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  pdftitle = {MNP: R Package for Fitting Multinomial Probit Model},
  pdfauthor = {Kosuke Imai and David A. van Dyk},
}


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\title{{\bf MNP: R Package for Fitting the \\ Multinomial Probit
    Model}\thanks{An earlier version of this paper appeared in {\it
      Journal of Statistical Software}, Vol. 14, No. 3 (May 2005),
    pp.1--32. We thank Jordan Vance for his valuable contribution to
    this project and Shigeo Hirano for providing the Japanese election
    dataset. We also thank Doug Bates for helpful advice on Lapack
    routines and Andrew Martin, Kevin Quinn, users of MNP, and
    anonymous reviewers and the associate editor for useful
    suggestions.  We gratefully acknowledge funding for this project
    partially provided by NSF grants DMS-01-04129, DMS-04-38240, and
    DMS-04-06085, and by the Committee on Research in the Humanities
    and Social Sciences at Princeton University.}}

\author{Kosuke Imai\thanks{Professor, Department of Government and Department of
      Statistics, Harvard University.  1737 Cambridge Street,
      Institute for Quantitative Social Science, Cambridge MA 02138.
      Email: \href{mailto:imai@harvard.edu}{imai@harvard.edu} URL:
      \href{https://imai.fas.harvard.edu}{https://imai.fas.harvard.edu}}\\
  David A. van Dyk\thanks{Professor, Statistics Section,
Department of Mathematics, Imperial College London,
London, UK SW7 2AZ. Email: \href{mailto:dvandyk@imperial.ac.uk}{dvandyk@imperial.ac.uk}}}

\date{Version 3.1--1}

\pdfbookmark[1]{Title Page}{Title Page}
\maketitle

\begin{abstract}
  MNP is a publicly available R package that fits the Bayesian
  multinomial probit model via Markov chain Monte Carlo. The
  multinomial probit model is often used to analyze the discrete
  choices made by individuals recorded in survey data. Examples where
  the multinomial probit model may be useful include the analysis of
  product choice by consumers in market research and the analysis of
  candidate or party choice by voters in electoral studies.  The MNP
  software can also fit the model with different choice sets for each
  individual, and complete or partial individual choice orderings of
  the available alternatives from the choice set.  The estimation is
  based on the efficient marginal data augmentation algorithm that is
  developed by \citet{imai:vand:05}.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage
\section{Introduction}
\label{sec:introduction}

This paper illustrates how to use MNP, a publicly available R
\citep{R:12} package, in order to fit the Bayesian multinomial probit
model via Markov chain Monte Carlo. The multinomial probit model is
often used to analyze the discrete choices made by individuals
recorded in survey data.
%Unlike the ordinal probit model, the
%multinomial probit model does not assume that the choice set is
%inherently ordered.
Examples where the multinomial probit model may be useful include the
analysis of product choice by consumers in market research and the
analysis of candidate or party choice by voters in electoral studies.
The MNP software can also fit the model with different choice sets for
each individual, and complete or partial individual choice orderings
of the available alternatives from the choice set.  We use Markov
chain Monte Carlo (MCMC) for estimation and computation. In
particular, we use the efficient marginal data augmentation MCMC
algorithm that is developed by \citet{imai:vand:05}.

MNP can be installed in the same way as other R packages via the
\texttt{install.packages("MNP")} command.  Appendix~\ref{sec:install}
gives instructions for obtaining R and installing MNP on Windows, Mac
OS X, and Linux/UNIX platforms. Only three commands are necessary to
use the MNP software; \texttt{mnp()} fits the multinomial probit
model, \texttt{summary()} summarizes the MCMC output, and
\texttt{predict()} gives posterior prediction based on the fitted
model (In addition, \texttt{coef()} and \texttt{vcov()} allow one
to extract the posterior draws of model coefficients and covariance
matrix).  To run an example script, start R and run the following
commands:

\spacingset{1}\begin{verbatim}
library(MNP)     # loads the MNP package
example(mnp)     # runs the example script
\end{verbatim}
\spacingset{1.5} Details of the example script are given in
Sections~\ref{sec:examp-soap}~and~\ref{sec:examp-japan}. Three
appendices describe installation, the commands, and version
history. We begin in Section~\ref{sec:mn_probit} with a brief
description of the multinomial probit model that MNP is designed to
fit.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The Method}
\label{sec:mn_probit}

MNP implements the marginal data augmentation algorithms for posterior
sampling in the multinomial probit model. The MCMC algorithm we
implement here is fully described in \citet{imai:vand:05}; we use
Scheme 1 of their Algorithm 1.

\subsection{The Multinomial Probit Model}
\label{sec:mnp}

Suppose we have a dataset of size $n$ with $p>2$ choices and $k$
covariates.  Here, choices refer to the number of classes in the
multinomial model. The word ``choices'' is used because the model is
often used to describe how individuals choose among a number of
alternatives, e.g., how a voter chooses which candidate to vote for
among four candidates running for a particular office.  We focus on
the case when $p>2$ because when $p=2$, the model reduces to the
standard binomial probit model, which can be fit via the \texttt{glm(,
  family = binomial(probit))} command in R.  The multinomial probit
model differs from the ordinal probit model in that the former does
not assume any inherent ordering on the choices. Thus, although the
individuals may have preferences among the available alternatives
these ordering are individual specific rather than being
characteristics of the alternatives themselves.  The ordinal probit
model can be fitted via an MCMC algorithm in R by installing a package
called MCMCpack \citep{mart:quin:park:13}.

