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\author{Hannah Frick\\Universit\"at Innsbruck \And
	Carolin Strobl\\Universit\"at Z\"urich \And
        Friedrich Leisch\\Universit\"at f\"ur\\Bodenkultur Wien \And
	Achim Zeileis\\Universit\"at Innsbruck}
\Plainauthor{Hannah Frick, Carolin Strobl, Friedrich Leisch, Achim Zeileis}

\title{Flexible Rasch Mixture Models with Package \pkg{psychomix}}
\Plaintitle{Flexible Rasch Mixture Models with Package psychomix}

\Keywords{mixed Rasch model, Rost model, mixture model, \pkg{flexmix}, \proglang{R}}
\Plainkeywords{mixed Rasch model, Rost model, mixture model, flexmix, R}

\Abstract{
  This introduction to the \proglang{R} package \pkg{psychomix} is a (slightly)
  modified version of \cite{psychomix:Frick+Strobl+Leisch:2012}, published in the
  \emph{Journal of Statistical Software}.

  Measurement invariance is an important assumption in the Rasch model and
  mixture models constitute a flexible way of checking for a violation of this
  assumption by detecting unobserved heterogeneity in item response data.
  Here, a general class of Rasch mixture models is established and implemented
  in \proglang{R}, using conditional maximum likelihood estimation of the item
  parameters (given the raw scores) along with flexible specification of two
  model building blocks: (1)~Mixture weights for the unobserved classes can be
  treated as model parameters or based on covariates in a concomitant variable
  model. (2)~The distribution of raw score probabilities can be parametrized
  in two possible ways, either using a saturated model or a
  specification through mean and variance. The function \fct{raschmix} in
  the \proglang{R} package \pkg{psychomix} provides these models, leveraging
  the general infrastructure for fitting mixture models in the
  \pkg{flexmix} package.
  Usage of the function and its associated methods is illustrated on
  artificial data as well as empirical data from a study of verbally
  aggressive behavior.
}

\Address{
  Hannah Frick, Achim Zeileis\\
  Department of Statistics\\
  Universit\"at Innsbruck\\
  Universit\"atsstr.~15\\
  6020 Innsbruck, Austria\\
  E-mail: \email{Hannah.Frick@uibk.ac.at}, \email{Achim.Zeileis@R-project.org}\\
  URL: \url{http://eeecon.uibk.ac.at/~frick/}, \url{http://eeecon.uibk.ac.at/~zeileis/}\\

  Carolin Strobl\\
  Department of Psychology\\
  Universit\"at Z\"urich\\
  Binzm\"uhlestr.~14\\
  8050 Z\"urich, Switzerland\\
  E-mail: \email{Carolin.Strobl@psychologie.uzh.ch}\\
  URL: \url{http://www.psychologie.uzh.ch/fachrichtungen/methoden.html}\\

  Friedrich Leisch\\
  Institute of Applied Statistics and Computing\\
  Universit\"at f\"ur Bodenkultur Wien\\
  Peter Jordan-Str.~82\\
  1180 Wien, Austria\\
  E-mail: \email{Friedrich.Leisch@R-project.org}\\
  URL: \url{http://www.rali.boku.ac.at/friedrichleisch.html}
}

%% Sweave/vignette information and metadata
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\SweaveOpts{engine = R, eps = FALSE, keep.source = TRUE}
%\VignetteIndexEntry{Flexible Rasch Mixture Models with Package psychomix}
%\VignetteDepends{graphics, stats, methods, Formula, flexmix, psychotools, psychomix}
%\VignetteKeywords{mixed Rasch model, Rost model, mixture model, flexmix, R}
%\VignettePackage{psychomix}

<<preliminaries, echo=FALSE, results=hide>>=
options(width = 70, prompt = "R> ", continue = "+  ", digits = 4)
library("psychomix")
library("lattice")
data("VerbalAggression", package = "psychotools")
suppressWarnings(RNGversion("3.5.0"))
set.seed(1090)
cache <- FALSE
@


\begin{document}

%\vspace*{-0.35cm}

\section{Introduction} \label{sec:intro}

%% Motivation

%%% Rasch model and its assumptions
In item response theory (IRT), latent traits are usually measured by 
employing probabilistic models for responses to sets of items.
One of the most prominent examples for such an approach is the
Rasch model \citep{psychomix:Rasch:1960} which captures the
difficulty (or equivalently easiness) of binary items and the
respondent's trait level on a single common scale.
Generally, a central assumption of most IRT models (including the Rasch model)
is measurement invariance, i.e., that all items measure the latent
trait in the same way for all subjects. If violated, measurements
obtained from such a model provide no fair comparisons of the
subjects. A typical violation of measurement invariance in the Rasch
model is differential item functioning (DIF), see \citet{psychomix:Ackerman:1992}.

%%% approaches to assess measurement invariance
Therefore, assessing the assumption of measurement invariance and checking
for DIF is crucial when establishing a Rasch model for measurements of
latent traits. Hence, various approaches have been suggested in the literature
that try to assess heterogeneity in (groups of) subjects either based on
observed covariates or unobserved latent classes. If covariates are available,
classical tests like the Wald or likelihood ratio test can be employed
to compare model fits between some reference group and one or more focal
groups \citep{psychomix:Glas+Verhelst:1995}. Typically, 
these groups are defined by the researcher based on categorical
covariates or arbitrary splits in either numerical covariates or the raw scores
\citep{psychomix:Andersen:1972}. More recently, extensions of these
classical tests have also been embedded into a mixed model
representation \citep{psychomix:VanDenNoortgate+DeBoeck:2005}. Another
recently suggested technique is to recursively define and assess
groupings in a data-driven way based on all available covariates (both
numerical and categorical) in so-called Rasch trees
\citep{psychomix:Strobl+Kopf+Zeileis:2011}.

%%% Rasch mixture models
Heterogeneity occurring in latent classes only (i.e., not observed or captured
by covariates), however, is typically addressed by mixtures of IRT models.
Specifically, \citet{psychomix:Rost:1990} combined a mixture model approach
with the Rasch model. If any covariates are present, they can be used
to predict the latent classes (as opposed to
the item parameters themselves) in a second step
\citep{psychomix:Cohen+Bolt:2005}. More recently, extensions to this
mixture model approach have been suggested that encompass this
prediction, see \citet{psychomix:Tay+Newman+Vermunt:2011} for a
unifying framework.

DIF cannot be separated from multidimensionality when items
measure several different latent traits which also differ in (groups
of) subjects \citep[cf., e.g.,][and the references
therein]{psychomix:Ackerman:1992}. In this sense, mixtures of Rasch
models can also be used to assess the assumption of unidimensionality, see
\citet{psychomix:Rijmen+DeBoeck:2005} on the relationship of
between-item multidimensional models and mixtures of Rasch models.

%% Software

%%% our contribution
In this paper, we introduce the \pkg{psychomix} package for the
\proglang{R} system for statistical computing \citep{psychomix:R:2011}
that provides software for fitting a general and flexible class of
Rasch mixture models along with comprehensive methods for model
selection, assessment, and visualization. The package leverages the
general and object-oriented infrastructure for fitting mixture models
from the \pkg{flexmix} package
\citep{psychomix:Leisch:2004,psychomix:Gruen+Leisch:2008}, combining
it with the function \fct{RaschModel.fit} from the \pkg{psychotools} package
\citep{psychomix:Zeileis+Strobl+Wickelmaier:2011} for the estimation of
Rasch models. All packages are freely available from the Comprehensive
\proglang{R} Archive Network at \url{http://CRAN.R-project.org/}.

