Index: trunk/docs/manuals/autodif/autodif.tex
===================================================================
--- trunk/docs/manuals/autodif/autodif.tex (revision 1072)
+++ trunk/docs/manuals/autodif/autodif.tex (revision 1073)
@@ -15,6 +15,7 @@
\newcommand\bs{$\backslash$}
\newcommand\adtypes{\texttt{dvariable}, \texttt{dvar\_array},
\texttt{dvar\_matrix}, \texttt{dvector}, and \texttt{dmatrix}}
+\newcommand\admbversion{11.1}
% Set the plot symbol size for PicTeX pictures.
\renewcommand\setplotsymbol{\fiverm .}
@@ -59,7 +60,8 @@
A \cplus\ Array Language Extension\\
with Automatic Differentiation\\
for Use in Nonlinear Modeling and Statistics}
- \vspace{4ex}\textsf{\textit{Version 11.1~~(2013-05-01)}}\vspace{3ex}
+ \vspace{4ex}\textsf{\textit{Version \admbversion~~(2013-05-01)\\[3pt]
+ Revised manual~~(2013-06-24)}}\vspace{3ex}
}
% Author definition.
\author{\textsf{\textit{David Fournier}}}
@@ -77,11 +79,11 @@
~\vfill
\noindent ADMB Foundation, Honolulu.\\\\
-\noindent This is the manual for AUTODIF version 10.0.\\\\
-\noindent Copyright \copyright\ 1991, 1992, 1993, 1994, 1996, 2000, 2008, 2011
-David Fournier\\\\
+\noindent This is the manual for AUTODIF version \admbversion.\\\\
+\noindent Copyright \copyright\ 1991, 1992, 1993, 1994, 1996, 2000, 2008, 2011,
+2013 David Fournier\\\\
\noindent The latest edition of the manual is available at:\\
-\url{http://admb-project.org/documentation/manuals/admb-user-manuals}
+\url{http://admb-project.org/documentation/manuals}
% We want a table of contents.
\tableofcontents
Index: trunk/docs/manuals/admb/admb.tex
===================================================================
--- trunk/docs/manuals/admb/admb.tex (revision 1072)
+++ trunk/docs/manuals/admb/admb.tex (revision 1073)
@@ -26,6 +26,7 @@
\newcommand\Pone{P_{t|t-1}}
\newcommand\diag{\textrm{diag}}
\newcommand\ep{\textrm{elem\_prod}}
+\newcommand\admbversion{11.1}
\makeindex
@@ -35,7 +36,8 @@
\smalltitlepart{An Introduction to}
\largetitlepart{AD MODEL BUILDER}
\smalltitlepart{for Use in Nonlinear Modeling and Statistics}
- \vspace{3ex}\textsf{\textit{Version 11.1~~(2013-05-01)}}\vspace{3ex}
+ \vspace{3ex}\textsf{\textit{Version \admbversion~~(2013-05-01)\\[3pt]
+ Revised manual~~(2013-06-24)}}\vspace{3ex}
}
\author{\textsf{\textit{David Fournier}}}
\manualname{AD Model Builder}
@@ -44,11 +46,12 @@
~\vfill
\noindent ADMB Foundation, Honolulu.\\\\
-\noindent This is the manual for AD Model Builder (ADMB) version 10.0.\\\\
+\noindent This is the manual for AD Model Builder (ADMB) version
+\admbversion.\\\\
\noindent Copyright \copyright\ 1993, 1994, 1996, 2000, 2001, 2004, 2007, 2008,
-2011 David Fournier\\\\
+2011, 2013 David Fournier\\\\
\noindent The latest edition of the manual is available at:\\
-\url{http://admb-project.org/documentation/manuals/admb-user-manuals}
+\url{http://admb-project.org/documentation/manuals}
\tableofcontents
@@ -163,8 +166,9 @@
render the estimation of parameters in such nonlinear models more tractable. The
\ADMS package is intended to organize these techniques in such a way that they
are easy to employ (where possible, employing them in a way that the user does
-not need to be aware of them), so that investigating nonlinear statistical models
-becomes---so far as possible---as simple as for linear statistical models.
