02-DichotomousRaschModel.tex

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\title{Rasch Measurement Theory Analysis in R: Illustrations and
Practical Guidance for Researchers and Practitioners}
\author{}
\date{\vspace{-2.5em}}

\begin{document}
\maketitle

\hypertarget{Dich_Rasch_model}{%
\section{Dichotomous Rasch Model}\label{Dich_Rasch_model}}

This chapter provides a basic overview of the dichotomous Rasch model,
along with guidance for analyzing data with the dichotomous Rasch model
using R. We use data from a transitive reasoning assessment presented by
{[}@Intro\_nonparametric{]} to illustrate the analysis using Conditional
Maximum Likelihood Estimation (CMLE) with the Extended Rasch Modeling
``eRm'' package {[}@eRm{]}. Then, we illustrate the application of the
dichotomous Rasch model using Marginal Maximum Likelihood Estimation
(MMLE) and Joint Maximum Likelihood Estimation (JMLE) with the Test
Analysis Modules (``TAM'') package {[}@TAM{]}. After the analyses are
complete, we present an example description of the results. The chapter
concludes with a challenge exercise and resources for further study.

\textbf{\emph{Overview of the Dichotomous Rasch Model}}

The \emph{dichotomous Rasch model} {[}@Rasch\_ori{]} is the simplest
model in the Rasch family of models {[}@Overview\_RM{]}. It was designed
for use with ordinal data that are scored in two categories (usually 0
or 1). The dichotomous Rasch model uses sum scores from these ordinal
responses to calculate interval-level estimates that represent person
locations (i.e., person ability or person achievement) and item
locations (i.e., the difficulty to provide a correct or positive
response) on a linear scale that represents the latent variable (the
log-odds or ``logit'' scale). The difference between person and item
locations can be used to calculate the probability for a correct or
positive response (\(x = 1\)), rather than an incorrect or negative
response (\(x = 0\)).

The equation for the dichotomous Rasch model can be expressed in
log-odds form as follows:

\begin{equation}\tag{2.1}\ln_{}{}\left[\frac{\phi_{n i 1}}{\phi_{n i 0}}\right]=\theta_{n}-\delta_{i}\end{equation}

The Rasch model predicts the probability that person \(n\) on item \(i\)
provides a correct or positive (\(x = 1\)), rather than an incorrect or
negative (\(x = 0\)) response, given person locations (i.e., ability,
achievement, \(\theta_n\)) and item locations (i.e., difficulty,
\(\delta_i\)), as expressed on the logit scale.

\textbf{\emph{Rasch Model Requirements}}

Estimates that are calculated using the dichotomous Rasch model can only
be meaningfully interpreted if there is evidence that the data
approximate the requirements for the model. Key among dichotomous Rasch
model requirements are the following:

\begin{itemize}
\tightlist
\item
  \emph{Unidimensionality}: A single latent variable is sufficient to
  explain most of the variation in item responses
\item
  \emph{Local independence}: After controlling for the latent variable,
  there is no substantial association between the responses to
  individual items
\item
  \emph{Person-invariant item estimates}: Item locations do not depend
  on (i.e., are independent from) the persons whose responses are used
  to estimate them
\item
  \emph{Item-invariant person estimates}: Person locations do not depend
  on (i.e., are independent from) the items used to estimate them
\end{itemize}

Evidence that data approximate these requirements provides support for
the meaningful interpretation and use of item and person estimates on
the logit scale as indicators of item and person locations on the latent
variable. In practice, many analysts evaluate some or all of these
requirements using various indicators of model-data fit for the facets
in a Rasch model (in this case, items and persons). In the current
chapter, we provide some basic code for calculating some popular
residual-based fit indices for items and persons. We explore issues
related to model requirements and evaluating model-data fit in more
detail in \protect\hyperlink{MD_fit}{Chapter 3}.

\hypertarget{example-data-transitive-reasoning-test}{%
\subsection{Example Data: Transitive Reasoning
Test}\label{example-data-transitive-reasoning-test}}

In this chapter, we will be working with data from a transitive
reasoning test designed to measure students' ability to reason about the
relationships among physical objects. The data were collected from a
one-on-one interactive assessment in which an experimenter presented
students with a set of objects, such as sticks, balls, cubes, and discs.
The following description is given in @Intro\_nonparametric, pp.~31-32:

\begin{quote}
The items for transitive reasoning had the following structure. A
typical item used three sticks, here denoted A, B, and C, of different
length, denoted Y, such that YA \textless{} YB \textless{} YC. The
actual test taking had the form of a conversation between experimenter
and child in which the sticks were identified by their colors rather
than letters. First, sticks A and B were presented to a child, who was
allowed to pick them up and compare their lengths, for example, by
placing them next to each other on a table.
\end{quote}

\begin{quote}
Next, sticks B and C were presented and compared. Then all three sticks
were displayed in a random order at large mutual distances so that their
length differences were imperceptible, and the child was asked to infer
the relation between sticks A and C from his or her knowledge of the
relationship in the other two pairs.
\end{quote}

The transitive reasoning items varied in terms of the property students
were asked to reason about (length, weight, area). The tasks also varied
in terms of the number of physical objects that students were asked to
reason about, and whether the comparison tasks involved equalities,
inequalities, or a mixture of equalities and inequalities. The
characteristics of the transitive reasoning data are summarized in the
following table:

\begin{longtable}[]{@{}lllll@{}}
\toprule
Task & Property & Format & Objects & Measures \\
\midrule
\endhead
1 & Length & YA \textgreater{} YB \textgreater{} YC & Sticks & 12, 11.5,
11 (cm) \\
2 & Length & YA = YB = YC = YD & Tubes & 12 (cm) \\
3 & Weight & YA \textgreater{} YB \textgreater{} YC & Tubes & 45, 25, 18
(g) \\
4 & Weight & YA = YB = YC = YD & Cubes & 65 (g) \\
5 & Weight & YA \textless{} YB \textless{} YC & Balls & 40, 50, 70
(g) \\
6 & Area & YA \textgreater{} YB\textgreater{} YC & Discs & 2.5, 7, 6.5
(diameter; cm) \\
7 & Length & YA \textgreater{} YB = YC & Sticks & 28.5, 27.5, 27.5
(cm) \\
8 & Weight & YA \textgreater YB = YC & Balls & 65, 40, 40 (g) \\
9 & Length & YA = YB = YC = YD & Sticks & 12.5, 12.5, 13, 13 (cm) \\
10 & Weight & YA = YB \textless{} YC = YD & Balls & 60, 60, 100, 100
(g) \\
\bottomrule
\end{longtable}

\hypertarget{dichotomous-rasch-model-analysis-with-cmle-in-erm}{%
\subsection{Dichotomous Rasch Model Analysis with CMLE in
eRm}\label{dichotomous-rasch-model-analysis-with-cmle-in-erm}}

In the next section, we provide a step-by-step demonstration of a
dichotomous Rasch model analysis using the \emph{Extended Rasch
Modeling}, or \emph{eRm} package {[}@eRm{]}, which uses Conditional
Maximum Likelihood Estimation (CMLE). We encourage readers to use the
example data set that is provided in the online supplement to conduct
the analysis along with us.

\textbf{\emph{Prepare for the Analyses}}

We selected eRm for the first illustration in the current chapter
because it includes functions for applying the dichotomous Rasch model
that are relatively straightforward to use and interpret. Please note
that the ``eRm'' package uses the Conditional Maximum Likelihood
Estimation (CMLE) method to estimate Rasch model parameters. As a
result, estimates from the eRm package are not directly comparable to
estimates obtained using other estimation methods. Later in this
chapter, we have included illustrations of dichotomous Rasch model
analyses with the \emph{Test Analysis Modules} or TAM package {[}@TAM{]}
with Marginal Maximum Likelihood Estimation (MMLE). We also provide an
illustration with TAM using Joint Maximum Likelihood Estimation (JMLE),
which produces comparable estimates to some popular standalone Rasch
software programs, such as Winsteps {[}@Winsteps{]} and Facets
{[}@Facets{]}.

To get started with the eRm package, install and load it into your R
environment using the following code.

\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\#install.packages("eRm")}
\FunctionTok{library}\NormalTok{(}\StringTok{"eRm"}\NormalTok{)}
\end{Highlighting}
\end{Shaded}

Now that we have installed and loaded the package to our R session, we
are ready to import the data.

In this book, we use the function \texttt{read.csv()} to import data
that are stored using comma separated values. We encourage readers to
use their preferred method for importing data files into R or R Studio.
Please note that if you use \texttt{read.csv()} you will need to first
specify the file path to the location at which the data file is stored
on your computer or set your working directory to the folder in which
you have saved the data.

First, we will import the data using \texttt{read.csv()} and store it in
an object called \texttt{transreas}.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{transreas }\OtherTok{\textless{}{-}} \FunctionTok{read.csv}\NormalTok{(}\StringTok{"transreas.csv"}\NormalTok{)}
\end{Highlighting}
\end{Shaded}

Next, we will explore the data using descriptive statistics using the
\texttt{summary()} function.

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{summary}\NormalTok{(transreas)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##     Student        Grade          task_01          task_02          task_03      
##  Min.   :  1   Min.   :2.000   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000  
##  1st Qu.:107   1st Qu.:3.000   1st Qu.:1.0000   1st Qu.:1.0000   1st Qu.:1.0000  
##  Median :213   Median :4.000   Median :1.0000   Median :1.0000   Median :1.0000  
##  Mean   :213   Mean   :4.005   Mean   :0.9412   Mean   :0.8094   Mean   :0.8847  
##  3rd Qu.:319   3rd Qu.:5.000   3rd Qu.:1.0000   3rd Qu.:1.0000   3rd Qu.:1.0000  
##  Max.   :425   Max.   :6.000   Max.   :1.0000   Max.   :1.0000   Max.   :1.0000  
##     task_04          task_05          task_06          task_07      
##  Min.   :0.0000   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000  
##  1st Qu.:1.0000   1st Qu.:1.0000   1st Qu.:1.0000   1st Qu.:1.0000  
##  Median :1.0000   Median :1.0000   Median :1.0000   Median :1.0000  
##  Mean   :0.7835   Mean   :0.8024   Mean   :0.9741   Mean   :0.8447  
##  3rd Qu.:1.0000   3rd Qu.:1.0000   3rd Qu.:1.0000   3rd Qu.:1.0000  
##  Max.   :1.0000   Max.   :1.0000   Max.   :1.0000   Max.   :1.0000  
##     task_08          task_09          task_10    
##  Min.   :0.0000   Min.   :0.0000   Min.   :0.00  
##  1st Qu.:1.0000   1st Qu.:0.0000   1st Qu.:0.00  
##  Median :1.0000   Median :0.0000   Median :1.00  
##  Mean   :0.9671   Mean   :0.3012   Mean   :0.52  
##  3rd Qu.:1.0000   3rd Qu.:1.0000   3rd Qu.:1.00  
##  Max.   :1.0000   Max.   :1.0000   Max.   :1.00
\end{verbatim}

From the summary of \texttt{transreas}, we can see there are no missing
data. We can also get a general sense of the scales, range, and
distribution of each variable in the dataset.

