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%%\VignetteIndexEntry{mvord: An R Package for Fitting Multivariate Ordinal Regression Models}
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\author{Rainer Hirk\\WU Wirtschafts-\\universit\"at Wien
   \And Kurt Hornik\\WU Wirtschafts-\\universit\"at Wien\\
   \And Laura Vana\\WU Wirtschafts-\\universit\"at Wien}
\Plainauthor{Rainer Hirk, Kurt Hornik and Laura Vana}

\title{\pkg{mvord}: An \proglang{R} Package for Fitting Multivariate Ordinal Regression Models}
\Plaintitle{mvord: An R Package for Fitting Multivariate Ordinal Regression Models}
\Shorttitle{\pkg{mvord}: Multivariate Ordinal Regression Models in \proglang{R}}

\Abstract{
  The \proglang{R} package \pkg{mvord} implements composite
  likelihood estimation in the class of multivariate ordinal
  regression models with a multivariate probit and a multivariate
  logit link.  A flexible modeling framework for multiple ordinal
  measurements on the same subject is set up, which takes into
  consideration the dependence among the multiple observations by
  employing different error structures.  Heterogeneity in the error
  structure across the subjects can be accounted for by the package,
  which allows for covariate dependent error structures.  In addition,
  different regression coefficients and threshold parameters for each
  response are supported. If a reduction of the parameter space is
  desired, constraints on the threshold as well as on the regression
  coefficients can be specified by the user.  The proposed
  multivariate framework is illustrated by means of a credit risk
  application.
}

\Keywords{composite likelihood estimation, correlated ordinal data, multivariate ordinal logit regression model, multivariate ordinal probit regression model, \proglang{R}}
\Plainkeywords{composite likelihood estimation, correlated ordinal data, multivariate ordinal logit regression model, multivariate ordinal probit regression model, R}

\Address{
  Rainer Hirk, Kurt Hornik, Laura Vana\\
  Department of Finance, Accounting and Statistics\\
  Institute for Statistics and Mathematics\\
  WU Wirtschaftsuniversit\"at Wien\\
  1020 Vienna, Austria\\
  E-mail: \email{Rainer.Hirk@wu.ac.at}, \email{Kurt.Hornik@wu.ac.at}, \email{Laura.Vana@wu.ac.at}
}

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\begin{document}
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library("mvord")
data("data_cr_panel")
data("data_cr")
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\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Introduction
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Preface}
This vignette corresponds to the article ``\pkg{mvord}: An \proglang{R} Package for Fitting Multivariate Ordinal Regression Models'' which is published in the \emph{Journal of Statistical Software}. When citing the paper and package please use \cite{mvord_JSS} by calling \code{citation("mvord")}.
\section{Introduction}

% Intro
% ------------
The analysis of ordinal data is an important task in various areas of
research. One of the most common settings is the modeling of
preferences or opinions (on a scale from, say, poor to
very good or strongly disagree to strongly
  agree).  The scenarios involved range from psychology (e.g.,
aptitude and personality testing), marketing (e.g., consumer
preferences research) and economics and finance (e.g., credit risk
assessment for sovereigns or firms) to information retrieval (where
documents are ranked by the user according to their relevance) and
medical sciences (e.g., modeling of pain severity or cancer stages).

Most of these applications deal with correlated ordinal data, as
typically multiple ordinal measurements or outcomes are
available for a collection of subjects or objects (e.g., interviewees
answering different questions, different raters assigning credit
ratings to a firm, pain levels being recorded for patients repeatedly
over a period of time, etc.).
% Approach
% ------------
In such a multivariate setting, models which are able to deal with the
correlation in the ordinal outcomes are desired. One possibility is to
employ a multivariate ordinal regression model where the
marginal distribution of the subject errors is assumed to be
multivariate. Other options are the inclusion of random effects in the
ordinal regression model and conditional models
\citep[see, e.g.,][]{fahrmeir2001multivariate}.

% R packages
% ------------
Several ordinal regression models can be employed for the analysis of
ordinal data, with cumulative link models being the most popular ones
\cite[e.g.,][]{Tutz12, ordinal}. Other approaches include
continuation-ratio or adjacent-category models
\cite[e.g.,][]{Agresti02, Agresti10}.  Different packages to analyze
and model ordinal data are available in \proglang{R}
\citep{RCoreTeam}.  For univariate ordinal regression models with
fixed effects the function \code{polr()} of the \pkg{MASS} package
\citep{MASSpkg}, the function \code{clm()} of the \pkg{ordinal}
package \citep{ordinalR}, which supports scale effects as well as
nominal effects, and the function \code{vglm()} of the \pkg{VGAM}
package \citep{VGAMpkg} are available. Another package which accounts
for heteroskedasticity is \pkg{oglmx} \citep{oglmx}. Package
\pkg{ordinalNet} \citep{ordinalNetpkg} offers tools for model
selection by using an elastic net penalty, whereas package
\pkg{ordinalgmifs} \citep{ordinalgmifs} performs variable selection by
using the generalized monotone incremental forward stagewise (GMIFS)
method. Moreover, ordinal logistic models can be fitted by the
functions \code{lms()} and \code{orm()} in package \pkg{rms}
\citep{rmspkg}, while ordinal probit models can be fitted by the
\code{MCMCoprobit()} function in package \pkg{MCMCpack}
\citep{MCMCpackpkg} which uses Markov chain Monte Carlo methods to fit
ordinal probit regression models.

An overview on ordinal regression models in other statistical software
packages like \proglang{Stata} \citep{Stata}, \proglang{SAS}
\citep{SAS} or \proglang{SPSS} \citep{SPSS} is provided by
\cite{Liu09}. These software packages include the \proglang{Stata}
procedure \code{OLOGIT}, the \proglang{SAS} procedure \code{PROC
  LOGISTIC} and the \proglang{SPSS} procedure \code{PLUM} which
perform ordinal logistic regression models. The software procedure
\code{PLUM} additionally includes other link functions like probit,
complementary log-log, cauchit and negative log-log. Ordinal models
for multinomial data are available in the \proglang{SAS} package
\code{PROC GENMOD}, while another implementation of ordinal logistic
regression is available in \proglang{JMP} \citep{JMP}. In
\proglang{Python} \citep{Python}, package \pkg{mord} \citep{mordpkg}
implements ordinal regression methods.

While there are sufficient software tools in \proglang{R} which deal
with the univariate case, the ready-to-use packages for dealing with
the multivariate case fall behind, mainly due to computational
problems or lack of flexibility in the model specification. However,
there are some \proglang{R} packages which support correlated ordinal
data. One-dimensional normally distributed random effects in ordinal
regression can be handled by the \code{clmm()} function of package
\pkg{ordinal} \citep{ordinalR}. Multiple possibly correlated random
effects are implemented in package \pkg{mixor} \citep{mixor}. Note
that this package uses multi-dimensional quadrature methods and
estimation becomes infeasible for increasing dimension of the random
effects. Bayesian multilevel models for ordinal data are implemented
in package \pkg{brms} \citep{brmspkg}. Multivariate ordinal probit
models, where the subject errors are assumed to follow a multivariate
normal distribution with a general correlation matrix, can be
estimated with package \pkg{PLordprob} \citep{PLordprob}, which uses
maximum composite likelihood methods estimation. This package works
well for standard applications but lacks flexibility. For example, the
number of levels of the ordinal responses needs to be equal across all
dimensions, threshold and regression coefficients are the same for all
multiple measurements and the package does not account for missing
observations in the outcome variable.  Polychoric correlations, which
are used to measure association among two ordinal outcomes, can be
estimated by the \code{polychor()} function of package \pkg{polycor}
\citep{polycor}, where a simple bivariate probit model without
covariates is estimated using maximum likelihood estimation.  None of
these packages support at the time of writing covariate dependent
error structures. A package which allows for different error
structures in non-linear mixed effects models is package \pkg{nlme}
\citep{nlme}, even though models dealing with ordinal data are not
supported.

% Motivation
% ----------
The original motivation for this package lies in a credit risk application,
where multiple credit ratings are assigned by various credit rating
agencies (CRAs) to firms over several years. CRAs have an important
role in financial markets, as they deliver subjective assessments or
opinions of an entity's creditworthiness, which are then used by the other players on the
market, such as investors and regulators, in their decision making
process.  Entities are assigned to rating classes by CRAs on an
ordinal scale by using both quantitative and qualitative criteria.
Ordinal credit ratings can be seen as a coarser version of an
underlying continuous latent process, which is related to the ability of the firm to meet its financial obligations. In the literature, this latent variable motivation
has been used in various credit rating models \citep[e.g.,][]{Blume98,
  afonso2009, Alp13, reusens2017sovereign}.

This setting is an example of an application where correlated ordinal
data arise naturally. On the one hand, multiple ratings assigned by
different raters to one firm at the same point in time can be assumed
to be correlated. On the other hand, given the longitudinal dimension
of the data, for each rater, there is serial dependence in the ratings
assigned over several periods.  Moreover, aside from the need of a
model class that can handle correlated ordinal data, additional
flexibility is desired due to the following characteristics of the
problem at hand:
% Firstly
Firstly, there is heterogeneity in the rating methodology. Raters use
different labeling as well as a different number of rating classes.
% Secondly
Secondly, the credit risk measure employed in assessing
creditworthiness can differ among raters (e.g., probability of default
versus recovery in case of default), which leads to heterogeneity in
the covariates, as raters might use different variables in their
rating process and assign different importance to the variables
employed.
% Thirdly
Thirdly, the data have missing values and are unbalanced, as firms can
leave the data set before the end of the observation period due to
various reasons such as default but also because of mergers and
acquisitions, privatizations, etc., or ratings can be withdrawn.
Moreover, there are missings in the multiple ratings, as not all firms
are rated by all raters at each time point.

The scope of the application of multivariate ordinal regression models reaches far beyond credit risk applications. For example, pain severity studies are a popular setting where repeated ordinal measurements occur. A migraine severity study was employed by \cite{Varin09}, where patients recorded their pain severity over some time period. In addition to a questionnaire with personal and clinical information, covariates describing the weather conditions were collected.
%
Another application area constitutes the field of customer satisfaction surveys, where questionnaires with ordinal items are often divided into two separate blocks \citep[e.g.,][]{Pagui15}. A first block contains questions regarding the general importance of some characteristics of a given service, and a second block relates more to the actual satisfaction on the same characteristics. An analysis of the dependence structure between and within the two blocks is of particular interest.
%
Furthermore, in the presence of multi-rater agreement data, where
several raters assign ordinal rankings to different individuals, the
influence of covariates on the ratings can be investigated and an
analysis and a comparison of the rater behavior can be conducted
\citep[e.g.,][]{DeYoreo2017}. In addition to these few examples
mentioned above, the class of multivariate ordinal regression models
implemented in \pkg{mvord} \citep{mvordpkg} can be applied to other
settings where multiple or repeated ordinal observations occur.

% Contribution
% ------------
This paper discusses package \pkg{mvord} for \proglang{R} which aims
at providing a flexible framework for analyzing correlated ordinal
data by means of the class of multivariate ordinal regression models
and which is available from the Comprehensive \proglang{R} Archive
Network (CRAN) at \url{https://CRAN.R-project.org/package=mvord}. In
this model class, each of the ordinal responses is modeled as a
categorized version of an underlying continuous latent variable which
is slotted according to some threshold parameters.  On the latent
scale we assume a linear model for each of the underlying continuous
variables and the existence of a joint distribution for the
corresponding error terms. A common choice for this joint distribution
is the multivariate normal distribution, which corresponds to the
multivariate probit link. We extend the available software in several
directions.  The flexible modeling framework allows imposing
constraints on threshold as well as regression coefficients.  In
addition, various assumptions about the variance-covariance structure
of the errors are supported, by specifying different types of error
structures.  These include a general correlation, a general
covariance, an equicorrelation and an $AR(1)$ error structure. The
general error structures can depend on a categorical covariate, while
in the equicorrelation and $AR(1)$ structures both numerical and
categorical covariates can be employed. Moreover, in addition to the
multivariate probit link, we implement a multivariate logit link for
the class of multivariate ordinal regression models.

%% Structure of the paper
This paper is organized as follows: Section~\ref{sect:model} provides
an overview of the model class and the estimation procedure, including
model specification and identifiability issues.
Section~\ref{sect:implementation} presents the main functions of the
package. A couple of worked examples are given in
Section~\ref{sect:examples}. Section~\ref{sect:conclusion} concludes.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Model class and estimation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Model class and estimation}\label{sect:model}
Multivariate ordinal regression models are an appropriate modeling
choice when a vector of correlated ordinal response variables,
together with covariates, is observed for each unit or subject in the
sample.  The response vector can be composed of different variables,
i.e., multiple measurements on the same subject (e.g., different
credit ratings assigned to a firm by different CRAs,
different survey questions answered by an interviewee, etc.) or
repeated measurements on the same variable at different time points.

In order to introduce the class of multivariate ordinal regression
models considered in this paper, we start with a brief overview on
univariate cumulative link models.

