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DISCRETE_AUTOREGRESSIVE_MODELS

DISCRETE_AUTOREGRESSIVE_MODELS - ESE 502 Tony E Smith...

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ESE 502 Tony E. Smith ________________________________________________________________________ 1 DISCRETE SPATIAL AUTOREGRESSIVE MODELS The standard logistic, binomial and Poisson regression models of discrete counting data have natural spatial generalizations in a manner similar to the conditional autoregressive (CAR) model. We consider each in turn. 1. Autologistic Model Suppose one considers the voting outcomes for Pennsylvania counties during a given Presidential election. Let 1 i v = if county i voted Democratic and 0 i v = otherwise. To model this voting behavior, one may consider a number of county covariates, 1 ( ,.., ) i i ik x x x = , say with 1 i x = “per capita income” in county i , 2 i x = “average years of education” for voters in county i , and so on. A standard logistic model of voting behavior would then take the form: (1) Pr( 1) 1 i i i v µ µ = = + where (2) ( ) 1 exp k i j ij j x µ α β = = + However, it may well be that there are voting similarities among adjacent counties. Hence, if we let 1 ij w = if county j is a contiguous neighbor of county i (with 0 ii w = ) and if we let 1 1 1 ( ,.., , ,.., ) i i i n v v v v v + = denote the voting outcomes of all counties other than i , then a natural spatial generalization of this model would be to hypothesize that the conditional distribution of i v given the voting outcomes, i v , of all other counties depends only on the voting behavior of its neighbors, say, (3) ( ) Pr( 1| ) 1 ( ) i i i i i i v v v v µ µ = = + with (4) ( ) 1 ( ) exp k i i j ij ij j j j i v x w v µ α β ρ = = + + To estimate this model, we start with the standard logistic model that has a likelihood function of the form: (5) 1 1 ( , | ) Pr( 1) [1 Pr( 1)] i i n v v i i i L v v v α β = = = =
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ESE 502 Tony E. Smith ________________________________________________________________________ 2 { } 1 log ( , | ) logPr( 1) (1 )log[1 Pr( 1)] n i i i i i L v v v v v α β = = = + = where 1 ( ,.., ) n v v v = and where Pr( 1) i v = is given by (1) and (2). This function is quite easy to maximize, and yields well behaved maximum-likelihood estimates of α and β .
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