Under the multinomial probit model, we assume a multivariate normal
distribution on the latent variables,
$W_i=(W_{i1},\ldots,W_{i,p-1})$.
\begin{equation}
  W_i = X_i \beta + e_i, \quad e_i \sim \N(0, \Sigma),
  \ \hbox{for} \ i=1,\ldots,n, \label{eq:mnp}
\end{equation}
where $X_i$ is a $(p-1) \times k$ matrix of covariates, $\beta$ is $k
\times 1$ vector of fixed coefficients, $e_i$ is $(p-1) \times 1$
vector of disturbances, and $\Sigma$ is a $(p-1) \times (p-1)$
positive definite matrix. For the model to be identified, the first
diagonal element of $\Sigma$ is constrained, $\sigma_{11}=1$.  Please note that starting with version 2.6-1, we use the restriction ${\rm trace}(\Sigma)=p$ as the default identification strategy following the recommendation of \citet{burg:nord:09}.  This avoids the arbitrariness of fixing one particular diagonal element.  The
response variable, $Y_i$, is the index of the choice of individual $i$
among the alternatives in the choice set and is modeled in terms of
this latent variable, $W_i$, via
\begin{eqnarray}
  Y_i(W_i) & = & \l\{ \begin{array}{ccl}
    0 & {\rm if} & {\rm max}(W_i)<0 \\
    j & {\rm if} & {\rm max}(W_i)=W_{ij}>0 \end{array}\r. ,
  \quad {\rm for} \ i=1,\ldots,n, \; {\rm and} \; j=1,\ldots,p-1,
\end{eqnarray}
where $Y_i$ equal to 0 corresponds to a base category.

The matrix $X_i$ may include both choice-specific and
individual-specific variables. A choice-specific variable is a
variable that has a value for each of the $p$ choices, and these $p$
values may be different for each individual (e.g., the price of a
product in a particular region where an individual lives).
Choice-specific variables are recorded relative to the baseline choice
and thus there are $p-1$ recorded values for each individual. In this
way a choice-specific variable is tabulated as a column in $X_i$.
Individual-specific variables, on the other hand, take on a value for
each individual, but are constant across the choices, e.g., the age or
gender of the individual. These variables are tabulated via their
interaction with each of the choice indicator variables. Thus, an
individual-specific variable corresponds to $p-1$ columns of $X_i$ and
$p-1$ components of $\beta$.


\subsection{The Multinomial Probit Model with Ordered Preferences}
\label{sec:mnpop}

In some cases, we observe a complete or partial ordering of $p$
alternatives. For example, we may observe the preferences of each
individual among different brands of a product. We denote the outcome
variable in such situations by ${\cal Y}_i=\{\Y_{i1}, \ldots,
\Y_{ip}\}$ where $i=1,\ldots,n$ indexes individuals and $j=1,\ldots,p$
represent alternatives. If $\Y_{ij} > \Y_{ij'}$ for some $j \ne j'$,
we say $j$ is preferred to $j'$.  If $\Y_{ij} = \Y_{ij'}$ for some $j
\ne j'$, we say individual $i$ is indifferent to the choice between
alternatives $j$ and $j'$, but treat the data as if the actual
ordering is unknown.  In other words, formally we insist on strict
inequalities among the preferences, but allow for some inequalities to
be unobserved.  The preference ordering is assumed to satisfy the
usual axioms of preference comparability.  Namely, preference is
connected: For any $j$ and $j'$, either $\Y_{ij} \le \Y_{ij'}$ or
$\Y_{ij} \ge \Y_{ij'}$.  Preference also must be transitive: for any
$j$,$j'$, and $j''$, $\Y_{ij} \le \Y_{ij'}$ and $\Y_{ij'} \le
\Y_{ij''}$ imply $\Y_{ij} \le \Y_{ij''}$. For notational simplicity
and without loss of generality, we assume that $\Y_{ij}$ takes an
integer value ranging from $0$ to $p-1$. We emphasize that we have not
changed the model from Section~\ref{sec:mnp}. Rather, we simply have
more observed data: the index of the choice of the individual $i$,
$Y_i$, can be computed from $\Y_i$. Thus, we continue to model the
preference ordering, $\Y_{i}$, in terms of a latent (multivariate
normal) random vector, $W_{i} = (W_{ij},\ldots,W_{i,p-1})$, via
\begin{eqnarray} \Y_{ij}(W_i) & = & \#\{W_{ij'}: W_{ij'} < W_{ij} \}
\quad {\rm for} \quad i=1,\ldots,n, \quad {\rm and} \quad j =
1,\ldots,p, \label{eq:mop}
\end{eqnarray}
where $W_{ip}=0$, the distribution of $W_i$ is specified in
equation~\ref{eq:mnp}, and $\#\{\cdots\}$ indicates the number of
elements in a finite set.  This model can be fitted via a slightly
modified version of the MCMC algorithm in \citet{imai:vand:05}. In
particular, we need only modify the way in which $W_{ij}$ is sampled
and use a truncation rule based on Equation~\ref{eq:mop}.

\subsection{Prior Specification}

Our prior distribution for the multinomial probit model is
\begin{equation}
\beta \sim \N(0,A\inv) \quad {\rm and} \quad
p(\Sigma)\propto
|\Sigma|^{-(\nu+p)/2}\l[\tr(S\Sigma\inv)\r]^{-\nu(p-1)/2},
\label{eq:priors}
\end{equation}
where $A$ is the prior precision matrix of $\beta$, $\nu$ is the prior
degrees of freedom parameter for $\Sigma$, and the $(p-1)\times(p-1)$
positive definite matrix $S$ is the prior scale for $\Sigma$; we
assume the first diagonal element of $S$ is one.  The prior
distribution on $\Sigma$ is proper if $\nu\geq p-1$, the prior mean of
$\Sigma$ is approximately equal to $S$ if $\nu> p-2$, and the prior
variance of $\Sigma$ increase as $\nu$ decreases as long as this
variance exists. We also allow for an improper prior on $\beta$, which
is $p(\beta) \propto 1$ (i.e., $A=0$).\footnote{Algorithm~2 of
  \citet{imai:vand:05} allows for a non zero prior mean for $\beta$.
  Because the update for $\Sigma$ in this sampler is not exactly its
  complete conditional distribution, however, this algorithm may
  exhibit undesirable convergence properties in some situations.}

Alternate prior specifications were introduced by \citet{mccu:ross:94}
and \citet{mccu:pols:ross:00}. The relative advantage of the various
prior distributions are discussed by \citet{mccu:pols:ross:00},
\citet{nobi:00}, and \citet{imai:vand:05}. We prefer our choice
because it allows us to directly specify the prior distribution on the
identifiable model parameters, allows us to specify an improper prior
distribution on regression coefficient, and results in a Monte Carlo
sampler that is relatively quick to converge. An implementation of of
the sampler proposed by \citet{mccu:ross:94} has recently been
released in the R package bayesm \citep{ross:mccu:05}.