\subsection{Univariate cumulative link models}
Cumulative link models are often motivated by the assumption that the
observed categories $Y_i$ are a categorized version of an underlying latent variable $\widetilde{Y}_i$ with
\begin{align*}
    	\widetilde{Y}_{i} &= \beta_{0} + \bm{x}_{i}^\top
        \bm{\beta} + \epsilon_{i},
\end{align*}
where $\beta_{0}$ is an intercept term, $\bm{x}_{i}$ is a $p \times 1$ vector of covariates, $\bm{\beta} = (\beta_{1},\,
\dots,\, \beta_{p})^\top$ is a vector of regression coefficients and $\epsilon_{i}$ is a mean zero error
term with distribution function $F$. The link between the observed variable $Y_i$ with $K$ categories and the latent variable $\widetilde{Y}_i$ is~given~by:
\begin{align*}
	Y_{i} = r_i \quad \Leftrightarrow \quad \theta_{r_i-1} <
        \widetilde{Y}_{i} \leq \theta_{r_i}, \qquad r_i \in
        \{1,\, \dots,\, K\},
\end{align*}
where
$ - \infty \equiv \theta_{0} < \theta_{1}< \dots < \theta_{K-1} <
\theta_{K} \equiv \infty$ are threshold parameters on the latent scale
\citep[see, e.g.,][]{Agresti10, Tutz12}. In such a setting the ordinal
response variable $Y_i$ follows a multinomial distribution with
parameter $\bm{\pi}_i$. Let denote by $\pi_{ir_i}$ the probability that
observation $i$ falls in category $r_i$. Then the cumulative link model
\citep{Mccullagh80} is specified by:
\begin{align*}
  \Prob(Y_i \leq r_i) = \Prob(\beta_{0} + \bm{x}_{i}^\top
        \bm{\beta} + \epsilon_{i} \leq \theta_{r_i}) = F(\theta_{r_i} - \beta_{0} - \bm{x}_{i}^\top
        \bm{\beta}) = \pi_{i1} + \cdots + \pi_{i{r_i}}.
\end{align*}
Typical choices for the distribution function~$F$ are the normal and the logistic distributions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Multivariate ordinal regression}\label{subsect:model}
Univariate cumulative link models can be extended to a multivariate setting by assuming the existence of several latent variables with a joint error distribution \citep[see, e.g.,][]{Varin09, Bhat10, Pagui15}.
Let $Y_{ij}$ denote an ordinal observation and $\bm x_{ij}$ be a
$p$-dimensional vector of covariates for subject~$i$ and outcome~$j$,
where $i=1,\, \dots,\, n$ and $j\in J_i$, for $J_i$ a subset of all
available outcomes $J$ in the data set. Moreover, we denote by $q=|J|$
and $q_i = |J_i|$ the number of elements in the sets $J$ and $J_i$,
respectively. Following the cumulative link modeling approach, the ordinal response $Y_{ij}$ is assumed to be a
coarser version of a latent continuous variable
$\widetilde Y_{ij}$.  The observable categorical outcome $Y_{ij}$ and
the unobservable latent variable $\widetilde Y_{ij}$ are connected by:
\begin{align*}
  Y_{ij} = r_{ij} \quad \Leftrightarrow \quad \theta_{j,r_{ij}-1} <
  \widetilde{Y}_{ij} \leq \theta_{j, r_{ij}}, \qquad r_{ij} \in
  \{1,\, \dots,\, K_j\},
\end{align*}
where $r_{ij}$ is a category out of $K_j$ ordered categories and
$\bm \theta_{j}$ is a vector of suitable threshold parameters for
outcome~$j$ with the following restriction:
$ - \infty \equiv \theta_{j,0} < \theta_{j,1}< \dots <
\theta_{j,K_j-1} < \theta_{j,K_j} \equiv \infty$. Note that in this
setting binary observations can be treated as ordinal observations
with two categories ($K_j = 2$).

The following linear model is assumed for the relationship between the
latent variable $\widetilde{Y}_{ij}$ and the vector of covariates
$\bm{x}_{ij}$:
\begin{align}\label{eqn:model}
  \widetilde{Y}_{ij} &= \beta_{j0} + \bm{x}_{ij}^\top
                       \bm{\beta}_j + \epsilon_{ij},
\end{align}
where $\beta_{j0}$ is an intercept term, $\bm{\beta}_j = (\beta_{j1},\,
\dots,\, \beta_{jp})^\top$ is a vector of regression coefficients, both
corresponding to outcome~$j$.
We further assume the $n$~subjects to be independent. Note that the number of ordered categories $K_j$ as well as the
threshold parameters $\bm\theta_j$ and the regression coefficients
$\bm\beta_j$ are allowed to vary across outcome dimensions $j \in J$
to account for possible heterogeneity across the response variables.

% Partial Prop. Odds
\paragraph{Category-specific regression coefficients.}
By employing one set of regression coefficients $\bm\beta_j$
for all categories of the $j$th outcome it is implied that the relationship
between the covariates and the responses does not depend on the category.
This assumption is called parallel regression or proportional odds
assumption \citep{Mccullagh80} and can be relaxed for one or more covariates by
allowing the corresponding regression coefficients to be category-specific \citep[see, e.g.,][]{Peterson90}.

%% Links
\paragraph{Link functions.}
The dependence among the different responses is accounted for by
assuming that, for each subject~$i$, the vector of error terms
$\bm{\epsilon}_i = [\epsilon_{ij}]_{j\in J_i}$ follows a suitable
multivariate distribution. We consider two multivariate distributions
which correspond to the multivariate probit and logit link
functions. For the multivariate probit link, we assume that the
errors follow a multivariate normal distribution: $\bm{\epsilon}_i
\sim \mathcal{N}_{q_i}(\bm 0,\bm\Sigma_i)$. A multivariate logit link is constructed by employing a multivariate logistic
distribution family with univariate logistic margins and a $t$ copula
with certain degrees of freedom as proposed by \cite{o2004bayesian}.  For a vector $\bm z=(z_1,\,\dots,\,
z_q)^\top$, the multivariate logistic distribution function with $\nu > 2$
degrees of freedom, location vector $\bm \mu$ and positive-definite dispersion matrix $\bm \Sigma$
is defined as:
\begin{align}\label{eqn:logistic}
  F_{\nu, \bm\mu,\bm\Sigma}(\bm z) = t_{\nu, \bm
    R}(\{g_\nu((z_1-\mu_1)/\sigma_1),\, \dots,\,
  g_\nu((z_q-\mu_q)/\sigma_q)\}^\top),
\end{align}
where $t_{\nu,\bm R}$ is the $q$-dimensional multivariate
$t$~distribution with $\nu$ degrees of freedom and correlation matrix
$\bm R$ implied by $\bm \Sigma$, $g_\nu(x) =
t^{-1}_{\nu}(\exp(x)/(\exp(x) + 1))$, with $t^{-1}_{\nu}$ the quantile
function of the univariate $t$ distribution with $\nu$ degrees of
freedom and $\sigma^2_1,\,\dots,\,\sigma^2_q$ the diagonal
elements of $\bm\Sigma$.

\cite{Hirk+Hornik+Vana:2018} employed this $t$~copula based
multivariate logistic family, while \cite{BIOM:BIOM12414} used a
multivariate $t$ distribution with $\nu = 8$ degrees of freedom as an
approximation for this multivariate logistic distribution. The
employed distribution family differs from the conventional
multivariate logistic distributions of \cite{gumbel1961bivariate} or
\cite{malik1973multivariate} in that it offers a more flexible
dependence structure through the correlation matrix of the $t$~copula,
while still keeping the log odds interpretation of the regression
coefficients through the univariate logistic margins.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% IDENTIFIABILTY
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Identifiability issues}\label{subsect:ident}
As the absolute scale and the absolute location are not identifiable
in ordinal models, further restrictions on the parameter set need to
be imposed.  Assuming $\bm\Sigma_i$ to be a covariance matrix with
diagonal elements $[\sigma^2_{ij}]_{j \in J_i}$, only the quantities
$\bm\beta_j/\sigma_{ij}$ and $(\theta_{j,r} -\beta_{j0})/\sigma_{ij}$
are identifiable in the model in Equation~\ref{eqn:model}. Hence, in
order to obtain an identifiable model the parameter set is typically
constrained in one of the following ways:
\begin{itemize}
  \item Fixing the intercept $\beta_{j0}$ (e.g., to zero), using
    flexible thresholds $\bm\theta_{j}$ and fixing $\sigma_{ij}$ (e.g.,
    to unity)  $\forall j \in J_i$, $\forall i \in \{1,\, \dots,\, n\}$.
  \item Leaving the intercept $\beta_{j0}$ unrestricted, fixing one
    threshold parameter (e.g., $\theta_{j,1}=0$) and fixing
    $\sigma_{ij}$ (e.g., to unity) $\forall j \in J_i$, $\forall i \in \{1,\, \dots,\, n\}$.
   \item Fixing the intercept $\beta_{j0}$ (e.g., to zero), fixing one
     threshold parameter (e.g., $\theta_{j,1}=0$) and leaving
     $\sigma_{ij}$ unrestricted $\forall j \in J_i$, $\forall i \in \{1,\, \dots,\, n\}$.
   \item Leaving the intercept $\beta_{j0}$ unrestricted, fixing two
     threshold parameters (e.g., $\theta_{j,1}=0$ and
     $\theta_{j,2}=1$) and leaving $\sigma_{ij}$ unrestricted $\forall
     j \in J_i$, $\forall i \in \{1,\, \dots,\, n\}$\footnote{Note that this
     parameterization cannot be applied to the binary case.}.
\end{itemize}
Note that the first two options are the most commonly used in the
literature. All of these alternative model parameterizations are
supported by the \pkg{mvord} package, allowing the user to choose the
most convenient one for each specific
application. Table~\ref{tab:Identifiability} in Section~\ref{sec:constraints.thresholds} gives an overview on
the identifiable parameterizations implemented in the package.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Error structures}
Different structures on the covariance matrix $\bm\Sigma_i$ can be imposed.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Basic model}
 The basic multivariate ordinal regression model assumes that
 the correlation (and possibly variance, depending on the
 parameterization) parameters in the distribution function of the
 $\bm\epsilon_i$ are constant for all subjects~$i$.
 %% Correlation
 \paragraph{Correlation.}
 The dependence between the multiple measurements or outcomes can be
 captured by different correlation structures. Among them, we
 concentrate on the following three:
 \begin{itemize}
   \item The general correlation structure assumes different correlation
     parameters between pairs of outcomes $\COR(\epsilon_{ik},
     \epsilon_{il}) = \rho_{kl}$. This error structure is among the most
  common in the literature \citep[e.g.,][]{Scott02, Bhat10, Pagui15}.
   \item The equicorrelation structure $\COR(\epsilon_{ik},
     \epsilon_{il}) = \rho$ implies that the correlation between
     all pairs of outcomes is constant.
   \item When faced with longitudinal data, especially when moderate
     to long subject-specific time series are available, an $AR(1)$
     autoregressive correlation model of order one can be employed. Given equally spaced time points this $AR(1)$ error structure implies an exponential decay in the correlation with the
     lag. If $k$ and $l$ are the time points when $Y_{ik}$
     and $Y_{il}$ are observed, then $\COR(\epsilon_{ik},
     \epsilon_{il}) = \rho^{|k-l|}$.
 \end{itemize}
 %% Variance
 \paragraph{Variance.}
 If a parameterization with identifiable variance is used (see
 Section~\ref{subsect:ident}), in the basic model we assume that for
 each multiple measurement the variance is constant across all
 subjects ($\VAR(\epsilon_{ij}) = \sigma^2_{j}$).

\subsubsection{Extending the basic model}
In some applications, the constant correlation (and variance)
structure across subjects may be too restrictive. We hence extend the
basic model by allowing the use of covariates in the
correlation (and variance) specifications.

\paragraph{Correlation.}
For each subject~$i$ and each pair $(k,l)$ from the set $J_i$, the
correlation parameter $\rho_{ikl}$ is assumed to depend on a vector
$\bm{s}_{i}$ of $m$ subject-specific covariates. In this paper we use the hyperbolic
tangent transformation to reparameterize the linear term
$\alpha_{0kl} + \bm{s}_{i}^\top \bm \alpha_{kl}$ in terms of a
correlation parameter:
\begin{align*}
\frac{1}{2} \log \left( \frac{1+ \rho_{ikl}}{1-\rho_{ikl}}\right) =
\alpha_{0kl} + \bm{s}_{i}^\top \bm\alpha_{kl}, \qquad \rho_{ikl} =
\frac{e^{2 (\alpha_{0kl} + \bm{s}_{i}^\top \bm\alpha_{kl})} -
  1}{e^{2 (\alpha_{0kl} + \bm{s}_{i}^\top \bm\alpha_{kl})} + 1}.
\end{align*}
If $\bm{\alpha}_{kl}=0$ for all $k,l\in J_i$, this model would
correspond to the general correlation structure in the basic
model. Moreover, if $\alpha_{0kl}=0$ and $\bm{\alpha}_{kl}=0$ for all $k,l\in
J_i$, the correlation matrix is the identity matrix and the responses
are uncorrelated.