Specifically, we can see that Student ID numbers range from 1 to 425,
student grade levels range from 2 to 6, and that all tasks have scores
in both of the dichotomous categories (0 and 1). We can also get a sense
for the range of item difficulty by examining the mean for each task,
which is the proportion-correct statistic (item difficulty estimate for
Classical Test Theory).

\textbf{\emph{Run the Dichotomous Rasch Model}}

To run the dichotomous Rasch Model using the eRm package, need to
isolate the item response matrix from the other variables in the data
(student IDs and grade level). To do this, we will create an object made
up of only the item responses by removing the first two variables from
the data. We will remove the descriptive variables using the
\texttt{subset()} function with the \texttt{select=} option. We will
save the response matrix in a new object called
\texttt{transreas.responses}.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{transreas.responses}\OtherTok{\textless{}{-}}\FunctionTok{subset}\NormalTok{(transreas,}\AttributeTok{select=}\SpecialCharTok{{-}}\FunctionTok{c}\NormalTok{(Student,Grade))}
\end{Highlighting}
\end{Shaded}

Next, we will use \texttt{summary()} to calculate descriptive statistics
for the \texttt{transreas.responses} object to check our work and ensure
that the responses are ready for analysis.

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{summary}\NormalTok{(transreas.responses)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##     task_01          task_02          task_03          task_04      
##  Min.   :0.0000   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000  
##  1st Qu.:1.0000   1st Qu.:1.0000   1st Qu.:1.0000   1st Qu.:1.0000  
##  Median :1.0000   Median :1.0000   Median :1.0000   Median :1.0000  
##  Mean   :0.9412   Mean   :0.8094   Mean   :0.8847   Mean   :0.7835  
##  3rd Qu.:1.0000   3rd Qu.:1.0000   3rd Qu.:1.0000   3rd Qu.:1.0000  
##  Max.   :1.0000   Max.   :1.0000   Max.   :1.0000   Max.   :1.0000  
##     task_05          task_06          task_07          task_08      
##  Min.   :0.0000   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000  
##  1st Qu.:1.0000   1st Qu.:1.0000   1st Qu.:1.0000   1st Qu.:1.0000  
##  Median :1.0000   Median :1.0000   Median :1.0000   Median :1.0000  
##  Mean   :0.8024   Mean   :0.9741   Mean   :0.8447   Mean   :0.9671  
##  3rd Qu.:1.0000   3rd Qu.:1.0000   3rd Qu.:1.0000   3rd Qu.:1.0000  
##  Max.   :1.0000   Max.   :1.0000   Max.   :1.0000   Max.   :1.0000  
##     task_09          task_10    
##  Min.   :0.0000   Min.   :0.00  
##  1st Qu.:0.0000   1st Qu.:0.00  
##  Median :0.0000   Median :1.00  
##  Mean   :0.3012   Mean   :0.52  
##  3rd Qu.:1.0000   3rd Qu.:1.00  
##  Max.   :1.0000   Max.   :1.00
\end{verbatim}

Now, we are ready to run the dichotomous Rasch model on the transitive
reasoning response data. We will use the \texttt{RM()} function to run
the model and store the results in an object called
\texttt{dichot.transreas}.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{dichot.transreas }\OtherTok{\textless{}{-}} \FunctionTok{RM}\NormalTok{(transreas.responses)}
\end{Highlighting}
\end{Shaded}

\textbf{\emph{Overall Model Summary}}

The next step is to request a summary of the model estimation results in
order to begin to understand the results from the analysis. We can do so
by applying the \texttt{summary()} function to the model object.

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{summary}\NormalTok{(dichot.transreas)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
## 
## Results of RM estimation: 
## 
## Call:  RM(X = transreas.responses) 
## 
## Conditional log-likelihood: -921.3465 
## Number of iterations: 18 
## Number of parameters: 9 
## 
## Item (Category) Difficulty Parameters (eta): with 0.95 CI:
##         Estimate Std. Error lower CI upper CI
## task_02    0.258      0.133   -0.003    0.518
## task_03   -0.416      0.157   -0.723   -0.109
## task_04    0.441      0.128    0.190    0.692
## task_05    0.309      0.131    0.052    0.567
## task_06   -2.175      0.292   -2.747   -1.604
## task_07   -0.025      0.141   -0.302    0.252
## task_08   -1.909      0.262   -2.423   -1.395
## task_09    2.923      0.130    2.668    3.179
## task_10    1.836      0.115    1.610    2.062
## 
## Item Easiness Parameters (beta) with 0.95 CI:
##              Estimate Std. Error lower CI upper CI
## beta task_01    1.243      0.204    0.842    1.643
## beta task_02   -0.258      0.133   -0.518    0.003
## beta task_03    0.416      0.157    0.109    0.723
## beta task_04   -0.441      0.128   -0.692   -0.190
## beta task_05   -0.309      0.131   -0.567   -0.052
## beta task_06    2.175      0.292    1.604    2.747
## beta task_07    0.025      0.141   -0.252    0.302
## beta task_08    1.909      0.262    1.395    2.423
## beta task_09   -2.923      0.130   -3.179   -2.668
## beta task_10   -1.836      0.115   -2.062   -1.610
\end{verbatim}

The summary of the dichotomous Rasch model output includes the
Conditional Log-likelihood statistic, details about the number of
iterations and model parameters, and a table with item parameters, their
standard errors, and confidence intervals. It is important to note that
the item parameters included in this preliminary output are \emph{item
easiness} parameters-- \emph{not} item difficulty parameters. We will
examine item difficulty parameters in detail later in our analysis.

\textbf{\emph{Wright Map}}

A useful feature of Rasch models is that when there is acceptable fit
between the model and the data (discussed in detail in
\protect\hyperlink{MD_fit}{Chapter 3}), it is possible to visualize and
compare item and person locations on a single linear continuum.
Professor Bejamin D. Wright popularized an approach to displaying Rasch
model results on a linear continuum, and Rasch measurement researchers
across disciplines have adopted this technique. In his honor, many
researchers refer to these displays as \emph{Wright Maps.} Researchers
also refer to these displays as \emph{Variable Maps}. Please see Wilson
(2011) for a discussion of the term \emph{Wright Map}.

As the next step in our analysis, we will create a Wright Map from our
model results. We will create the plot using the function
\texttt{PlotPImap()} on the model object (\texttt{dichot.transreas}). We
will set the option for displaying threshold labels as \texttt{FALSE},
because we are working with dichotomous data. We also used the
\texttt{main.title=}option to customize the title of the plot.

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{plotPImap}\NormalTok{(dichot.transreas, }\AttributeTok{main =} \StringTok{"Transitive Reasoning Assessment Wright Map"}\NormalTok{)}
\end{Highlighting}
\end{Shaded}

\includegraphics{02-DichotomousRaschModel_files/figure-latex/unnamed-chunk-7-1.pdf}

In this \emph{Wright Map} display, the results from the dichotomous
Rasch model analysis of the Transitive Reasoning data are summarized
graphically. The figure is organized as follows:

Starting at the bottom of the figure, the horizontal axis (labeled
\emph{Latent Dimension}) is the logit scale that represents the latent
variable. In the application of the Transitive Reasoning data, lower
numbers indicate less transitive reasoning ability, and higher numbers
indicate more transitive reasoning ability.

The central panel of the figure shows item difficulty locations on the
logit scale for the 10 transitive reasoning tasks that were included in
the analysis; the y-axis for this panel shows the item labels. By
default in eRm, the items are ordered according to their original order
in the response matrix. The items can be ordered by difficulty by adding
\texttt{sorted\ =\ TRUE} as an argument in the \texttt{plotPImap()}
call. For each item, a solid circle plotting symbol shows the location
estimate.

The upper panel of the figure shows a histogram of person (in this case,
student) location estimates on the logit scale. Small vertical lines on
the x-axis of this histogram show the points on the logit scale at which
information (variance) is maximized for the sample of persons and items
in the analysis. These lines can be omitted by adding
\texttt{irug\ =\ FALSE} as an argument in the \texttt{plotPImap()} call.

Even though it is not appropriate to fully interpret item and person
locations on the logit scale until there is evidence of acceptable
model-data fit, we recommend examining the Wright Map during the
preliminary stages of an item analysis to get a general sense of the
model results and to identify any potential scoring or data entry
errors.

A quick glance at the Wright Map suggests that, on average, the persons
(students) are located higher on the logit scale compared to the average
item (task) locations. In addition, there appears to be a relatively
wide spread of person and item locations on the logit scale, such that
the transitive reasoning test appears to be a useful tool for
identifying differences in person locations and item locations on the
latent variable. We will return to this display for more exploration
after we check for acceptable model-data fit, among other psychometric
properties.

\textbf{\emph{Item Parameters}}

As the next step in our analysis, we will examine the item parameters in
detail. The eRm package provides several options with which analysts can
find and examine item and item location parameters. For example, one way
to obtain the overall item location parameters (\(\delta\)) is to
extract the \emph{eta parameters} from the model object using the
\texttt{\$} operator. We extract these parameters, print them to the
console, and calculate summary statistics for them with the following
code.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{difficulty }\OtherTok{\textless{}{-}}\NormalTok{ dichot.transreas}\SpecialCharTok{$}\NormalTok{etapar}
\NormalTok{difficulty}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##     task_02     task_03     task_04     task_05     task_06     task_07 
##  0.25775001 -0.41581384  0.44093780  0.30941151 -2.17531698 -0.02481129 
##     task_08     task_09     task_10 
## -1.90884851  2.92343872  1.83581566
\end{verbatim}

Because of the nature of the estimation process used in eRm, the
item.locations object that we just created does not include the location
estimate for the first item. One can calculate the location for item 1
by subtracting the sum of the item locations from zero. In the following
code, we find the location for item 1, and then create a new object with
all 10 item locations.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{n.items }\OtherTok{\textless{}{-}} \FunctionTok{ncol}\NormalTok{(transreas.responses)}
\NormalTok{i1 }\OtherTok{\textless{}{-}} \DecValTok{0} \SpecialCharTok{{-}} \FunctionTok{sum}\NormalTok{(difficulty[}\DecValTok{1}\SpecialCharTok{:}\NormalTok{(n.items }\SpecialCharTok{{-}} \DecValTok{1}\NormalTok{)])}
\NormalTok{difficulty.all }\OtherTok{\textless{}{-}} \FunctionTok{c}\NormalTok{(i1, difficulty[}\FunctionTok{c}\NormalTok{(}\DecValTok{1}\SpecialCharTok{:}\NormalTok{(n.items }\SpecialCharTok{{-}} \DecValTok{1}\NormalTok{))])}
\NormalTok{difficulty.all}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##                 task_02     task_03     task_04     task_05     task_06 
## -1.24256308  0.25775001 -0.41581384  0.44093780  0.30941151 -2.17531698 
##     task_07     task_08     task_09     task_10 
## -0.02481129 -1.90884851  2.92343872  1.83581566
\end{verbatim}