+not need to be aware of them), so that investigating nonlinear statistical
+models becomes---so far as possible---as simple as for linear statistical
+models.
\section{Installing the software}
@@ -3052,8 +3056,8 @@
$\infty$ for this example). The integer argument \texttt{nsteps} determines how
accurate the integration will be. Higher values of \texttt{nsteps} will be more
accurate, but greatly increase the amount of time necessary to fit the model.
-The basic strategy is to use a moderate value for \texttt{nsteps}, such as~6, and
-then to increase this value to see if the parameter estimates change much.
+The basic strategy is to use a moderate value for \texttt{nsteps}, such as~6,
+and then to increase this value to see if the parameter estimates change much.
\begin{lstlisting}
FUNCTION dvariable h(const dvariable& z)
\end{lstlisting}
@@ -3423,9 +3427,9 @@
default behavior of \ADM\ is to read in initial parameter values for the
parameters from a \texttt{PAR} file, if it finds one. Otherwise, they are given
default values consistent with their type. The quantity~\texttt{f} is a vector
-of four coefficients for the autoregressive process. \texttt{Pcoff} is a $2\times
-2$ matrix used to parameterize the transition matrix \texttt{P} for the Markov
-process. Its values are restricted to lie between~$0.01$ and~$0.99$.
+of four coefficients for the autoregressive process. \texttt{Pcoff} is a
+$2\times 2$ matrix used to parameterize the transition matrix \texttt{P} for the
+Markov process. Its values are restricted to lie between~$0.01$ and~$0.99$.
\texttt{smult} is a number used to parameterize \texttt{sigma} and \texttt{var}
(which is the variance) as a multiple of the mean-squared residuals. This
reparameterization undimensionalizes the calculation and is a good technique to
@@ -3616,9 +3620,9 @@
interest. The matrix~\texttt{z} is calculated using a transformed matrix,
because \ADM\ deals with vector rows better than columns. The probability
distribution for the states in period~1, \texttt{qb1}, is set equal to the
-unconditional distribution for a Markov process in terms of its transition matrix
-\texttt{P}, as discussed in~\cite{hamilton1994}. The transition matrix is used
-to compute the probability distribution of the states in periods $(2,1)$,
+unconditional distribution for a Markov process in terms of its transition
+matrix \texttt{P}, as discussed in~\cite{hamilton1994}. The transition matrix is
+used to compute the probability distribution of the states in periods $(2,1)$,
$(3,2,1)$, $(4,3,2,1)$, and finally, $(5,4,3,2,1)$. For the last quintuplet,
this is the probability distribution before observing~\texttt{y(5)}. The
quantities \texttt{eps} in the code correspond to the possible realized values
Index: trunk/docs/manuals/admb-re/admbre.tex
===================================================================
--- trunk/docs/manuals/admb-re/admbre.tex (revision 1072)
+++ trunk/docs/manuals/admb-re/admbre.tex (revision 1073)
@@ -16,6 +16,7 @@
\newcommand{\scGLM}{\textsc{glm}}
\newcommand{\scGLMM}{\textsc{glmm}}
\newcommand{\scLIDAR}{\textsc{lidar}}
+\newcommand\admbversion{11.1}
\makeindex
@@ -24,7 +25,8 @@
\title{%
\largetitlepart{Random Effects in\\ \ADM}
\smalltitlepart{ADMB-RE User Guide}
- \vspace{4.5ex}\textsf{\textit{Version 11.1~~(2013-05-01)}}\vspace{3ex}
+ \vspace{4.5ex}\textsf{\textit{Version \admbversion~~(2013-05-01)\\[3pt]
+ Revised manual~~(2013-06-24)}}\vspace{3ex}
}
% Author definition.