For the more parsimonious error structures of equicorrelation and
$AR(1)$, in the extended model the correlation parameters are modeled
as:
\begin{align*}
  \frac{1}{2} \log \left( \frac{1+ \rho_{i}}{1-\rho_{i}}\right) =
  \alpha_{0} + \bm{s}_{i}^\top \bm{\alpha}, \qquad \rho_{i} =
  \frac{e^{2 (\alpha_{0} + \bm{s}_{i}^\top \bm{\alpha})} - 1}{e^{2
      (\alpha_{0} + \bm{s}_{i}^\top \bm{\alpha})} + 1}.
\end{align*}

\paragraph{Variance.}
Similarly, one can model the heterogeneity among the subjects through the variance parameters $\VAR(\epsilon_{ij}) = \sigma^2_{ij}$
by employing the following linear model on the log-variance:
\begin{align*}
  \text{log}(\sigma^2_{ij}) = \gamma_{0j} + \bm{s}_{i}^\top \bm\gamma_{j}.
\end{align*}

Note that other suitable link functions for the correlation and variance parameterizations could also be applied.
The positive-semi-definiteness of the correlation (or covariance) matrix $\bm\Sigma_i$ can be ensured by the use of special algorithms such as the one proposed by \cite{higham1988computing}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Composite Likelihood Estimation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Composite likelihood estimation} \label{sect:estimation}
In order to estimate the model parameters we use a composite
likelihood approach, where the full likelihood is approximated by a
pseudo-likelihood which is constructed from lower dimensional
marginal distributions, more specifically by ``aggregating'' the
likelihoods corresponding to pairs of observations \citep{varin_overview}.

For a given parameter vector $\bm\delta$, which contains the threshold
parameters, the regression coefficients and the parameters of the
error structure, the likelihood is given by:
\begin{align*}
\mathscr{L} (\bm\delta) &= \prod_{i=1}^n \Prob\bigg(\bigcap_{j\in
  J_{i}}\{Y_{ij} = r_{ij}\}\bigg)^{w_{i}} =\prod_{i=1}^n \bigg(
\int_{D_{i}} f_{i,q_{i}}(\bm{\widetilde{Y}}_{i}; \bm
\delta)d^{q_{i}}\widetilde{\bm{Y}}_{i}\bigg)^{w_{i}},
 \end{align*}
where $D_{i} =
\prod_{j\in J_{i}} (\theta_{j,r_{ij}-1}, \theta_{j,r_{ij}})$ is a
  Cartesian product, $w_{i}$ are subject-specific non-negative
  weights (which are set to one in the default case) and $f_{i,q_{i}}$
  is the $q_{i}$-dimensional density of the error terms
  $\bm\epsilon_i$. We approximate this full likelihood by a pairwise likelihood which is
constructed from bivariate marginal distributions. If the number of
observed outcomes for subject~$i$ is less than two ($q_i<2$), the
univariate marginal distribution enters the likelihood. The pairwise
log-likelihood function is obtained by:
\begin{align}\label{eqn:logpl}
  p\ell(\bm\delta)= \sum_{i=1}^n w_i \biggl[& \mathbbm{1}_{\{q_i \geq
      2\}} \sum_{k=1}^{q-1} \sum_{l=k+1}^{q} \mathbbm{1}_{\{k, l \in J_i\}}\log\left(\Prob(Y_{ik} = r_{ik},
    Y_{il} = r_{il})\right)+ \nonumber \\
    & \mathbbm{1}_{\{q_i = 1\}} \sum_{k=1}^{q}
    \mathbbm{1}_{\{k \in J_i\}}\biggl.\log\left(\Prob(Y_{ik} = r_{ik})
    \right)\biggr].
\end{align}
Denoting by $f_{i,1}$ and $f_{i,2}$ the uni- and bivariate density functions
corresponding to the error distribution, the uni- and bivariate
probabilities are given by:
\begin{align*}
\Prob(Y_{ik} &= r_{ik}, Y_{il} = r_{il})
=\displaystyle\int_{\theta_{k,r_{ik}-1}}^{\theta_{k,r_{ik}}}
\displaystyle\int_{\theta_{l,r_{il}-1}}^{\theta_{l,r_{il}}} f_{i,2}(\widetilde{Y}_{ik},\widetilde{Y}_{il};\bm\delta)
d\widetilde{Y}_{ik}d\widetilde{Y}_{il},\\ \Prob(Y_{ik} &= r_{ik}) =\displaystyle
\int_{\theta_{k,r_{ik}-1}}^{\theta_{k,r_{ik}}} f_{i,1}(\widetilde{Y}_{ik}; \bm\delta) d\widetilde{Y}_{ik}.
\end{align*}

The maximum pairwise likelihood estimates
$\hat{\bm\delta}_{p\ell}$ are obtained by direct maximization of
the composite likelihood given in Equation~\ref{eqn:logpl}. The
threshold and error structure parameters to be estimated are
reparameterized such that unconstrained optimization can be
performed. Firstly, we reparameterize the threshold parameters in
order to achieve monotonicity. Secondly, for all unrestricted
correlation (and covariance) matrices we use the spherical
parameterization of \cite{Pinheiro96}. This parameterization has the
advantage that it can be easily applied to correlation
matrices. Thirdly, for equicorrelated or $AR(1)$
errors, we use the hyperbolic tangent transformation.

Computation of the standard errors is needed in order to quantify the
uncertainty of the maximum pairwise likelihood estimates. Under
certain regularity conditions, the maximum pairwise likelihood
estimates are consistent as the number of responses is fixed and
$n\rightarrow \infty$. In addition, the maximum pairwise likelihood
estimator is asymptotically normal with asymptotic mean $\bm\delta$
and a covariance matrix which equals the inverse of the Godambe
information matrix:
\begin{align*}
  G(\bm\delta)^{-1} = H(\bm\delta)^{-1}V(\bm\delta) H(\bm\delta)^{-1},
\end{align*}
where $H(\bm\delta)$ is the Hessian (sensitivity matrix) and $V(\bm\delta)$ the
variability matrix. The variability matrix $V(\bm\delta)$ and the
Hessian $H(\bm\delta)$ can be estimated as:
\begin{align*}
\widehat V(\bm\delta)= \frac{1}{n} \sum_{i=1}^n \left(\frac{\partial
  p\ell_i(\bm\delta)}{\partial \bm\delta}\right)
\left(\frac{\partial p\ell_i(\bm\delta)}{\partial
  \bm\delta}\right)^{\top}\biggr\rvert_{\bm\delta = \hat{\bm\delta}_{p\ell}},
\end{align*}
and
\begin{align*}
\widehat H(\bm\delta) = -\frac{1}{n} \sum_{i=1}^n \frac{\partial^2
  p\ell_i(\bm\delta)}{\partial
  \bm\delta\partial\bm\delta^\top} \biggr\rvert_{\bm\delta = \hat{\bm\delta}_{p\ell}} =\frac{1}{n} \sum_{i=1}^n
\sum_{k=1}^q \sum_{l=k+1}^{q-1} \mathbbm{1}_{\{k, l \in J_i\}} \left(\frac{\partial
  p\ell_{ikl}(\bm\delta)}{\partial \bm\delta}\right)
\left(\frac{\partial
  p\ell_{ikl}(\bm\delta)}{\partial
  \bm\delta}\right)^\top\biggr\rvert_{\bm\delta = \hat{\bm\delta}_{p\ell}},
\end{align*}
where $p\ell_i(\bm\delta)$ is the component of the pairwise
log-likelihood corresponding to subject~$i$ and $p\ell_{ikl}(\bm\delta)$
corresponds to subject~$i$ and pair $(k,l)$.

In order to compare different models, the composite likelihood
information criterion by \cite{varin2005note} can be used:
$\text{CLIC}(\bm\delta) = -2\ p\ell(\hat{\bm\delta}_{p\ell}) +
k\ \mathrm{tr}(\widehat V(\bm\delta)\widehat H(\bm\delta)^{-1})$ (where $k=2$
corresponds to CLAIC and $k=\text{log}(n)$ corresponds to CLBIC). A
comprehensive overview and further details on the properties of the
maximum composite likelihood estimates are provided in \cite{Varin08}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Interpretation of the coefficients}
Unlike in linear regression models, the interpretation of the
regression coefficients and of the threshold parameters in ordinal
models is not straightforward. Estimated thresholds and coefficients
represent only signal to noise ratios and cannot be interpreted
directly (see Section~\ref{subsect:ident}). For one particular
outcome~$j$, the coefficients can be interpreted in the same way as in
univariate cumulative link models. Let us assume without loss of
generality that a higher latent score leads to better ratings on the
ordinal scale. This implies that the first category is the worst and
category $K_j$ is the best category. In this section we assume for sake of notational simplicity that $\bm \Sigma_i$ is a correlation matrix implying that marginally the errors of subject~$i$ have variance one and univariate marginal distribution function $F_{1}$ for each outcome~$j$.
In the more general case with non-constant variances $\sigma_{ij}^2$, $F_{i,1}^j$ should be used instead of $F_1$.
The marginal cumulative
probabilities implied by the model in Equation~\ref{eqn:model} are
then given by the following relationship:
\begin{align*}
\Prob(Y_{ij} \leq r_{ij}|\bm x_{ij}) = \Prob(\bm{x}_{ij}^\top \bm{\beta}_j +
\epsilon_{ij} \leq \theta_{j,r_{ij}}) = \Prob(\epsilon_{ij} \leq \theta_{j,r_{ij}} -
\bm{x}_{ij}^\top \bm{\beta}_j)  = F_{1}(\theta_{j,r_{ij}} - \bm{x}_{ij}^\top
\bm{\beta}_j).
\end{align*}

%marginal/partial effects
One natural way to interpret ordinal regression models is to analyze
partial effects, where one is interested in how a marginal change in one
variable $x_{ijv}$ changes the outcome distribution. The partial
probability effects in the cumulative model are given by:
\begin{align*}
  \delta_{r_{ij},v}^j(\bm x_{ij}) = \frac{\partial \Prob(Y_{ij} = r_{ij}|
    \bm x_{ij})}{\partial x_{ijv}} = -\left(f_{1}(\theta_{j,r_{ij}} -
  \bm{x}_{ij}^\top \bm{\beta}_j) - f_{1}(\theta_{j,r_{ij}-1} -
  \bm{x}_{ij}^\top \bm{\beta}_j)\right)\beta_{jv},
\end{align*}
where $f_{1}$ is the density corresponding to $F_{1}$, $x_{ijv}$ is the
$v$th element in $\bm x_{ij}$ and $\beta_{jv}$ is the $v$th element
in $\bm \beta_{j}$. In case of discrete variables it is more
appropriate to consider the changes in probability before and after
the change in the variable instead of the partial effects using:
\begin{align*}
\Delta \Prob(Y_{ij} = r_{ij}|\bm x_{ij}, \tilde{\bm x}_{ij}) =
\Prob(Y_{ij} = r_{ij}|\tilde{\bm x}_{ij}) - \Prob(Y_{ij} = r_{ij} |
\bm x_{ij}),
\end{align*}
where all elements of $\tilde{\bm x}_{ij}$ are equal to $\bm x_{ij}$
except for the $v$th element, which is equal to $\tilde{x}_{ijv} =
x_{ijv} + \Delta x_{ijv}$ for the change $\Delta x_{ijv}$ in
the variable $x_v$. We refer to \cite{Hensher10} and \cite{Boes06} for
further discussion of the interpretation of partial effects in ordered
response models.

In the presence of the probit link function, we have the following
relationship between the cumulative probabilities and the latent
process:
\begin{align*}
\Phi^{-1}\left(\Prob(Y_{ij} \leq r_{ij}|\bm{x}_{ij})\right) = \theta_{j,r_{ij}} -
\bm{x}_{ij}^\top \bm{\beta}_j.
\end{align*}

An increase of one unit in variable $v$ of outcome $j$ (given that all other
variables are held constant) changes the probit of the probability
that category $r_{ij}$ or lower is observed by the value of the coefficient
$\beta_{jv}$ of this variable. In other words $\Prob(Y_{ij} \leq
r_{ij}|\bm{x}_{ij})$, the probability that category $r_{ij}$ or lower
is observed, changes by the increase/decrease in the distribution
function.  Moreover, predicted probabilities for all ordered response
categories can be calculated and compared for given sets of
explanatory variables.