Alternatively, we could find the item difficulty parameters by
extracting the item \emph{easiness} parameters from the model object
(\emph{beta parameters}) and multiplying them by -1 to find item
difficulty.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{difficulty2 }\OtherTok{\textless{}{-}}\NormalTok{ dichot.transreas}\SpecialCharTok{$}\NormalTok{betapar }\SpecialCharTok{*} \SpecialCharTok{{-}}\DecValTok{1}
\NormalTok{difficulty2}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
## beta task_01 beta task_02 beta task_03 beta task_04 beta task_05 beta task_06 
##  -1.24256308   0.25775001  -0.41581384   0.44093780   0.30941151  -2.17531698 
## beta task_07 beta task_08 beta task_09 beta task_10 
##  -0.02481129  -1.90884851   2.92343872   1.83581566
\end{verbatim}

The item difficulty parameters (\texttt{xsi}) are the item location
estimates on the logit scale that represents the latent variable.
Assuming that the responses are scored such that lower scores
(\(x = 0\)) indicate lower locations on the latent variable (e.g.,
incorrect, negative, or absent responses), \emph{lower} item estimates
on the logit scale indicate items that are \emph{more difficult} or
require persons to have relatively \emph{higher locations} on the
construct to provide a correct response. On the other hand,
\emph{higher} item estimates on the logit scale indicate items that are
\emph{easier} or require persons to have relatively \emph{lower
locations} on the construct to provide a correct response. In our
analysis, Task 9 is the most difficult item (\(\delta\) = 2.92), whereas
Task 6 is the easiest item (\(\delta\) = -2.18).

Next, we will examine item standard errors. The standard errors are
reported for all of the items as part of the beta (item easiness)
parameters.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{dichot.transreas}\SpecialCharTok{$}\NormalTok{se.beta}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##  [1] 0.2043488 0.1327967 0.1566695 0.1282289 0.1314316 0.2916385 0.1413876
##  [8] 0.2622114 0.1302767 0.1152173
\end{verbatim}

The standard error for each item is an estimate of the precision of the
item difficulty estimates, where larger standard errors indicate
less-precise estimates. Standard errors are reported on the same logit
scale as item locations. In our analysis, the standard errors range from
0.11 for Task 9 and Task 10, which were the items with the most precise
estimates, to 0.31 for Task 6, which was the item with the least precise
estimate. These differences largely reflect differences in item
targeting to the person locations (see the Wright Map above).

Next, let's calculate descriptive statistics to better understand the
distribution of the item locations and standard errors. We will do so
using the \texttt{summary()} function and the \texttt{sd()} function.

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{summary}\NormalTok{(difficulty.all)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -2.1753 -1.0359  0.1165  0.0000  0.4081  2.9234
\end{verbatim}

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{sd}\NormalTok{(difficulty.all)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
## [1] 1.576441
\end{verbatim}

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{summary}\NormalTok{(dichot.transreas}\SpecialCharTok{$}\NormalTok{se.beta)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.1152  0.1306  0.1371  0.1694  0.1924  0.2916
\end{verbatim}

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{sd}\NormalTok{(dichot.transreas}\SpecialCharTok{$}\NormalTok{se.beta)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
## [1] 0.06206381
\end{verbatim}

We can also visualize the item difficulty estimates using a simple
histogram by using the \texttt{hist()} function. In our plot, we
specified a custom title using \texttt{main\ =} and a custom x-axis
label using \texttt{xlab\ =}:

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{hist}\NormalTok{(difficulty.all, }\AttributeTok{main =} \StringTok{"Histogram of Item Difficulty Estimates for the }\SpecialCharTok{\textbackslash{}n}\StringTok{Transitive Reasoning Data"}\NormalTok{,}
     \AttributeTok{xlab =} \StringTok{"Item Difficulty Estimates in Logits"}\NormalTok{)}
\end{Highlighting}
\end{Shaded}

\includegraphics{02-DichotomousRaschModel_files/figure-latex/unnamed-chunk-13-1.pdf}

\textbf{\emph{Item Response Functions}}

We will examine graphical displays of item difficulty using item
response functions (IRFs). With the dichotomous Rasch model, the eRm
package creates plots of the probability for a correct or positive
response (\(x = 1\)), conditional on person locations on the latent
variable. In the following code, we use \texttt{plotICC()} from eRm to
create IRF plots for the items in our analysis. We included
\texttt{ask\ =\ FALSE} in our function call to generate all of the plots
at once. For brevity, we have only included plots for the first three
items in this book. The specific items to be plotted can be controlled
by changing the items included in the \texttt{items.to.plot} object.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{items.to.plot }\OtherTok{\textless{}{-}} \FunctionTok{c}\NormalTok{(}\DecValTok{1}\SpecialCharTok{:}\DecValTok{3}\NormalTok{)}
\FunctionTok{plotICC}\NormalTok{(dichot.transreas, }\AttributeTok{ask =} \ConstantTok{FALSE}\NormalTok{, }\AttributeTok{item.subset =}\NormalTok{ items.to.plot)}
\end{Highlighting}
\end{Shaded}

\includegraphics{02-DichotomousRaschModel_files/figure-latex/unnamed-chunk-14-1.pdf}

This code generates IRFs for the first three items in our analysis. In
each item-specific plot, the x-axis is the logit scale that represents
the latent variable; this scale represents transitive reasoning ability
in our example. The y-axis is the probability for a correct or positive
response, conditional on person locations on the latent variable.

\textbf{\emph{Person Parameters and Item Fit}}

In the eRm package, it is necessary to calculate person parameters
\emph{before} item fit statistics can be calculated. Accordingly, we
will proceed with a brief examination of person parameters before we
conduct item fit analyses. In practice, we recommend examining item fit
before examining and interpreting item locations in detail.

\textbf{\emph{Person Parameters}}

In the following code, we use the \texttt{person.parameter()} function
to estimate student locations and save the results in an object called
\texttt{student.locations}. Then, we save the theta estimates (i.e.,
achievement estimates) in a data.frame object called
\texttt{achievement}, and add student identification numbers for
reference.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{student.locations }\OtherTok{\textless{}{-}} \FunctionTok{person.parameter}\NormalTok{(dichot.transreas)}
\NormalTok{achievement }\OtherTok{\textless{}{-}}\NormalTok{ student.locations}\SpecialCharTok{$}\NormalTok{theta.table}
\NormalTok{achievement}\SpecialCharTok{$}\NormalTok{id }\OtherTok{\textless{}{-}} \FunctionTok{rownames}\NormalTok{(achievement)}
\end{Highlighting}
\end{Shaded}

The estimation procedure in eRm does not directly produce parameter
estimates for persons with extreme scores. In our example, 51 students
earned extreme scores on the transitive reasoning assessment, so
achievement estimates are reported for the remaining 374 students with
non-extreme scores. The achievement estimates for the 51 students with
extreme scores are extrapolated, and standard errors are not calculated.

We can add standard errors for the student achievement estimates to our
\texttt{achievement} object as follows.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{se }\OtherTok{\textless{}{-}} \FunctionTok{as.data.frame}\NormalTok{(student.locations}\SpecialCharTok{$}\NormalTok{se.theta}\SpecialCharTok{$}\NormalTok{NAgroup1)}
\NormalTok{se}\SpecialCharTok{$}\NormalTok{id }\OtherTok{\textless{}{-}} \FunctionTok{rownames}\NormalTok{(se)}
\FunctionTok{names}\NormalTok{(se) }\OtherTok{\textless{}{-}} \FunctionTok{c}\NormalTok{(}\StringTok{"person\_se"}\NormalTok{, }\StringTok{"id"}\NormalTok{)}
\NormalTok{achievement.with.se }\OtherTok{\textless{}{-}} \FunctionTok{merge}\NormalTok{(achievement, se, }\AttributeTok{by =} \StringTok{"id"}\NormalTok{ )}
\end{Highlighting}
\end{Shaded}

As a final step in examining the person parameters, we will calculate
descriptive statistics and use a histogram to examine the distribution
of student locations on the logit scale.

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{summary}\NormalTok{(achievement)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##  Person Parameter    NAgroup  Interpolated         id           
##  Min.   :-2.916   Min.   :1   Mode :logical   Length:425        
##  1st Qu.: 1.160   1st Qu.:1   FALSE:374       Class :character  
##  Median : 1.932   Median :1   TRUE :51        Mode  :character  
##  Mean   : 2.010   Mean   :1                                     
##  3rd Qu.: 3.028   3rd Qu.:1                                     
##  Max.   : 4.201   Max.   :1
\end{verbatim}

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{hist}\NormalTok{(achievement}\SpecialCharTok{$}\StringTok{\textasciigrave{}}\AttributeTok{Person Parameter}\StringTok{\textasciigrave{}}\NormalTok{, }\AttributeTok{main =} \StringTok{"Histogram of Person Achievement Estimates }\SpecialCharTok{\textbackslash{}n}\StringTok{for the Transitive Reasoning Data"}\NormalTok{,}
     \AttributeTok{xlab =} \StringTok{"Person Achievement Estimates in Logits"}\NormalTok{)}
\end{Highlighting}
\end{Shaded}

\includegraphics{02-DichotomousRaschModel_files/figure-latex/unnamed-chunk-17-1.pdf}

\textbf{\emph{Item Fit}}

Next, we will conduct a brief exploration of item fit statistics. We
explore item fit in more detail in \protect\hyperlink{MD_fit}{Chapter
3}.

To calculate numeric item fit statistics, we will use the function
\texttt{itemfit()} from eRm on the person parameter object
(\texttt{person.locations.estimate}). This function produces several
item fit statistics, including infit mean square error (\(MSE\)), outfit
\(MSE\), and standardized infit and outfit \(MSE\) statistics. We will
store the item fit results in a new object called \texttt{item.fit}, and
then format this object as a data.frame object for easy manipulation and
exporting.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{student.locations }\OtherTok{\textless{}{-}} \FunctionTok{person.parameter}\NormalTok{(dichot.transreas)}
\NormalTok{item.fit }\OtherTok{\textless{}{-}} \FunctionTok{itemfit}\NormalTok{(student.locations)}
\NormalTok{item.fit}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
## 
## Itemfit Statistics: 
##           Chisq  df p-value Outfit MSQ Infit MSQ Outfit t Infit t Discrim
## task_01 245.550 373   1.000      0.657     0.761   -1.327  -1.622   0.455
## task_02 421.904 373   0.041      1.128     1.087    1.071   1.187   0.054
## task_03 218.155 373   1.000      0.583     0.745   -2.742  -2.688   0.583
## task_04 478.894 373   0.000      1.280     1.158    2.442   2.268  -0.014
## task_05 346.015 373   0.839      0.925     0.991   -0.625  -0.109   0.184
## task_06  65.613 373   1.000      0.175     0.611   -2.789  -1.781   0.620
## task_07 244.494 373   1.000      0.654     0.804   -2.768  -2.475   0.493
## task_08 125.478 373   1.000      0.336     0.687   -2.205  -1.569   0.563
## task_09 394.207 373   0.216      1.054     0.904    0.462  -1.727   0.022
## task_10 326.534 373   0.960      0.873     0.898   -1.822  -2.408   0.226
\end{verbatim}

Next, we will request a summary of the numeric fit statistics using the
\texttt{summary()} function.

