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\newcommand{\scGLM}{\textsc{glm}}

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\newcommand{\scGLMM}{\textsc{glmm}}

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\newcommand{\scLIDAR}{\textsc{lidar}}

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\newcommand\admbversion{11.1}

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\makeindex

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\title{%

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    \largetitlepart{Random Effects in\\ \ADM}

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    \smalltitlepart{ADMB-RE User Guide}

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    \vspace{4.5ex}\textsf{\textit{Version 11.1~~(2013-05-01)}}\vspace{3ex}

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    \vspace{4.5ex}\textsf{\textit{Version \admbversion~~(2013-05-01)\$3pt]  29  Revised manual~~(2013-06-24)}}\vspace{3ex}  28 30 }  29 31 % Author definition.  30 32 \author{\textsf{\textit{Hans Skaug \& David Fournier}}}  ...... 35 37 ~\vfill  36 38 \noindent ADMB Foundation, Honolulu.\\\\  37 39 \noindent This is the manual for AD Model Builder with Random Effects (ADMB-RE)  38 version 11.1.\\\\  39 \noindent Copyright \copyright\ 2004, 2006, 2008, 2009, 2011 Hans Skaug \& David  40 version \admbversion.\\\\  41 \noindent Copyright \copyright\ 2004, 2006, 2008, 2009, 2011, 2013 Hans Skaug \& David  40 42 Fournier\\\\  41 43 \noindent The latest edition of the manual is available at:\\  42 \url{http://admb-project.org/documentation/manuals/admb-user-manuals}  44 \url{http://admb-project.org/documentation/manuals}  43 45 44 46 \tableofcontents  45 47 ...... 140 142  \item Hyper-parameters (variance components, etc.) estimated by maximum  141 143  likelihood.  142 144 143  \item Marginal likelihood evaluated by the Laplace approximation, (adaptive) importance  144  sampling or Gauss-Hermite integration.  145  \item Marginal likelihood evaluated by the Laplace approximation, (adaptive)  146  importance sampling or Gauss-Hermite integration.  145 147 146 148  \item Exact derivatives calculated using Automatic Differentiation.  147 149 ...... 385 387 term borrowed from Bayesian statistics.  386 388 387 389 A central concept that originates from generalized linear models is that of a  388 linear predictor.'' Let x_{1},\ldots ,x_{p} denote observed covariates  389 (explanatory variables), and let \beta _{1},\ldots ,\beta _{p} be the  390 linear predictor.'' Let x_{1},\ldots,x_{p} denote observed covariates  391 (explanatory variables), and let \beta _{1},\ldots,\beta _{p} be the  390 392 corresponding regression parameters to be estimated. Many of the examples in  391 393 this manual involve a linear predictor \eta_{i}=\beta_{1}x_{1,i}+\cdots  392 394 +\beta_{p}x_{p,i}, which we also will write in vector form as  ...... 414 416 exemplify the use of random effects. The statistical model underlying this  415 417 example is the simple linear regression  416 418 \[  417 Y_i=ax_i+b+\varepsilon_i,\qquad i=1,\ldots ,n,  419 Y_i=ax_i+b+\varepsilon_i,\qquad i=1,\ldots,n,  418 420 $

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where $Y_i$ and $x_i$ are the data, $a$ and $b$ are the unknown parameters to be

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estimated, and $\varepsilon_i\sim N(0,\sigma ^{2})$ is an error term.

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know the value of $\sigma_e$, so we shall pretend that~$\sigma_e=0.5$.

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Because $x_i$ is not observed, we model it as a random effect with $x_i\sim  434 N(\mu ,\sigma_{x}^{2})$. In \scAR, you are allowed to make such definitions

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N(\mu,\sigma_{x}^{2})$. In \scAR, you are allowed to make such definitions  435 437 through the new parameter type called \texttt{random\_effects\_vector}.  436 438 \index{random effects!random effects vector} (There is also a  437 439 \texttt{random\_effects\_matrix}, which allows you to define a matrix of random  ...... 446 448  a better term.  447 449 448 450  \item The unknown parameters in our model are:$a$,$b$,$\mu$,$\sigma$,  449 $\sigma_{x}$, and$x_{1},\ldots ,x_{n}$. We have agreed to call  451 $\sigma_{x}$, and$x_{1},\ldots,x_{n}$. We have agreed to call  450 452 $x_{1},\ldots,x_{n}$random effects.'' The rest of the parameters are called  451 453  hyper-parameters.'' Note that we place no prior distribution on the  452 454  hyper-parameters.  ...... 573 575 574 576 \section{The flexibility of \scAR\label{lognormal}}  575 577 576 Say that you doubt the distributional assumption$x_i\sim N(\mu ,\sigma

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Say that you doubt the distributional assumption $x_i\sim N(\mu,\sigma  577 579 _{x}^{2})$ made in \texttt{simple.tpl}, and that you want to check if a skewed

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distribution gives a better fit. You could, for instance,~take

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\[

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\section{The random effects distribution (prior)}

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In \texttt{simple.tpl}, we declared $x_{1},\ldots ,x_{n}$ to be of type

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In \texttt{simple.tpl}, we declared $x_{1},\ldots,x_{n}$ to be of type

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