\subsection{Prediction under the Multinomial Probit Model}

Predictions of individual preferences given particular values of the
covariates can be useful in interpreting the fitted model. Consider a
value of the $(p-1)\times k$ matrix of covariates, $X^\star$, that may
or may not correspond to the values for one of the observed
individuals. We are interested in the distribution of the preferences
among the alternatives in the choice set given this value of the
covariates. Let $Y^\star$ be the preferred choice and ${\cal
  Y}^\star=({\cal Y}^\star_1, \ldots, {\cal Y}^\star_p)$ indicate the
ordering of the preferences among the available alternatives. As an
example, one might be interested in $\Pr(Y^\star = j \mid X^\star)$
for some $j$.  By varying $X^\star$, one could explore how preferences
are expected to change with covariates.  Similarly, one might be
interested in how relative preferences such as $\Pr( {\cal
  Y}_{j}^\star > {\cal Y}_{j^\prime}^\star \mid X^\star)$ are expected
to change with the covariates.

In the context of a Bayesian analysis, such predictive probabilities
are computed via the posterior predictive distribution. This
distribution conditions on the observed data, $Y=(Y_1,\ldots, Y_n)$ or
${\cal Y}= ({\cal Y}_1, \ldots, {\cal Y}_n)$, but averages over the
uncertainty in the model parameters. For example,
\begin{eqnarray}
\Pr(Y^\star = j \mid X^\star, Y) & = &
\int \Pr(Y^\star = j \mid X^\star, \beta, \Sigma, Y)\, p(\beta, \Sigma \mid Y)
\; d(\beta, \Sigma).
\end{eqnarray}
Thus, the posterior predictive distribution accounts for both variability
in the response variable given the model parameters (i.e., the likelihood
or sampling distribution) and the uncertainty in the model parameters
as quantified in the posterior distribution.  Monte Carlo evaluation of
the posterior predictive distribution is easy once we obtain a Monte
Carlo sample of the model parameters from the posterior distribution: We
simply sample according to the likelihood for each Monte Carlo sample from
the posterior distribution. This involves sampling the latent variable
under the model in (1) and computing the preferred choice using (2) or
the ordering of preferences using (3).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Example 1: Detergent Brand Choice}
\label{sec:examp-soap}

In this and the next section, we describe the details of two examples
of MNP. In this section we use a market research dataset to illustrate
the fitting of the multinomial probit model. In
Section~\ref{sec:examp-japan} we fit the multinomial probit model with
ordered preference to a Japanese election dataset.  We also describe
how to perform convergence diagnostics of the MCMC sampler and
analysis of the Monte Carlo output of MNP using an existing R
package. Additional examples of MNP can be found in
\citet{imai:vand:05}.

\subsection{Preliminaries}
\label{subsec:soap-prelim}

Our first example analyzes a typical dataset in market research.  The
 dataset contains information about the brand and price of the laundry
 detergent purchased by 2657 households originally analyzed by
 \citet{chin:pras:98}. The dataset contains the log prices of six
 detergent brands -- Tide, Wisk, EraPlus, Surf, Solo, and All -- as
 well as the brand chosen by each household (Type {\tt
 help(detergent)} in R for details about the dataset).

We fit the multinomial probit model by using \texttt{choice} as the
outcome variable and the other six variables as choice-specific
covariates. After loading the MNP package, this can be accomplished
using the following three commands,

\spacingset{1}
\begin{verbatim}

data(detergent)
res <- mnp(choice ~ 1, choiceX = list(Surf=SurfPrice, Tide=TidePrice,
                                      Wisk=WiskPrice, EraPlus=EraPlusPrice,
                                      Solo=SoloPrice, All=AllPrice),
           cXnames = c("price"), data = detergent, n.draws = 10000,
           burnin = 2000, thin = 3, verbose = TRUE)
summary(res)

\end{verbatim}
\spacingset{1.5} The first command loads the example dataset and
stores it as the data frame called \texttt{detergent}. The second
command fits the multinomial probit model. The default base category
in this case is \texttt{All}. (The default base category in MNP is the
first factor level of the outcome variable, $Y$.)  Each household
chooses among the six brands of laundry detergent, i.e., $p=6$. We
specify the choice-specific variables, \texttt{choiceX}, using a named
list. The elements of the list are the log price of each detergent
brand and they are named after the levels of factor variable,
\texttt{choice}.  We also name the coefficient for this set of
choice-specific variables by using \texttt{cXnames}. The argument
\texttt{data} allows us to specify the name of the data frame where the
data are stored.  The model estimates five intercepts and the price
coefficient as well as 14 parameters in the covariance matrix,
$\Sigma$.

We use the default prior distribution; an improper prior distribution
for $\beta$ and a diffuse prior distribution for $\Sigma$ with $\nu =
p = 6$ and $S=I$.  We sample 10,000 replications of the parameter from
the resulting posterior distribution, saving every fourth sample after
discarding the first 2,000 samples as specified by the arguments,
\texttt{n.draws}, \texttt{thin}, and \texttt{burnin}.  The argument
\texttt{verbose = TRUE} specifies that a progress report and other
useful messages be printed while the MCMC sampler is running.  The
\texttt{summary(res)} command gives a summary of the output including
the posterior means and standard deviations of the parameters. The
summary is based on the single MCMC chain produced with this call of
MNP.  Before we can reliably draw conclusions based on these results,
we must be sure the chain has converged.  Convergence diagnostics are
discussed and illustrated in Section~\ref{sec:coda}. The
result of the call of \texttt{summary(res)} are as follows.