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{summary}\NormalTok{(item.fit}\SpecialCharTok{$}\NormalTok{i.infitMSQ)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.6114  0.7490  0.8506  0.8645  0.9689  1.1575
\end{verbatim}

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{summary}\NormalTok{(item.fit}\SpecialCharTok{$}\NormalTok{i.infitZ)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -2.6885 -2.2509 -1.6743 -1.0924 -0.4742  2.2680
\end{verbatim}

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{summary}\NormalTok{(item.fit}\SpecialCharTok{$}\NormalTok{i.outfitMSQ)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.1754  0.6009  0.7648  0.7665  1.0218  1.2805
\end{verbatim}

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{summary}\NormalTok{(item.fit}\SpecialCharTok{$}\NormalTok{i.outfitZ)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -2.7885 -2.6080 -1.5748 -1.0304  0.1904  2.4418
\end{verbatim}

The \texttt{item.fit} object includes mean square error (\(MSE\)) and
standardized (\(Z\)) versions of the outfit and infit statistics for
each item included in the analysis.These statistics are summaries of the
residuals associated with each item. When data fit Rasch model
expectations, the \(MSE\) versions of outfit and infit are expected to
be close to 1.00 and the standardized versions of outfit and infit are
expected to be around 0.00. Please refer to
\protect\hyperlink{MD_fit}{Chapter 3} for a more-detailed discussion of
item fit.

As a final step in examining item fit, we will calculate the
\emph{reliability of separation statistic for items} using the following
procedure.

\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# Get Item scores}
\NormalTok{ItemScores }\OtherTok{\textless{}{-}} \FunctionTok{colSums}\NormalTok{(transreas.responses)}

\CommentTok{\# Get Item Standard Deviation (SD)}
\NormalTok{ItemSD }\OtherTok{\textless{}{-}} \FunctionTok{apply}\NormalTok{(transreas.responses,}\DecValTok{2}\NormalTok{,sd)}

\CommentTok{\# Calculate the Standard Error (SE) for the Item}
\NormalTok{ItemSE }\OtherTok{\textless{}{-}}\NormalTok{ ItemSD}\SpecialCharTok{/}\FunctionTok{sqrt}\NormalTok{(}\FunctionTok{length}\NormalTok{(ItemSD))}

\CommentTok{\# Calculate the Observed Variance (also known as Total Person Variability or Squared Standard Deviation)}
\NormalTok{SSD.ItemScores }\OtherTok{\textless{}{-}} \FunctionTok{var}\NormalTok{(ItemScores)}

\CommentTok{\# Calculate the Mean Square Measurement error (also known as Model Error variance)}
\NormalTok{Item.MSE }\OtherTok{\textless{}{-}} \FunctionTok{sum}\NormalTok{((ItemSE)}\SpecialCharTok{\^{}}\DecValTok{2}\NormalTok{) }\SpecialCharTok{/} \FunctionTok{length}\NormalTok{(ItemSE)}

\CommentTok{\# Calculate the Item Separation Reliability}
\NormalTok{item.separation.reliability }\OtherTok{\textless{}{-}}\NormalTok{ (SSD.ItemScores}\SpecialCharTok{{-}}\NormalTok{Item.MSE) }\SpecialCharTok{/}\NormalTok{ SSD.ItemScores}
\NormalTok{item.separation.reliability}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
## [1] 0.9999984
\end{verbatim}

Briefly, the \emph{item reliability of separation statistic} describes
the degree to which items have unique locations on the logit-scale.
Please refer to \protect\hyperlink{MD_fit}{Chapter 3} for more details
about item reliability analysis.

\textbf{\emph{Person Fit}}

Next, we will conduct a brief exploration of person fit. To calculate
numeric person fit statistics, we will use the function
\texttt{personfit()} from eRm on the person parameter object
(\texttt{person.locations.estimate}). This function produces several
person fit statistics, including infit mean square error (\(MSE\)),
outfit \(MSE\), and standardized infit and outfit \(MSE\) statistics. We
will store the person fit results in a new object called
\texttt{person.fit}, and then format this object as a data.frame for
easy manipulation and exporting. Then, we request summaries of the infit
\(MSE\) and outfit \(MSE\) statistics using \texttt{summary()}.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{person.fit }\OtherTok{\textless{}{-}} \FunctionTok{personfit}\NormalTok{(student.locations)}
\FunctionTok{summary}\NormalTok{(person.fit}\SpecialCharTok{$}\NormalTok{p.infitMSQ)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.4060  0.4910  0.7487  0.9102  1.1740  2.2483
\end{verbatim}

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{summary}\NormalTok{(person.fit}\SpecialCharTok{$}\NormalTok{p.outfitMSQ)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.1723  0.2399  0.5468  0.7665  0.9592  7.3060
\end{verbatim}

We can calculate the \emph{reliability of person separation statistic}
using eRm. This value is interpreted similarly to Cronbach's alpha
{[}@Cronbach{]} when there is good fit between the data and the Rasch
model {[}@Person\_Sep{]}. However, it is important to note that these
coefficients are not equivalent because alpha is based on an assumption
of linearity and the Rasch reliabilty of separation statistic is based
on a linear, interval-level scale when there is evidence of good
model-data fit (discussed in \protect\hyperlink{MD_fit}{Chapter 3}) is
observed.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{person\_rel }\OtherTok{\textless{}{-}} \FunctionTok{SepRel}\NormalTok{(student.locations)}
\NormalTok{person\_rel}\SpecialCharTok{$}\NormalTok{sep.rel}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
## [1] 0.2061478
\end{verbatim}

\textbf{\emph{Summarize the Results in Tables}}

As a final step, we will create tables that summarize the calibrations
of the items and persons from our dichotomous Rasch model analysis with
eRm.

Table 1 is an overall model summary table that provides an overview of
the logit scale locations, standard errors, fit statistics, and
reliability statistics for items and persons. This type of table is
useful for reporting the results from Rasch model analyses because it
provides a concise overview of the location estimates and numeric
model-data fit statistics for the items and persons in the analysis.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{summary.table.statistics }\OtherTok{\textless{}{-}} \FunctionTok{c}\NormalTok{(}\StringTok{"Logit Scale Location Mean"}\NormalTok{,}
                              \StringTok{"Logit Scale Location SD"}\NormalTok{,}
                              \StringTok{"Standard Error Mean"}\NormalTok{,}
                              \StringTok{"Standard Error SD"}\NormalTok{,}
                              \StringTok{"Outfit MSE Mean"}\NormalTok{,}
                              \StringTok{"Outfit MSE SD"}\NormalTok{,}
                              \StringTok{"Infit MSE Mean"}\NormalTok{,}
                              \StringTok{"Infit MSE SD"}\NormalTok{,}
                              \StringTok{"Std. Outfit Mean"}\NormalTok{,}
                              \StringTok{"Std. Outfit SD"}\NormalTok{,}
                              \StringTok{"Std. Infit Mean"}\NormalTok{,}
                              \StringTok{"Std. Infit SD"}\NormalTok{,}
                             \StringTok{"Reliability of Separation"}\NormalTok{)}

\NormalTok{item.summary.results }\OtherTok{\textless{}{-}} \FunctionTok{rbind}\NormalTok{(}\FunctionTok{mean}\NormalTok{(difficulty.all),}
                              \FunctionTok{sd}\NormalTok{(difficulty.all),}
                              \FunctionTok{mean}\NormalTok{(dichot.transreas}\SpecialCharTok{$}\NormalTok{se.beta),}
                              \FunctionTok{sd}\NormalTok{(dichot.transreas}\SpecialCharTok{$}\NormalTok{se.beta),}
                              \FunctionTok{mean}\NormalTok{(item.fit}\SpecialCharTok{$}\NormalTok{i.outfitMSQ),}
                              \FunctionTok{sd}\NormalTok{(item.fit}\SpecialCharTok{$}\NormalTok{i.outfitMSQ),}
                              \FunctionTok{mean}\NormalTok{(item.fit}\SpecialCharTok{$}\NormalTok{i.infitMSQ),}
                              \FunctionTok{sd}\NormalTok{(item.fit}\SpecialCharTok{$}\NormalTok{i.infitMSQ),}
                              \FunctionTok{mean}\NormalTok{(item.fit}\SpecialCharTok{$}\NormalTok{i.outfitZ),}
                              \FunctionTok{sd}\NormalTok{(item.fit}\SpecialCharTok{$}\NormalTok{i.outfitZ),}
                              \FunctionTok{mean}\NormalTok{(item.fit}\SpecialCharTok{$}\NormalTok{i.infitMSQ),}
                              \FunctionTok{sd}\NormalTok{(item.fit}\SpecialCharTok{$}\NormalTok{i.infitZ),}
\NormalTok{                              item.separation.reliability)}


\NormalTok{person.summary.results }\OtherTok{\textless{}{-}} \FunctionTok{rbind}\NormalTok{(}\FunctionTok{mean}\NormalTok{(achievement.with.se}\SpecialCharTok{$}\StringTok{\textasciigrave{}}\AttributeTok{Person Parameter}\StringTok{\textasciigrave{}}\NormalTok{),}
                                \FunctionTok{sd}\NormalTok{(achievement.with.se}\SpecialCharTok{$}\StringTok{\textasciigrave{}}\AttributeTok{Person Parameter}\StringTok{\textasciigrave{}}\NormalTok{),}
                                \FunctionTok{mean}\NormalTok{(achievement.with.se}\SpecialCharTok{$}\NormalTok{person\_se, }\AttributeTok{na.rm =} \ConstantTok{TRUE}\NormalTok{),}
                                \FunctionTok{sd}\NormalTok{(achievement.with.se}\SpecialCharTok{$}\NormalTok{person\_se, }\AttributeTok{na.rm =} \ConstantTok{TRUE}\NormalTok{),}
                                \FunctionTok{mean}\NormalTok{(person.fit}\SpecialCharTok{$}\NormalTok{p.outfitMSQ),}
                                \FunctionTok{sd}\NormalTok{(person.fit}\SpecialCharTok{$}\NormalTok{p.outfitMSQ),}
                                \FunctionTok{mean}\NormalTok{(person.fit}\SpecialCharTok{$}\NormalTok{p.infitMSQ),}
                                \FunctionTok{sd}\NormalTok{(person.fit}\SpecialCharTok{$}\NormalTok{p.infitMSQ),}
                                \FunctionTok{mean}\NormalTok{(person.fit}\SpecialCharTok{$}\NormalTok{p.outfitZ),}
                                \FunctionTok{sd}\NormalTok{(person.fit}\SpecialCharTok{$}\NormalTok{p.outfitZ),}
                                \FunctionTok{mean}\NormalTok{(person.fit}\SpecialCharTok{$}\NormalTok{p.infitZ),}
                                \FunctionTok{sd}\NormalTok{(person.fit}\SpecialCharTok{$}\NormalTok{p.infitZ),}
\NormalTok{                                person\_rel}\SpecialCharTok{$}\NormalTok{sep.rel)}