The reason for using \fct{RaschModel.fit} as opposed to other
previously existing (and much more powerful and flexible)
\proglang{R}~packages for Rasch modeling -- such as \pkg{ltm} \citep{psychomix:Rizopoulos:2006}
or \pkg{eRm} \citep{psychomix:Mair+Hatzinger:2007} -- is reduced computational
complexity: \fct{RaschModel.fit} is intended to provide a ``no frills''
implementation of simple Rasch models, useful when refitting a model
multiple times in mixtures or recursive partitions
\citep[see also][]{psychomix:Strobl+Kopf+Zeileis:2011}. 

%% While \pkg{psychomix} was under development, another \proglang{R} implementation
%% of the \cite{psychomix:Rost:1990} model became available in package
%% \pkg{mRm} \citep{psychomix:Preinerstorfer+Formann:2011}. As this builds on
%% specialized \proglang{C++} code, it runs considerably faster than the generic
%% \pkg{flexmix} approach -- however, it only covers this one particular type
%% of model and offers fewer methods for specifying, inspecting, and assessing
%% (fitted) models. In \pkg{psychomix}, both approaches are reconciled by
%% optionally employing the \pkg{mRm} solution as an input to the \pkg{flexmix}
%% routines.


%% overview

In the following, we first briefly review both Rasch and mixture models
and combine them in a general Rasch mixture framework
(Section~\ref{sec:theory}). Subsequently, the \proglang{R}
implementation in \pkg{psychomix} is introduced
(Section~\ref{sec:implementation}), illustrated by means of simulated
data, and applied in practice to a study of verbally aggressive
behavior (Section~\ref{sec:application}). Concluding remarks are
provided in Section~\ref{sec:summary}.


%\vspace*{-0.2cm}


\section{Rasch mixture models} \label{sec:theory}

In the following, we first provide a short introduction to the Rasch model,
subsequently outline the basics of mixture models in general, and finally
introduce a general class of Rasch mixture models along with the
corresponding estimation techniques.

\subsection{The Rasch model} \label{sec:Rasch}

Latent traits can be measured through a set of items to which binary
responses are given. Success of solving an item or agreeing with it is
coded as \dquote{1}, while \dquote{0} codes the opposite response. The
model suggested by \citet{psychomix:Rasch:1960} uses the person's ability
$\theta_i$ ($i = 1, \dots, n$) and the item's difficulty $\beta_j$ ($j
= 1, \dots, m$) to model the response $y_{ij}$ of person $i$ to item $j$:
%
\begin{equation}
  P(Y_{ij} = y_{ij} | \theta_i, \beta_j) ~=~
  \frac{\exp \{y_{ij}(\theta_i - \beta_j)\}}{1 + \exp \{\theta_i -
    \beta_j\}}. \label{eq:RM:model}
\end{equation}
%

Under the assumption of independence -- both across persons and items
within persons \citep[see][]{psychomix:Molenaar:1995}
-- the likelihood for the whole sample $y = (y_{ij})_{n \times m}$ can
be written as the product of the likelihood contributions from
Equation~\ref{eq:RM:model} for all combinations of subjects and
items. It is parametrized by the vector of all person parameters
$\theta = (\theta_1, \dots, \theta_n)^\top$ and the vector of all item
parameters $\beta = (\beta_1, \dots, \beta_m)^\top$ (see
Equation~\ref{eq:RM:L:fac:f}). 
Since the probability of solving an item is modeled only
through the logit of the difference between person ability and
item difficulty, these parameters are on an interval scale with an
arbitrary zero point. To obtain a fixed reference point, usually
one item difficulty (e.g., the first $\beta_1$) or the sum of 
item difficulties is restricted to zero. See
\citet{psychomix:Fischer:1995} for details.

On the basis of the number of correctly
solved items, the so-called ``raw'' scores $r_i =
\sum_{j=1}^{m}{y_{ij}}$, the likelihood for the full sample can be
factorized into a conditional likelihood of the item parameters
$h(\cdot)$ and the score probabilities $g(\cdot)$
(Equation~\ref{eq:RM:L:fac:theta}). Because the scores $r$ are
sufficient statistics for the person parameters $\theta$, the
likelihood of the item parameters $\beta$ conditional on the
scores~$r$ does not depend on the person parameters~$\theta$
(Equation~\ref{eq:RM:L:fac}).
\begin{eqnarray}
  L(\theta, \beta) &=& f(y | \theta, \beta) \label{eq:RM:L:fac:f} \\
  &=& h(y | r, \theta, \beta) g(r | \theta, \beta) \label{eq:RM:L:fac:theta} \\
  &=& h(y | r, \beta) g(r | \theta, \beta) \label{eq:RM:L:fac}.
\end{eqnarray}
The conditional likelihood of the item parameters takes the form
\begin{equation}
  h(y| r, \beta) ~=~ \prod_{i = 1}^{n}{\frac
    {\exp\{- \sum_{j = 1}^{m}{y_{ij} \beta_j}\}}
    {\gamma_{r_i}(\beta)}} \label{eq:RM:condL}
\end{equation}
where $\gamma_{r_i}(\cdot)$ is the elementary symmetric function of order
$r_i$, capturing all possible response patterns leading to a certain
score \citep[see][for details]{psychomix:Molenaar:1995a}.

There are several approaches to estimating the Rasch model:
\emph{Joint maximum likelihood (ML)} estimation of $\beta$ and
$\theta$ is inconsistent, thus two other approaches have been
established. Both are two-step approaches but differ in the way the
person parameters $\theta$ are handled. For \emph{marginal ML}
estimation a distribution for $\theta$ is assumed and integrated out
in~$L(\theta, \beta)$, or equivalently in~$g(r | \theta, \beta)$. 
In the \emph{conditional ML} approach only the conditional likelihood
of the item parameters~$h(y | r, \beta)$ from
Equation~\ref{eq:RM:condL} is maximized for estimating the item
parameters. Technically, this is equivalent to maximizing $L(\theta, \beta)$
with respect to $\beta$ if one assumes that $g(r | \delta) = g(r |
\theta, \beta)$ does not depend on $\theta$ or $\beta$, but
potentially other parameters $\delta$.

In \proglang{R}, the \pkg{ltm} package
\citep{psychomix:Rizopoulos:2006} uses the marginal ML approach while
the \pkg{eRm} package \citep{psychomix:Mair+Hatzinger:2007} employs
the conditional ML approach, i.e., uses and reports only the conditional part
of the likelihood in the estimation of $\beta$.
The latter approach is also taken by the \fct{RaschModel.fit} function in the
\pkg{psychotools} package \citep{psychomix:Zeileis+Strobl+Wickelmaier:2011}.


\subsection{Mixture models} \label{sec:mixtureModels}

Mixture models are a generic approach for modeling data that is assumed to
stem from different groups (or clusters) but group membership is unknown.

The likelihood $f(\cdot)$ of such a mixture model is a weighted sum (with prior
weights $\pi_k$) of the likelihood from several components $f_k(\cdot)$
representing the different groups:
%
\begin{equation*}
  f(y_i) ~=~ \sum_{k = 1}^{K}{\pi_k f_k(y_i)}.
\end{equation*}
%
Generally, the components $f_k(\cdot)$ can be densities or
(regression) models. Typically, all\linebreak $K$~components $f_k(\cdot)$ are
assumed to be of the same type $f(y | \xi_k)$, distinguished through
their component-specific parameter vector $\xi_k$.