\spacingset{1}
\begin{verbatim}
Call:
mnp(formula = choice ~ 1, data = detergent, choiceX = list(Surf = SurfPrice,
    Tide = TidePrice, Wisk = WiskPrice, EraPlus = EraPlusPrice,
    Solo = SoloPrice, All = AllPrice), cXnames = c("price"),
    n.draws = 10000, burnin = 2000, thin = 3, verbose = TRUE)


Coefficients:
                       mean std.dev.    2.5%  97.5%
(Intercept):EraPlus   2.567    0.238   2.123   3.03
(Intercept):Solo      1.722    0.247   1.248   2.25
(Intercept):Surf      1.572    0.163   1.259   1.91
(Intercept):Tide      2.716    0.252   2.269   3.22
(Intercept):Wisk      1.620    0.162   1.328   1.96
price               -82.102    8.952 -99.896 -66.32

Covariances:
                    mean std.dev.     2.5% 97.5%
EraPlus:EraPlus  1.00000  0.00000  1.00000  1.00
EraPlus:Solo     0.82513  0.26942  0.31029  1.36
EraPlus:Surf     0.17021  0.16115 -0.15810  0.48
EraPlus:Tide     0.24872  0.12956  0.00253  0.52
EraPlus:Wisk     0.88170  0.16614  0.54500  1.20
Solo:Solo        2.56481  0.68678  1.53276  4.25
Solo:Surf        0.45246  0.34572 -0.28018  1.13
Solo:Tide        0.50836  0.32706 -0.09069  1.22
Solo:Wisk        1.46997  0.44596  0.65506  2.45
Surf:Surf        1.69005  0.50978  0.92334  2.82
Surf:Tide        0.80762  0.30381  0.34019  1.44
Surf:Wisk        1.01614  0.36503  0.44121  1.85
Tide:Tide        1.32024  0.41669  0.62898  2.25
Tide:Wisk        1.05396  0.30137  0.59323  1.74
Wisk:Wisk        2.58761  0.55076  1.68773  3.82

Base category: All
Number of alternatives: 6
Number of observations: 2657
Number of stored MCMC draws: 2000
\end{verbatim}
\spacingset{1.5} We emphasize that these results are preliminary
because convergence has not yet been assessed. Thus, we delay
interpretation of the fit until Section~\ref{sec:examp-soap-final},
after we discuss convergence diagnostics in Section~\ref{sec:coda}.
Note that \texttt{coef(res)} and \texttt{vcov(res)} allow one to
extract the posterior draws of model coefficients and covariance
matrix if desired.  Type {\tt help(mnp)} in R for details.

\subsection{Using coda for Convergence Diagnostics and Output Analysis}
\label{sec:coda}

It is possible to use coda \citep*{plum:best:cowl:vine:05}, to
perform various convergence diagnostics, as well as to summarize
results.  The coda package requires a matrix of posterior draws for
relevant parameters to be saved as an \texttt{mcmc} object. Here, we
illustrate how to use coda to calculate the Gelman-Rubin convergence
diagnostic statistic \citep{gelm:rubi:92}. This diagnostic is based on
multiple independent Markov chains initiated at over-dispersed
starting values.  Here, we obtain these chains by independently running
the \texttt{mnp()} command three times, specifying different starting
values for each time.  This can be accomplished by typing the
following commands at the R prompt,

\spacingset{1}
\begin{verbatim}

data(detergent)
res1 <- mnp(choice ~ 1, choiceX = list(Surf=SurfPrice, Tide=TidePrice,
                                       Wisk=WiskPrice, EraPlus=EraPlusPrice,
                                       Solo=SoloPrice, All=AllPrice),
            cXnames = c("price"), data = detergent, n.draws = 50000,
            verbose = TRUE)

res2 <- mnp(choice ~ 1, choiceX = list(Surf=SurfPrice, Tide=TidePrice,
                                       Wisk=WiskPrice, EraPlus=EraPlusPrice,
                                       Solo=SoloPrice, All=AllPrice),
            coef.start = c(1, -1, 1, -1, 1, -1)*10,
            cov.start = matrix(0.5, ncol=5, nrow=5) + diag(0.5, 5),
            cXnames = c("price"), data = detergent, n.draws = 50000,
            verbose = TRUE)

res3 <- mnp(choice ~ 1, choiceX = list(Surf=SurfPrice, Tide=TidePrice,
                                       Wisk=WiskPrice, EraPlus=EraPlusPrice,
                                       Solo=SoloPrice, All=AllPrice),
            coef.start=c(-1, 1, -1, 1, -1, 1)*10,
            cov.start = matrix(0.9, ncol=5, nrow=5) + diag(0.1, 5),
            cXnames = c("price"), data = detergent, n.draws = 50000,
            verbose = TRUE)
\end{verbatim}
\spacingset{1.5} where we save the output of each chain separately as
 \texttt{res1}, \texttt{res2}, and \texttt{res3}. The first chain is
 initiated at the default starting values for all parameters; i.e., a
 vector of zeros for $\beta$ and an identity matrix for $\Sigma$. The
 second chain is run starting from a vector of three $10$'s and three
 $-10$'s for $\beta$ and a matrix with all diagonal elements equal to
 1 and all correlations equal to 0.5 for $\Sigma$. Finally, the third
 chain is run starting from a permutation of the starting value used
 for $\beta$ in the second chain, and a matrix with all diagonal
 elements equal to 1 and all correlations equal to 0.9 for
 $\Sigma$. We again use the default prior specification and obtain
 50,000 draws for each chain.

We store the output from each of the three chains as an object of class
\texttt{mcmc}, and then combine them into a single list using the
following commands,

\spacingset{1}
\begin{verbatim}
library(coda)
res.coda <- mcmc.list(chain1=mcmc(res1$param[,-7]),
                      chain2=mcmc(res2$param[,-7]),
                      chain3=mcmc(res3$param[,-7]))
\end{verbatim}
\spacingset{1.5} where the first command loads the coda
package\footnote{If you have not used the \texttt{coda} package
  before, you must install it. At the R prompt, type
  \texttt{install.packages("coda")}.} and the second command saves the
results as an object of class \texttt{mcmc.list}, which is called
\texttt{res.coda}. We exclude the 7th column of each chain, because
this column corresponds to the first diagonal element of the
covariance matrix which is always equal to 1.  The following command
computes the Gelman-Rubin statistic from these three chains,

\spacingset{1}
\begin{verbatim}
gelman.diag(res.coda, transform = TRUE)
\end{verbatim}
\spacingset{1.5} where \texttt{transform = TRUE} applies log or logit
transformation as appropriate to improve the normality of each of the
marginal distributions. \citet{gelm:carl:ster:rubi:04} suggest
computing the statistic for each scalar estimate of interest, and to
continue to run the chains until the statistics are all less than 1.1.
Inference is then based on the Monte Carlo sample obtained by
combining the second half of each of the chains.  The output of the
coda command lists the value and a 97.5\% upper limit of the
Gelman-Rubin statistic for each parameter.

