\hfill \thepage} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \title{Mixed logistic regression; A comparison with BUGS} \author{} \maketitle \paragraph{Model description} Let $\mathbf{y}=(y_{1},\ldots ,y_{n})$ be a vector of dichotomous observations ($y_{i}\in \{0,1\}$), and let $\mathbf{u}=(u_{1},\ldots ,u_{q})$ be a vector of independent random effects, each with Gaussian distribution (expectation $0$ and variance $\sigma ^{2}$). Define the success probability $\pi _{i}=\Pr (y_{i}=1)$. The following relationship between $\pi _{i}$ and explanatory variables (contained in matrices $\mathbf{X}$ and $\mathbf{Z}$) is assumed:% \[ \log \left( \frac{\pi _{i}}{1-\pi _{i}}\right) =\mathbf{X}_{i}\mathbf{\beta }% +\mathbf{Z}_{i}\mathbf{u}, \]% where $\mathbf{X}_{i}$ and $\mathbf{Z}_{i}$ are the $i$'th rows of the known covariates matrices $\mathbf{X}$ ($n\times p$) and $\mathbf{Z}$ ($n\times q$% ), respectively, and $\mathbf{\beta }$ is a $p$-vector of regression parameters. Thus, the vector of fixed-effects vector is $\mathbf{\theta }=(% \mathbf{\beta },\log \left( \sigma \right) )$. \bigskip \paragraph{Results} The goal here is to compare computation times with BUGS on a simulated data set. For this purpose we use $n=200$, $p=5$, $q=30$, and values of the the hyper parameters as showd in the table below (`True values'). The matrices $% \mathbf{X}$ and $\mathbf{Z}$ were generated randomly with each element uniformly distributed on $[-2,2]$. As start values for both AD Model Builder and BUGS we used $\beta _{\text{init},j}=-1$ and $\sigma _{\text{init}}=4.5$% . In BUGS we used a uniform $[-10,10]$ prior on $\beta _{j}$ and a standard (in the BUGS literature) noninformative gamma prior on $\tau =\sigma ^{-2}$. In AD Model Builder the parameter bounds $\beta _{j}\in \lbrack -10,10]$ and $\log (\sigma )\in \lbrack -5,3]$ were used in the optimization process. \begin{center} \begin{tabular}{lllllll} & $\beta _{1}$ & $\beta _{2}$ & $\beta _{3}$ & $\beta _{4}$ & $\beta _{5}$ & $\sigma $ \\ \hline True values & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.1 \\ \hline ADMB-RE & 0.03 & -0.07 & 0.08 & 0.08 & -0.11 & 0.17 \\ Std. dev. & 0.15 & 0.15 & 0.15 & 0.14 & 0.16 & 0.05 \\ WinBUGS & 0.03895 & -0.07871 & 0.07727 & 0.08405 & -0.1041 & 0.1862% \end{tabular} \quad \end{center} On the simulated dataset AD Model Builder used $27$ seconds to converge to the optimum of likelihood surface. On the same dataset we first ran WinBUGS (Version 1.4) for $5,000$ iterations. The recommended convergence dianostic in WinBUGS is the Gelman-Rubin plot (see the help files available from the menues in WinBUGS) which require that two Markov chains are run in paralell. From the Gelman-Rubin plot it was clear that convergence appeared after approximately $2,000$ iterations. The time taken by WinBUGS to perform generate the first $2,000$ was approximately $700$ seconds. \end{document}