\author{\textsf{\textit{Hans Skaug \& David Fournier}}}
@@ -35,11 +37,11 @@
~\vfill
\noindent ADMB Foundation, Honolulu.\\\\
\noindent This is the manual for AD Model Builder with Random Effects (ADMB-RE)
-version 11.1.\\\\
-\noindent Copyright \copyright\ 2004, 2006, 2008, 2009, 2011 Hans Skaug \& David
+version \admbversion.\\\\
+\noindent Copyright \copyright\ 2004, 2006, 2008, 2009, 2011, 2013 Hans Skaug \& David
Fournier\\\\
\noindent The latest edition of the manual is available at:\\
-\url{http://admb-project.org/documentation/manuals/admb-user-manuals}
+\url{http://admb-project.org/documentation/manuals}
\tableofcontents
@@ -140,8 +142,8 @@
\item Hyper-parameters (variance components, etc.) estimated by maximum
likelihood.
- \item Marginal likelihood evaluated by the Laplace approximation, (adaptive) importance
- sampling or Gauss-Hermite integration.
+ \item Marginal likelihood evaluated by the Laplace approximation, (adaptive)
+ importance sampling or Gauss-Hermite integration.
\item Exact derivatives calculated using Automatic Differentiation.
@@ -385,8 +387,8 @@
term borrowed from Bayesian statistics.
A central concept that originates from generalized linear models is that of a
-``linear predictor.'' Let $x_{1},\ldots ,x_{p}$ denote observed covariates
-(explanatory variables), and let $\beta _{1},\ldots ,\beta _{p}$ be the
+``linear predictor.'' Let $x_{1},\ldots,x_{p}$ denote observed covariates
+(explanatory variables), and let $\beta _{1},\ldots,\beta _{p}$ be the
corresponding regression parameters to be estimated. Many of the examples in
this manual involve a linear predictor $\eta_{i}=\beta_{1}x_{1,i}+\cdots
+\beta_{p}x_{p,i}$, which we also will write in vector form as
@@ -414,7 +416,7 @@
exemplify the use of random effects. The statistical model underlying this
example is the simple linear regression
\[
-Y_i=ax_i+b+\varepsilon_i,\qquad i=1,\ldots ,n,
+Y_i=ax_i+b+\varepsilon_i,\qquad i=1,\ldots,n,
\]
where $Y_i$ and $x_i$ are the data, $a$ and $b$ are the unknown parameters to be
estimated, and $\varepsilon_i\sim N(0,\sigma ^{2})$ is an error term.
@@ -431,7 +433,7 @@
know the value of $\sigma_e$, so we shall pretend that~$\sigma_e=0.5$.
Because $x_i$ is not observed, we model it as a random effect with $x_i\sim
-N(\mu ,\sigma_{x}^{2})$. In \scAR, you are allowed to make such definitions
+N(\mu,\sigma_{x}^{2})$. In \scAR, you are allowed to make such definitions
through the new parameter type called \texttt{random\_effects\_vector}.
\index{random effects!random effects vector} (There is also a
\texttt{random\_effects\_matrix}, which allows you to define a matrix of random
@@ -446,7 +448,7 @@
a better term.
\item The unknown parameters in our model are: $a$, $b$, $\mu$, $\sigma$,
- $\sigma_{x}$, and $x_{1},\ldots ,x_{n}$. We have agreed to call
+ $\sigma_{x}$, and $x_{1},\ldots,x_{n}$. We have agreed to call
$x_{1},\ldots,x_{n}$ ``random effects.'' The rest of the parameters are called
``hyper-parameters.'' Note that we place no prior distribution on the
hyper-parameters.