If variables are present which do not influence the components
$f_k(\cdot)$ themselves but rather the prior class membership probabilities
$\pi_k$, they can be incorporated in the model as so-called
\emph{concomitant variables}
\citep{psychomix:Dayton+Macready:1988}. In the psychometric 
literature, such covariates predicting latent information are
also employed, e.g., by \citet{psychomix:Tay+Newman+Vermunt:2011} who
advocate a unifying IRT framework that also optionally encompasses
concomitant information (labeled \emph{MM-IRT-C} for mixed-measurement IRT
with covariates). To embed such concomitant variables $x_i$ into the
general mixture model notation, a model for the component membership probability
$\pi(k | x_i, \alpha)$ with parameters~$\alpha$ is employed:
%
\begin{equation}
  f(y_i | x_i, \alpha, \xi_1, \ldots, \xi_K)
    ~=~ \sum_{k = 1}^{K}{\pi(k | x_i, \alpha) f(y_i | \xi_k)}, \label{eq:MM:conc}
\end{equation}
%
where commonly a multinomial logit model is chosen to parametrize
$\pi(k | x_i, \alpha)$ \citep[see e.g.,][]{psychomix:Gruen+Leisch:2008,psychomix:Tay+Newman+Vermunt:2011}.
Note that the multinomial model collapses to separate $\pi_k$ ($k = 1, \dots, K$)
if there is only an intercept and no real concomitants in $x_i$.

\subsection{Flavors of Rasch mixture models}\label{sec:flavors}

When combining the general mixture model framework from
Equation~\ref{eq:MM:conc} with the Rasch model based on
Equation~\ref{eq:RM:model}, several options are conceivable for two of
the building blocks. First, the component weights can be estimated via
a separate parameter $\pi_k$ for each component or via a
concomitant variable model $\pi(k | x_i, \alpha)$ with parameters $\alpha$.
Second, the full likelihood function $f(y_i | \xi_k)$ of the components needs
to be defined. If a conditional ML approach is adopted, it is clear
that the conditional likelihood $h(y_i | r_i, \beta)$ from
Equation~\ref{eq:RM:condL} should be one part, but various choices for
modeling the score probabilities are available. One option is to model
each score probability with its own parameter $g(r_i) = \psi_{r_i}$,
while another (more parsimonious) option would be to adopt a
parametric distribution of the score probabilities with fewer
parameters \citep{psychomix:Rost+VonDavier:1995}.

Note that while for a single-component model, the
estimates of the item parameters $\hat \beta$ are invariant to the choice
of the score probabilities (as long as it is independent from
$\beta$), this is no longer the case for a mixture model with $K \ge
2$. The estimation of the item parameters in the mixture is invariant
given the weights $\pi(k | x_i, \alpha)$ but the weights and thus the
estimates of $\beta$ may depend on the score distribution in a mixture
model. (This dependency is introduced through the
posterior probabilities calculated in the E-step of the algorithm
that is explained Section~\ref{sec:estimation}.)
Also note that the conditional ML approach employed here uses a model
$g(r | \delta)$ directly on the score probabilities rather than a
distribution on the person parameters $\theta$ as does the marginal
ML approach.


\subsubsection{Rost's original parametrization}

One of these possible mixtures --~the so-called ``mixed Rasch model''
introduced by \citet{psychomix:Rost:1990}~-- is already well-established
in the psychometric literature. It models the score probabilities
through separate parameters $g(r_i) = \psi_{r_i}$ (under the
restriction that they sum to unity) and does not employ concomitant
variables. The likelihood of Rost's mixture model can thus be written
as
%
\begin{equation}
  f(y | \pi, \psi, \beta) = 
  \prod_{i=1}^{n}{\sum_{k = 1}^{K}{\pi_k h(y_i | r_i, \beta_k) \psi_{r_{i},k}}}.
  \label{eq:Rost}
\end{equation}
%
%% This particular parametrization is implemented in the
%% \proglang{R} package \pkg{mRm} \citep{psychomix:Preinerstorfer+Formann:2011}.

Since subjects who solve either none or all items (i.e., $r_i = 0$ or
$m$, respectively) do not contribute to the conditional likelihood of
the item parameters they cannot be allocated to any of the components
in this parametrization. Hence, \citet{psychomix:Rost:1990} proposed
to remove those \dquote{extreme scorers} from the analysis entirely
and fix the corresponding score probabilities $\psi_0$ and $\psi_m$ at
0. However, if one wishes to include these extreme scorers in the
analysis, the corresponding score probabilities can be estimated through their
relative frequency (across all components) and the remaining score
probabilities within each component are rescaled to sum to unity
together with those extreme score probabilities. Nevertheless, the
extreme scorers still do not contribute to the estimation of the mixture itself.

\subsubsection{Other score distributions}

As noted by \cite{psychomix:Rost+VonDavier:1995}, the disadvantage of
this saturated model for the raw score probabilities is that many
parameters need to be estimated ($K \times (m-2)$, not counting
potential extreme scorers) that are typically not of interest. To
check for DIF, the item parameters are of prime importance while the
raw score distribution can be regarded as a nuisance term. This 
problem can be alleviated by embedding the model from Equation~\ref{eq:Rost}
into a more general framework that also encompasses more parsimonious
parametrizations. More specifically, a conditional logit model can be
established
% 
\begin{equation}
  \label{eq:scoreProbs:condLogit}
  g(r | \delta) ~=~
      \frac{\exp\{ z_{r}^\top \delta \}}
           {\sum_{j=1}^{m-1}{\exp\{z_{j}^\top \delta \}}},
\end{equation}
%
containing some auxiliary regressors $z_i$ with coefficients $\delta$.

The saturated $g(r_i) = \psi_{r_i}$ model is a special case
when constructing the auxiliary regressor from indicator/dummy variables
for the raw scores $2, \dots, m-1$: $z_i = (I_2(r_i), \dots, I_{m-1}(r_i))^\top$.
Then $\delta = (\log(\psi_{2}) - \log(\psi_1), \dots, \log(\psi_{m-1})
- \log(\psi_1))^\top$ is a simple logit transformation of $\psi$.

As an alternative \citet{psychomix:Rost+VonDavier:1995} suggests a specification
with only two parameters that link to mean and variance of the score 
distribution, respectively. More specifically, the auxiliary regressor
is $z_i = (r_i/m, 4 r_i (m - r_i)/m^2)^\top$ so that $\delta$
pertains to the vector of location and dispersion parameters
of the score distribution. Thus, for $m > 4$ items, this
parametrization is more parsimonious than the saturated model.

\subsubsection{General Rasch mixture model}

Combining all elements of the likelihood this yields a more general
specification of the Rasch mixture model
%
\begin{equation}
  f(y | \alpha, \beta, \delta) = 
  \prod_{i=1}^{n}{\sum_{k = 1}^{K}{\pi(k | x_i, \alpha)
    ~ h(y_i | r_i, \beta_k) ~ g(r_i | \delta_k)}} 
  \label{eq:RMMgeneral}
\end{equation}
%
with (a)~the concomitant model $\pi(k | x_i, \alpha)$ for modeling component
membership, (b)~the component-specific conditional likelihood of the
item parameters given the scores $h(y_i | r_i, \beta_k)$, and (c)~the
component-specific score distribution $g(r_i | \delta_k)$.

%\pagebreak

\subsection{Parameter estimation} \label{sec:estimation}

Parameter estimation for mixture models is usually done via the
expectation-maximization (EM) algorithm
\citep{psychomix:Dempster+Laird+Rubin:1977}. It treats group 
membership as unknown and optimizes the complete-data log-likelihood
including the group membership on basis of the observed values
only. It iterates between two steps until convergence: estimation of
group membership (E-step) and estimation of the components (M-step).

In the E-step, the posterior probabilities of each observation for the
$K$~components is estimated through:
\begin{equation}
  \label{eq:Estep:post}
  \hat{p}_{ik} ~=~ \frac{\hat{\pi}_k f(y_i | \hat{\xi}_k)}{\sum_{g=1}^{K}{\hat{\pi}_gf(y_i | \hat{\xi}_g)}}
\end{equation}
using the parameter estimates from the previous iteration for the
component weights $\pi$ and the model parameters $\xi$ which encompass
$\beta$ and $\delta$. In the case of concomitant variables, the
weights are $\hat \pi_{ik} = \pi(k | x_i, \hat \alpha)$. 