\spacingset{1}\begin{verbatim}
Potential scale reduction factors:

                    Point est. 97.5% quantile
(Intercept):EraPlus       1.01           1.02
(Intercept):Solo          1.03           1.08
(Intercept):Surf          1.01           1.05
(Intercept):Tide          1.01           1.02
(Intercept):Wisk          1.01           1.04
price                     1.01           1.02
EraPlus:Solo              1.02           1.03
EraPlus:Surf              1.02           1.04
EraPlus:Tide              1.03           1.08
EraPlus:Wisk              1.04           1.13
Solo:Solo                 1.01           1.04
Solo:Surf                 1.01           1.02
Solo:Tide                 1.02           1.07
Solo:Wisk                 1.00           1.00
Surf:Surf                 1.00           1.00
Surf:Tide                 1.01           1.04
Surf:Wisk                 1.02           1.06
Tide:Tide                 1.02           1.06
Tide:Wisk                 1.02           1.08
Wisk:Wisk                 1.01           1.04

Multivariate psrf

1.07+0i
\end{verbatim}
\spacingset{1.5} The Gelman-Rubin statistics are all less than 1.1,
suggesting satisfactory convergence has been achieved.  (Note that the
97.5% quantile for {\tt EraPlus:Wisk} is greater than 1.1; a more
conservative user might want to obtain a set of longer Markov chains
and recompute the Gelman-Rubin statistics.)  It may also be useful to
examine the change in the value of the Gelman-Rubin statistic over the
iterations.  The following commands produce a graphical summary of the
progression of the statistics over iterations.

\spacingset{1}
\begin{verbatim}
gelman.plot(res.coda, transform = TRUE, ylim = c(1,1.2))
\end{verbatim}
\spacingset{1.5} where \texttt{ylim = c(1,1.2)} specifies the range of
the vertical axis of the plot. The results appear in
Figure~\ref{fg:rhat}, as a cumulative evaluation of the Gelman-Rubin
statistic over iterations for nine selected parameters. (Three
coefficients appear in the first row; three covariance parameters
appear in the second row; and three variance parameters appear in the
third row.)
\begin{figure}[p]
  \spacingset{1}
  \includegraphics[scale=1.15]{rhat}
  \caption{The Gelman-Rubin Statistic Computed with Three Independent
    Markov Chains for Selected Parameters in the Detergent
    Example. The first row represents three coefficients, the second
    row represents three covariances, and the third row represents
    three variance parameters.}
  \label{fg:rhat}
\end{figure}

The coda package can also be used to produce univariate time-series
plots of the three chains and univariate density estimate of the
posterior distribution.  The following commands create these graphs
for the price coefficient.

%pdf("coda.pdf", width=9, height=4.5)
\spacingset{1}\begin{verbatim}
res.coda <- mcmc.list(chain1=mcmc(res1$param[25001:50000, "price"], start=25001),
                      chain2=mcmc(res2$param[25001:50000, "price"], start=25001),
                      chain3=mcmc(res3$param[25001:50000, "price"], start=25001))
plot(res.coda, ylab = "price coefficient")
\end{verbatim}
%dev.off()
  \spacingset{1.5}
Figure \ref{fg:trace} presents the resulting plots. The left panel
overlays the time-series plot for each chain with a different color
representing each chain. The right panel shows the kernel-smoothed
density estimate of the posterior distribution. One can also apply an
array of other functions to \texttt{res.coda}.  See the coda
homepage,
\href{https://CRAN.R-project.org/package=coda}{https://CRAN.R-project.org/package=coda}, for
details.

\begin{figure}
  \spacingset{1}
  \begin{center}
    \includegraphics[scale=0.75]{coda-small}
  \end{center}
  \vspace{-0.25in}
  \caption{Time-series Plot of Three Independent Markov Chains (Left
    Panel) and A Density Estimate of the Posterior Distribution of the
    Price Coefficient (Right Panel). The time-series plot overlays the
    three chains, each in a different color. A lowess smoothed line is
    also plotted for each of the three chains.  The density estimate
    is based on all three chains.}
  \label{fg:trace}
\end{figure}


\subsection{Final Analysis and Conclusions}
\label{sec:examp-soap-final}

In the final analysis, we combine the second half of each of the three
chains. This is accomplished using the following command that saves
the last 25,000 draws from each chain as an \texttt{mcmc} object and
combines the \texttt{mcmc} objects into a list,
% pdf("rhat.pdf")
%  par(mex=0.5)
% res.coda <- mcmc.list(chain1=mcmc(res1$param[,c(1:2,6,9,14,18,12,16,19,)]),
%                      chain2=mcmc(res2$param[,c(1:2,6,9,14,18,12,16,19,)]),
%                      chain3=mcmc(res3$param[,c(1:2,6,9,14,18,12,16,19,)]))
% gelman.plot(res.coda, transform = TRUE, ylim=c(1,1.2))
% dev.off()

\spacingset{1}\begin{verbatim}

res.coda <- mcmc.list(chain1=mcmc(res1$param[25001:50000,-7], start=25001),
                      chain2=mcmc(res2$param[25001:50000,-7], start=25001),
                      chain3=mcmc(res3$param[25001:50000,-7], start=25001))
summary(res.coda)
\end{verbatim}
\spacingset{1.5}

The second command produces the following summary of the posterior
distribution for each parameter based on the combined Monte Carlo
sample.