@@ -573,7 +575,7 @@
\section{The flexibility of \scAR\label{lognormal}}
-Say that you doubt the distributional assumption $x_i\sim N(\mu ,\sigma
+Say that you doubt the distributional assumption $x_i\sim N(\mu,\sigma
_{x}^{2})$ made in \texttt{simple.tpl}, and that you want to check if a skewed
distribution gives a better fit. You could, for instance,~take
\[
@@ -658,11 +660,11 @@
\section{The random effects distribution (prior)}
-In \texttt{simple.tpl}, we declared $x_{1},\ldots ,x_{n}$ to be of type
+In \texttt{simple.tpl}, we declared $x_{1},\ldots,x_{n}$ to be of type
\texttt{random\_effects\_vector}. This statement tells \scAB\ that $x_{1},\ldots
,x_{n}$ should be treated as random effects (i.e., be the targets for the
Laplace approximation), but it does not say anything about what distribution the
-random effects should have. We assumed that $x_i\sim N(\mu ,\sigma_{x}^{2})$,
+random effects should have. We assumed that $x_i\sim N(\mu,\sigma_{x}^{2})$,
and (without saying it explicitly) that the $x_i$s were statistically
independent. We know that the corresponding prior contribution to the
log-likelihood is
@@ -683,7 +685,7 @@
wrongly specified. The following trick can make the code easier to read, and has
the additional advantage of being numerically stable for small values of
$\sigma_{x}$. From basic probability theory, we know that if $u\sim N(0,1)$,
-then $x=\sigma_{x}u+\mu$ will have a $N(\mu ,\sigma_{x}^{2})$ distribution. The
+then $x=\sigma_{x}u+\mu$ will have a $N(\mu,\sigma_{x}^{2})$ distribution. The
corresponding \scAB\ code would be
\begin{lstlisting}
f += - 0.5*norm2(u);
@@ -872,9 +874,9 @@
\section{Built-in data likelihoods}
In the simple \texttt{simple.tpl}, the mathematical expressions for all
-log-likelihood contributions where written out in full detail. You may have hoped
-that for the most common probability distributions, there were functions written
-so that you would not have to remember or look up their log-likelihood
+log-likelihood contributions where written out in full detail. You may have
+hoped that for the most common probability distributions, there were functions
+written so that you would not have to remember or look up their log-likelihood
expressions. If your density is among those given in
Table~\ref{tab:distributions}, you are lucky. More functions are likely to be
implemented over time, and user contributions are welcomed!
@@ -1042,7 +1044,7 @@
penalized-likelihood version. In \texttt{simple.tpl}, we would simply redefine
the random effects vector \texttt{x} to be of type \texttt{init\_vector}. The
parameters would then be $a$, $b$, $\mu $, $\sigma $, $\sigma_{x}$, and
-$x_{1}$,\ldots , $x_{n}$. It is not recommended, or even possible, to estimate
+$x_{1}$,\ldots, $x_{n}$. It is not recommended, or even possible, to estimate
all of these simultaneously, so you should fix~$\sigma_{x}$ (by giving it a
phase~$-\kern-.1em1$) at some reasonable value. The actual value at which you
fix~$\sigma_{x}$ is not critically important, and you could even try a range of
@@ -1074,7 +1076,7 @@
boundaries usually makes one feel uncomfortable, but with random effects, the
interpretation of $\sigma_{x}=0$ is clear and unproblematic. All it really means
is that data do not support a random effect, and the natural consequence is to
-remove (or inactivate) $x_{1},\ldots ,x_{n}$, together with the corresponding
+remove (or inactivate) $x_{1},\ldots,x_{n}$, together with the corresponding
prior (and hence $\sigma_{x}$), from the model.
\section{\scMCMC}
@@ -1254,12 +1256,12 @@
If your model has special structure, such as
grouped or nested random effects, state-space structure,
crossed random effects or a general Markov strucure
-you will benefit greatly from using the techniques
+you will benefit greatly from using the techniques
described in Section~\ref{separability} below.
In this case the memory options in Table~\ref{tab:temporary-files}
are less relevant (although sometimes useful), and instead the
-memory use can be controlled with the classical \ADM~command line options
-\texttt{-cbs}, \texttt{-gbs} etc.
+memory use can be controlled with the classical \ADM~command line options
+\texttt{-cbs}, \texttt{-gbs} etc.