In the M-step, the parameters of the mixture are re-estimated with
the posterior probabilities as weights. Thus, observations deemed
unlikely to belong to a certain component have little influence on
estimation within this component.
For each component, the weighted ML estimation can be written as
%
\begin{eqnarray}
  \label{eq:MM:Mstep:comp}
  \hat{\xi}_{k}
    & = & \argmax_{\xi_k} \sum_{i=1}^{n}{\hat{p}_{ik}
          \log f(y_i | \xi_k)} \quad (k = 1, \ldots, K) \\
    & = & \left\{
          \argmax_{\beta_k} \sum_{i=1}^{n}{\hat{p}_{ik} \log h(y_i | r_i, \beta_k)};~
          \argmax_{\delta_k} \sum_{i=1}^{n}{\hat{p}_{ik} \log g(r_i | \delta_k)}
	  \right\} \nonumber    
\end{eqnarray}
which for the Rasch model amounts to separately maximizing the
weighted conditional log-likelihood for the item parameters and the
weighted score log-likelihood.

The concomitant model can be estimated separately from the posterior
probabilities:
\begin{equation}
  \label{eq:MM:Mstep:conc}
  \hat{\alpha} = \argmax_{\alpha} \sum_{i = 1}^{n}{\sum_{k =
      1}^{K}{\hat{p}_{ik} \log (\pi(k | x_i, \alpha))}},
\end{equation}

where $\pi(k | x_i, \alpha)$ could be, e.g, a multinomial model.

Finally, note that the number of components $K$ is not a standard model
parameter (because the likelihood regularity conditions do not apply)
and thus it is not estimated through the EM algorithm. Either it needs
to be chosen by the practitioner or by model selection techniques such
as information criteria, as illustrated in the following examples.
%% The standard approach of a likelihood ratio test is not applicable
%% here because regularity conditions are not fulfilled for mixture
%% models, thus the distribution under $H_0$ is unknown. It can only be
%% approximated through bootstrap procedures. Another approach is to
%% establish the number of components through information criteria. While
%% no p-values are available, also no bootstrapping is required. This
%% approach will be used here.



\section[Implementation in R]{Implementation in \proglang{R}}
\label{sec:implementation}

\subsection{User interface} \label{sec:interface}

The function \fct{raschmix} can be used to fit the different
flavors of Rasch mixture models described in Section~\ref{sec:flavors}: 
with or without concomitant variables in $\pi(k | x_i, \alpha)$,
and with different score distributions $g(r_i | \delta_k)$
(saturated vs.\ mean/variance parametrization). The function's synopsis is
%
\begin{Code}
raschmix(formula, data, k, subset, weights, scores = "saturated",
  nrep = 3, cluster = NULL, control = NULL,
  verbose = TRUE, drop = TRUE, unique = FALSE, which = NULL,
  gradtol = 1e-6, deriv = "sum", hessian = FALSE, ...)
\end{Code}
%
where the lines of arguments pertain to (1)~data/model specification
processed within\linebreak \fct{raschmix},
(2)~control arguments for fitting a single mixture model,
(3)~control arguments for iterating across mixtures over a range of
numbers of components $K$, all passed to \fct{stepFlexmix}, and
(4)~control arguments for fitting each model component within a mixture
(i.e., the M-step) passed to \fct{RaschModel.fit}. Details are
provided below, focusing on usage in practice first.

A formula interface with the usual \code{formula}, \code{data}, \code{subset},
and \code{weights} arguments is used: The left-hand side of the formula sets up
the response matrix $y$ and the right-hand side the concomitant variables
$x$ (if any). The response may be provided by a single matrix
or a set of individual dummy vectors, both of which
may be contained in an optional data frame. Example usages are
\code{raschmix(resp ~ 1, ...)} if the matrix \code{resp} is an object
in the environment of the formula, typically the calling environment,
or \code{raschmix(item1 + item2 + item3 ~ 1, data = d, ...)} if the
\code{item*} vectors are in the data frame \code{d}. In both cases,
\code{~ 1} signals that there are no concomitant variables -- if there
were, they could be specified as \code{raschmix(resp ~ conc1 + conc2,
  ...)}. As an additional convenience, the formula may be omitted
entirely if there are no concomitant variables, i.e.,
\code{raschmix(data = resp, ...)} or alternatively \code{raschmix(resp, ...)}.

The \code{scores} of the model can be set to either \code{"saturated"} 
(see Equation~\ref{eq:Rost}) or \code{"meanvar"} for the mean/variance
specification of \cite{psychomix:Rost+VonDavier:1995}. Finally, the number of
components $K$ of the mixture is specified through \code{k}, which may
be a vector resulting in a mixture model being fitted for each element.

To control the EM algorithm for fitting the specified mixture models,
\code{cluster} may optionally specify starting probabilities $\hat p_{ik}$
and \code{control} can set certain control arguments through a named list
or an object of class \class{FLXcontrol}.
One of these control arguments named \code{minprior} sets the
minimum prior probability for every component. If in an iteration of
the EM algorithm, any component has a prior probability smaller then
\code{minprior}, it is removed from the mixture in the next
iteration. The default is \code{0}, i.e., avoiding such shrinkage of the model.
If \code{cluster} is not provided, \code{nrep} different random
initializations are employed, keeping only the
best solution (to avoid local optima). 
%% Finally, \code{cluster} can be
%% set to \code{"mrm"} in which case the fast \proglang{C++} implementation
%% from \pkg{mRm} \citep{psychomix:Preinerstorfer+Formann:2011} can be leveraged
%% to generate optimized starting values. Again, the best solution of
%% \code{nrep} runs of \fct{mrm} is used. Note that as of version~1.0
%% of \pkg{mRm} only the model from Equation~\ref{eq:Rost} is supported
%% in \fct{mrm}, resulting in suboptimal -- but potentially still useful --
%% posterior probabilities $\hat p_{ik}$ for any other model flavor.

Internally, \fct{stepFlexmix} is called to fit all individual mixture
models and takes control arguments \code{verbose}, \code{drop}, and
\code{unique}. If \code{k} is a vector, the whole set of models is
returned by default but one may choose to select only the best model
according to an information criterion. For example,
\code{raschmix(resp, k = 1:3, which = "AIC", ...)} or
\code{raschmix(resp ~ 1, data = d, k = 1:4, which = "BIC", ...)}. 

The arguments \code{gradtol}, \code{deriv} and
\code{hessian} are used to control the estimation of the item
parameters in each M-step (Equation~\ref{eq:MM:Mstep:comp}) carried out
via \fct{RaschModel.fit}.

%\pagebreak
Function \fct{raschmix} returns objects of class \class{raschmix}
or \class{stepRaschmix}, respectively, depending on whether a single
or multiple mixture models are fitted. These classes extend
\class{flexmix} and \class{stepFlexmix}, respectively, for more technical
details see the next section. For standard methods for extracting or displaying
information, either for \class{raschmix} directly or by inheritance, see
Table~\ref{tab:methods} for an overview.