\CommentTok{\# Round the values for presentation in a table:}
\NormalTok{item.summary.results\_rounded }\OtherTok{\textless{}{-}} \FunctionTok{round}\NormalTok{(item.summary.results, }\AttributeTok{digits =} \DecValTok{2}\NormalTok{)}

\NormalTok{person.summary.results\_rounded }\OtherTok{\textless{}{-}} \FunctionTok{round}\NormalTok{(person.summary.results, }\AttributeTok{digits =} \DecValTok{2}\NormalTok{)}

\NormalTok{Table1 }\OtherTok{\textless{}{-}} \FunctionTok{cbind.data.frame}\NormalTok{(summary.table.statistics,}
\NormalTok{                           item.summary.results\_rounded,}
\NormalTok{                           person.summary.results\_rounded)}

\CommentTok{\# Add descriptive column labels:}
\FunctionTok{names}\NormalTok{(Table1) }\OtherTok{\textless{}{-}} \FunctionTok{c}\NormalTok{(}\StringTok{"Statistic"}\NormalTok{, }\StringTok{"Items"}\NormalTok{, }\StringTok{"Persons"}\NormalTok{)}

\NormalTok{Table1}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##                                             Statistic Items Persons
## X                           Logit Scale Location Mean  0.00    1.71
## X.1                           Logit Scale Location SD  1.58    1.09
## X.2                               Standard Error Mean  0.17    0.95
## X.3                                 Standard Error SD  0.06    0.16
## X.4                                   Outfit MSE Mean  0.77    0.77
## X.5                                     Outfit MSE SD  0.35    0.80
## X.6                                    Infit MSE Mean  0.86    0.91
## X.7                                      Infit MSE SD  0.18    0.47
## X.8                                  Std. Outfit Mean -1.03    0.08
## X.9                                    Std. Outfit SD  1.83    0.64
## X.10                                  Std. Infit Mean  0.86   -0.12
## X.11                                    Std. Infit SD  1.67    0.91
## item.separation.reliability Reliability of Separation  1.00    0.21
\end{verbatim}

Table 2 is a table that summarizes the calibrations of individual items.
For data sets with manageable item sample sizes such as the transitive
reasoning data example in this chapter, we recommend reporting details
about each item in a table similar to this one.

\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# Calculate the proportion correct for each task:}
\NormalTok{PropCorrect }\OtherTok{\textless{}{-}} \FunctionTok{apply}\NormalTok{(transreas.responses, }\DecValTok{2}\NormalTok{, mean)}

\CommentTok{\# Combine item calibration results in a table:}
\NormalTok{Table2 }\OtherTok{\textless{}{-}} \FunctionTok{cbind.data.frame}\NormalTok{(}\FunctionTok{colnames}\NormalTok{(transreas.responses),}
\NormalTok{                           PropCorrect,}
\NormalTok{                           difficulty.all,}
\NormalTok{                           dichot.transreas}\SpecialCharTok{$}\NormalTok{se.beta,}
\NormalTok{                           item.fit}\SpecialCharTok{$}\NormalTok{i.outfitMSQ,}
\NormalTok{                           item.fit}\SpecialCharTok{$}\NormalTok{i.outfitZ,}
\NormalTok{                           item.fit}\SpecialCharTok{$}\NormalTok{i.infitMSQ,}
\NormalTok{                           item.fit}\SpecialCharTok{$}\NormalTok{i.infitZ)}

\FunctionTok{names}\NormalTok{(Table2) }\OtherTok{\textless{}{-}} \FunctionTok{c}\NormalTok{(}\StringTok{"Task ID"}\NormalTok{, }\StringTok{"Proportion Correct"}\NormalTok{, }\StringTok{"Item Location"}\NormalTok{,}\StringTok{"Item SE"}\NormalTok{,}\StringTok{"Outfit MSE"}\NormalTok{,}\StringTok{"Std. Outfit"}\NormalTok{, }\StringTok{"Infit MSE"}\NormalTok{,}\StringTok{"Std. Infit"}\NormalTok{)}

\CommentTok{\# Sort Table 2 by Item difficulty:}
\NormalTok{Table2 }\OtherTok{\textless{}{-}}\NormalTok{ Table2[}\FunctionTok{order}\NormalTok{(}\SpecialCharTok{{-}}\NormalTok{Table2}\SpecialCharTok{$}\StringTok{\textasciigrave{}}\AttributeTok{Item Location}\StringTok{\textasciigrave{}}\NormalTok{),]}

\CommentTok{\# Round the numeric values (all columns except the first one) to 2 digits:}
\NormalTok{Table2[, }\SpecialCharTok{{-}}\DecValTok{1}\NormalTok{] }\OtherTok{\textless{}{-}} \FunctionTok{round}\NormalTok{(Table2[,}\SpecialCharTok{{-}}\DecValTok{1}\NormalTok{], }\AttributeTok{digits =} \DecValTok{2}\NormalTok{)}

\NormalTok{Table2}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##         Task ID Proportion Correct Item Location Item SE Outfit MSE Std. Outfit
## task_09 task_09               0.30          2.92    0.13       1.05        0.46
## task_10 task_10               0.52          1.84    0.12       0.87       -1.82
## task_04 task_04               0.78          0.44    0.13       1.28        2.44
## task_05 task_05               0.80          0.31    0.13       0.93       -0.62
## task_02 task_02               0.81          0.26    0.13       1.13        1.07
## task_07 task_07               0.84         -0.02    0.14       0.65       -2.77
## task_03 task_03               0.88         -0.42    0.16       0.58       -2.74
## task_01 task_01               0.94         -1.24    0.20       0.66       -1.33
## task_08 task_08               0.97         -1.91    0.26       0.34       -2.21
## task_06 task_06               0.97         -2.18    0.29       0.18       -2.79
##         Infit MSE Std. Infit
## task_09      0.90      -1.73
## task_10      0.90      -2.41
## task_04      1.16       2.27
## task_05      0.99      -0.11
## task_02      1.09       1.19
## task_07      0.80      -2.47
## task_03      0.75      -2.69
## task_01      0.76      -1.62
## task_08      0.69      -1.57
## task_06      0.61      -1.78
\end{verbatim}

Finally, Table 3 provides a summary of the person calibrations. When
there is a relatively large person sample size, it may be more useful to
present the results as they relate to individual persons or subsets of
the person sample as they are relevant to the purpose of the analysis.
In the following code, we create Table 3 and print the first 6 lines to
the console using the \texttt{head()} function.

\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# Calculate proportion correct for persons:}
\NormalTok{PersonPropCorrect }\OtherTok{\textless{}{-}} \FunctionTok{apply}\NormalTok{(student.locations}\SpecialCharTok{$}\NormalTok{X.ex, }\DecValTok{1}\NormalTok{, mean)}

\CommentTok{\# Combine person calibration results in a table:}
\NormalTok{Table3 }\OtherTok{\textless{}{-}} \FunctionTok{cbind.data.frame}\NormalTok{(achievement.with.se}\SpecialCharTok{$}\NormalTok{id,}
\NormalTok{                           PersonPropCorrect,}
\NormalTok{                           achievement.with.se}\SpecialCharTok{$}\StringTok{\textasciigrave{}}\AttributeTok{Person Parameter}\StringTok{\textasciigrave{}}\NormalTok{,}
\NormalTok{                           achievement.with.se}\SpecialCharTok{$}\NormalTok{person\_se,}
\NormalTok{                           person.fit}\SpecialCharTok{$}\NormalTok{p.outfitMSQ,}
\NormalTok{                           person.fit}\SpecialCharTok{$}\NormalTok{p.outfitZ,}
\NormalTok{                           person.fit}\SpecialCharTok{$}\NormalTok{p.infitMSQ,}
\NormalTok{                           person.fit}\SpecialCharTok{$}\NormalTok{p.infitZ)}

\FunctionTok{names}\NormalTok{(Table3) }\OtherTok{\textless{}{-}} \FunctionTok{c}\NormalTok{(}\StringTok{"Person ID"}\NormalTok{, }\StringTok{"Proportion Correct"}\NormalTok{, }\StringTok{"Person Location"}\NormalTok{,}\StringTok{"Person SE"}\NormalTok{,}\StringTok{"Outfit MSE"}\NormalTok{,}\StringTok{"Std. Outfit"}\NormalTok{, }\StringTok{"Infit MSE"}\NormalTok{,}\StringTok{"Std. Infit"}\NormalTok{)}

\CommentTok{\# Round the numeric values (all columns except the first one) to 2 digits:}
\NormalTok{Table3[, }\SpecialCharTok{{-}}\DecValTok{1}\NormalTok{] }\OtherTok{\textless{}{-}} \FunctionTok{round}\NormalTok{(Table3[,}\SpecialCharTok{{-}}\DecValTok{1}\NormalTok{], }\AttributeTok{digits =} \DecValTok{2}\NormalTok{)}

\FunctionTok{head}\NormalTok{(Table3)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##   Person ID Proportion Correct Person Location Person SE Outfit MSE Std. Outfit
## 1         1                0.8            1.93      0.94       0.24       -0.52
## 2        10                0.6            1.16      0.83       1.10        0.36
## 3       100                0.4            1.16      0.83       5.62        3.50
## 5       101                0.7            3.03      1.19       0.56       -0.37
## 6       102                0.9            0.52      0.77       0.17        0.12
## 8       103                0.7            3.03      1.19       0.73       -0.09
##   Infit MSE Std. Infit
## 1      0.41      -1.19
## 2      1.13       0.47
## 3      2.10       2.55
## 5      0.71      -0.57
## 6      0.49      -0.61
## 8      0.80      -0.34
\end{verbatim}

\hypertarget{dichotomous-rasch-model-analysis-with-mmle-in-tam}{%
\subsection{Dichotomous Rasch Model Analysis with MMLE in
TAM}\label{dichotomous-rasch-model-analysis-with-mmle-in-tam}}

Next, we will use the \emph{``Test Analysis Modules''}, or ``TAM''
package {[}@TAM{]} to run the dichotomous Rasch model analyses. By
default, the TAM package applies marginal maximum likelihood estimation
(MMLE) to estimate the dichotomous Rasch model. Please keep this
estimation approach in mind when comparing the results between TAM and R
packages or software programs that use other estimation techniques, such
as Winsteps {[}@Winsteps{]} or Facets {[}@Facets{]}.

Except where there are significant differences between the eRm and TAM
procedures, we provide fewer details about the analysis procedures and
interpretations in this section compared to the first illustration.

