Revision 1073 trunk/docs/manuals/admbre/admbre.tex
admbre.tex (revision 1073)  

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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{ADMBRE User Guide} 
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\vspace{4.5ex}\textsf{\textit{Version 11.1~~(20130501)}}\vspace{3ex} 

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\vspace{4.5ex}\textsf{\textit{Version \admbversion~~(20130501)\\[3pt] 

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Revised manual~~(20130624)}}\vspace{3ex} 

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} 
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% Author definition. 
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\author{\textsf{\textit{Hans Skaug \& David Fournier}}} 
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~\vfill 
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\noindent ADMB Foundation, Honolulu.\\\\ 
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\noindent This is the manual for AD Model Builder with Random Effects (ADMBRE) 
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version 11.1.\\\\


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\noindent Copyright \copyright\ 2004, 2006, 2008, 2009, 2011 Hans Skaug \& David 

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version \admbversion.\\\\


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\noindent Copyright \copyright\ 2004, 2006, 2008, 2009, 2011, 2013 Hans Skaug \& David


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Fournier\\\\ 
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\noindent The latest edition of the manual is available at:\\ 
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\url{http://admbproject.org/documentation/manuals/admbusermanuals}


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\url{http://admbproject.org/documentation/manuals} 

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\tableofcontents 
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\item Hyperparameters (variance components, etc.) estimated by maximum 
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likelihood. 
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\item Marginal likelihood evaluated by the Laplace approximation, (adaptive) importance


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sampling or GaussHermite integration.


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\item Marginal likelihood evaluated by the Laplace approximation, (adaptive) 

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importance sampling or GaussHermite integration.


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\item Exact derivatives calculated using Automatic Differentiation. 
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term borrowed from Bayesian statistics. 
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A central concept that originates from generalized linear models is that of a 
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``linear predictor.'' Let $x_{1},\ldots ,x_{p}$ denote observed covariates


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(explanatory variables), and let $\beta _{1},\ldots ,\beta _{p}$ be the


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``linear predictor.'' Let $x_{1},\ldots,x_{p}$ denote observed covariates 

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(explanatory variables), and let $\beta _{1},\ldots,\beta _{p}$ be the 

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corresponding regression parameters to be estimated. Many of the examples in 
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this manual involve a linear predictor $\eta_{i}=\beta_{1}x_{1,i}+\cdots 
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+\beta_{p}x_{p,i}$, which we also will write in vector form as 
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exemplify the use of random effects. The statistical model underlying this 
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example is the simple linear regression 
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\[ 
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Y_i=ax_i+b+\varepsilon_i,\qquad i=1,\ldots ,n,


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Y_i=ax_i+b+\varepsilon_i,\qquad i=1,\ldots,n, 

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\] 
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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 $x_i\sim 
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N(\mu ,\sigma_{x}^{2})$. 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 

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through the new parameter type called \texttt{random\_effects\_vector}. 
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\index{random effects!random effects vector} (There is also a 
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\texttt{random\_effects\_matrix}, which allows you to define a matrix of random 
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a better term. 
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\item The unknown parameters in our model are: $a$, $b$, $\mu$, $\sigma$, 
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$\sigma_{x}$, and $x_{1},\ldots ,x_{n}$. We have agreed to call


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$\sigma_{x}$, and $x_{1},\ldots,x_{n}$. We have agreed to call 

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$x_{1},\ldots,x_{n}$ ``random effects.'' The rest of the parameters are called 
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``hyperparameters.'' Note that we place no prior distribution on the 
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hyperparameters. 
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\section{The flexibility of \scAR\label{lognormal}} 
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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 

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_{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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