\begin{table}[t!]
  \centering
  \begin{tabular}[b]{|l|l|p{9cm}|}
    \hline
    Function & Class & Description\\
    \hline
    \fct{summary} & \class{raschmix} & display information about the posterior
    probabilities and item parameters; returns an object of class
    \class{summary.raschmix} containing the relevant summary
    statistics (which has a \fct{print} method)\\
    \fct{parameters} & \class{raschmix} & extract estimated parameters
    of the model for all or specified components, extract either
    parameters $\alpha$ of the concomitant model or item parameters
    $\beta$ and/or score parameters $\delta$\\
    \fct{worth} & \class{raschmix} & extract the item parameters $\beta$ under
    the restriction $\sum_{j = 1}^{m}{\beta_j} = 0$\\
    \fct{scoreProbs} & \class{raschmix} & extract the score
    probabilities $g(r | \delta)$\\
    \fct{plot} & \class{raschmix} & base graph of item parameter
    profiles in all or specified components\\
    \fct{xyplot} & \class{raschmix} & lattice graph of item parameter
    profiles of all or specified components in a single or multiple panels\\
    \fct{histogram} & \class{raschmix} & lattice rootogram or
    histogram of posterior probabilities\\
    \hline
    \fct{print} & \class{stepFlexmix} & simple printed display of
    number of components, log-likelihoods, and information criteria\\
    \fct{plot} & \class{stepFlexmix} & plot information criteria
    against number of components\\
    \fct{getModel} & \class{stepFlexmix} & select model according to
    either an information criterion or the number of components\\
    \fct{print} & \class{flexmix} & simple printed display with
    cluster sizes and convergence\\
    \fct{clusters} & \class{flexmix} & extract predicted class memberships\\
    \fct{posterior} & \class{flexmix} & extract posterior class probabilities\\
    \fct{logLik} & \class{flexmix} & extract fitted log-likelihood\\
    \fct{AIC}; \fct{BIC} & \class{flexmix} & compute
    information criteria AIC, BIC\\
    \hline
  \end{tabular}
  \caption{\label{tab:methods}Methods for objects of classes
    \class{raschmix}, \class{flexmix}, and \class{stepFlexmix}.}
\end{table}


\subsection{Internal structure} \label{sec:flexmix}

As briefly mentioned above, \fct{raschmix} leverages the \pkg{flexmix}
package \citep{psychomix:Leisch:2004,psychomix:Gruen+Leisch:2008} and
particularly its \fct{stepFlexmix} function for the estimation of (sets
of) mixture models.

The \pkg{flexmix} package is designed specifically to
provide the infrastructure for flexible mixture modeling via the EM algorithm,
where the type of a mixture model is determined through the model employed
in the components. In the estimation process, this component model
definition corresponds to the definition of the M-step
(Equation~\ref{eq:MM:Mstep:comp}). 
Consequently, the \pkg{flexmix} package provides the framework
for fitting mixture models by leveraging the modular structure of the
EM algorithm.  Provided with the right M-step, \pkg{flexmix} takes care of the
data handling and iterating estimation through both E-step and M-step.

% The M-step needs to be provided in the form of a \code{flexmix}
% driver inheriting from class \class{FLXM} \citep[see][for details]{psychomix:Gruen+Leisch:2008}.
% The \pkg{psychomix} package includes two such driver functions:
% \fct{FLXMCraschrost} for fitting Rost's version (Equation~\ref{eq:Rost})
% and \fct{FLXMCraschcond} for fitting the conditional version (Equation~\ref{eq:Cond})
% of the Rasch mixture model, respectively. Both drivers rely on
% the function \fct{RaschModel.fit} from the \pkg{psychotools} package
% for estimation of the item parameters (i.e., maximization of the 
% conditional likelihood from Equation~\ref{eq:RM:condL}) but add
% different estimates of raw score probabilities. 
The M-step needs to be provided in the form of a \code{flexmix}
driver inheriting from class \class{FLXM} \citep[see][for
details]{psychomix:Gruen+Leisch:2008}. The \pkg{psychomix} package
includes such a driver function: \fct{FLXMCrasch} relies on
the function \fct{RaschModel.fit} from the \pkg{psychotools} package
for estimation of the item parameters (i.e., maximization of the 
conditional likelihood from Equation~\ref{eq:RM:condL}) and adds
different estimates of raw score probabilities depending on their
parametrization. The driver can also be used directly with
functions \fct{flexmix} and \fct{stepFlexmix}. The differences in
model syntax and functionality for the classes of the resulting
objects are illustrated in the Appendix~\ref{sec:appendix}.
As noted in the introduction, the reason for employing
\fct{RaschModel.fit} rather than one of the more established Rasch
model packages such as \pkg{eRm} or \pkg{ltm} is speed.
% \fct{RaschModel.fit} has been designed with reduced flexibility in
% order to save time when refitted multiple times as in Rasch mixture
% models or also Rasch trees in the \pkg{psychotree} package
% \citep{psychomix:Strobl+Kopf+Zeileis:2011}.

In the \pkg{flexmix} package, two fitting functions are
provided. \fct{flexmix} is designed for fitting one model
once and returns an object of class \class{flexmix}. \fct{stepFlexmix}
extends this so that either a single model or several models can be
fitted. It also provides the functionality to fit each model
repeatedly to avoid local optima. 

When fitting models repeatedly, only the solution with the highest likelihood is
returned. Thus, if \fct{stepFlexmix} is used to repeatedly fit a
single model, it returns an object of class \class{flexmix}. If
\fct{stepFlexmix} is used to fit several different models (repeatedly or just
once), it returns an object of class \class{stepFlexmix}.

This principle extends to \fct{raschmix}: If it is used to fit a
single model, the returned object is of class \class{raschmix}. If
used for fitting multiple models, \fct{raschmix} returns an object of
class \class{stepRaschmix}. Both classes extend their \pkg{flexmix}
counterparts.


\subsection{Illustrations} \label{sec:illustration}

For illustrating the flexible usage of \fct{raschmix}, we employ an artificial
data set drawn from one of the three data generating processes (DGPs) suggested
by \citet{psychomix:Rost:1990} for the introduction of Rasch mixture models.
All three DGPs are provided in the function \fct{simRaschmix} setting
the \code{design} to \code{"rost1"}, \code{"rost2"}, or
\code{"rost3"}, respectively. The DGPs contain mixtures of $K = 1$, and
2, and 3 components, respectively, all with $m = 10$ items.

The DGP \code{"rost1"} is intended to illustrate the model's capacity
to correctly determine when no DIF is present. Thus, it includes only
one latent class with item parameters $\beta^{(1)} = (2.7, 2.1, 1.5,
0.9, 0.3, -0.3, -0.9, -1.5, -2.1, -2.7)^\top$. (Rost originally used
opposite signs to reflect item easiness parameters but since 
difficulty parameters are estimated by \fct{raschmix} the signs have
been reversed.) The DGP \code{"rost2"} draws observations from
two latent classes of equal sizes with item parameters of opposite signs:
$\beta^{(1)}$ and $\beta^{(2)} = -\beta^{(1)}$, respectively
(see Figure~\ref{fig:itemCompPlot} for an example). 
Finally, for the DGP \code{"rost3"} a third component is added with
item parameters $\beta^{(3)} = (-0.5, 0.5, -0.5, 0.5, -0.5, 0.5,
-0.5, 0.5, -0.5, 0.5)^\top$. The prior probabilities for the latent
classes with item parameters $\beta^{(1)}$, $\beta^{(2)}$, and
$\beta^{(3)}$ are $4/9$, $2/9$, and $3/9$ respectively.
In all three DGPs, the person parameters $\theta$ are drawn from a discrete
uniform distribution on $\{ 2.7,\;  0.9, -0.9, -2.7 \}$, except for
the third class of DGP \code{"rost3"} which uses only one level of
ability, drawn from the before-mentioned set of four ability
levels.
In all DGPs, response vectors for 1800~subjects are initially
drawn but the extreme scorers who solved either none or all items are
excluded.