In the presence of the logit link function, the regression
coefficients of the underlying latent process are scaled in terms of
marginal log odds \citep{Mccullagh80}:
\begin{align*}
  \log\left(\frac{\Prob(Y_{ij} \leq r_{ij}|\bm{x}_{ij})}{\Prob(Y_{ij}
    > r_{ij}|\bm{x}_{ij})} \right) = \theta_{j,r_{ij}} -
  \bm{x}_{ij}^\top \bm{\beta}_j.
\end{align*}
For a one unit increase in variable $v$ of outcome $j$ holding all the
others constant, we expect a change of size of the coefficient
$\beta_{jv}$ of this variable in the expected value on the log odds
scale.  Due to the fact that the marginal effects of the odds ratios
do not depend on the category, one often exponentiates the
coefficients in order to obtain the following convenient
interpretation in terms of odds ratios:
\begin{align*}
\frac{\Prob(Y_{ij} \leq r_{ij}|\bm{x}_{ij})/\Prob(Y_{ij} >
r_{ij}|\bm{x}_{ij})}{\Prob(Y_{ij} \leq r_{ij}|\tilde{\bm{x}}_{ij})/\Prob(Y_{ij}
> r_{ij}|\tilde{\bm{x}}_{ij})} = \exp((\tilde{\bm{x}}_{ij} -
\bm{x}_{ij})^\top \bm{\beta}_j).
\end{align*}
This means for a one
unit increase in variable $v$ of outcome $j$, holding all the other variables constant, changes the
odds ratio by $\exp(\beta_{jv})$. In other words, the odds after a one unit
change in variable $v$ of outcome $j$ are the odds before the change multiplied by $\exp(-\beta_{jv})$:
\begin{align*}
  \frac{\Prob(Y_{ij} \leq r_{ij}|\bm{x}_{ij})}{\Prob(Y_{ij} > r_{ij}|\bm{x}_{ij})}
\exp(-\beta_{jv}) = \frac{\Prob(Y_{ij} \leq
r_{ij}|\tilde{\bm{x}}_{ij})}{\Prob(Y_{ij} > r_{ij}|\tilde{\bm{x}}_{ij})}.
\end{align*}

If the regression coefficients vary across the multiple responses,
they cannot be compared directly due to the fact that the measurement
units of the underlying latent processes differ. Nevertheless, one
possibility to compare coefficients is through the concept of
importance. \cite{reusens2017sovereign} extend an approach
for comparing coefficients of probit and logit models by
\cite{Hoetker07} in order to compare the coefficients across repeated
measurements. They analyze the importance ratio
\begin{align*}
  R_{jv} = \frac{\beta_{jv}}{\beta_{j,\mathit{base}}},
\end{align*}
where $\beta_{j,\mathit{base}}$ is the coefficient of a base variable
and $v$ is one of the remaining $p-1$ variables. This ratio can be
interpreted as follows: A one unit increase in the variable $v$ has in
expectation the same effect in the $\mathit{base}$ variable multiplied
by the ratio $R_{jv}$.  Another interpretation is the so called
compensation variation: The ratio is the required increase in the
$\mathit{base}$ variable that is necessary to compensate a one unit
decrease in the variable $v$ in a way that the score of the outcome
remains the same. It is to be noted that the importance ratio $R_{jv}$
depends on the scale of the $\mathit{base}$ variable and variable $v$ of outcome $j$. This implies that the comparison among the
measurements~$j$ should be done only if the scales of these variables
are equal across the multiple measurements.  For this purpose,
standardization of the covariates for each measurement should be
employed.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% IMPLEMENTATION
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Implementation}\label{sect:implementation}
The \pkg{mvord} package contains six data sets and the built-in
functions presented in Table~\ref{tab:Functions}.
\begin{table}[t!]
  \centering
\setlength{\tabcolsep}{11pt}
\begin{tabularx}{\textwidth}{p{5.1cm}X}
\hline
 Function & Description \\
\hline
\multirow{2}{*}{\emph{Fitting function}} & \\
&\\
\code{mvord(formula, data, ...)} & Estimates the multivariate ordinal regression model.\\
%%%%%%
\hline
\multirow{2}{*}{\emph{Prediction functions}} & \\
&\\
\multirow{2}{*}{\shortstack[l]{\code{predict(object, type,}\\ \code{\hspace{12pt}...)}}} & Obtains different types of predicted or fitted values from the joint distribution of the responses for objects of class \class{mvord}.\\
\multirow{2}{*}{\shortstack[l]{\code{marginal\_predict(object,}\\ \code{\hspace{12pt}type, ...)}}} & Obtains different types of predictions or fitted values from the marginal distributions of the responses for objects of class \class{mvord}.\\
\multirow{2}{*}{\shortstack[l]{\code{joint\_probabilities(object,}\\ \code{\hspace{12pt}response.cat, ...)}}} & For each subject, the joint probability of observing a predefined configuration of responses \code{response.cat} is computed for objects of class \class{mvord}.\\
%%%%%%%%%%%%%%%%%%%%%%%%
\hline
\multirow{2}{*}{\emph{Utility functions}} & \\
&\\
\code{coef(object, ...)} & Extracts the estimated regression coefficients. \\
\code{thresholds(object, ...)} & Extracts the estimated threshold coefficients.\\
\multirow{2}{*}{\shortstack[l]{\code{error\_structure(object,}\\ \code{\hspace{12pt}type, ...)}}} & Extracts for each subject the estimated parameters of the error structure.\\
\code{constraints(object)} & Extracts the constraint matrices corresponding to each regression coefficient.\\
\multirow{2}{*}{\shortstack[l]{\code{names\_constraints(formula,}\\ \code{\hspace{12pt}data, ...)}}} & Extracts the names of the regression coefficients in the model matrix.\\
\multirow{2}{*}{\shortstack[l]{\code{pseudo\_R\_squared(object,}\\ \code{\hspace{12pt}...)}}} & Computes McFadden's Pseudo $R^2$.\\
&\\
\hline
\multirow{2}{*}{\emph{Other methods for objects of class \class{mvord}.}} & \\
&\\
\multicolumn{2}{l}{
\code{summary()}, \code{print()},
\code{vcov()},
\code{fitted()}, \code{model.matrix()}, \code{terms()}, \code{nobs()},
\code{logLik()}}\\
\hline
\end{tabularx}
\caption{This table summarizes fitting, prediction, utility functions and other methods implemented in \pkg{mvord}.}\label{tab:Functions}
\end{table}
Multivariate ordinal regression models in the \proglang{R} package
\pkg{mvord} can be fitted using the main function \code{mvord()}.  Two
different data structures can be passed on to the \code{mvord()}
function through the use of two different multiple measurement objects
\code{MMO} and \code{MMO2} in the left-hand side of the model formula.
\code{MMO} uses a long data format, which has the advantage that it
allows for varying covariates across multiple measurements. This
flexibility requires to specify a subject index as well as a multiple
measurement index.  In contrast to \code{MMO}, the multiple
measurement object \code{MMO2} has a simplified data structure but is
only applicable in settings where the covariates do not vary between
the multiple measurements. In this case, the multiple ordinal
observations as well as the covariates are stored in different columns
of a `\code{data.frame}'. We refer to this data structure as wide data
format.

For illustration purposes we use a worked example based on a simulated
data set consisting of 100 subjects for which two multiple ordinal
responses (\code{Y1} and \code{Y2}), two continuous covariates
(\code{X1} and \code{X2}) and two factor covariates (\code{f1} and
\code{f2}) are available. The ordinal responses each have three
categories labeled with 1, 2 and 3.
<<>>=
data("data_mvord_toy", package = "mvord")
str(data_mvord_toy)
@

<<echo=FALSE>>=
data_toy_long <- cbind.data.frame(i = rep(1:100,2),
  j = rep(1:2,each = 100), Y = c(data_mvord_toy$Y1, data_mvord_toy$Y2),
  X1 = rep(data_mvord_toy$X1, 2), X2 = rep(data_mvord_toy$X2, 2),
  f1 = rep(data_mvord_toy$f1, 2), f2 = rep(data_mvord_toy$f2, 2))
@
%
The data set \code{data_mvord_toy} has a wide format. We convert the
data set into the long format, where the first column contains the
subject index~$i$ and the second column the multiple measurement
index~$j$.
%
<<>>=
str(data_toy_long)
@
\subsection[Implementation MMO]{Implementation \code{MMO}}\label{subsect:mmo}
The fitting function \code{mvord()} requires two compulsory input
arguments, a \code{formula} argument and a \code{data} argument:
<<eval=F>>=
res <- mvord(formula = MMO(Y, i, j) ~ 0 + X1 + X2, data = data_toy_long)
@
<<echo = FALSE, results = hide>>=
FILE <- "res1.rda"
if (cache & file.exists(FILE)) {
  load(FILE)
} else {
  if (cache) {
res <- mvord(formula = MMO(Y, i, j) ~ 0 + X1 + X2, data = data_toy_long)
res <- mvord:::reduce_size.mvord(res)
  save(res, file  = FILE)
  } else {
      if(file.exists(FILE)) file.remove(FILE)
  }
}
@
(runtime \Sexpr{round(res$rho$runtime[[1]],2)} seconds).\footnote{Computations have been performed with \proglang{R} version 3.4.4 on a machine with an Intel Core i5-4200U CPU 1.60GHz processor and 8GB RAM.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Data structure}
In \code{MMO} we use a long format for the input of \code{data}, where
each row contains a subject index~\code{i}, a multiple measurement
index~\code{j}, an ordinal observation \code{Y} and all the covariates
(\code{X1} to \code{Xp}).  This long format data structure is
internally transformed to an $n\times
q$ matrix of responses which contains \code{NA} in the case of missing
entries and a list of covariate matrices $\bm X_j$ for all $j \in
J$. This is performed by the multiple measurement object \code{MMO(Y,
  i, j)} which specifies the column names of the subject index and the
multiple measurement index in \code{data}.  The column containing the
ordinal observations can contain integer or character values or
inherit from class (ordered) \class{factor}.  When using the long data
structure, this column is basically a concatenated vector of each of
the multiple ordinal responses. Internally, this vector is then split
according to the measurement index. Then the ordinal variable
corresponding to each measurement index is transformed into an ordered
\class{factor}. For an integer or a character vector the natural
ordering is used (ascending, or alphabetical).  If for character
vectors the alphabetical order does not correspond to the ordering of
the categories, the optional argument \code{response.levels} allows to
specify the levels for each response explicitly. This is performed by
a list of length
$q$, where each element contains the names of the levels of the
ordered categories in ascending (or if desired descending) order. If
all the multiple measurements use the same number of classes and same
labeling of the classes, the column \code{Y} can be stored as an
ordered \class{factor} (as it is often the case in longitudinal
studies).

The order of the multiple measurements is needed when specifying constraints on the threshold or regression parameters (see Sections~\ref{sec:constraints.thresholds} and \ref{sec:coef.constraints}). This order is based on the type of the multiple measurement index column in \code{data}. For \class{integer}, \class{character} or \class{factor} the natural ordering is used (ascending, or alphabetical).  If a different order of the multiple responses is desired, the multiple measurement index column should be an ordered factor with a corresponding ordering of the levels.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Formula}
The multiple measurement object \code{MMO} including the ordinal
responses \code{Y}, the subject index~\code{i} and the multiple
measurement index~\code{j} is passed on the left-hand side of a
`\code{formula}' object. The covariates \code{X1}, \ldots, \code{Xp} are
passed on the right-hand side. In order to ensure identifiability
intercepts can be included or excluded in the model depending on the
chosen model parameterization.
\paragraph{Model without intercept.}
If the intercept should be removed, the \code{formula} can be specified in the
following ways:
%
\begin{Code}
formula = MMO(Y, i, j) ~ 0 + X1 + ... + Xp
\end{Code}
%
or
%
\begin{Code}
formula = MMO(Y, i, j) ~ -1 + X1 + ... + Xp
\end{Code}
\paragraph{Model with intercept.}
If one wants to include an intercept in the model, there are two
equivalent possibilities to specify the model formula. Either the
intercept is included explicitly by:
%
\begin{Code}
formula = MMO(Y, i, j) ~ 1 + X1 + ... + Xp
\end{Code}
%
or by
%
\begin{Code}
formula = MMO(Y, i, j) ~ X1 + ... + Xp
\end{Code}
%
\paragraph{Note on the intercept in the formula.}
We differ in our approach of specifying the model formula from the
model formula specification in, e.g., \code{MASS::polr()} or
\code{ordinal::clm()}, in that we allow the user to specify models
without an intercept. This option is not supported in the \pkg{MASS}
and \pkg{ordinal} packages, where an intercept is always specified in
the model formula as the threshold parameters are treated as
intercepts.  We choose to allow for this option, in order to have a
correspondence to the identifiability constraints presented in
Section~\ref{subsect:ident}.

Even so, the user should be aware that the threshold parameters are
basically category- and outcome-specific intercepts.  This implies
that, even if the intercept is explicitly removed from the model
through the `\code{formula}' object and hence set to zero, the rest of
the covariates should be specified in such a way that
multicollinearity does not arise.  This is of primary importance when
including categorical covariates, where one category will be taken as
baseline by default.