\subsection{Limited memory Newton optimization}
\index{limited memory quasi-Newton}
@@ -1277,29 +1279,30 @@
A model is said to be ``separable'' if the likelihood can be written
as a product of terms, each involving only a small number of random effects.
Not all models are separable, and for small toy examples (less than 50~random
-effects, say), we do not need to care about separability. You
-need to care about separability both to reduce memory requirements and
-computation time. Examples of separable models are
+effects, say), we do not need to care about separability. You
+need to care about separability both to reduce memory requirements and
+computation time. Examples of separable models are
\begin{itemize}
\item Grouped or nested random effects
\item State-space models
\item Crossed random effects
\item Latent Markov random fields
\end{itemize}
-The presence of separability allows \scAB\ to calculate the ``Hessian'' matrix
-very efficiently.The Hessian~$H$ is defined as
-the (negative) Fisher information matrix (inverse covariance matrix) of the
-posterior distribution of the random~effects , and is a key component of the Laplace
-approximation.
+The presence of separability allows \scAB\ to calculate the ``Hessian'' matrix
+very efficiently.The Hessian~$H$ is defined as the (negative) Fisher information
+matrix (inverse covariance matrix) of the posterior distribution of the
+random~effects, and is a key component of the Laplace approximation.
-How do we inform \scAR\ that the model is separable? We define \texttt{SEPARABLE\_FUNCTION}s
-in the \texttt{PROCEDURE\_SECTION} to specify the individual terms in the
-product that defines the likelihood function. Typically, a \texttt{SEPARABLE\_FUNCTION}
-is invoked many times, with a small subset of the random effects each time.
+How do we inform \scAR\ that the model is separable? We define
+\texttt{SEPARABLE\_FUNCTION}s in the \texttt{PROCEDURE\_SECTION} to specify the
+individual terms in the product that defines the likelihood function. Typically,
+a \texttt{SEPARABLE\_FUNCTION} is invoked many times, with a small subset of the
+random effects each time.
-For separable models the Hessian is a sparse matrix which means that it contains mostly zeros.
-Sparsity can be exploited by \scAB\ when manipulating the matrix $H$,
-such as calculating its determinant. The actual sparsity pattern depends on the model type:
+For separable models the Hessian is a sparse matrix which means that it contains
+mostly zeros. Sparsity can be exploited by \scAB\ when manipulating the matrix
+$H$, such as calculating its determinant. The actual sparsity pattern depends on
+the model type:
\begin{itemize}
\item \textit{Grouped or nested random effects:} $H$ is block diagonal.
\item \textit{State-space models:} $H$ is a banded matrix with a narrow band.
@@ -1307,25 +1310,27 @@
\item \textit{Latent Markov random fields:} often banded, but with a wide
band.
\end{itemize}
-For block diagonal and banded $H$, \scAR\ automatically will detect the structure from
-the \texttt{SEPARABLE\_FUNCTION} specification, and will print out a message such as:
+For block diagonal and banded $H$, \scAR\ automatically will detect the
+structure from the \texttt{SEPARABLE\_FUNCTION} specification, and will print
+out a message such as:
\begin{lstlisting}
Block diagonal Hessian (Block size = 3)
\end{lstlisting}
at the beginning of the phase when the random effects are becoming active
parameters. For general sparsity pattern the command line option \texttt{-shess}
-can be used to invoke the sparse matrix libraries for manipulation of the matrix $H$.
+can be used to invoke the sparse matrix libraries for manipulation of the matrix
+$H$.