Here, a dataset from the second DGP is generated
along with two artificial covariates \code{x1} and
\code{x2}. Covariate \code{x1} is an informative binary variable
(i.e., correlated with the true group membership) while \code{x2} is
an uninformative continuous variable.
%
<<data-rost2>>=
set.seed(1)
r2 <- simRaschmix(design = "rost2")
d <- data.frame(
  x1 = rbinom(nrow(r2), prob = c(0.4, 0.6)[attr(r2, "cluster")], size = 1),
  x2 = rnorm(nrow(r2))
)
d$resp <- r2
@ 
% 
The \cite{psychomix:Rost:1990} version of the Rasch mixture model --
i.e., with a saturated score model and without concomitant variables
-- is fitted for one to three components. As no concomitants are
employed in this model flavor, the matrix \code{r2} can be passed to
\fct{raschmix} without formula:
%
<<fitRost0, echo = TRUE, eval = FALSE>>=
set.seed(2)
m1 <- raschmix(r2, k = 1:3)
m1
@ 
<<fitRost, echo = FALSE, eval = TRUE, results=hide>>=
if(cache & file.exists("m1.rda")) {
  load("m1.rda")
} else {
<<fitRost0>>
if(cache) {
  save(m1, file = "m1.rda")
} else {
  if(file.exists("m1.rda")) file.remove("m1.rda")
}
}
@
<<printRost, echo = FALSE, eval = TRUE>>=
m1
@
% 
\begin{figure}[t!]
\centering
\setkeys{Gin}{width=0.7\textwidth}
<<plotRost, echo = FALSE, fig = TRUE, height = 4, width = 5.5>>=
plot(m1)
@ 
\caption{\label{fig:r2RostNComp} Information criteria for Rost's model
  with $K = 1, 2, 3$ components for the artificial scenario~2 data.}
\end{figure}
%
To inspect the results, the returned object can either be printed, as
illustrated above, or plotted yielding a visualization of information
criteria (see Figure~\ref{fig:r2RostNComp}). Both printed display
and visualization show a big difference in information criteria across
number of components $K$, with the minimum always being assumed for $K = 2$,
thus correctly recovering the two latent classes constructed in the
underlying DGP.

The values of the information criteria can also be accessed directly
via the functions of the corresponding names. 
To select a certain model from a \class{stepRaschmix} object, the
\fct{getModel} function from the \pkg{flexmix} package can be
employed. The specification of \code{which} model is to be selected
can either be an information criterion, or the number of components as
a string, or the index of the model in the original vector \code{k}.
In this particular case, \code{which = "BIC"}, \code{which = "2"},
and \code{which = 2} would all return the model with $K = 2$ components.
%
<<selectRost, echo = TRUE>>=
BIC(m1)
m1b <- getModel(m1, which = "BIC")
summary(m1b)
@
%
To inspect the main properties of the model, \fct{summary} can be
called. The information about the components of the mixture
includes a priori component weights $\pi_k$ and sizes as well as the estimated
item parameters $\hat \beta$ per component. Additionally, the fitted
log-likelihood and the information criteria AIC and BIC are reported.
As one of the item parameters in the Rasch model is not identified, a
restriction needs to be applied to the item parameters.
In the output of the \fct{summary} function, the item parameters of each
component are scaled to sum to zero.

Two other functions, \fct{worth} and \fct{parameters}, can be used to
access the item parameters. The sum restriction employed in the
\fct{summary} output is also applied by \fct{worth}. Additionally,
\fct{worth} provides the possibilities to select several or just one
specific \code{component} and to transform item \code{difficulty} parameters
to item easiness parameters. The function \fct{parameters} also offers
these two options but restricts the first item parameter to be zero (rather
than to restrict the sum of item parameters), as this restriction is used in the internal
computations. 
Thus, for the illustrative dataset with 10~items, \fct{parameters}
returns 9 item parameters, leaving out the first item parameter
restricted to zero while \fct{worth} returns 10 item parameters summing to zero.
The latter corresponds to the parametrization employed by
\citet{psychomix:Rost:1990} and \fct{simRaschmix}. For convenience reasons, the
true parameters are attached to the simulated dataset as an attribute
named \code{"difficulty"}. These are printed below and visualized in
Figure~\ref{fig:itemCompPlot} (left), showing that all item parameters
are recovered rather well. Note that the ordering of the components in mixture
models is generally arbitrary.
%
<<itemParaRost, echo = TRUE>>=
parameters(m1b, "item")
worth(m1b)
attr(r2, "difficulty")
@ 
%
In addition to the item parameters, the \fct{parameters} function 
can also return the parameters of the \code{"score"} model and the
\code{"concomitant"} model (if any). The type of parameters can be set
via the \code{which} argument. Per default \fct{parameters} returns
both item and score parameters. 

A comparison between estimated and true class membership can be
conducted using the \fct{clusters} function and the
corresponding attribute of the data, respectively. As already
noticeable from the item parameters, the first component of the
mixture matches the second true group of the data and vice versa. This
label-switching property of mixture models in general can also be seen
in the cross-table of class memberships. We thus have
\Sexpr{sum(diag(table(model = clusters(m1b), true = attr(r2, "cluster"))))}~misclassifications among the \Sexpr{nrow(r2)}
observations.
%
<<clustersRost, echo = TRUE>>=
table(model = clusters(m1b), true = attr(r2, "cluster"))
@ 
%
For comparison, a Rasch mixture model with mean/variance parametrization
for the score probabilities, as introduced in
Section~\ref{sec:flavors}, is fitted with one to three components and
the best BIC model is selected.
%
<<fitMeanVar0, echo = TRUE, eval = FALSE>>=
set.seed(3)
m2 <- raschmix(data = r2, k = 1:3, scores = "meanvar")
@ 
<<fitMeanVar, echo = FALSE, eval = TRUE, results=hide>>=
if(cache & file.exists("m2.rda")) {
  load("m2.rda")
} else {
<<fitMeanVar0>>
if(cache) {
  save(m2, file = "m2.rda")
} else {
  if(file.exists("m2.rda")) file.remove("m2.rda")
}
}
@ 
%
<<selectMeanVar, echo = TRUE>>=
m2
m2b <- getModel(m2, which = "BIC")
@
% 
\begin{figure}[t!]
\centering
\setkeys{Gin}{width=\textwidth}
<<itemCompPlot, echo = FALSE, fig = TRUE, width = 8.5, height = 4.5>>=
par(mfrow = c(1,2))
plot(m1b, pos = "top")
for(i in 1:2) lines(attr(r2, "difficulty")[,i], lty = 2, type = "b")
plot(m2b, pos = "top")
for(i in 1:2) lines(attr(r2, "difficulty")[,i], lty = 2, type = "b")
@ 
\caption{\label{fig:itemCompPlot}
  True (black) and estimated (blue/red) item parameters for the two
  model specifications, \code{"saturated"} (left) and \code{"meanvar"} (right),
  for the artificial scenario~2 data.}  
\end{figure}
%
As in the saturated version of the Rasch mixture model, all three
information criteria prefer the two-component model. Thus, this
version of a Rasch mixture model is also capable of recognizing the
two latent classes in the data while using a more 
parsimonious parametrization with \Sexpr{attr(logLik(m2b), "df")}
instead of \Sexpr{attr(logLik(m1b), "df")} parameters.
%
<<compNPara, echo = TRUE>>=
logLik(m2b)
logLik(m1b)
@ 
%
The estimated parameters of the distribution of the score
probabilities can be accessed through \fct{parameters} while the
full set of score probabilities is returned by
\fct{scoreProbs}. The estimated score probabilities of the illustrative
model are approximately equal across components and roughly uniform.
%
<<scoreProbs:Rost2, echo = TRUE>>=
parameters(m2b, which = "score")
scoreProbs(m2b)
@ 
%
The resulting item parameters for this particular data set are
virtually identical to those from the saturated version, as can be
seen in Figure~\ref{fig:itemCompPlot}.