\textbf{\emph{Prepare for the Analyses}}

To get started with the ``TAM'' package, install and load it into your R
environment using the following code:

\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# install.packages("TAM")}
\FunctionTok{library}\NormalTok{(}\StringTok{"TAM"}\NormalTok{)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
## Loading required package: CDM
\end{verbatim}

\begin{verbatim}
## Loading required package: mvtnorm
\end{verbatim}

\begin{verbatim}
## **********************************
## ** CDM 7.5-15 (2020-03-10 14:19:21)      
## ** Cognitive Diagnostic Models  **
## **********************************
\end{verbatim}

\begin{verbatim}
## * TAM 3.7-16 (2021-06-24 14:31:37)
\end{verbatim}

To facilitate the analysis, we will also use the \emph{``WrightMap''}
package {[}@WrightMap{]}:

WrightMap:

\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# install.packages("WrightMap")}
\FunctionTok{library}\NormalTok{(}\StringTok{"WrightMap"}\NormalTok{) }
\end{Highlighting}
\end{Shaded}

If you have not already imported the Transitive Reasoning data and
prepared it for analysis as described earlier in this chapter by
isolating the response matrix, please do so before continuing with the
TAM analyses.

\textbf{\emph{Run the Dichotomous Rasch Model}}

Now, we are ready to run the dichotomous Rasch model on the transitive
reasoning response data. We will use the \texttt{tam.mml()} function to
run the model and store the results in an object called
\texttt{dichot.transreas}. We specified \texttt{constraint\ =\ items} to
obtain item locations that are centered at zero logits for ease of
interpretation.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{dichot.transreas\_MMLE }\OtherTok{\textless{}{-}} \FunctionTok{tam.mml}\NormalTok{(transreas.responses, }\AttributeTok{constraint =} \StringTok{"items"}\NormalTok{, }\AttributeTok{verbose =} \ConstantTok{FALSE}\NormalTok{)}
\end{Highlighting}
\end{Shaded}

For brevity, we do not include all of the output from the model function
in this book. However, after you run the model, you should see some
output in the R console that includes information about each iteration
in the estimation process.

\textbf{\emph{Overall Model Summary}}

The next step is to request a summary of the model estimation results to
begin to understand the results from the analysis. We can do so by
applying the \texttt{summary()} function to the model object.

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{summary}\NormalTok{(dichot.transreas\_MMLE)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
## ------------------------------------------------------------
## TAM 3.7-16 (2021-06-24 14:31:37) 
## R version 4.1.0 (2021-05-18) x86_64, darwin17.0 | nodename=Chengs-iMac | login=root 
## 
## Date of Analysis: 2021-10-22 21:25:19 
## Time difference of 0.08837509 secs
## Computation time: 0.08837509 
## 
## Multidimensional Item Response Model in TAM 
## 
## IRT Model: PCM2
## Call:
## tam.mml(resp = transreas.responses, constraint = "items", verbose = FALSE)
## 
## ------------------------------------------------------------
## Number of iterations = 72 
## Numeric integration with 21 integration points
## 
## Deviance = 3354.91 
## Log likelihood = -1677.46 
## Number of persons = 425 
## Number of persons used = 425 
## Number of items = 10 
## Number of estimated parameters = 11 
##     Item threshold parameters = 9 
##     Item slope parameters = 0 
##     Regression parameters = 1 
##     Variance/covariance parameters = 1 
## 
## AIC = 3377  | penalty=22    | AIC=-2*LL + 2*p 
## AIC3 = 3388  | penalty=33    | AIC3=-2*LL + 3*p 
## BIC = 3421  | penalty=66.57    | BIC=-2*LL + log(n)*p 
## aBIC = 3386  | penalty=31.56    | aBIC=-2*LL + log((n-2)/24)*p  (adjusted BIC) 
## CAIC = 3432  | penalty=77.57    | CAIC=-2*LL + [log(n)+1]*p  (consistent AIC) 
## AICc = 3378  | penalty=22.64    | AICc=-2*LL + 2*p + 2*p*(p+1)/(n-p-1)  (bias corrected AIC) 
## GHP = 0.39728     | GHP=( -LL + p ) / (#Persons * #Items)  (Gilula-Haberman log penalty) 
## 
## ------------------------------------------------------------
## EAP Reliability
## [1] 0.503
## ------------------------------------------------------------
## Covariances and Variances
##       [,1]
## [1,] 0.938
## ------------------------------------------------------------
## Correlations and Standard Deviations (in the diagonal)
##       [,1]
## [1,] 0.968
## ------------------------------------------------------------
## Regression Coefficients
##         [,1]
## [1,] 1.93558
## ------------------------------------------------------------
## Item Parameters -A*Xsi
##       item   N     M xsi.item AXsi_.Cat1 B.Cat1.Dim1
## 1  task_01 425 0.941   -1.233     -1.233           1
## 2  task_02 425 0.809    0.235      0.235           1
## 3  task_03 425 0.885   -0.432     -0.432           1
## 4  task_04 425 0.784    0.418      0.418           1
## 5  task_05 425 0.802    0.286      0.286           1
## 6  task_06 425 0.974   -2.133     -2.133           1
## 7  task_07 425 0.845   -0.047     -0.047           1
## 8  task_08 425 0.967   -1.875     -1.875           1
## 9  task_09 425 0.301    2.938      2.938           1
## 10 task_10 425 0.520    1.842      1.842           1
## 
## Item Parameters in IRT parameterization
##       item alpha   beta
## 1  task_01     1 -1.233
## 2  task_02     1  0.235
## 3  task_03     1 -0.432
## 4  task_04     1  0.418
## 5  task_05     1  0.286
## 6  task_06     1 -2.133
## 7  task_07     1 -0.047
## 8  task_08     1 -1.875
## 9  task_09     1  2.938
## 10 task_10     1  1.842
\end{verbatim}

From these results, we suggest taking a quick look at the item
parameters as reported in the table labeled \emph{Item Parameters in IRT
parameterization.}

Because we ran a Rasch model, the \emph{alpha} (discrimination)
parameters are fixed to a constant value of 1. The \emph{beta}
parameters are the item locations on the latent variable. We will
explore the item locations in more detail later in the analysis.

\textbf{\emph{Wright Map}}

As the next step in our analysis, we will use the WrightMap package to
create a Wright Map from our model results. We will create the plot
using the function \texttt{IRT.WrightMap()} on the model object
(\texttt{dichot.transreas}). As before, we will set the option for
displaying threshold labels as \texttt{FALSE}, because we are working
with dichotomous data. We also used the \texttt{main.title\ =} option to
customize the title of the plot.

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{IRT.WrightMap}\NormalTok{(dichot.transreas\_MMLE, }\AttributeTok{show.thr.lab=}\ConstantTok{FALSE}\NormalTok{, }\AttributeTok{main.title =} \StringTok{"Transitive Reasoning Wright Map (MMLE)"}\NormalTok{)}
\end{Highlighting}
\end{Shaded}

\includegraphics{02-DichotomousRaschModel_files/figure-latex/unnamed-chunk-30-1.pdf}

In this \emph{Wright Map} display, the results from the dichotomous
Rasch model analysis of the transitive reasoning data are summarized
graphically. The figure is organized as follows:

The left panel of the plot shows a histogram of respondent (person)
locations on the logit scale that represents the latent variable. Units
on the logit scale are shown on the far-right axis of the plot (labeled
\emph{Logits}). In the panel with the person locations, the label
``Dim1'' means that the distribution of person locations is specific to
one dimension. In multi-dimensional Rasch models {[}@Multidi\_RM{]} and
multidimensional IRT analyses {[}@Multi\_IRT\_Wes{]} or
{[}@Multi\_IRT{]}, the Wright Map can show multiple distributions of
person locations that correspond to each dimension in the model.

\begin{figure}
\centering
\includegraphics{screenshots/Cha2_WM.png}
\caption{Wright Map illustration}
\end{figure}

The large central panel of the plot shows the item locations (item
difficulty estimates) on the logit scale that represents the latent
variable. Light grey diamond shapes show the logit scale location of
each item, as labeled on the x-axis.

Even though it is not appropriate to fully interpret item and person
locations on the logit scale until there is evidence of acceptable
model-data fit, we recommend examining the Wright Map during the
preliminary stages of an item analysis to get a general sense of the
model results and to identify any potential scoring or data entry
errors.

A quick glance at the Wright Map suggests that, on average, the persons
are located higher on the logit scale compared to the average item
locations. In addition, there appears to be a relatively wide spread of
person and item locations on the logit scale, such that the transitive
reasoning test appears to be a useful tool for identifying differences
in person locations and item locations on the latent variable. We will
return to this display for more exploration after checking for
acceptable model-data fit, among other psychometric properties.

\textbf{\emph{Item Parameters}}

As the next step in our analysis, we will examine the item parameters in
detail. We will use the \texttt{\$} operator to request the item
location estimates that are stored in the \texttt{item\_irt} table
within the \texttt{dichot.transreas\_MMLE} object. Then, we will save
the item location estimates in a new object called
\texttt{difficulty\_MMLE}.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{difficulty\_MMLE }\OtherTok{\textless{}{-}} \FunctionTok{as.data.frame}\NormalTok{(dichot.transreas\_MMLE}\SpecialCharTok{$}\NormalTok{item\_irt}\SpecialCharTok{$}\NormalTok{beta)}
\NormalTok{difficulty\_MMLE}\SpecialCharTok{$}\NormalTok{se }\OtherTok{\textless{}{-}}\NormalTok{ dichot.transreas\_MMLE}\SpecialCharTok{$}\NormalTok{se.AXsi}
\FunctionTok{names}\NormalTok{(difficulty\_MMLE) }\OtherTok{\textless{}{-}} \FunctionTok{c}\NormalTok{(}\StringTok{"item\_difficulty"}\NormalTok{, }\StringTok{"item\_se"}\NormalTok{)}
\end{Highlighting}
\end{Shaded}

By running this code, we have created a data.frame object that includes
two variables: item difficulty estimates, and standard errors. The row
labels show the item numbers from our response matrix. The item
difficulty parameters and standard errors are interpreted in the same
way as in the eRm analysis presented earlier in this chapter.

Next, let's calculate descriptive statistics to better understand the
distribution of the item locations and standard errors. We will do so
using the \texttt{summary()} function and the \texttt{sd()} function.

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{summary}\NormalTok{(difficulty\_MMLE)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##  item_difficulty         item_se.V1            item_se.V2     
##  Min.   :-2.13288   Min.   :0             Min.   :0.07786662  
##  1st Qu.:-1.03275   1st Qu.:0             1st Qu.:0.08245890  
##  Median : 0.09391   Median :0             Median :0.08679307  
##  Mean   : 0.00000   Mean   :0             Mean   :0.10578784  
##  3rd Qu.: 0.38541   3rd Qu.:0             3rd Qu.:0.09815207  
##  Max.   : 2.93795   Max.   :0             Max.   :0.26524247
\end{verbatim}

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{sd}\NormalTok{(difficulty\_MMLE}\SpecialCharTok{$}\NormalTok{item\_difficulty)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
## [1] 1.567331
\end{verbatim}

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{sd}\NormalTok{(difficulty\_MMLE}\SpecialCharTok{$}\NormalTok{item\_se)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
## [1] 0.06678788
\end{verbatim}

We can also visualize the item difficulty estimates using a simple
histogram by using the \texttt{hist()} function. In our plot, we
specified a custom title using \texttt{main\ =} and a custom x-axis
label using \texttt{xlab\ =}.