\section{The first example}
A simple example is the one-way variance component model
\[
y_{ij}=\mu +\sigma_u u_{i}+\varepsilon_{ij},
- \qquad i=1,\ldots ,q,\quad j=1,\ldots ,n_{i}
+ \qquad i=1,\ldots,q,\quad j=1,\ldots,n_{i}
\]
where $u_{i}\sim N(0,1)$ is a random effect and $\varepsilon_{ij}\sim
-N(0,\sigma^2)$ is an error term. The straightforward (non-separable) implementation of this
-model (shown only in part)~is
+N(0,\sigma^2)$ is an error term. The straightforward (non-separable)
+implementation of this model (shown only in part)~is
\begin{lstlisting}
PARAMETER_SECTION
random_effects_vector u(1,q)
@@ -1402,13 +1407,13 @@
There are other rules that have to be obeyed:
\begin{itemize}
-\item[$\bigstar$]
- No calculations of the log-likelihood, except calling \texttt{SEPARABLE\_FUNCTION},
- are allowed in \texttt{PROCEDURE\_SECTION}. Hence, the only allowed
- use of parameters defined in \texttt{PARAMETER\_SECTION} is to pass them
- as arguments to \texttt{SEPARABLE\_FUNCTION}s. However,
- evaluation of \texttt{sdreport} numbers during the \texttt{sd\_phase},
- as well as MCMC calculations, are allowed.
+ \item[$\bigstar$] No calculations of the log-likelihood, except calling
+ \texttt{SEPARABLE\_FUNCTION}, are allowed in \texttt{PROCEDURE\_SECTION}.
+ Hence, the only allowed use of parameters defined in
+ \texttt{PARAMETER\_SECTION} is to pass them as arguments to
+ \texttt{SEPARABLE\_FUNCTION}s. However, evaluation of \texttt{sdreport}
+ numbers during the \texttt{sd\_phase}, as well as MCMC calculations, are
+ allowed.
\end{itemize}
This rule implies that all the action has to take place inside the
\texttt{SEPARABLE\_FUNCTION}s. To minimize the number of parameters that have be
@@ -1440,7 +1445,7 @@
following model:
\[
y_{ijk}= \sigma_v v_{i} + \sigma_u u_{ij}+\varepsilon_{ijk},
- \qquad i=1,\ldots ,q,\quad j=1,\ldots ,m,\quad k=1,\ldots ,n_{ij},
+ \qquad i=1,\ldots,q,\quad j=1,\ldots,m,\quad k=1,\ldots,n_{ij},
\]
where the random effects $v_{i}$ and $u_{ij}$ are independent
$N(0,1)$~distributed, and $\varepsilon_{ijk}\sim N(0,\sigma^2)$ is still the
@@ -1461,11 +1466,11 @@
for(i=1;i<=q;i++)
g_cluster(v(i),u(i),sigma,sigma_u,sigma_v,i);
\end{lstlisting}
-Each element of \texttt{v} and each row (\texttt{u(i)}) of the matrix \texttt{u}
-are passed only once to the separable function \texttt{g\_cluster}.
-This is the criterion \scAB\ uses to detect the block diagonal Hessian structure.
-Note that \texttt{v(i)} is passed as a single value while \texttt{u(i)}
-is passed as a vector to the \texttt{SEPARABLE\_FUNCTION} as follows:
+Each element of \texttt{v} and each row (\texttt{u(i)}) of the matrix \texttt{u}
+are passed only once to the separable function \texttt{g\_cluster}. This is the
+criterion \scAB\ uses to detect the block diagonal Hessian structure. Note that
+\texttt{v(i)} is passed as a single value while \texttt{u(i)} is passed as a
+vector to the \texttt{SEPARABLE\_FUNCTION} as follows:
\begin{lstlisting}
SEPARABLE_FUNCTION void g_cluster(const dvariable& v,const dvar_vector& u,...)
g -= -0.5*square(v);
@@ -1516,17 +1521,18 @@
\index{Gauss-Hermite quadrature}
In the situation where the model is separable of type ``Block diagonal
-Hessian,'' (see Section~\ref{separability}),
-Gauss-Hermite quadrature is available as an option to improve upon the Laplace
-approximation. It is invoked with the command line option \texttt{-gh N}, where \texttt{N}
-is the number of quadrature points determining the accuracy of the integral approximation.
-For a block size of~1, the default choice should be $N=10$ or greater, but for larger
-block sizes the computational and memory requirements very quickly limits the range
-of $N$. If $N$ is chosen too large \scAR\ will crash witout giving a meaningful
-error message. To avoid \scAR\ creating large temporary files the command
-line options \texttt{-cbs} and \texttt{-gbs} can be used.