To demonstrate the use of a concomitant variable model for the weights
of the mixture, the two artificial variables \code{x1} and \code{x2}
are employed. They are added on the right-hand side of the formula, yielding
a multinomial logit model for the weights (only if \code{k = 2} or
more components are specified).
%
<<fitMeanVarConc0, echo = TRUE, eval = FALSE>>=
set.seed(4)
cm2 <- raschmix(resp ~ x1 + x2, data = d, k = 1:3, scores = "meanvar")
@ 
<<fitMeanVarConc, echo = FALSE, eval = TRUE, results=hide>>=
if(cache & file.exists("cm2.rda")) {
  load("cm2.rda")
} else {
<<fitMeanVarConc0>>
if(cache) {
  save(cm2, file = "cm2.rda")
} else {
  if(file.exists("cm2.rda")) file.remove("cm2.rda")
}
}
@ 
%
The BIC is used to compare the models with and without concomitant
variables and different number of components. The two true groups are recognized
correctly with and without concomitant variables, while the model with
concomitants manages to employ the additional information and reaches
a somewhat improved model fit.
%
<<selectMeanVarConc, echo = TRUE>>=
rbind(m2 = BIC(m2), cm2 = BIC(cm2))
@ 
%
<<pValueMeanVarConc, echo = FALSE>>=
cm2b <- getModel(cm2, which = "BIC")
tStat <- 2 * (logLik(cm2b) - logLik(m2b))
pValue <- pchisq(tStat, attr(logLik(cm2b), "df") - attr(logLik(m2b), "df"), lower.tail = FALSE)
if(pValue < 0.001) pValue <- "< 0.001"
@ 
While the likelihood ratio (LR) test cannot be employed to choose the number
of components in a mixture model, it can be used to assess the
concomitant variable model for a mixture model with a fixed number of
components. Testing the 3-component model with concomitant
variables against the 3-component model without concomitant variables
yields a test statistic of \Sexpr{round(tStat, 2)} ($p \Sexpr{pValue}$).


As mentioned above, the parameters of the concomitant model can be
accessed via the\linebreak \fct{parameters} function, setting
\code{which = "concomitant"}. The influence of the informative
covariate \code{x1} is reflected in the large absolute coefficient
while the estimated coefficient for the noninformative covariate
\code{x2} is close to zero. 
%
<<concPara, echo = TRUE>>=
cm2b <- getModel(cm2, which = "BIC")
parameters(cm2b, which = "concomitant")
@ 
%
%However, despite the improvement in BIC, 
The corresponding estimated item parameters \code{parameters(cm2b,
  "item")} are not very different from the previous models (and are
hence not shown here). This illustrative application shows that the inclusion of
concomitant variables can provide additional information, e.g., that
\code{x1} but not \code{x2} is associated with the class membership.
% Thus, in this illustrative application, the main merit of the
% concomitant part of the model is to bring out that the clustering
% into mixture components is associated with \code{x1} but not \code{x2}. 
Note also that this is picked up although a rather weak association
was simulated here.
<<concTable>>=
table(x1 = d$x1, clusters = clusters(cm2b))
@

\pagebreak

\section{Empirical application: Verbal aggression}
\label{sec:application}

The verbal aggression dataset \citep{psychomix:Boeck+Wilson:2004}
contains item response data from 316 first-year psychology students
along with gender and trait anger (assessed by the Dutch adaptation of
the state-trait anger scale) as covariates
\citep{psychomix:Smits+Boeck+Vansteelandt:2004}.
The \Sexpr{table(VerbalAggression$gender)[1]}~women and
\Sexpr{table(VerbalAggression$gender)[2]}~men responded to 24~items constructed
the following way: Following the description of a frustrating
situation, subjects are asked to agree or disagree with a
possible reaction. The situations are described by the following four sentences:
\begin{itemize}
 \item S1: A bus fails to stop for me.
 \item S2: I miss a train because a clerk gave me faulty information.
 \item S3: The grocery store closes just as I am about to enter.
 \item S4: The operator disconnects me when I had used up my last 10 cents
  for a call.
\end{itemize}
Each reaction begins with either \dquote{I want to} or \dquote{I do}
and is followed by one of the three verbally
aggressive reactions \dquote{curse}, \dquote{scold}, or \dquote{shout},
e.g., \dquote{I want to curse}, \dquote{I do curse}, \dquote{I want to
  scold}, or \dquote{I do scold}.

For our illustration, we use only the first two sentences which describe
situations in which the others are to blame. Extreme-scoring subjects agreeing
with either none or all responses are removed.
%
<<dataVerbal>>=
data("VerbalAggression", package = "psychotools")
VerbalAggression$resp2 <- VerbalAggression$resp2[, 1:12]
va12 <- subset(VerbalAggression,
  rowSums(resp2) > 0 & rowSums(resp2) < 12)
colnames(va12$resp2)
@ 
%
We fit Rasch mixture models with the mean/variance score model,
one to four components, and with and without the two concomitant variables,
respectively (the single component model being only fitted without covariates).
%
<<vMeanVarFit0, echo = TRUE, eval = FALSE>>=
set.seed(1)
va12_mix1 <- raschmix(resp2 ~ 1, data = va12, k = 1:4, scores = "meanvar")
set.seed(2)
va12_mix2 <- raschmix(resp2 ~ gender + anger, data = va12, k = 1:4,
  scores = "meanvar")
@ 
<<vMeanVarFit, echo = FALSE, eval = TRUE, results=hide>>=
if(cache & file.exists("va12_mix.rda")) {
  load("va12_mix.rda")
} else {
<<vMeanVarFit0>>
if(cache) {
  save(va12_mix1, va12_mix2, file = "va12_mix.rda")
} else {
  if(file.exists("va12_mix.rda")) file.remove("va12_mix.rda")
}
}
@
%
The corresponding BIC for all considered models can be computed by
% 
<<selectVMeanVar, echo = TRUE>>=
rbind(BIC(va12_mix1), BIC(va12_mix2))
va12_mix3 <- getModel(va12_mix2, which = "3")
@ 
%
<<VMeanVarPValue, echo = FALSE>>=
va12_mix1b <- getModel(va12_mix1, which = "3")
va12_mix2b <- getModel(va12_mix2, which = "3")

tStatVA <- 2 * (logLik(va12_mix2b) - logLik(va12_mix1b))
pValueVA <- pchisq(tStatVA, attr(logLik(va12_mix2b), "df") - 
   attr(logLik(va12_mix1b), "df"), lower.tail = FALSE)
if(pValueVA < 0.001) pValueVA <- "< 0.001"
@ 

showing that three components are preferred regardless of whether
or not concomitant variables are used. The difference in BIC between
the models with and without concomitant variables is very small 
and
the LR test yields a test statistic of \Sexpr{round(tStatVA, 2)} 
($p \Sexpr{pValueVA}$), thus the 3-component model with
concomitant variables is chosen.

% 
\begin{figure}[t!]
\centering
\setkeys{Gin}{width=\textwidth}
<<plotVroot, echo = FALSE, fig = TRUE, width = 10, height = 4.3>>=
trellis.par.set(theme = standard.theme(color = FALSE))
print(histogram(va12_mix3))
@   
\caption{\label{fig:verbalRoot} Rootogram of posterior probabilities in the
  3-component Rasch mixture model on verbal aggression data.}
\end{figure}
%

The posterior probabilities for the three components can be visualized
via\linebreak \code{histogram(va12_mix3)} -- by default using a
square-root scale, yielding a so-called rootogram -- as shown in
Figure~\ref{fig:verbalRoot}. In the ideal case, posterior probabilities
of the observations for each component are either high or low,
yielding a U-shape in all panels. In this case here, the components
are separated acceptably well.

\begin{figure}[t!]
\centering
\setkeys{Gin}{width=\textwidth}
<<plotVMeanVar, echo = FALSE, fig = TRUE, width = 10, height = 4.3>>=
trellis.par.set(theme = standard.theme(color = FALSE))
print(xyplot(va12_mix3))
@ 
\caption{\label{fig:verbalProfile} Item profiles for the 3-component
  Rasch mixture model on verbal aggression data. Items~1--6
  pertain to situation S1 (bus), items~7--12 to situation S2 (train),
  each in the following order: 
  want to curse, do curse, want to scold, do scold, want to shout, do shout.}
\end{figure}

The item profiles in the three components can be visualized via
\code{plot(va12_mix3)} or\linebreak \code{xyplot(va12_mix3)} with the
output of the latter being shown in Figure~\ref{fig:verbalProfile}. 
The first six items are responses to the first sentence (bus), the remaining
six refer to the second sentence (train). The six reactions are grouped in
\dquote{want}/\dquote{do} pairs: first for \dquote{curse}, then \dquote{scold},
and finally \dquote{shout}.