\spacingset{1}\begin{verbatim}

Iterations = 25001:50000
Thinning interval = 1
Number of chains = 3
Sample size per chain = 25000

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

                        Mean     SD  Naive SE Time-series SE
(Intercept):EraPlus   2.5398 0.2300 0.0008400       0.014332
(Intercept):Solo      1.7218 0.2227 0.0008131       0.012972
(Intercept):Surf      1.5634 0.1663 0.0006072       0.010462
(Intercept):Tide      2.6971 0.2374 0.0008670       0.015153
(Intercept):Wisk      1.6155 0.1594 0.0005822       0.010221
price               -80.9097 8.4292 0.0307791       0.556483
EraPlus:Solo          0.8674 0.2954 0.0010787       0.021698
EraPlus:Surf          0.1226 0.1991 0.0007269       0.014043
EraPlus:Tide          0.2622 0.1525 0.0005568       0.009833
EraPlus:Wisk          0.9062 0.1912 0.0006982       0.012893
Solo:Solo             2.6179 0.7883 0.0028785       0.055837
Solo:Surf             0.5348 0.4307 0.0015728       0.030113
Solo:Tide             0.5570 0.3544 0.0012941       0.024548
Solo:Wisk             1.5442 0.4643 0.0016954       0.031574
Surf:Surf             1.6036 0.4758 0.0017374       0.031269
Surf:Tide             0.7689 0.2992 0.0010926       0.020253
Surf:Wisk             0.9949 0.3548 0.0012955       0.022963
Tide:Tide             1.2841 0.3660 0.0013364       0.024095
Tide:Wisk             1.0658 0.3147 0.0011492       0.020229
Wisk:Wisk             2.5801 0.5523 0.0020167       0.034974

2. Quantiles for each variable:

                         2.5%       25%      50%      75%    97.5%
(Intercept):EraPlus   2.09105   2.38514   2.5321   2.6926   3.0022
(Intercept):Solo      1.28315   1.57316   1.7219   1.8705   2.1639
(Intercept):Surf      1.24132   1.45272   1.5583   1.6701   1.9023
(Intercept):Tide      2.23562   2.53443   2.6886   2.8544   3.1721
(Intercept):Wisk      1.31120   1.50841   1.6104   1.7191   1.9429
price               -97.62406 -86.61736 -80.7783 -75.1013 -64.9720
EraPlus:Solo          0.32811   0.65755   0.8480   1.0694   1.4666
EraPlus:Surf         -0.24159  -0.01596   0.1131   0.2507   0.5491
EraPlus:Tide         -0.02109   0.16081   0.2571   0.3527   0.5957
EraPlus:Wisk          0.54089   0.77643   0.9035   1.0331   1.2917
Solo:Solo             1.37468   2.02720   2.5225   3.0985   4.4160
Solo:Surf            -0.31143   0.25493   0.5251   0.8180   1.3880
Solo:Tide            -0.05172   0.30518   0.5297   0.7740   1.3106
Solo:Wisk             0.74682   1.21192   1.5131   1.8380   2.5445
Surf:Surf             0.86883   1.25552   1.5377   1.8834   2.6935
Surf:Tide             0.30606   0.55218   0.7285   0.9413   1.4556
Surf:Wisk             0.40231   0.74907   0.9579   1.1957   1.8026
Tide:Tide             0.69841   1.02206   1.2396   1.4999   2.1100
Tide:Wisk             0.51732   0.84824   1.0411   1.2556   1.7579
Wisk:Wisk             1.60662   2.18760   2.5407   2.9238   3.7722
\end{verbatim}
\spacingset{1.5}

The output shows the mean, standard deviation, and various percentiles
of the posterior distributions of the coefficients and the elements of
the variance-covariance matrix. The base category is the detergent
\texttt{All}.  Separate intercepts are estimated for each detergent.
The price coefficient is negative and highly statistically significant,
agreeing with the standard economic expectation that consumers are
less likely to buy more expensive goods.

MNP also allows one to calculate the posterior predictive
probabilities of each alternative being most preferred given a
particular value of the covariates.  For example, one can calculate
the posterior predictive probabilities using the covariate values of
the first two observations by using the \texttt{predict()} command,

\spacingset{1}
\begin{verbatim}
predict(res1, newdata = detergent[1:2,],
        newdraw = rbind(res1$param[25001:50000,],
                        res2$param[25001:50000,],
                        res3$param[25001:50000,]), type = "prob")
\end{verbatim}
\spacingset{1.5} where \texttt{res1} is the output object from the
\texttt{mnp()} command, and we set \texttt{newdata} to the first two
observations of the detergent data set and \texttt{newdraw} to the
combined draws from the second half of three chains. Setting
\texttt{type = "prob"} causes the function \texttt{predict()} to
return the posterior predictive probabilities. Moreover, a new {\tt
  n.draws} option in \texttt{predict()} command allows one to compute
the uncertainty estimates about these predicted probabilities. It is
also possible to return a Monte Carlo sample of the the alternative
that is most preferred (\texttt{type = "choice"}), a Monte Carlo
sample of the latent variables (\texttt{type = "latent"}), or a Monte
Carlo sample of the preference-ordered alternatives (\texttt{type =
  "order"}).  (Type \texttt{help(predict.mnp)} in R for more details
about the \texttt{predict()} function in MNP.) The above command
yields the following output,

\spacingset{1}
\begin{verbatim}

            All   EraPlus    Solo       Surf      Tide       Wisk
[1,] 0.01281333 0.1946400 0.12292 0.46208000 0.1401733 0.06737333
[2,] 0.04649333 0.1262133 0.05996 0.03169333 0.3589867 0.37665333
\end{verbatim}
\spacingset{1.5} The result indicates that the posterior predictive
probability of purchasing \texttt{Surf} is the largest for households
with covariates equal to those in the first household in the data set.
Under the model, approximately 46\% of such households will purchase
\texttt{Surf}.  On the other hand, \texttt{All} is the brand least
likely to be purchased by these households.  The households with
covariates equal to the second household are most likely to buy
\texttt{Wisk}. Also, they are almost equally likely to purchase
\texttt{Tide}.  (The posterior predictive probabilities of buying
\texttt{Wisk} and \texttt{Tide} are both around 0.35)


\section{Example 2: Voters' Preference of Political Parties}
\label{sec:examp-japan}

Our second example illustrates how to fit the multinomial probit model
with ordered preferences (see Section~\ref{sec:mnpop}).

