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A Change in Attitudes Toward Muslims? A Bayesian Investigation of Pre and Post 9/11 Public Opinion
Unformatted Document Text:  persion (Crouchley 1995; Agresti 1999). The formal expression of the model is: y it ∼ Categorical (π it ) , i = 1, 2, . . . , 8842, t = 1, 2, 3, 4 π it = logit −1 (z it,κ−1 ) − logit −1 (z it,κ ) , κ = 1, 2, 3 z it,κ = X it β − c t,κ β ∼ N (0, 1000) c t,1 ∼ N (0, 1000)I(, c t,2 ) c t,2 ∼ N (0, 1000)I(c t,1 , c t,3 ) c t,3 ∼ N (0, 1000)I(c t,2 , ) where we assign uninformative priors β ∼ N (0, 1000) for the coefficients of the pre- dictors and constrained uninformative priors for the cutpoints c’s. 4 For each imputed dataset, we run 10000 iterations, discard first 5000 as the burnin, and thin the re- maining iterates by 5. Strong evidence of convergence of these Markov chains are provided by standard diagnostics (graphic methods and Gelman and Rubin (1992)). We combine the estimates by pooling the estimates across three imputed datasets, thus: ˆ β = 1 m 3 m=1 ˆ β m , m = 1, 2, 3 V β = V within + (1 + 1 m )V between where V within = 1 m 3m=1 s 2 and V between = 1 m 3m=1 ( ˆ β − ˆ β m ) 2 (Gelman and Hill 2007). Figure 7 displays posterior means ˆ β and 95% intervals (±2 V β ) for each of the coefficients, including the cutpoints. The estimates of predictors are similar to those of the pooled model or the model with no pooling. Likewise, the major quantities 4 We implement the analysis using WinBUGS (Spiegelhalter, Thomas, Best, Gilks, and Lunn 1996,2003) and do the postestimation using R package R2WinBUGS (Sibylle, Uwe, and Gelman 2005). 17

Authors: Kalkan, Kerem. and Su, Yu-Sung.
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persion (Crouchley 1995; Agresti 1999). The formal expression of the model is:
y
it
∼ Categorical (π
it
) , i = 1, 2, . . . , 8842, t = 1, 2, 3, 4
π
it
= logit
−1
(z
it,κ−1
) − logit
−1
(z
it,κ
) , κ = 1, 2, 3
z
it,κ
= X
it
β − c
t,κ
β ∼ N (0, 1000)
c
t,1
∼ N (0, 1000)I(, c
t,2
)
c
t,2
∼ N (0, 1000)I(c
t,1
, c
t,3
)
c
t,3
∼ N (0, 1000)I(c
t,2
, )
where we assign uninformative priors β ∼ N (0, 1000) for the coefficients of the pre-
dictors and constrained uninformative priors for the cutpoints c’s.
4
For each imputed
dataset, we run 10000 iterations, discard first 5000 as the burnin, and thin the re-
maining iterates by 5. Strong evidence of convergence of these Markov chains are
provided by standard diagnostics (graphic methods and Gelman and Rubin (1992)).
We combine the estimates by pooling the estimates across three imputed datasets,
thus:
ˆ
β =
1
m
3
m=1
ˆ
β
m
, m = 1, 2, 3
V
β
= V
within
+ (1 +
1
m
)V
between
where V
within
=
1
m
3
m=1
s
2
and V
between
=
1
m
3
m=1
( ˆ
β − ˆ
β
m
)
2
Figure 7 displays posterior means ˆ
β and 95% intervals (±2
V
β
) for each of the
coefficients, including the cutpoints. The estimates of predictors are similar to those
of the pooled model or the model with no pooling. Likewise, the major quantities
4
We implement the analysis using WinBUGS (Spiegelhalter, Thomas, Best, Gilks, and Lunn 1996,
2003)
and do the postestimation using R package R2WinBUGS (Sibylle, Uwe, and Gelman 2005).
17


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