The first component displays a zigzag pattern which indicates that
subjects in this component always find it easier or less extreme to
\dquote{want to} react a certain way rather than to actually
\dquote{do} react that way. In the other two components this want/do
relationship is reversed, except for the shouting response (to either
situation) and the scolding response to the train situation~(S2) in
the second component.

In the third component, there are no big differences in the estimated
item parameters. Neither any situation (S1 or S2) nor any type of
verbal response (curse, scold, or shout) is perceived as particularly
extreme by subjects in this component. In components~1 and~2, the
situation is also not very relevant but subjects differentiate between
the three verbal responses. This is best visible in component~2 where item
difficulty is clearly increasing from response \dquote{curse} to
response \dquote{shout}. Thus, shouting is perceived as the most
extreme verbal response while cursing is considered a comparably
moderate response. In component~1 this pattern is also visible albeit
not as prominently as in component~2.

% One could also consider the 3-component model \emph{with} concomitant
% variables as its BIC was almost equivalent to that of the model
% \emph{without} concomitant variables. The estimated item parameters are
% virtually identical between both models and are hence not shown here.
% Nevertheless, the link between the concomitant variables and the
% latent classes may still be of interest:
The link between the covariates and the latent classes is described
through the concomitant variable model:
%
<<concomitantsV>>=
parameters(getModel(va12_mix2, which = "3"), which = "concomitant")
@ 
%
The absolute sizes of the coefficients reflect that there may be
some association with gender but less with the anger score. However,
as there is a slight increase in BIC compared to the model without
concomitants, the association with the covariates appears to be
relatively weak. In comparison to other approaches exploring the
association of class membership with covariates \emph{ex post}
\cite[e.g., as in][]{psychomix:Cohen+Bolt:2005}, the main advantage of
the concomitant variables model lies in the \emph{simultaneous}
estimation of the mixture and the influence of covariates.


% <<clustersV, echo = TRUE>>=
% ## comparison with concomitant model, also consider parameters
% table(
%    without = clusters(getModel(va12_mix1, which = "3")),
%    with = clusters(getModel(va12_mix2, which = "3")))
% @ 
% <<paraV, echo = TRUE>>=
% parameters(va12_mix3)
% parameters(getModel(va12_mix2, which = "3"))[,c(2,1,3)]
% parameters(getModel(va12_mix2, which = "3"), which = "concomitant")
% @ 
% <<propTableV, echo = TRUE>>=
% prop.table(xtabs(~ clusters(va12_mix3) + gender, data = va12), 1)
% prop.table(xtabs(~ clusters(va12_mix3) + cut(anger, fivenum(anger)),
%                  data = va12), 1)
% @ 

\section{Summary} \label{sec:summary}

Mixtures of Rasch models are a flexible means of
checking measurement invariance and testing for differential
item functioning. Here, we establish a rather general unifying
conceptual framework for Rasch mixture models along with the
corresponding computational tools in the \proglang{R} package
\pkg{psychomix}. In particular, this includes the original model
specification of \citet{psychomix:Rost:1990} as well as more parsimonious
parametrizations \citep{psychomix:Rost+VonDavier:1995}, along with
the possibility to incorporate concomitant variables predicting the
latent classes \citep[as in][]{psychomix:Tay+Newman+Vermunt:2011}.

The \proglang{R} implementation is based on the infrastructure provided by
the \pkg{flexmix} package, allowing for convenient model specification
and selection. The rich set of methods for \pkg{flexmix} objects is
complemented by additional functions specifically designed for Rasch
models, e.g., extracting different types of parameters in different
transformations and visualizing the estimated component-specific
item parameters in various ways. 
%% Optionally, speed gains can be obtained
%% from utilizing the \proglang{C++} implementation in the \pkg{mRm}
%% package for selecting optimal starting values.
Thus, \pkg{psychomix} provides a comprehensive and convenient toolbox
for the application of Rasch mixture models in psychometric research practice.


\section*{Acknowledgments}

We are thankful to the participants of the ``Psychoco~2011'' workshop
at Universit\"at T\"ubingen for helpful feedback and discussions.


\nocite{psychomix:Fischer+Molenaar:1995}
\bibliography{psychomix}

\newpage
\begin{appendix}
  
\section[Using the FLXMCrasch() driver directly with stepFlexmix()]{Using the \fct{FLXMCrasch} driver directly with \fct{stepFlexmix}}
\label{sec:appendix}

The \class{FLXM} driver \fct{FLXMCrasch}, underlying the \fct{raschmix} function,
can also be used directly with
\fct{flexmix} or \fct{stepFlexmix}, respectively, via the
\code{model} argument. To do so, essentially the same arguments
as in \fct{raschmix} (see Section~\ref{sec:interface}) can be used,
however, they need to be arranged somewhat differently. The \code{formula}
just specifies the item responses, while the concomitant variables
need to be passed to the \code{concomitant} argument in
a suitable \class{FLXP} driver \citep[see][]{psychomix:Gruen+Leisch:2008}.
Also some arguments, such as the \code{scores}
distribution have to be specified in the \fct{FLXMCrasch} driver
for the \code{model} argument. As an example, consider replication
of the \code{cm2} model fit on the artificial data from
Section~\ref{sec:illustration}:
%
<<appendixFit0, eval=FALSE>>=
set.seed(4)
fcm2 <- stepFlexmix(resp ~ 1, data = d, k = 1:3,
  model = FLXMCrasch(scores = "meanvar"),
  concomitant = FLXPmultinom(~ x1 + x2))
@ 
<<appendixFit, echo = FALSE, eval = TRUE, results=hide>>=
if(cache & file.exists("fcm2.rda")) {
  load("fcm2.rda")
} else {
<<appendixFit0>>
if(cache) {
  save(fcm2, file = "fcm2.rda")
} else {
  if(file.exists("fcm2.rda")) file.remove("fcm2.rda")
}
}
@
%
Thus, there is some more flexibility but somewhat less convenience
when using \fct{flexmix} or \fct{stepFlexmix} directly as opposed through
the \fct{raschmix} interface. This is also reflected in the objects
returned which are of class \class{flexmix} or \class{stepFlexmix}, not class
\class{raschmix} or \class{stepRaschmix}, respectively. Thus, only the
generic functions for those objects apply and not the additional ones
specific to Rasch mixture models. In various cases, the methods are inherited
or reused from \pkg{flexmix} and thus behave identically, e.g., for \fct{BIC} or \fct{getModel}.
%
<<appendixSelectModel>>=
rbind(cm2 = BIC(cm2), fcm2 = BIC(fcm2))
fcm2b <- getModel(fcm2, which = "BIC")
@ 
%
For other methods, such as \fct{parameters}, the methods in \pkg{psychomix}
offer more convenience. For example, the concomitant model coefficients
can be accessed in the same way
%
<<appendixParametersConc>>=
cbind(parameters(cm2b, which = "concomitant"),
  parameters(fcm2b, which = "concomitant"))
@ 
%
while the item and score parameters cannot be accessed separately (as it is possible for
\class{raschmix} objects). They can only be accessed jointly as the 
\code{"model"} parameters. The method for \class{raschmix} objects also excludes 
non-identified or aliased parameters, e.g., the parameter for the achor item.
%
<<appendixParametes>>=
parameters(fcm2b, which = "model")
@ 
%
To sum up, some (convenience) functionality which is specific for Rasch mixture models
is only available for objects of class \class{raschmix}, e.g., the
score probabilities via \fct{scoreProbs} or the item profiles via the
default \fct{plot} method among others. On the other hand, functionality
which is applicable to mixture models in general is inherited or preserved as part
of the additional methods.

\end{appendix}

\end{document}