\subsection{Preliminaries}

We analyze a survey dataset describing the preferences of individual
voters in Japan among the political parties.  Political scientists may
be interested in using the gender, age and education level of voters
to predict their party preferences (Type {\tt help(japan)} in R
for details about the dataset).  The outcome variable is a vector of
relative preferences for each of the four parties, i.e., $p=4$. Each
of 418 voters is asked to give a score between 0 and 100 to each
party. For example, the first voter in the dataset has the following
preferences.
\begin{verbatim}
  LDP NFP SKG JCP
   80  75  80   0
\end{verbatim}
That is, this voter prefers \texttt{LDP} and \texttt{SKG} to
\texttt{NFP} and \texttt{JCP}, and between the latter two, she prefers
\texttt{NFP} to \texttt{JCP}.  Although \texttt{LDP} and \texttt{SKG}
have the same preference, we do not constrain the estimated
preferences to be the same for these two alternatives.  Under the
Gaussian random utility model, the probability that the two
alternatives having exactly the same preferences is zero.  Therefore,
inequality constraints are respected, but equality constraints are
not.

Furthermore, we only preserve the ranking, not the relative numerical
values.  Therefore, the following coding of the variables, for our
purposes, is equivalent to that given above,

\spacingset{1}
\begin{verbatim}
  LDP NFP SKG JCP
    3   2   3   1
\end{verbatim}
\spacingset{1.5}
Finally, it is possible to have non-response for one of the
categories; e.g., no candidate from a particular party may run in a
certain district.  If \texttt{NFP = NA}, we have no information about
the relative ranking of \texttt{NFP}.

\spacingset{1}
\begin{verbatim}
  LDP NFP SKG JCP
    3  NA   3   1
\end{verbatim}
\spacingset{1.5}
In this case, there is no constraint when estimating the preference
for this alternative; only the inequality constraint, \texttt{(LDP,
  SKG) > JCP}, is imposed.

All three covariates -- gender, education, and age of voters -- are
individual-specific variables rather than choice-specific ones. The
model estimates three intercepts and 9 coefficients along with 6
parameters in the covariance matrix. The following commands fit the
model,

\spacingset{1}
\begin{verbatim}
data(japan)
res <- mnp(cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan,
           n.draws = 10000, verbose = TRUE)
summary(res)
\end{verbatim}
\spacingset{1.5}
The first command loads the dataset, and the second command fits the
model. The base category is \texttt{JCP}, which is the last column of
the outcome matrix. The default prior distribution is used as in the
previous example: an improper prior distribution for $\beta$ and a
diffuse prior distribution for $\Sigma$ with $\nu = p = 4$ and $S=I$.
10,000 draws are obtained with no burnin or thinning. The final
command summarizes the Monte Carlo sample and gives the following
output,

\spacingset{1}
\begin{verbatim}

Call:
mnp(formula = cbind(LDP, NFP, SKG, JCP) ~ gender + education + age,
    data = japan, n.draws = 10000, verbose = TRUE)


Coefficients:
                     mean  std.dev.      2.5% 97.5%
(Intercept):LDP  0.615184  0.517157 -0.386151  1.61
(Intercept):NFP  0.689753  0.568109 -0.419521  1.79
(Intercept):SKG  0.133961  0.455960 -0.758883  1.02
gendermale:LDP   0.099748  0.152323 -0.194786  0.40
gendermale:NFP   0.216824  0.166103 -0.102108  0.54
gendermale:SKG   0.132661  0.134605 -0.127145  0.40
education:LDP   -0.107038  0.074792 -0.253483  0.04
education:NFP   -0.107222  0.082324 -0.270127  0.05
education:SKG   -0.003728  0.066429 -0.132496  0.13
age:LDP          0.013518  0.006122  0.001492  0.03
age:NFP          0.006948  0.006783 -0.006572  0.02
age:SKG          0.009653  0.005431 -0.000812  0.02

Covariances:
          mean std.dev.   2.5% 97.5%
LDP:LDP 1.0000   0.0000 1.0000  1.00
LDP:NFP 1.0502   0.0585 0.9373  1.16
LDP:SKG 0.7070   0.0622 0.5822  0.82
NFP:NFP 1.4068   0.1359 1.1682  1.70
NFP:SKG 0.7452   0.0864 0.5800  0.91
SKG:SKG 0.6913   0.0874 0.5296  0.87

Base category: JCP
Number of alternatives: 4
Number of observations: 418
Number of stored MCMC draws: 10000

\end{verbatim}
\spacingset{1.5}

\subsection{Convergence Diagnostics, Final Analysis, and Conclusions}

In order to evaluate convergence of the MCMC sampler, we again obtain
three independent Markov chains by running the \texttt{mnp()} command
three times with three sets of different starting values.  We use
starting values that are relatively dispersed given the preliminary
analysis of the previous section.  Note that when fitting the
multinomial probit model with ordered preferences, the algorithm
requires the starting values of the latent variable to respect the
order constraints of equation (\ref{eq:mop}). Therefore, the starting
values of the parameters cannot be too far away from the posterior
mode.  The following commands fits the model with the default starting
value and two sets of overdispersed starting values,

\spacingset{1}
\begin{verbatim}

res1 <- mnp(cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan,
            n.draws = 50000, verbose = TRUE)

res2 <- mnp(cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan,
            coef.start = c(1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1),
            cov.start = matrix(0.5, ncol=3, nrow=3) + diag(0.5, 3),
            n.draws = 50000, verbose = TRUE)

res3 <- mnp(cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan,
            coef.start = c(-1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1),
            cov.start = matrix(0.9, ncol=3, nrow=3) + diag(0.1, 3),
            n.draws = 50000, verbose = TRUE)
\end{verbatim}
\spacingset{1.5}
%res.coda <- mcmc.list(chain1=mcmc(res1$param[,-13]),
%                      chain2=mcmc(res2$param[,-13]),
%                      chain3=mcmc(res3$param[,-13]))
%gelman.plot(res.coda, transform = TRUE, ylim = c(1,1.2))
%res.coda <- mcmc.list(chain1=mcmc(res1$param[25001:50000,-13], start=25001),
%                      chain2=mcmc(res2$param[25001:50000,-13], start=25001),
%                      chain3=mcmc(res3$param[25001:50000,-13], start=25001))
%summary(res.coda)
We follow the commands used in Section~\ref{sec:coda} and compute the
Gelman-Rubin statistic for each parameter. Upon examination of the
resulting statistics, we determined that satisfactory convergence has
been achieved. For example, the value of the Gelman-Rubin statistic is
less than 1.01 for all the parameters.  Hence, we base our final
analysis on the combined draws from the second half of the three
chains (i.e., a total of 75,000 draws using 25,000 draws from each
chain). Posterior summaries can be obtained using the coda package as
before,

