Index: admbre.tex
===================================================================
 admbre.tex (revision 876)
+++ admbre.tex (revision 877)
@@ 739,7 +739,7 @@
PROCEDURE_SECTION
 int k=0;
+ int k=1;
L(1,1) = 1.0;
for(i=2;i<=4;i++)
{
@@ 872,7 +872,7 @@
\section{Builtin data likelihoods}
In the simple \texttt{simple.tpl}, the mathematical expressions for all
loglikehood contributions where written out in full detail. You may have hoped
+loglikelihood 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 loglikelihood
expressions. If your density is among those given in
@@ 1502,57 +1502,57 @@
\index{GaussHermite quadrature}
In the situation where the model is separable of type ``Block diagonal
Hessian,'' with only a single random effect in each block (see
Section~\ref{separability}), GaussHermite quadrature is available as an option
+Hessian,'' (see Section~\ref{separability}),
+GaussHermite quadrature is available as an option
to the Laplace approximation and to the \texttt{is} (importance sampling)
option. It is invoked with command line option \texttt{gh N}, where \texttt{N}
is the number of quadrature points.
\subsection{Frequency weighting for multinomial likelihoods}
+%\subsection{Frequency weighting for multinomial likelihoods}
+%
+%In situations where the response variable only can take on a finite number of
+%different values, it is possibly to reduce the computational burden enormously.
+%As an example, consider a situation where observation $y_{i}$ is binomially
+%distributed with parameters $N=2$ and $p_{i}$. Assume that
+%\begin{equation*}
+% p_{i}=\frac{\exp (\mu +u_{i})}{1+\exp (\mu +u_{i})},
+%\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 loglikelihood function for
+%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}
+%In \scAR, $p(x_{i};\theta)$ is approximated using the Laplace approximation.
+%However, since $y_i$ only can take the values~$0$, $1$, and~$2$, we can rewrite
+%the loglikelihood~as
+%$$
+%l(\theta )=\sum_{j=0}^{2}n_{j}\log \left[ p(j;\theta )\right],
+%$$
+%where $n_j$ is the number $y_i$s being equal to $j$. Still, the Laplace
+%approximation must be used to approximate $p(j;\theta )$, but now only for
+%$j=0,1,2$, as opposed to $n$~times above. For large~$n$, this can give large a
+%large reduction in computing time.
+%
+%To implement the weighted loglikelihood~(\ref{l_w}), we define a weight vector
+%$(w_1,w_2,w_3)=(n_{0},n_{1},n_{2})$. To read the weights from file, and to tell
+%\scAR\ that~\texttt{w} is a weights vector, the following code is used:
+%\begin{lstlisting}
+%DATA_SECTION
+% init_vector w(1,3)
+%
+%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
+%\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.
+%
+%See a full example
+%\href{http://www.otterrsch.com/admbre/examples/weights/weights.html}{here}.
In situations where the response variable only can take on a finite number of
different values, it is possibly to reduce the computational burden enormously.
As an example, consider a situation where observation $y_{i}$ is binomially
distributed with parameters $N=2$ and $p_{i}$. Assume that
\begin{equation*}
 p_{i}=\frac{\exp (\mu +u_{i})}{1+\exp (\mu +u_{i})},
\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 loglikelihood function for
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}
In \scAR, $p(x_{i};\theta)$ is approximated using the Laplace approximation.
However, since $y_i$ only can take the values~$0$, $1$, and~$2$, we can rewrite
the loglikelihood~as
$$
l(\theta )=\sum_{j=0}^{2}n_{j}\log \left[ p(j;\theta )\right],
$$
where $n_j$ is the number $y_i$s being equal to $j$. Still, the Laplace
approximation must be used to approximate $p(j;\theta )$, but now only for
$j=0,1,2$, as opposed to $n$~times above. For large~$n$, this can give large a
large reduction in computing time.

To implement the weighted loglikelihood~(\ref{l_w}), we define a weight vector
$(w_1,w_2,w_3)=(n_{0},n_{1},n_{2})$. To read the weights from file, and to tell
\scAR\ that~\texttt{w} is a weights vector, the following code is used:
\begin{lstlisting}
DATA_SECTION
 init_vector w(1,3)

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
\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.

See a full example
\href{http://www.otterrsch.com/admbre/examples/weights/weights.html}{here}.

\section{Statespace models: banded $H$}
\label{sec:statespace}\index{statespace models}
@@ 1819,7 +1819,7 @@
\item Since equation~(\ref{eqn:first_order}) does not penalize the mean
of~$\mathbf{u}$, we impose the restriction that $\sum_{k=1}u_{k}=0$ (see
\texttt{union.tpl} for details). Without this restriction, the model would be
 overparameterized, since we allready have an overall mean~$\mu $ in
+ overparameterized, since we already have an overall mean~$\mu $ in
equation~(\ref{eqn:gam}).
\item To speed up computations, the parameter $\mu $ (and other regression
@@ 1830,7 +1830,7 @@
\begin{figure}[h]
\centering\hskip1pt
\includegraphics[width=6in]{union_fig.pdf}
\caption{Probablity of membership as a function of covariates. In each plot,
+\caption{Probability of membership as a function of covariates. In each plot,
the remaining covariates are fixed at their sample means. The effective
degrees of freedom (df) are also given \protect\cite{hast:tibs:1990}.}
\label{fig:union}
@@ 2127,7 +2127,7 @@
\subsubsection{Model description}
Let $X_{i}$ be binomially distributed with paramters $N=2$ and $p_{i}$, and
+Let $X_{i}$ be binomially distributed with parameters $N=2$ and $p_{i}$, and
further assume that
\begin{equation}
p_{i}=\frac{\exp (\mu +u_{i})}{1+\exp (\mu +u_{i})},
@@ 2299,7 +2299,7 @@
\subsection{Multilevel Rasch model}
The multilevel Rasch model can be implented using random effects in \scAB. As an
+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