2 2 Table: Suppose a 3-way table is formed by the combinations of levels of X, Y, Z where each variable has 2-levels. 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 log-linear model. The constant log-odds 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 sum-to-zero constraints. For the set-the-last-level-to-zero 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.

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lec3 - For the two-way table, the log-linear model can be...

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