\spacingset{1}
\begin{verbatim}

Iterations = 25001:50000
Thinning interval = 1
Number of chains = 3
Sample size per chain = 25000

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

                    Mean      SD Naive SE Time-series SE
(Intercept):LDP  0.60167 0.51421 1.88e-03       8.05e-03
(Intercept):NFP  0.68294 0.56867 2.08e-03       7.95e-03
(Intercept):SKG  0.12480 0.45680 1.67e-03       7.25e-03
gendermale:LDP   0.10668 0.15448 5.64e-04       2.95e-03
gendermale:NFP   0.22240 0.16983 6.20e-04       2.91e-03
gendermale:SKG   0.13897 0.13753 5.02e-04       2.70e-03
education:LDP   -0.10517 0.07643 2.79e-04       1.35e-03
education:NFP   -0.10634 0.08448 3.08e-04       1.28e-03
education:SKG   -0.00258 0.06766 2.47e-04       1.18e-03
age:LDP          0.01361 0.00617 2.25e-05       9.90e-05
age:NFP          0.00698 0.00680 2.48e-05       1.01e-04
age:SKG          0.00972 0.00547 2.00e-05       9.13e-05
LDP:NFP          1.05535 0.05508 2.01e-04       1.15e-03
LDP:SKG          0.71199 0.06125 2.24e-04       1.59e-03
NFP:NFP          1.41860 0.13540 4.94e-04       2.45e-03
NFP:SKG          0.75391 0.08262 3.02e-04       2.12e-03
SKG:SKG          0.70007 0.08488 3.10e-04       2.16e-03

2. Quantiles for each variable:

                     2.5%      25%      50%     75%  97.5%
(Intercept):LDP -0.405757  0.25198  0.60033  0.9476 1.6172
(Intercept):NFP -0.428421  0.30016  0.68058  1.0657 1.7981
(Intercept):SKG -0.769018 -0.18476  0.12335  0.4303 1.0258
gendermale:LDP  -0.197199  0.00328  0.10643  0.2105 0.4096
gendermale:NFP  -0.110730  0.10856  0.22135  0.3361 0.5566
gendermale:SKG  -0.131661  0.04702  0.13905  0.2307 0.4096
education:LDP   -0.254631 -0.15718 -0.10519 -0.0533 0.0447
education:NFP   -0.271657 -0.16329 -0.10654 -0.0496 0.0595
education:SKG   -0.135829 -0.04833 -0.00248  0.0429 0.1306
age:LDP          0.001591  0.00941  0.01361  0.0177 0.0257
age:NFP         -0.006336  0.00240  0.00697  0.0115 0.0203
age:SKG         -0.000947  0.00603  0.00967  0.0134 0.0206
LDP:NFP          0.944577  1.01919  1.05564  1.0924 1.1623
LDP:SKG          0.587135  0.67120  0.71364  0.7544 0.8266
NFP:NFP          1.181667  1.32454  1.40803  1.5028 1.7125
NFP:SKG          0.590798  0.69778  0.75463  0.8104 0.9125
SKG:SKG          0.538858  0.64219  0.69806  0.7564 0.8711
\end{verbatim}
\spacingset{1.5} Here, one of the findings is that older voters tend
to prefer \texttt{LDP} as indicated by the statistically significant
positive age coefficient for \texttt{LDP}.  This is consistent with
the conventional wisdom of Japanese politics that the stronghold of
\texttt{LDP} is elderly voters.

To further investigate the marginal effect of age, we calculate the
posterior predictive probabilities of party preference under two
scenarios. First, we choose the 10th individual in the survey data and
compute the predictive probability that a voter with this set of
covariates prefers each of the parties. This can be accomplished by
the following commands,

\spacingset{1}\begin{verbatim}
japan10a <- japan[10,]
predict(res1, newdata = japan10a,
        newdraw = rbind(res1$param[25001:50000,],
                        res2$param[25001:50000,],
                        res3$param[25001:50000,]), type = "prob")
\end{verbatim}
\spacingset{1.5}
where the first command extracts the 10th observation from the Japan
data, and the second command computes the predictive probabilities.
Note that this individual has the following attributes,

\spacingset{1}
\begin{verbatim}
gender education age
  male         4  50
\end{verbatim}
\spacingset{1.5}
The resulting posterior predictive probabilities of being the most
preferred party are,

\spacingset{1}
\begin{verbatim}
          JCP      LDP      NFP      SKG
[1,] 0.107707 0.359267 0.324613 0.208413
\end{verbatim}
\spacingset{1.5} The result indicates that under the model, we should
expect 36\% of voters with these covariates to prefer \texttt{LDP},
32\% to prefer \texttt{NFP}, 21\% to prefer \texttt{SKG}, and 11\% to
prefer \texttt{JCP}.  Next, we change the value of the age variable of
this voter from 50 to 75, while holding the other variables constant.
We then recompute the posterior predictive probabilities and examine
how they change. This can be accomplished using the following
commands,

\spacingset{1}
\begin{verbatim}
japan10b <- japan10a
japan10b[,"age"] <- 75
predict(res1, newdata = japan10b,
        newdraw = rbind(res1$param[25001:50000,],
                        res2$param[25001:50000,],
                        res3$param[25001:50000,]), type = "prob")
\end{verbatim}
\spacingset{1.5}
where the first two commands recode the age variable for the voter and
the second command makes the prediction.  We obtain the following
results,

\spacingset{1}
\begin{verbatim}
         JCP      LDP      NFP      SKG
[1,] 0.06548 0.485467 0.249667 0.199387
\end{verbatim}
\spacingset{1.5}
The comparison of the two results shows that changing the value of the
age variable from 50 to 75 increases the estimated posterior
predictive probability of preferring \texttt{LDP} most and by more
than 10 percentage points.  Interestingly, the predictive probability
for \texttt{SKG} changes very little, while that of \texttt{NFP}
decreases significantly.  This suggests that older voters tend to
prefer \texttt{LDP} over \texttt{NFP}.


\clearpage
\pdfbookmark{References}{References}
\bibliography{my,imai}

\end{document}