\subsection[Implementation MMO2]{Implementation \code{MMO2}}\label{subsect:mmo2}
We use the same worked example as above to show the usage of
\code{mvord()} with the multiple measurement object \code{MMO2}. The
data set \code{data_mvord_toy} has already the required data structure
with each response and all the covariates in separate columns. The
multiple measurement object \code{MMO2} combines the different
response columns on the left-hand side of the \code{formula} object:
<<eval=F>>=
res <- mvord(formula = MMO2(Y1, Y2) ~ 0 + X1 + X2, data = data_mvord_toy)
@
<<echo = FALSE, results = hide>>=
FILE <- "res2.rda"
if (cache & file.exists(FILE)) {
  load(FILE)
} else {
  if (cache) {
res <- mvord(formula = MMO2(Y1, Y2) ~ 0 + X1 + X2, data = data_mvord_toy)
res <- mvord:::reduce_size2.mvord(res)
  save(res, file  = FILE)
  } else {
      if(file.exists(FILE)) file.remove(FILE)
  }
}
@
(runtime \Sexpr{round(res$rho$runtime[[1]],2)} seconds).

The multiple measurement object \code{MMO2} is only applicable for
settings where the covariates do not vary between the multiple
measurements.

\subsubsection{Data structure}
The data structure applied by \code{MMO2} is slightly simplified,
where the multiple ordinal observations as well as the covariates are
stored as columns in a `\code{data.frame}'. Each subject~$i$ corresponds
to one row of the data frame, where all outcomes
$Y_{i1}, \dots, Y_{iq}$ (with missing observations set to \code{NA})
and all the covariates $x_{i1}, \dots, x_{ip}$ are stored in different
columns.  Ideally each outcome column is of type ordered
\class{factor}. If columns of the responses have types like
\class{integer}, \class{character} or \class{factor} a warning is
displayed and the natural ordering is used (ascending, or
alphabetical).

\subsubsection{Formula}
In order to specify the model we use a
multivariate `\code{formula}' object of the form:
%
\begin{Code}
formula = MMO2(Y1, ..., Yq) ~ 0 + X1 + ... + Xp
\end{Code}
%
The ordering of the responses is given by the ordering in the
left-hand side of the model formula. \code{MMO2} performs like
\code{cbind()} in fitting multivariate models in, e.g., \code{lm()} or
\code{glm()}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Link functions}
The multivariate link functions are specified as objects of class
\class{mvlink}, which is a list with
elements specifying the distribution function of the errors, functions
for computing the corresponding univariate and bivariate
probabilities, as well as additional arguments specific to each
link. If gradient functions are passed on, these will be used in the
computation of the standard errors.  This design was inspired by
the design of the \class{family} class in package \pkg{stats} and
facilitates the addition of new link functions to the package.

We offer two different multivariate link functions, the
multivariate probit link and a multivariate logit link.  For the
multivariate probit link a multivariate normal distribution for the
errors is applied. The bivariate normal probabilities which enter the
pairwise log-likelihood are computed with package \pkg{pbivnorm}
\citep{pbivnorm}. The multivariate probit link is the default link function and can be specified by:
%
\begin{Code}
link = mvprobit()
\end{Code}
%
For the multivariate logit link a $t$~copula based multivariate
distribution with logistic margins is used (as explained in
Section~\ref{subsect:model}) and can be specified by:
%
\begin{Code}
link = mvlogit(df = 8L)
\end{Code}
%
The \code{mvlogit()} function has an optional integer valued argument
\code{df} which specifies the degrees of freedom to be used for the
$t$~copula.  The default value of the degrees of freedom parameter is
8. When choosing $\nu \approx 8$, the multivariate logistic
distribution in Equation~\ref{eqn:logistic} is well approximated by a
multivariate $t$ distribution \citep{o2004bayesian}. This is also the
value chosen by \cite{BIOM:BIOM12414} in their
analysis. We restrict the degrees of freedom to be integer valued
because the most efficient routines for computing bivariate
$t$~probabilities do not support non-integer degrees of freedom. We
use the \proglang{Fortran} code from Alan Genz \citep{Genz09} to
compute the bivariate $t$~probabilities.  As the degrees of freedom
parameter is integer valued, we do not estimate it in the optimization
procedure. If the optimal degrees of freedom are of interest, we leave
the task of choosing an appropriate grid of values of \code{df} to the
user, who could then estimate a separate model for each value in the
grid. The best model can be chosen by CLAIC or CLBIC.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Error structures}
Different error structures are implemented in \pkg{mvord} and can be specified
through the argument \code{error.structure}. The error structure objects are
of class \class{error\_struct}. This approach slightly differs from the approach in package \pkg{nlme},
where the error structure is defined by two classes: \class{varFunc} for the variance function
and \class{corStruct} for the correlation structure.
We also define the following subclasses for the error structures: \class{cor\_general}
(similar to  \pkg{nlme}'s \class{corSymm}),  \class{cor\_equi} (similar to \class{corCompSymm}),
\class{cor\_ar1}  (similar to \class{corAR1}) and
\class{cov\_general} (similar to \class{corSymm} with variance function \class{varIdent}).
The different error structures are chosen through the argument \code{error.structure}.
\subsubsection{Basic model}
In the basic model we support three different correlation structures
and one covariance structure.
\paragraph{Correlation.}
For the basic model specification the following correlation structures are implemented in \pkg{mvord}:
\begin{itemize}
  \item \code{cor_general(formula = ~ 1)} -- A general error structure, where the
  correlation matrix of the error terms is unrestricted and constant
  across all subjects: $\COR(\epsilon_{ik},
  \epsilon_{il})=\rho_{kl}$.
\item \code{cor_equi(formula = ~ 1)} -- An equicorrelation structure
  is used with $\COR(\epsilon_{ik}, \epsilon_{il})=\rho$.
  \item \code{cor_ar1(formula = ~ 1)} -- An autoregressive error
    structure of order one is used with
    $\COR(\epsilon_{ik}, \epsilon_{il}) = \rho^{|k-l|}$.
\end{itemize}
\paragraph{Variance.}
A model with variance parameters $\VAR(\epsilon_{ij})=\sigma^2_j$
corresponding to each outcome, when the identifiability requirements
are fulfilled, can be specified in the following way:
\begin{itemize}
  \item \code{cov_general(formula = ~ 1)} -- The estimation of $\sigma^2_j$ is only implemented in
    combination with the general correlation structure.
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Extending the basic model}
The basic model can be extended by allowing covariate dependent error structures.
%\begin{itemize}
\paragraph{Correlation.}
The following correlation structures are implemented in \pkg{mvord}:
\begin{itemize}
  \item \code{cor_general(formula = ~ f)} -- For the heterogeneous general correlation structure, the
    current implementation only allows the use of one \class{factor}
    variable \code{f} as covariate. As previously mentioned, this
    factor variable should be subject-specific and hence should not
    vary across the multiple responses.  This implies that a
    correlation matrix will be estimated for each factor level.
\item \code{cor_equi(formula = ~ S1 + ... + Sm)} -- Estimating an equicorrelation structure depending on $m$
  subject-specific covariates \code{S1}, \ldots, \code{Sm}.
\item \code{cor_ar1(formula = ~ S1 + ... + Sm)} -- Estimating an $AR(1)$ correlation structure depending on $m$
  subject-specific covariates \code{S1}, \ldots, \code{Sm}.
\end{itemize}
\paragraph{Variance.}
The following variance structure is implemented in \pkg{mvord}:
\begin{itemize}
\item \code{cov_general(formula = ~ f)} -- As in the basic model, the estimation of the heterogeneous
  variance parameters can be performed for the
  general covariance structure. A subject-specific \class{factor}
  \code{f} can be used as a covariate in the log-variance equation. In addition to the correlation matrices, which are estimated for
each factor level of \code{f}, a vector of dimension $q$ of variance
parameters will be estimated for each factor level.
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Constraints on thresholds} \label{sec:constraints.thresholds}
The package supports constraints on the threshold
parameters. Firstly, the user can specify whether the threshold
parameters should be equal across some or all response dimensions.
Secondly, the values of some of the threshold parameters can be fixed.
This feature is important for users who wish to further restrict
the parameter space of the thresholds or who wish to specify values
for the threshold parameters other than the default values used in the
package.  Note that some of the thresholds have to be fixed for some
of the parameterizations presented in
Table~\ref{tab:Identifiability} in order to ensure identifiability
of the model.
%%%%%%%%%%%%%%%%%%
\subsubsection{Threshold constraints across responses}
Such constraints can be imposed by a vector of positive integers
\code{threshold.constraints}, where dimensions with equal threshold
parameters get the same integer. When restricting two outcome
dimensions to be equal, one has to be careful that the number of
categories in the two outcome dimensions must be the same. In the worked example,
if one wishes to restrict the
threshold parameters of the two outcomes \code{Y1} and \code{Y2} to be equal
($\bm\theta_1 = \bm\theta_2$), this can be specified as:
%
\begin{Code}
threshold.constraints = c(1, 1)
\end{Code}
%
where the first value corresponds to the first response \code{Y1} and the
second to the second response \code{Y2}. This order of the responses is defined
as explained in Sections~\ref{subsect:mmo} and~\ref{subsect:mmo2}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Fixing threshold values}
Values for the threshold parameters can be specified by the argument
\code{threshold.values}. For this purpose the user can pass a
`\code{list}' with $q$ elements, where each element is a `\code{vector}'
of length $K_j - 1$ (where $K_j$ is the number of ordered categories for
ordinal outcome~$j$). A numeric value in this vector fixes the
corresponding threshold parameter to a specified value while
\code{NA} leaves the parameter flexible and indicates it should be
estimated.

After specifying the error structure (through the
\code{error.structure} argument) and the inclusion/exclusion of an intercept in the \code{formula} argument, the user can choose
among five possible options for fixing the thresholds:
\begin{itemize}
  \item Leaving all thresholds flexible.
  \item Fixing the first threshold $\theta_{j,1}$ to a constant $a_j$ for all $j\in J$.
  \item Fixing the first and second thresholds
    $\theta_{j,1}=a_j$, $\theta_{j,2}=b_j$ for all outcomes with $K_j > 2$.
  \item Fixing the first and last thresholds
     $\theta_{j,1}=a_j$, $\theta_{j,K_j-1}=b_j$ for all outcomes with $K_j > 2$.
  \item An extra option is fixing all of the threshold parameters, for all $j\in J$.
\end{itemize}
Note that the option chosen needs to be consistent across the
different outcomes (e.g., it is not allowed to fix the first and the last
threshold for one outcome and the first and the second threshold for a
different outcome). Table~\ref{tab:Identifiability} provides information
about the options available for each combination of error structure and
intercept, as well as about the default values in case the user does
not specify any threshold values.
%
In the presence of binary observations ($K_j = 2$), if a \code{cov_general} error structure is used,
the intercept has always to be fixed to some value due to identifiability constraints. In a correlation
structure setting no further restrictions are required.

For example, if the following restrictions should apply to the worked example:
\begin{itemize}
\item $\theta_{11} = -1 \leq \theta_{12}$,
\item $\theta_{21} = -1 \leq \theta_{22}$,
\end{itemize}
this can be specified as:
%
\begin{Code}
threshold.values = list(c(-1, NA), c(-1, NA))
\end{Code}
%
\begin{table}[t!]
\setlength{\tabcolsep}{2.5pt}
\begin{tabularx}{\textwidth}{CCCCCCC}
\hline
\multirow{4}{*}{\parbox{2cm}{\centering Error structure}}&\multirow{4}{*}{\centering Intercept} & \multicolumn{5}{c}{Threshold parameters}\\
\cline{3-7}
 & & All flexible & One fixed & Two fixed & Two fixed & All fixed \\
&   & & $\theta_{j,1}=a_j$ & $\theta_{j,1}=a_j$  &$\theta_{j,1}=a_j$  & \\
 &  &&& $\theta_{j,2}=b_j$ & $\theta_{j,K_j-1}=b_j$ & \\
\hline
\multirow{2}{*}{\code{cor}}  &no     &\textcolor{Green}{\checkmark} &\checkmark&\checkmark&\checkmark&\checkmark\\
 &yes    &   &\textcolor{Green}{\checkmark} & \checkmark &\checkmark&\checkmark\\
\hline
\multirow{2}{*}{\code{cov}}  & no &   & \textcolor{Green}{\checkmark } & \checkmark &  \checkmark &  \checkmark\\
 &yes&  &  & \textcolor{Green}{\checkmark }&\checkmark &\checkmark\\
\hline
\end{tabularx}
\caption{This table displays different model parameterizations in the presence of ordinal observations ($K_j > 2 \; \forall j \in J$). The
  row \code{cor} includes error structures \code{cor\_general},
  \code{cor\_equi} and \code{cor\_ar1}, while row \code{cov} includes the
  error structure \code{cov\_general}. The minimal restrictions (default) to
  ensure identifiability are given in green. The default threshold values (in
  case \code{threshold.values = NULL}) are always $a_j = 0$ and $b_j = 1$.}
  \label{tab:Identifiability}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Constraints on coefficients}\label{sec:coef.constraints}
The package supports
constraints on the regression coefficients. Firstly, the
user can specify whether the regression coefficients should be equal
across some or all response dimensions. Secondly, values of some
of the regression coefficients can be fixed.