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{hist}\NormalTok{(difficulty\_MMLE}\SpecialCharTok{$}\NormalTok{item\_difficulty, }\AttributeTok{main =} \StringTok{"Histogram of Item Difficulty Estimates }\SpecialCharTok{\textbackslash{}n}\StringTok{for the Transitive Reasoning Data (MMLE)"}\NormalTok{,}
     \AttributeTok{xlab =} \StringTok{"Item Difficulty Estimates in Logits"}\NormalTok{) }
\end{Highlighting}
\end{Shaded}

\includegraphics{02-DichotomousRaschModel_files/figure-latex/unnamed-chunk-33-1.pdf}

\textbf{\emph{Item Fit}}

To calculate numeric item fit statistics, we will use the function
\texttt{tam.fit()} from TAM on the model object
(\texttt{dichot.transreas\_MMLE}). We will store the item fit results in
a new object called \texttt{item.fit\_MMLE}, and then format this object
as a data.frame for easy manipulation and exporting.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{item.fit\_MMLE }\OtherTok{\textless{}{-}} \FunctionTok{tam.fit}\NormalTok{(dichot.transreas\_MMLE) }
\NormalTok{item.fit\_MMLE }\OtherTok{\textless{}{-}} \FunctionTok{as.data.frame}\NormalTok{(item.fit\_MMLE}\SpecialCharTok{$}\NormalTok{itemfit)}
\end{Highlighting}
\end{Shaded}

Next, we will request a summary of the numeric fit statistics using the
\texttt{summary()} function:

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{summary}\NormalTok{(item.fit\_MMLE)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##   parameter             Outfit          Outfit_t          Outfit_p        
##  Length:9           Min.   :0.8323   Min.   :-5.7303   Min.   :0.0000000  
##  Class :character   1st Qu.:0.9033   1st Qu.:-2.3356   1st Qu.:0.0002126  
##  Mode  :character   Median :0.9793   Median :-0.4544   Median :0.0195102  
##                     Mean   :0.9933   Mean   :-0.2944   Mean   :0.2777366  
##                     3rd Qu.:0.9891   3rd Qu.:-0.2882   3rd Qu.:0.6495455  
##                     Max.   :1.2746   Max.   : 7.0373   Max.   :0.7732039  
##   Outfit_pholm          Infit           Infit_t           Infit_p         
##  Min.   :0.000000   Min.   :0.8310   Min.   :-5.7768   Min.   :0.0000000  
##  1st Qu.:0.001488   1st Qu.:0.9007   1st Qu.:-2.3934   1st Qu.:0.0001533  
##  Median :0.097551   Median :0.9832   Median :-0.3632   Median :0.0166951  
##  Mean   :0.456209   Mean   :0.9948   Mean   :-0.2534   Mean   :0.2990287  
##  3rd Qu.:1.000000   3rd Qu.:0.9936   3rd Qu.:-0.1661   3rd Qu.:0.7164407  
##  Max.   :1.000000   Max.   :1.2738   Max.   : 7.0249   Max.   :0.8680834  
##   Infit_pholm      
##  Min.   :0.000000  
##  1st Qu.:0.001073  
##  Median :0.083475  
##  Mean   :0.455035  
##  3rd Qu.:1.000000  
##  Max.   :1.000000
\end{verbatim}

The \texttt{item.fit\_MMLE} object includes mean square error (\(MSE\))
and standardized (\(t\)) versions of the outfit and infit statistics for
Rasch models. TAM also reports a \(p\) value for the standardized fit
statistics (\texttt{Outfit\_p} and \texttt{Infit\_p}), along with
adjusted significance values (\texttt{Infit\_pholm} and
\texttt{Outfit\_pholm}).

With the item centering constraint that we used, the TAM package does
not report fit statistics for all ten items. We can find approximate
item fit statistics for all ten items by running the model without the
centering constraint and saving the fit statistics.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{dichot.transreas\_MMLE2 }\OtherTok{\textless{}{-}} \FunctionTok{tam}\NormalTok{(transreas.responses, }\AttributeTok{verbose =} \ConstantTok{FALSE}\NormalTok{)}
\NormalTok{item.fit\_MMLE2 }\OtherTok{\textless{}{-}} \FunctionTok{tam.fit}\NormalTok{(dichot.transreas\_MMLE2) }
\NormalTok{item.fit\_MMLE2 }\OtherTok{\textless{}{-}} \FunctionTok{as.data.frame}\NormalTok{(item.fit\_MMLE2}\SpecialCharTok{$}\NormalTok{itemfit)}
\end{Highlighting}
\end{Shaded}

\textbf{\emph{Person Parameters}}

When the default MMLE method is used to estimate the dichotomous Rasch
model in TAM, person parameters are calculated after the item locations
are estimated. As a result, the person estimation procedure for this
model requires additional iterations.

In the following code, we calculate person locations that correspond to
our model using the \texttt{tam.wle()}function with the dichotomous
Rasch model object (\texttt{dichot.transreas\_MMLE}). We stored the
results in a new data frame called \texttt{achievement\_MMLE}.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{achievement\_MMLE }\OtherTok{\textless{}{-}} \FunctionTok{as.data.frame}\NormalTok{(}\FunctionTok{tam.wle}\NormalTok{(dichot.transreas\_MMLE))}
\end{Highlighting}
\end{Shaded}

The \texttt{achievement\_MMLE} data frame includes the following
variables:

\begin{itemize}
\tightlist
\item
  \emph{pid}: Person identification number (based on person ordering in
  the item response matrix)
\item
  \emph{N.items}: Number of scored item responses for each person
\item
  \emph{PersonScores}: Sum of scored item responses for each person
\item
  \emph{PersonMax}: Maximum possible score for item responses
\item
  \emph{theta}: Person location estimate on the logit scale that
  represents the latent variable.
\item
  \emph{error}: Standard error for person location estimate, as
  described earlier in this chapter.
\item
  \emph{WLE.rel}: Reliability of person separation statistic, as
  described earlier in this chapter.
\end{itemize}

Next, we will calculate descriptive statistics to better understand the
distribution of the person estimates. We will do so using the
\texttt{summary()} function with the \texttt{achievement\_MMLE}object.

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{summary}\NormalTok{(achievement\_MMLE)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##       pid         N.items    PersonScores      PersonMax      theta       
##  Min.   :  1   Min.   :10   Min.   : 1.000   Min.   :10   Min.   :-2.592  
##  1st Qu.:107   1st Qu.:10   1st Qu.: 7.000   1st Qu.:10   1st Qu.: 1.058  
##  Median :213   Median :10   Median : 8.000   Median :10   Median : 1.772  
##  Mean   :213   Mean   :10   Mean   : 7.828   Mean   :10   Mean   : 1.880  
##  3rd Qu.:319   3rd Qu.:10   3rd Qu.: 9.000   3rd Qu.:10   3rd Qu.: 2.727  
##  Max.   :425   Max.   :10   Max.   :10.000   Max.   :10   Max.   : 4.283  
##      error           WLE.rel      
##  Min.   :0.7508   Min.   :0.3082  
##  1st Qu.:0.8191   1st Qu.:0.3082  
##  Median :0.9172   Median :0.3082  
##  Mean   :1.0227   Mean   :0.3082  
##  3rd Qu.:1.1106   3rd Qu.:0.3082  
##  Max.   :1.7628   Max.   :0.3082
\end{verbatim}

We can also visualize the person location estimates using a simple
histogram by using the \texttt{hist()} function. In our plot, we
specified a custom title using \texttt{main\ =} and a custom x-axis
label using \texttt{xlab\ =}.

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{hist}\NormalTok{(achievement\_MMLE}\SpecialCharTok{$}\NormalTok{theta, }\AttributeTok{main =} \StringTok{"Histogram of Person Achievement Estimates }\SpecialCharTok{\textbackslash{}n}\StringTok{for the Transitive Reasoning Data }\SpecialCharTok{\textbackslash{}n}\StringTok{ (MMLE)"}\NormalTok{,}
     \AttributeTok{xlab =} \StringTok{"Person Achievement Estimates in Logits"}\NormalTok{, }\AttributeTok{cex.main =}\NormalTok{ .}\DecValTok{9}\NormalTok{) }
\end{Highlighting}
\end{Shaded}

\includegraphics{02-DichotomousRaschModel_files/figure-latex/unnamed-chunk-39-1.pdf}

\textbf{\emph{Person Fit}}

As a final step in our preliminary person analysis, we will conduct a
brief exploration of person fit statistics. We explore person fit in
more detail in \protect\hyperlink{MD_fit}{Chapter 3}.

To calculate numeric person fit statistics, we will use the function
\texttt{tam.personfit()} from TAM on the model object
(\texttt{dichot.transreas\_MMLE}). We will store the person fit results
in a new object called \texttt{person.fit\_MMLE}, which is a data.frame
object.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{person.fit\_MMLE }\OtherTok{\textless{}{-}} \FunctionTok{tam.personfit}\NormalTok{(dichot.transreas\_MMLE)}
\end{Highlighting}
\end{Shaded}

Next, we will request a summary of the numeric person fit statistics
using the \texttt{summary()} function.

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{summary}\NormalTok{(person.fit\_MMLE)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##   outfitPerson     outfitPerson_t      infitPerson     infitPerson_t    
##  Min.   :0.04345   Min.   :-0.95709   Min.   :0.1569   Min.   :-1.5008  
##  1st Qu.:0.16335   1st Qu.:-0.63496   1st Qu.:0.3613   1st Qu.:-0.9720  
##  Median :0.46295   Median :-0.06839   Median :0.6428   Median :-0.5789  
##  Mean   :0.61001   Mean   : 0.05723   Mean   :0.7734   Mean   :-0.2966  
##  3rd Qu.:0.84557   3rd Qu.: 0.68920   3rd Qu.:1.0647   3rd Qu.: 0.2983  
##  Max.   :4.21347   Max.   : 2.84202   Max.   :2.2003   Max.   : 2.5764
\end{verbatim}

The \texttt{person.fit\_MMLE} object includes mean square error
(\(MSE\)) and standardized (\(t\)) versions of the person outfit and
infit statistics for Rasch models.

\hypertarget{dichotomous-rasch-model-analysis-with-jmle-in-tam}{%
\subsection{Dichotomous Rasch Model Analysis with JMLE in
TAM}\label{dichotomous-rasch-model-analysis-with-jmle-in-tam}}

We can also use the \texttt{tam.jml()} function to estimate the
dichotomous Rasch model using Joint Maximum Likelihood Estimation
(JMLE), which is used in the Facets software program {[}@Facets{]} and
the Winsteps software program {[}@Winsteps{]}, which are popular
standalone software programs for Rasch analyses.

