Dear all!
I am a new WINBUGS user.
The new disposition models proposed by George E. Bonney (1998)
for correlated binary outcomes are based on the logistic regression
model.
The proposed likelihood for m regions each of size n_{r}is given by:
L(\theta) = \prod_{r=1}^{m} \log
[(1-\alpha_{r})\prod_{j=1}{n_{r}}(1-y_{rj})+
\alpha_{r}\prod_{j=1}^{n_{r}}d_{rj}^{y_{rj}}(1-d_{rj})^{1-y_{rj}}]
where the parameters are as defined below. Note that when \alpha_{r}=1,
we
get the standard logistic regression model (the term after the +).
Following Bayesian principles, we assume that \theta has a normal prior
(obvious reasons for a normal prior for the logistic regression since
there
is no congugate prior for the logistic regression!) and build up the
log-posterior
-likelihood function as below.
I am requesting assistant regarding the estimation of the the following
log-posterior-likelihood function denoted \log_{p}L(\theta) with the
bugs
software:
\log_{p}L(\theta) =
\sum_{r=1}^{m} \log [(1-\alpha_{r})\prod_{j=1}{n_{r}}(1-y_{rj})+
\alpha_{r}\prod_{j=1}^{n_{r}}d_{rj}^{y_{rj}}(1-d_{rj})^{1-y_{rj}}]
- ||\theta - \mu||^{2}/2\sigma^{2}
where y_{rj} is a binary response variable, r=1,...,m (regions),
j=1,...,n_{r} (individuals) with n_{r} the sample size of region r;
\theta = (\gamma_{0}, \gamma_{1}, ... ,\gamma_{k}; \lambda;
\beta_{1}, ... ,\beta_{s})
i.e. \theta \in R^{k+s+2}
\theta has a normal prior with mean = \mu and variance
\sigms^{2}I_{k+s+2},
Take: \mu = 0_{k+s+2}, \sigma^{2}=1;
e^{M(G_{r})+N(G_{r})+W(X_{rj})}
d_{rj} = ------------------------------------------
1+e^{M(G_{r})+N(G_{r})+W(X_{rj})}
M(G_{r}) = \gamma_{0}+\gamma_{1}G_{r1}+...+\gamma_{k}G_{rk}
N(G_{r}) = \lambda (constant, say 0.5)
W(X_{rj}) = \beta_{1}G_{1r1}+ ... +\beta_{sG_{srn_{r}}}
1+e^{-[M(G_{r})+N(G_{r})+(W(X_{rj})]}
\alpha_{r} = ----------------------------------------------
1+e^{-M(G_{r})}
Small sample size test:
Health Status
of a tree labeling index tempertaure
Y Li T
REGION ONE
1 1.9 .996
1 1.4 .992
0 .8 .982
0 .7 .986
1 1.3 .980
REGION TWO
0 .6 .982
1 1.0 .992
0 1.9 1.020
0 .8 .990
0 .5 1.038
REGION THREE
1 1.0 1.002
0 1.6 .998
1 1.7 .990
1 .9 .986
0 .7 .986
Grateful for any assistance/tips.
With best wishes,
Osman Sankoh
Department of Statisitcs
University of Dortmund
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