As there is no unanimous way to specify such constraints, we offer
two options. The first option is similar to the specification of constraints on the thresholds.
The constraints can be specified in this case as a vector or matrix of integers, where coefficients getting the same integer value are set equal. Values of the regression coefficients can be fixed through a matrix.
Alternatively, constraints on the regression coefficients can be specified
by using the design employed by the \pkg{VGAM} package.
The constraints in this setting are set through a named list,
where each element of the list contains a matrix of full-column rank.
If the values of some regression coefficients should be fixed, offsets can be used.
This design has the advantage that it supports
constraints on outcome-specific as well as category-specific
regression coefficients. While the first option has the advantage of requiring a more concise input, it does not support category-specific coefficients.
The second option offers a more flexible design in this respect.
%
\subsubsection{Coefficient constraints across responses}
Such constraints can be specified by the argument
\code{coef.constraints}, which can be either a vector or a matrix of
integer values. If vector constraints of the type $\bm \beta_{k} = \bm \beta_{l}$ are desired, which
should hold for all regression coefficients corresponding to
outcome $k$ and $l$, the easiest way to specify this is by means of a
vector of integers of dimension~$q$, where outcomes with equal vectors
of regression coefficients get the same integer.

Consider the following specification of the latent processes in the worked example:
\begin{align*}
\widetilde Y_{i1} = \beta_{1} x_{i1} + \beta_{2}  x_{i2} + \epsilon_{i1},\quad
\widetilde Y_{i2} = \beta_{1}  x_{i1} + \beta_{2}  x_{i2} + \epsilon_{i2},
\end{align*}
where the regression coefficients for variables \code{X1} and \code{X2}
are set to be equal across the two outcomes ($\bm\beta_{1} =
\bm\beta_{2}$) by:
%
\begin{Code}
coef.constraints = c(1, 1)
\end{Code}
%
A more flexible framework allows the user to specify constraints for
each of the regression coefficients of the
$p$~covariates\footnote{Note that if categorical covariates or
  interaction terms are included in the \code{formula}, $p$ denotes
  the number of columns of the design matrix.} and not only for the
whole vector. Such constraints will be specified by means of a matrix
of dimension $q\times p$, where each column specifies constraints for
one of the $p$~covariates in the same way as presented
above. Moreover, a value of \code{NA} indicates that the corresponding
coefficient is fixed (as we will show below) and should not be
estimated.

Consider the following specification of the latent processes in the worked example:
\begin{align}\label{eqn:m1}
\widetilde Y_{i1} = \beta_{11} x_{i1} + \beta_{3} \mathbbm{1}_{\{f_{i2}=c2\}} + \epsilon_{i1},\quad
\widetilde Y_{i2} = \beta_{21} x_{i1} + \beta_{22} x_{i2} + \beta_{3}\mathbbm{1}_{\{f_{i2}=c2\}}+\epsilon_{i2},
\end{align}
where $\mathbbm{1}_{\{f_{i2}=c2\}}$ is the indicator function which equals one in case the categorical covariate \code{f2} is equal to class \code{c2}. Class \code{c1} is taken as the baseline category.
These restrictions on the regression coefficients are imposed by:
%
\begin{Code}
coef.constraints = cbind(c(1, 2), c(NA, 1), c(1, 1))
\end{Code}
%
Specific values of coefficients can be fixed
through the \code{coef.values} argument, as we will show in the
following.
\subsubsection{Fixing coefficient values}
In addition, specific values on the regression coefficients can be set in
the $q\times p$ matrix \code{coef.values}. Again each column
corresponds to the regression coefficients of one covariate. This
feature is to be used if some of the covariates have known slopes, but
also for excluding covariates from the mean model of some of the
outcomes (by fixing the regression coefficient to zero). Fixed coefficients are treated internally as offsets and are not displayed in the model output.

By default, if no \code{coef.values} are passed by the user, all the
regression coefficients which receive an \code{NA} in
\code{coef.constraints} will be set to zero. \code{NA} in the
\code{coef.values} matrix indicates the regression coefficient ought
to be estimated.  Setting \code{coef.values} in accordance with the
\code{coef.constraints} from above (not needed as this is the default case):
%
\begin{Code}
coef.values = cbind(c(NA, NA), c(0, NA), c(NA, NA))
\end{Code}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Constraints on category-specific coefficients} \label{sec:VGAM}
If the parallel regression or proportional odds assumption ought to be relaxed,
the constraint design of package \pkg{VGAM} can be employed.
Let us consider the model specification in Equation~\ref{eqn:m1}. For illustration purposes we now relax the parallel regression assumption partially for covariates \code{X1} and \code{X2} in the following way:
\begin{itemize}
\item $\beta_{11, 1} \neq \beta_{11, 2}$,

\item $\beta_{22, 1} \neq \beta_{22, 2}$,
\end{itemize}
where $\beta_{jk, r}$ denotes the regression coefficient of
covariate~$k$ in the linear predictor of the $r$th cumulative probit
or logit for measurement~$j$. By the first restriction, for the first
outcome two regression coefficients are employed for covariate
\code{X1}: $\beta_{11, 1}$ for the first linear predictor and
$\beta_{11, 2}$ for the second linear predictor. Covariate \code{X2}
only appears in the model for the second outcome. For each of the two
linear predictors a different regression coefficient is estimated:
$\beta_{22, 1}$ and $\beta_{22, 2}$.

The constraints are set up as a named list where the names correspond
to the names of all covariates in the model matrix. To check
the name of the covariates in the model matrix one can use the
auxiliary function \code{names_constraints()} available in \pkg{mvord}
(see also next subsection):
<<eval=T>>=
names_constraints(formula = Y ~ 0 + X1 + X2 + f2, data = data_mvord_toy)
@
The number of rows is equal to the total number of linear predictors
$\sum_j (K_j - 1)$ of the ordered responses; in the example above
$2+2=4$ rows. The number of columns represents the number of
parameters to be estimated for each covariate:
%
\begin{Code}
coef.constraints = list(
  X1 = cbind(c(1, 0, 0, 0), c(0, 1, 0, 0),  c(0, 0, 1, 1)),
  X2 = cbind(c(0, 0, 1, 0), c(0, 0, 0, 1)), f2c2 = cbind(rep(1, 4)))
\end{Code}
%
For more details we refer the reader to the documentation of the \pkg{VGAM} package.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Interaction terms and categorical covariates}
When constraints on the regression coefficients should be specified in
models with interaction terms or categorical covariates, the
\code{coef.constraints} matrix has to be constructed appropriately. If
the order of the terms in the covariate matrix is not clear to the
user, it is helpful to call the function \code{names_constraints()}
before constructing the \code{coef.constraints} and \code{coef.values}
matrices.  The names of each column in the covariate matrix can be obtained by:
<<eval=T>>=
formula <- MMO2(Y1, Y2) ~ 1 + X1 : X2 + f1 + f2 * X1
names_constraints(formula, data = data_mvord_toy)
@
%
This should be used when setting up the coefficient
constraints. Please note that by default category \code{A} for factor
\code{f1} and category \code{c1} for factor \code{f2} are taken as
baseline categories. This can be changed by using the optional
argument \code{contrasts}. In models without intercept, the estimated
threshold parameters relate to the baseline category and the
coefficients of the remaining factor levels can be interpreted as a
shift of the thresholds.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\subsection{Additional arguments}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection[weights.name]{\code{weights.name}}
Weights on each subject~$i$ are chosen in a way that they are constant
across multiple measurements. Weights should be stored in a column of
\code{data}.  The column name of the weights in \code{data} should be
passed as a character string to this argument by \code{weights.name =
  "weights"}.  If \code{weights.name = NULL} all weights are set to
one by default. Negative weights are not allowed.
%
\subsubsection[offset]{\code{offset}}
If offsets are not specified in the model \code{formula}, a list with a vector of offsets for each multiple measurement can be passed.
%
\subsubsection[contrasts]{\code{contrasts}}
\code{contrasts} can be specified by a named list as in the argument
\code{contrasts.arg} of the default method of \code{model.matrix()}.
%% PL.lag
\subsubsection[PL.lag]{\code{PL.lag}}
In longitudinal studies, where $q_i$ is possibly large, the pairwise
likelihood estimation can be time consuming as it is built from all
two-dimensional combinations of $j,k \in J_i$. To overcome this
difficulty, one can construct the likelihood using only the bivariate
probabilities for pairs of observations less than $\mathit{lag}$ in ``time
units'' apart. A similar approach was proposed by
\cite{Varin09}. Assuming that, for each subject~$i$, we have a
time series of consecutive ordinal observations, the $i$th component
of the pairwise likelihood has the following form:
\begin{align*}
p\ell^{\mathit{lag}}_i(\bm\delta)=
 w_i  \left[ \sum_{k = 1}^{q_i-1} \sum_{l =
k+1}^{q_i}\mathbbm{1}_{\{|k-l|\leq \mathit{lag}\}}  \log \Prob(Y_{ik} = r_{ik},
Y_{il} = r_{il}) \right].
\end{align*}
The $\mathit{lag}$ can be fixed by a positive integer argument
\code{PL.lag} and it can only be used along with \code{error.structure
  = cor_ar1()}. The use of this argument is, however, not recommended
if there are missing observations in the time series, i.e., if the
ordinal variables are not observed in consecutive years. Moreover, one
should also proceed with care if the observations are not missing at
random.
%
%
\subsection[Control function mvord.control]{Control function
  \code{mvord.control()}}
Control arguments can be passed by the argument \code{control} and are
hidden in the sub-function \code{mvord.control()} with the following
arguments.
\subsubsection[solver]{\code{solver}}
This argument can either be a character string or a function.  All
general purpose optimizers of the \proglang{R} package \pkg{optimx}
\citep{optimx1, optimx2} can be used for maximization of the composite
log-likelihood by passing the name of the solver as a character string
to the \code{solver} argument. The available solvers in \pkg{optimx}
are, at the time of writing, \code{"Nelder-Mead"}, \code{"BFGS"},
\code{"CG"}, \code{"L-BFGS-B"}, \code{"nlm"}, \code{"nlminb"},
\code{"spg"}, \code{"ucminf"}, \code{"newuoa"}, \code{"bobyqa"},
\code{"nmkb"}, \code{"hjkb"}, \code{"Rcgmin"} and \code{"Rvmmin"}.
The default in \pkg{mvord} is solver \code{"newuoa"}.  The
\code{"BFGS"} solver performs well in terms of computation time, but
it suffers from convergence problems, especially for the
\code{mvlogit()} link.

Alternatively, the user has the possibility of applying other solvers
by using a wrapper function with arguments \code{starting.values} and
\code{objFun} of the following form:
%
\begin{Code}
solver = function(starting.values, objFun) {
  optRes <- solver.function(...)
  list(optpar = optRes$optpar, objvalue = optRes$objvalue,
    convcode = optRes$convcode, message = optRes$message)
}
\end{Code}
%
The \code{solver.function()} should return a list with the following
three elements: \code{optpar}, \code{objvalue} and \code{convcode}.
The element \code{optpar} should be a vector of length equal to the
number of parameters to optimize containing the estimated parameters,
while the element \code{objvalue} should contain the value of the
objective function after the optimization procedure.  The convergence
status of the optimization procedure should be returned in element
\code{convcode} with \code{0} indicating successful
convergence. Moreover, an optional solver message can be returned in
element \code{message}.
%
\subsubsection[solver.optimx.control]{\code{solver.optimx.control}}
A list of control arguments that are to be passed to the function \code{optimx()}. For further details see \cite{optimx1}.
%
\subsubsection[se]{\code{se}}
If \code{se = TRUE} standard errors are computed analytically using the Godambe
information matrix (see Section~\ref{sect:estimation}).
%
\subsubsection[start.values]{\code{start.values}}
A list of starting values for threshold as well as regression
coefficients can be passed by the argument \code{start.values}.
This list contains a list (with a vector of starting values for each
dimension) \code{theta} of all flexible threshold parameters and a
list \code{beta} of all flexible regression parameters.
%
\subsection[Output and methods for class mvord]{Output and methods for class \class{mvord}}
The function \code{mvord()} returns an object of class
\class{mvord},
which is a list containing the following
components:
\begin{itemize}
\item \code{beta}: A named `\code{matrix}' of regression coefficients.
\item \code{theta}: A named `\code{list}' of threshold parameters.
\item \code{error.struct}: An object of class \class{error\_struct}.
\item \code{sebeta}: A named `\code{matrix}' of standard errors of the regression coefficients.
\item \code{setheta}: A named `\code{list}' of standard errors of the threshold parameters.
\item \code{seerror.struct}: A `\code{vector}' of standard errors for the parameters of the error structure.
\item \code{rho}: A `\code{list}' of objects that are used in \code{mvord()}.
\end{itemize}

Several methods are implemented for the class \class{mvord}. These
methods include a \code{summary()} and a \code{print()} function to
display the estimation results, a \code{coef()} function to extract
the regression coefficients, a \code{thresholds()} function to extract
the threshold coefficients and a function \code{error_structure()} to
extract the estimated parameters of the correlation/covariance
structure of the errors. The pairwise log-likelihood can be extracted
by the function \code{logLik()}, function \code{vcov()} extracts the
variance-covariance matrix of the parameters and \code{nobs()}
provides the number of subjects.  Other standard methods such as
\code{terms()} and \code{model.matrix()} are also available. Functions
\code{AIC()} and \code{BIC()} can be used to extract the composite
likelihood information criteria CLAIC and CLBIC.