+Hessian,'' (see Section~\ref{separability}), Gauss-Hermite quadrature is
+available as an option to improve upon the Laplace approximation. It is invoked
+with the command line option \texttt{-gh N}, where \texttt{N} is the number of
+quadrature points determining the accuracy of the integral approximation. For a
+block size of~1, the default choice should be $N=10$ or greater, but for larger
+block sizes the computational and memory requirements very quickly limits the
+range of $N$. If $N$ is chosen too large \scAR\ will crash witout giving a
+meaningful error message. To avoid \scAR\ creating large temporary files the
+command line options \texttt{-cbs} and \texttt{-gbs} can be used.
-The \texttt{-gh N} option should be preferred over importance sampling (\texttt{-is}).
+The \texttt{-gh N} option should be preferred over importance sampling
+(\texttt{-is}).
%\subsection{Frequency weighting for multinomial likelihoods}
@@ -1540,7 +1546,7 @@
%\end{equation*}
%where $\mu$ is a parameter and $u_{i}\sim N(0,\sigma ^{2})$ is a random effect.
%For independent observations $y_1,\ldots,y_n$, the log-likelihood function for
-%the parameter $\theta =(\mu ,\sigma )$ can be written as
+%the parameter $\theta =(\mu,\sigma )$ can be written as
%\begin{equation}
% l(\theta )=\sum_{i=1}^{n}\log \bigl[\, p(x_{i};\theta )\bigr] .
%\end{equation}
@@ -1565,8 +1571,8 @@
%PARAMETER_SECTION
% !! set_multinomial_weights(w);
%\end{lstlisting}
-%In addition, it is necessary to explicitly multiply the likelihood contributions
-%in~(\ref{l_w}) by~$w$. The program must be written with
+%In addition, it is necessary to explicitly multiply the likelihood
+%contributions in~(\ref{l_w}) by~$w$. The program must be written with
%\texttt{SEPARABLE\_FUNCTION}, as explained in Section~\ref{sec:nested}. For the
%likelihood~(\ref{l_w}), the \texttt{SEPARABLE\_FUNCTION} will be invoked three
%times.
@@ -2155,7 +2161,7 @@
\end{equation}%
where $\mu $ is a parameter and $u_{i}\sim N(0,\sigma ^{2})$ is a random effect.
Assuming independence, the log-likelihood function for the parameter $\theta
-=(\mu ,\sigma )$ can be written as
+=(\mu,\sigma )$ can be written as
\begin{equation}
l(\theta )=\sum_{i=1}^{n}\log \bigl[\, p(x_{i};\theta )\bigr] .
\end{equation}%
@@ -2163,7 +2169,7 @@
However, since $x_{i}$ only can take the values~$0$, $1$, and~$2$, we can
rewrite the log-likelihood as
\begin{equation}
-l(\theta )=\sum_{j=0}^{2}n_{j}\log \bigl[\, p(j;\theta )\bigr] , \label{l_w}
+l(\theta )=\sum_{j=0}^{2}n_{j}\log \bigl[\, p(j;\theta )\bigr], \label{l_w}
\end{equation}%
where $n_{j}$ is the number $x_{i}$ being equal to~$j$. Still, the Laplace
approximation must be used to approximate $p(j;\theta )$, but now only for
@@ -2320,8 +2326,8 @@
\subsection{Multilevel Rasch model}
-The multilevel Rasch model can be implemented using random effects in \scAB. As an
-example, we use data on the responses of 2042~soldiers to a total of 19~items
+The multilevel Rasch model can be implemented using random effects in \scAB. As
+an example, we use data on the responses of 2042~soldiers to a total of 19~items
(questions), taken from~\cite{doran2007estimating} %Doran et al.\ (2007).
This illustrates the use of crossed random effects in \scAB. Furthermore, it is
shown how the model easily can be generalized in \scAB. These more general