In addition, joint probabilities can be extracted by the
\code{predict()} or \code{fitted()} functions:
<<eval=T>>=
predict(res, subjectID =  1:6)
@
as well as joint cumulative probabilities:
<<eval=T>>=
predict(res, type = "cum.prob", subjectID =  1:6)
@
and classes:
<<eval=T>>=
predict(res, type = "class", subjectID =  1:6)
@
The function \code{marginal_predict()} provides marginal predictions
for the types probability, cumulative probability and class, while
\code{joint_probabilities()} extracts fitted joint probabilities (or
cumulative probabilities) for given response categories from a fitted
model.

\pagebreak

\section{Examples}\label{sect:examples}
In credit risk applications, multiple credit ratings from different
credit rating agencies are available for a panel of firms. Such a data
set has been analyzed in \cite{Hirk+Hornik+Vana:2018}, where a
multivariate model of corporate credit ratings has been
proposed. Unfortunately, this original data set is not freely
re-distributable. Therefore, we resort to the simulation of
illustrative data sets by taking into consideration key features of
the original data.

We simulate relevant covariates corresponding to firm-level and market
financial ratios in the original data set. The following covariates
are chosen in line with literature on determinants of credit ratings
\citep[e.g.,][]{campbell2008search,sp}: \code{LR} (liquidity ratio
relating the current assets to current liabilities), %R5
\code{LEV} (leverage ratio relating debt to earnings before interest
and taxes), %R13 Debt to EBITDA
\code{PR} (profitability ratio of retained earnings to assets), % R18
\code{RSIZE} (logarithm of the relative size of the company in the
market), %RSIZE
\code{BETA} (a measure of systematic risk). %SIGMA
We fit a distribution to each covariate using the function
\code{fitdistr()} of the \pkg{MASS} package. The best fitting
distribution among all available distributions in \code{fitdistr()}
has been chosen by AIC.

We generate two data sets for illustration purposes. The first data
set \code{data_cr} consists of multiple ordinal rating observations at
the same point in time for a collection of $690$ firms. We generate
ratings from four rating sources \code{rater1}, \code{rater2},
\code{rater3} and \code{rater4}. Raters \code{rater1} and
\code{rater2} assign ordinal ratings on a five-point scale (from best
to worst \code{A}, \code{B}, \code{C}, \code{D} and \code{E}),
\code{rater3} on a six-point scale (from best to worst, \code{F},
\code{G}, \code{H}, \code{I}, \code{J} and \code{K}) and \code{rater4}
distinguishes between investment and speculative grade firms (from
best to worst, \code{L} and \code{M}). The panel of ratings in the
original data set is unbalanced, as not all firms receive ratings from
all four sources. We therefore keep the missingness pattern and remove
the simulated ratings that correspond to missing observations in the
original data set. For \code{rater1} we remove $5$\%, for
\code{rater2} $30$\%, and for \code{rater3} $50$\% of the
observations. This data set has a wide data format.

The second data set \code{data_cr_panel} contains ordinal rating
observations assigned by one rater to a panel of 1415 firms over a
period of eight years on a yearly basis. In addition to the
covariates described above, a business sector variable (\code{BSEC})
with eight levels is included for each firm. This data set has a long
format, with 11320 firm-year observations.

\subsection{Example 1: A simple model of firm ratings assigned by multiple raters}\label{subsect:example1}
%
The first example presents a multivariate ordinal regression model
with probit link and a general correlation error structure
\code{cor_general(~ 1)}.  The simulated data set contains the ratings
assigned by raters \code{rater1}, \code{rater2}, \code{rater3} and
\code{rater4} and the five covariates \code{LR}, \code{LEV},
\code{PR}, \code{RSIZE} and \code{BETA} for a cross-section of 690
firms. A value of \code{NA} indicates a missing observation in the
corresponding outcome variable.
%<
<<eval=T>>=
data("data_cr", package = "mvord")
head(data_cr, n = 3)
str(data_cr, vec.len = 2.9)
@

<<plot_data, include = FALSE, fig = TRUE, height = 5, width = 12, echo = FALSE, eval = T>>=
op <- par(mfrow = c(1,4),
          oma = c(0,1.1,0,0),
          mar = c(2,3,5,1))
cexf <- 1.8
barplot(table(data_cr$rater1),  ylim = c(0, 500), las=1, main="rater1",
        cex.lab = cexf, cex.names = cexf, cex.main = 2, cex.axis = cexf, col=rgb(0.2,0.4,0.6,0.6))
barplot(table(data_cr$rater2), ylim = c(0, 500),las=1, main="rater2",
        cex.lab = cexf, cex.names = cexf, cex.main = 2, cex.axis = cexf, col=rgb(0.2,0.4,0.6,0.6))
barplot(table(data_cr$rater3), ylim = c(0, 500),las=1,main="rater3",
        cex.lab = cexf, cex.names = cexf, cex.main = 2, cex.axis = cexf, col=rgb(0.2,0.4,0.6,0.6))
barplot(table(data_cr$rater4), las=1, ylim = c(0, 500), main="rater4",
        cex.lab = cexf, cex.names = cexf, cex.main = 2, cex.axis = cexf, col=rgb(0.2,0.4,0.6,0.6))
par(op)
@

\begin{figure}[t!]
\centering
\includegraphics[width=1\textwidth]{vignette_mvord-plot_data.pdf}
\caption{This figure displays the rating distribution of all the raters.}
\end{figure}

We include five financial ratios as covariates in the model without
intercept through the following \code{formula}:
%
\begin{Code}
formula = MMO2(rater1, rater2, rater3, rater4) ~ 0 + LR + LEV + PR + RSIZE +
  BETA
\end{Code}
%
We are dealing with a wide data format, as the covariates do not vary
among raters. Hence, the estimation can be performed by applying
multiple measurement object \code{MMO2} in the fitting function
\code{mvord()}. A model with multivariate probit link (default) is
fitted by:
<<eval = FALSE>>=
res_cor_probit_simple <- mvord(formula = MMO2(rater1, rater2, rater3,
  rater4) ~ 0 + LR + LEV + PR + RSIZE + BETA, data = data_cr)
@
<<echo = FALSE, results = hide,eval=T>>=
FILE <- "res_cor_probit_simple.rda"
if (cache & file.exists(FILE)) {
  load(FILE)
} else {
  if (cache) {
res_cor_probit_simple <- mvord(
    formula =
    MMO2(rater1, rater2, rater3, rater4) ~ 0 + LR + LEV + PR + RSIZE + BETA, data = data_cr)
res_cor_probit_simple <- mvord:::reduce_size.mvord(res_cor_probit_simple)
  save(res_cor_probit_simple, file  = FILE)
  } else {
      if(file.exists(FILE)) file.remove(FILE)
  }
}
@
(runtime \Sexpr{round(res_cor_probit_simple$rho$runtime[[1]]/60,0)} minutes).

%% Then discuss output
The results are displayed by the function \code{summary()}:
<<eval=TRUE>>=
summary(res_cor_probit_simple, call = FALSE)
@
%
The threshold parameters are labeled with the name of the
corresponding outcome and the two adjacent categories which are
separated by a vertical bar \code{|}.
%% coefs
For each covariate the estimated coefficients are labeled with the
covariate name and a number. This number is from the sequence along
the number of columns in the list element of \code{constraints()}
which corresponds to the covariate. Note that if no constraints are
set on the regression coefficients, this number of the coefficient
corresponds to the outcome dimension.  If constraints are set on the
parameter space, we refer the reader to
Section~\ref{subsect:example2}.
% error struct
The last part of the summary contains the estimated error structure
parameters. For error structures \code{cor_general} and
\code{cov_general} the correlations (and variances) are displayed. The
coefficients corresponding to the error structure are displayed for
\code{cor_ar1} and \code{cor_equi}. Correlations and Fisher-$z$ scores
for each subject are obtained by function \code{error_structure()}.

Another option to display the results is the function \code{print()}.
The threshold coefficients can be extracted by the function \code{thresholds()}:
<<eval=T>>=
thresholds(res_cor_probit_simple)
@
The regression coefficients are obtained by the function \code{coef()}:
<<eval=T>>=
coef(res_cor_probit_simple)
@
The error structure for firm with \code{firm_id = 11} is displayed by
\code{error_structure()}:
<<eval=T>>=
error_structure(res_cor_probit_simple)[[11]]
@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\subsection{Example 2: A more elaborate model of ratings by multiple raters}\label{subsect:example2}
In the second example, we extend the setting of Example~1 by imposing
constraints on the regression as well as on the threshold parameters
and changing the link function to the multivariate logit link.  We
include the following features in the model:
\begin{itemize}
\item We assume that \code{rater1} and \code{rater2} use the same
  rating methodology. This means that they use the same rating classes
  with the same labeling and the same thresholds on the latent
  scale. Hence, we set the following constraints on the threshold
  parameters:
%
\begin{Code}
threshold.constraints = c(1, 1, 2, 3)
\end{Code}
%
\item We assume that some covariates are equal for some of the
  raters. We assume that the coefficients of \code{LR} and \code{PR}
  are equal for all four raters, that the coefficients of \code{RSIZE}
  are equal for the raters \code{rater1}, \code{rater2} and
  \code{rater3} and the coefficients of \code{BETA} are the same for
  the raters \code{rater1} and \code{rater2}. The coefficients of
  \code{LEV} are assumed to vary for all four raters. These
  restrictions are imposed by:
%
\begin{Code}
coef.constraints = cbind(LR = c(1, 1, 1, 1), LEV = c(1, 2, 3, 4),
  PR = c(1, 1, 1, 1), RSIZE = c(1, 1, 1, 2), BETA = c(1, 1, 2, 3))
\end{Code}
%
\end{itemize}
The estimation can now be performed by the function \code{mvord()}:
<<eval = FALSE>>=
res_cor_logit <- mvord(formula = MMO2(rater1, rater2, rater3, rater4) ~
  0 + LR + LEV + PR + RSIZE + BETA, data = data_cr, link = mvlogit(),
  coef.constraints = cbind(LR = c(1, 1, 1, 1), LEV = c(1, 2, 3, 4),
    PR = c(1, 1, 1, 1), RSIZE = c(1, 1, 1, 2), BETA = c(1, 1, 2, 3)),
  threshold.constraints = c(1, 1, 2, 3))
@
%
<<echo = FALSE, results = hide>>=
FILE <- "res_cor_logit.rda"
if (cache & file.exists(FILE)) {
  load(FILE)
} else {
  if (cache) {
res_cor_logit <- mvord(formula = MMO2(rater1, rater2, rater3, rater4) ~
    0 + LR + LEV + PR + RSIZE + BETA, data = data_cr, link = mvlogit(),
  coef.constraints = cbind(
    c(1,1,1,1),
    c(1,2,3,4),
    c(1,1,1,1),
    c(1,1,1,2),
    c(1,1,2,3)),
    threshold.constraints = c(1, 1, 2, 3))
res_cor_logit <- mvord:::reduce_size2.mvord(res_cor_logit)
  save(res_cor_logit, file  = FILE)
  } else {
      if(file.exists(FILE)) file.remove(FILE)
  }

}
@
(runtime \Sexpr{round(res_cor_logit$rho$runtime[[1]]/60,0)} minutes).

The results are displayed by the function \code{summary()}:
<<>>=
summary(res_cor_logit, call = FALSE)
@
If constraints on the threshold or regression coefficients are
imposed, duplicated estimates are not displayed. If thresholds are set
equal for two outcome dimensions only the thresholds for the former
dimension are shown. In the example above only the thresholds for
\code{rater1} are displayed. For each covariate the estimated
coefficients are labeled with the covariate name and a number. This
number is from a sequence along the number of columns in the list
element of the corresponding covariate in \code{constraints()} (see
Section~\ref{sec:coef.constraints}). The auxiliary function \code{constraints()}
can be used to extract the constraints on the coefficients. The column
names of the constraint matrices for each outcome correspond to the
coefficient names displayed in the \code{summary}. For each covariate
the coefficients to be estimated are numbered consecutively.  In the
above example this means that for covariates \code{LR} and \code{PR}
only one covariate is estimated, a coefficient for each outcome is
estimated for \code{LEV}, while for covariate \code{RSIZE} two and for
covariate \code{BETA} three coefficients are estimated. For example,
the coefficient \code{BETA 1} is used by \code{rater1} and
\code{rater2}, the coefficient \code{BETA 2} is used by \code{rater3}
while \code{BETA 3} is the coefficient for \code{rater4}. The
constraints for covariate \code{BETA} can be extracted by:
<<>>=
constraints(res_cor_logit)$BETA
@
<<plot_fit1, include = FALSE, fig = TRUE, height = 7, width = 7, echo = FALSE, eval=T>>=
cols <- rev(colorspace::sequential_hcl(round(200),
                                   h = 260, c = c(80, 0),
                                   l = c(30, 90),
                                   power = 0.7))
 scatterplot.mvord <- function(tab,
                               zlim = NULL,
                               col = cols,
                               xlab = NULL, ylab = NULL, main = NULL,
                               percent = FALSE,
                               col.one.to.one.line=grey(0.4),
                               col.bar.legend=TRUE,
                               ...){
  if(percent == "all") tab <- tab/sum(tab)*100
  if(percent == "row") tab <- sweep(tab, 1, rowSums(tab), "/")
  if(percent == "col") tab <- sweep(tab, 2, colSums(tab), "/")

  if(percent %in% c("all", "row", "col")){  zlim = c(0,100)
    tab <- round(tab*100,4)
  }

  tab[tab==0] <- NA

  if (is.null(zlim))  zlim <- range(tab, na.rm=T)

    plot.seq.x <- seq_len(nrow(tab))
    plot.seq.y <- seq_len(ncol(tab))
    labels.x <- rownames(tab)
    labels.y <- colnames(tab)

    if(is.null(xlab)) xlab <- ""
    if(is.null(ylab)) ylab <- ""

  image(x=plot.seq.x, y=plot.seq.y, z=tab, zlim=zlim, col=col,
        xlab= "", ylab= "",
        main=main, axes = FALSE,
        xlim = c(min(plot.seq.x) - 1,max(plot.seq.x) + 1),
        ylim = c(min(plot.seq.y) - 1,max(plot.seq.y) + 1), ...)
  axis(1, at = plot.seq.x, line = 0.5, labels = labels.x)
  axis(2, at = plot.seq.y, line = 0.5, labels = labels.y, las = 1)
  title(xlab= xlab)
  title(ylab= ylab, line = 4)

  if (!is.null(col.one.to.one.line))
    segments(min(plot.seq.x) - 0.5, min(plot.seq.y) - 0.5,
          max(plot.seq.x) + 0.5, max(plot.seq.y) + 0.5, lty = 3,
          col=col.one.to.one.line)

  starting.par.settings <- par(no.readonly = TRUE)
    mai <- par("mai")
    fin <- par("fin")
    x.legend.fig <- c(1 - (mai[4]/fin[1]), 1)
    y.legend.fig <- c(mai[1]/fin[2], 1 - (mai[3]/fin[2]))
    x.legend.plt <- c(x.legend.fig[1] + (0.08 * (x.legend.fig[2] -
        x.legend.fig[1])), x.legend.fig[2] - (0.6 * (x.legend.fig[2] -
        x.legend.fig[1])))
    y.legend.plt <- y.legend.fig
    cut.pts <- seq(zlim[1], zlim[2], length = length(col) + 1)
    z <- (cut.pts[1:length(col)] + cut.pts[2:(length(col) + 1)])/2
    par(new = TRUE, pty = "m", plt = c(x.legend.plt, y.legend.plt))
    image(x = 1, y = z, z = matrix(z, nrow = 1, ncol = length(col)),
        col = col, xlab = "", ylab = "", xaxt = "n", yaxt = "n")
    axis(4, mgp = c(3, 0.2, 0), las = 2, cex.axis = 0.8, tcl = -0.1)
    box()
    mfg.settings <- par()$mfg
    par(starting.par.settings)
    par(mfg = mfg.settings, new = FALSE)
}

op <- par(mfrow = c(2,2),
          oma = c(1,1,0,0),
          mar = c(4,5,2,3))
op <- par(mfrow = c(2,2),
          oma = c(0,0,0,0),
          mar = c(4,5,2,3))
scatterplot.mvord(
    table(res_cor_logit$rho$y[,1], marginal_predict(res_cor_logit, type = "class")[,1]),
                  main = "rater 1", ylab = "predicted", xlab = "observed", percent = "row")#row of table
scatterplot.mvord(
    table(res_cor_logit$rho$y[,2], marginal_predict(res_cor_logit, type = "class")[,2]),
                  main = "rater 2", ylab = "predicted", xlab = "observed", percent = "row")
scatterplot.mvord(
    table(res_cor_logit$rho$y[,3], marginal_predict(res_cor_logit, type = "class")[,3]),
                  main = "rater 3", ylab = "predicted", xlab = "observed", percent = "row")
scatterplot.mvord(
    table(res_cor_logit$rho$y[,4], marginal_predict(res_cor_logit, type = "class")[,4]),
                  main = "rater 4", ylab = "predicted", xlab = "observed", percent = "row")
par(op)
@

\begin{figure}[t!]
\centering
\includegraphics[width=1\textwidth]{vignette_mvord-plot_fit1.pdf}
\caption{This figure displays agreement plots of the predicted categories of the model \code{res\_cor\_logit} against the observed rating categories for all raters. For each observed rating class the distribution of the predicted ratings is displayed.}
\end{figure}

\subsubsection{Comparing the model fits of examples one and two}
Note that the composite likelihood information criteria can be used
for model comparison. For objects of class \class{mvord} CLAIC and
CLBIC are computed by \code{AIC()} and \code{BIC()}, respectively. The
value of the pairwise log-likelihood of the two models can be
extracted by \code{logLik()}. The model fit of examples one and two are
compared by means of BIC and AIC. From Table~\ref{tab:bic} we observe
that the model of Example~2 has a lower BIC and AIC, which indicates a
better model fit.
%
\begin{table}[t!]
\begin{center}
\begin{tabular}{lZ{2.2cm}Z{2.2cm}Z{2.2cm}}
\hline
 & \code{logLik()} & \code{BIC()} & \code{AIC()}\\
<<echo=FALSE, results=tex>>=
BICvec <- c(BIC(res_cor_probit_simple), BIC(res_cor_logit))
AICvec <- c(AIC(res_cor_probit_simple), AIC(res_cor_logit))
loglikvec <- c(logLik(res_cor_probit_simple), logLik(res_cor_logit))
tab <- cbind(loglikvec, BICvec, AICvec)
colnames(tab) <- c("logLik", "BIC", "AIC")
rownames(tab) <- c("Example 1", "Example 2")
print(xtable::xtable(tab),
   only.contents = TRUE, math.style.negative = TRUE,
   include.colnames = FALSE)
@
\end{tabular}
\caption{This table displays measures of fit for the multivariate probit model in Example~1 (presented in Section~\ref{subsect:example1}) and the multivariate logit model in Example~2 (presented in Section~\ref{subsect:example2}).}\label{tab:bic}
\end{center}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Example 3: Ratings assigned by one rater to a panel of firms}
%% From chapter 4
In the third example, we present a longitudinal multivariate ordinal
probit regression model with a covariate dependent $AR(1)$ error
structure using the data set \code{data_cr_panel}:
<<>>=
data("data_cr_panel")
str(data_cr_panel, vec.len = 3)
head(data_cr_panel, n = 3)
@
The simulated data set has a long data format and contains the credit
risk measure \code{rating} and six covariates for a panel of 1415
firms over eight years. The number of firm-year observations is 11320.

We include five financial ratios as covariates in the model with
an intercept by a \code{formula} with multiple measurement object
\code{MMO}:
%
\begin{Code}
formula = MMO(rating, firm_id, year) ~ LR + LEV + PR + RSIZE + BETA
\end{Code}
%
Additionally, the model has the following features:
\begin{itemize}
\item The threshold parameters are constant over the years. This can be specified through the argument \code{threshold.constraints}:
%
\begin{Code}
threshold.constraints = rep(1, nlevels(data_cr_panel$year))
\end{Code}
%
\item In order to ensure identifiability in a model with intercept,
  some thresholds need to be fixed. We fix the first thresholds for all
  outcome dimensions to zero by the argument \code{threshold.values}:
%
\begin{Code}
threshold.values = rep(list(c(0, NA, NA, NA)), 8)
\end{Code}
%
\item We assume that there is a break-point in the regression coefficients after \code{year4} in the sample. This break-point could correspond to the beginning of a crisis in a real case application.
Hence, we use one set of regression coefficients for years \code{year1}, \code{year2}, \code{year3} and \code{year4} and a different set for \code{year5}, \code{year6}, \code{year7} and \code{year8}.
This can be specified through the argument \code{coef.constraints}:
%
\begin{Code}
coef.constraints = c(rep(1, 4),  rep(2, 4))
\end{Code}
%
\item Given the longitudinal aspect of the data, an $AR(1)$
  correlation structure is an appropriate choice. Moreover, we use the
  business sector as a covariate in the correlation structure. The
  dependence of the correlation structure on the business sector is
  motivated by the fact that in some sectors, such as manufacturing,
  ratings tend to be more ``sticky'', i.e., do not change often over
  the years, while in more volatile sectors like IT there is less
  ``stickiness'' in the ratings.
%
\begin{Code}
error.structure = cor_ar1(~ BSEC)
\end{Code}
%
\end{itemize}
The estimation is performed by calling the function \code{mvord()}:
%
<<eval=F>>=
res_AR1_probit <- mvord(formula = MMO(rating, firm_id, year) ~ LR + LEV +
  PR + RSIZE + BETA, error.structure = cor_ar1(~ BSEC), link = mvprobit(),
  data = data_cr_panel, coef.constraints = c(rep(1, 4), rep(2, 4)),
  threshold.constraints = rep(1, 8), threshold.values = rep(list(c(0, NA,
    NA, NA)),8), control = mvord.control(solver = "BFGS"))
@
<<echo = FALSE, results = hide>>=
FILE <- "res_AR1_probit.rda"
if (cache & file.exists(FILE)) {
  load(FILE)
} else {
  if (cache) {
res_AR1_probit <- mvord(
  formula = MMO(rating, firm_id, year) ~ LR + LEV + PR + RSIZE +  BETA,
  data = data_cr_panel,
  error.structure = cor_ar1(~ BSEC),
  coef.constraints = c(rep(1,4), rep(2,4)),
  threshold.constraints = c(rep(1,8)),
  threshold.values = rep(list(c(0,NA,NA,NA)),8),
  link = mvprobit(),
  control = mvord.control(solver = "BFGS",  solver.optimx.control = list(trace = TRUE)))
   res_AR1_probit <- mvord:::reduce_size.mvord(res_AR1_probit)
  save(res_AR1_probit, file  = FILE)
  } else {
      if(file.exists(FILE)) file.remove(FILE)
  }

}
@
(runtime \Sexpr{round(res_AR1_probit$rho$runtime[[1]]/60,0)} minutes).

%% Then discuss output
The results of the model can be presented by the function \code{summary()}:
<<>>=
summary(res_AR1_probit, short = TRUE, call = FALSE)
@
For the fixed threshold coefficient \code{year1 A|B}, the
$z$~values and the corresponding $p$~values are set to \code{NA}.

The default \code{error_structure()} method for a \class{cor\_ar1} gives:
<<>>=
error_structure(res_AR1_probit)
@
In addition, the correlation parameters $\rho_i$ for each firm are obtained by choosing \code{type = "corr"} in \code{error_structure()}:
<<>>=
head(error_structure(res_AR1_probit, type = "corr"), n = 3)
@
Moreover, the correlation matrices for each specific firm are obtained by choosing \code{type = "sigmas"} in \code{error_structure()}:
<<>>=
head(error_structure(res_AR1_probit, type = "sigmas"), n = 1)
@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Conclusion
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion} \label{sect:conclusion}
The present paper is meant to provide a general overview on the
\proglang{R} package \pkg{mvord}, which implements the estimation of
multivariate ordinal probit and logit regression models using the
pairwise likelihood approach. We offer the following features which (to the best
of our knowledge) enhance the currently available software for
multivariate ordinal regression models in \proglang{R}:
\begin{itemize}
\item  Different error structures like a
general correlation and a covariance structure, an equicorrelation structure and an $AR(1)$ structure are available.
\item We account for heterogeneity in the error structure among
  the subjects by allowing the use of subject-specific covariates in
  the specification of the error structure.
\item We allow for outcome-specific threshold parameters.
\item We allow for outcome-specific regression parameters.
\item The user can impose further restrictions on the threshold and
  regression parameters in order to achieve a more parsimonious model
  (e.g., using one set of thresholds for all outcomes).
\item We offer the possibility to choose different
  parameterizations, which are needed in ordinal models to ensure
  identifiability.
\end{itemize}
Additional flexibility is achieved by allowing the user to implement alternative multivariate link functions or error structures (e.g., alternative transformations for the variance or correlation parameters can be implemented).
Furthermore, the long as well as the wide data format are supported by either applying \code{MMO} or \code{MMO2} as a multiple measurement object to estimate the model parameters. The functionality of the package is illustrated by a credit risk application. Further examples from different areas of application are presented in the package vignette.

Further research and possible extensions of \pkg{mvord} could consist
of the implementation of variable selection procedures in multivariate
ordinal regression models and the inclusion of multivariate semi- or
non-parametric ordinal models.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibliography{mvord}
\end{document}