With the exception of the model estimation code, most of the code for
the JMLE approach is the same as the previous analysis with MMLE.
Accordingly, we provide fewer explanations in our presentation of this
code.

\textbf{\emph{Estimate the Dichotomous Rasch Model}}

We already prepared the transitive reasoning data for analysis when we
checked the data and isolated the item response matrix
(\texttt{transreas.responses}) earlier in this chapter. We can begin our
analysis using the \texttt{tam.jml()} function with this response
matrix.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{dichot.transreas\_JMLE }\OtherTok{\textless{}{-}} \FunctionTok{tam.jml}\NormalTok{(transreas.responses, }\AttributeTok{constraint =} \StringTok{"items"}\NormalTok{, }\AttributeTok{verbose =} \ConstantTok{FALSE}\NormalTok{)}
\end{Highlighting}
\end{Shaded}

\textbf{\emph{Overall Model Summary}}

\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# Request a summary of the model results}
\FunctionTok{summary}\NormalTok{(dichot.transreas\_JMLE)}
\end{Highlighting}
\end{Shaded}

\textbf{\emph{Wright Map}}

When the JMLE estimation method is used, we need to specify the item
location estimates and the person location estimates as vectors in the
plotting function for the Wright Map. We will also use a different
function to plot the Wright Map: \texttt{wrightMap()}, which takes
arguments for the item and person locations.

In the following code, we saved the item locations in a vector called
\texttt{difficulty\_JMLE}, and the person locations in a vector called
\texttt{theta\_JMLE}. Then, we used these vectors as arguments in the
\texttt{wrightMap()} function to create the Wright Map.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{difficulty\_JMLE }\OtherTok{\textless{}{-}}\NormalTok{ dichot.transreas\_JMLE}\SpecialCharTok{$}\NormalTok{item}\SpecialCharTok{$}\NormalTok{xsi.item}
\NormalTok{theta\_JMLE }\OtherTok{\textless{}{-}}\NormalTok{ dichot.transreas\_JMLE}\SpecialCharTok{$}\NormalTok{theta}
\FunctionTok{wrightMap}\NormalTok{(}\AttributeTok{thetas =}\NormalTok{ theta\_JMLE, }\AttributeTok{thresholds =}\NormalTok{ difficulty\_JMLE, }\AttributeTok{show.thr.lab=}\ConstantTok{FALSE}\NormalTok{, }\AttributeTok{main.title =} \StringTok{"Transitive Reasoning Wright Map: JMLE"}\NormalTok{)}
\end{Highlighting}
\end{Shaded}

\includegraphics{02-DichotomousRaschModel_files/figure-latex/unnamed-chunk-44-1.pdf}

\begin{verbatim}
##              [,1]
##  [1,] -1.22764983
##  [2,]  0.19372234
##  [3,] -0.46058464
##  [4,]  0.37626060
##  [5,]  0.24499876
##  [6,] -2.04579773
##  [7,] -0.08402885
##  [8,] -1.81784893
##  [9,]  2.99257136
## [10,]  1.82835693
\end{verbatim}

\textbf{\emph{Item Parameters}}

When we plotted the Wright Map, we created an object with the item
difficulty parameters. We can find the standard errors for the item
estimates by using the \texttt{\$} operator to extract \texttt{errorP}
from the model results.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{jmle.item.se }\OtherTok{\textless{}{-}}\NormalTok{ dichot.transreas\_JMLE}\SpecialCharTok{$}\NormalTok{errorP}
\NormalTok{jmle.item.se}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
## [1] 0.10134872 0.08825844 0.09427051 0.08677937 0.08783074 0.10718132 0.09071165
## [8] 0.10578694 0.08445690
\end{verbatim}

Because of the item centering constraint that we included in our
analysis, the standard error for item 10 is not reported.

\textbf{\emph{Person Parameters}}

Unlike the MMLE estimation approach, the JMLE approach calculates item
parameters and person parameters in the same step. As a result, we do
not need to use a separate function to find the person parameter
estimates from our model.

When we plotted the Wright Map, we already saved the theta estimates in
a vector called \texttt{theta\_JMLE}. We can find the standard errors
for the person estimates by using the \texttt{\$} operator to extract
\texttt{errorWLE} from the model results.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{jmle.person.se }\OtherTok{\textless{}{-}}\NormalTok{ dichot.transreas\_JMLE}\SpecialCharTok{$}\NormalTok{errorWLE}
\end{Highlighting}
\end{Shaded}

\textbf{\emph{Item and Person Fit}}

When JMLE is used, TAM calculates both person and item fit using a
single function: \texttt{tam.jml.fit()}.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{jmle.fit }\OtherTok{\textless{}{-}} \FunctionTok{tam.jml.fit}\NormalTok{(dichot.transreas\_JMLE)}
\end{Highlighting}
\end{Shaded}

Now that we have calculated fit statistics for items and persons, we
will save the item fit statistics in a data frame called
\texttt{jmle.item.fit}. We can examine the item fit statistics in more
detail by printing the \texttt{jmle.item.fit} object to the console or
preview window and calculating summary statistics.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{jmle.item.fit }\OtherTok{\textless{}{-}} \FunctionTok{as.data.frame}\NormalTok{(jmle.fit}\SpecialCharTok{$}\NormalTok{fit.item)}
\NormalTok{jmle.item.fit}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##            item outfitItem outfitItem_t infitItem infitItem_t
## task_01 task_01  0.4789486   -1.3892725 0.7618074  -1.6224741
## task_02 task_02  1.0961304    0.5264972 1.1485741   1.8556014
## task_03 task_03  0.5213243   -1.9455120 0.7764883  -2.2436599
## task_04 task_04  1.2730392    1.4251000 1.2202071   2.9097277
## task_05 task_05  0.8907676   -0.5080772 1.0439175   0.5973591
## task_06 task_06  0.1379772   -2.0288954 0.5391578  -2.3667045
## task_07 task_07  0.6110629   -1.8512903 0.8437433  -1.8383988
## task_08 task_08  0.2378325   -1.7907958 0.6393029  -1.9529903
## task_09 task_09  0.8683358   -1.0803307 0.8585916  -2.5264478
## task_10 task_10  0.7954148   -2.2413359 0.8964146  -2.1916767
\end{verbatim}

\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{summary}\NormalTok{(jmle.item.fit)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##      item             outfitItem      outfitItem_t       infitItem     
##  Length:10          Min.   :0.1380   Min.   :-2.2413   Min.   :0.5392  
##  Class :character   1st Qu.:0.4895   1st Qu.:-1.9220   1st Qu.:0.7655  
##  Mode  :character   Median :0.7032   Median :-1.5900   Median :0.8512  
##                     Mean   :0.6911   Mean   :-1.0884   Mean   :0.8728  
##                     3rd Qu.:0.8852   3rd Qu.:-0.6511   3rd Qu.:1.0070  
##                     Max.   :1.2730   Max.   : 1.4251   Max.   :1.2202  
##   infitItem_t     
##  Min.   :-2.5264  
##  1st Qu.:-2.2307  
##  Median :-1.8957  
##  Mean   :-0.9380  
##  3rd Qu.: 0.0424  
##  Max.   : 2.9097
\end{verbatim}

Next, we will save the person fit statistics in a data frame called
\texttt{jmle.person.fit}. We can examine the person fit statistics in
more detail by viewing it in the preview window and calculating summary
statistics.

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{jmle.person.fit }\OtherTok{\textless{}{-}} \FunctionTok{as.data.frame}\NormalTok{(jmle.fit}\SpecialCharTok{$}\NormalTok{fit.person)}
\FunctionTok{summary}\NormalTok{(jmle.person.fit)}
\end{Highlighting}
\end{Shaded}

\begin{verbatim}
##   outfitPerson     outfitPerson_t     infitPerson     infitPerson_t     
##  Min.   :0.02708   Min.   :-0.9354   Min.   :0.1067   Min.   :-1.51822  
##  1st Qu.:0.17379   1st Qu.:-0.5101   1st Qu.:0.4296   1st Qu.:-0.71666  
##  Median :0.49116   Median : 0.2006   Median :0.6515   Median :-0.31480  
##  Mean   :0.69108   Mean   : 0.2507   Mean   :0.8560   Mean   :-0.05799  
##  3rd Qu.:0.89769   3rd Qu.: 0.7786   3rd Qu.:1.1756   3rd Qu.: 0.56986  
##  Max.   :4.95982   Max.   : 3.0980   Max.   :2.3619   Max.   : 2.54349
\end{verbatim}

\hypertarget{example-results-section}{%
\subsection{Example Results Section}\label{example-results-section}}

Table 1 presents a summary of the results from the analysis of the
transitive reasoning data {[}@Intro\_nonparametric{]}
(\url{https://methods.sagepub.com/book/introduction-to-nonparametric-item-response-theory})
using the dichotomous Rasch model {[}@Rasch\_ori{]}. Specifically, the
calibration of students (\(N = 425\)) and tasks (\(N = 10\)) are
summarized using average logit-scale calibrations, standard errors, and
model-data fit statistics. Examination of the results indicates that, on
average, the students were located higher on the logit scale
(\(M = 1.71\), \(SD = 1.09\)), compared to tasks, whose locations were
centered at zero logits (\(M = 0.00\), \(SD = 1.58\)). This finding
suggests that the items were relatively easy for the sample of students
who participated in this transitive reasoning test. However, average
values of the Standard Error (\(SE\)) were slightly larger for students
(\(M = 0.95\)) compared to Tasks (\(M = 0.17\)), indicating that there
may be some issues related to targeting for some of the students who
participated in the assessment. Average values of model-data fit
statistics indicate overall adequate fit to the model, with average
outfit and infit mean square statistics around 1.00, and average
standardized outfit and infit statistics near the expected value of 0.00
when data fit the model. This finding of overall adequate fit to the
model supports the interpretation of item and person calibrations on the
logit scale as indicators of their locations on the latent variable
measured by the test.

\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# Print Table 1:}
\NormalTok{knitr}\SpecialCharTok{::}\FunctionTok{kable}\NormalTok{(}
\NormalTok{  Table1, }\AttributeTok{booktabs =} \ConstantTok{TRUE}\NormalTok{,}
  \AttributeTok{caption =} \StringTok{\textquotesingle{}Model Summary Table\textquotesingle{}}
\NormalTok{)}
\end{Highlighting}
\end{Shaded}

\begin{longtable}[]{@{}llrr@{}}
\caption{Model Summary Table}\tabularnewline
\toprule
& Statistic & Items & Persons \\
\midrule
\endfirsthead
\toprule
& Statistic & Items & Persons \\
\midrule
\endhead
X & Logit Scale Location Mean & 0.00 & 1.71 \\
X.1 & Logit Scale Location SD & 1.58 & 1.09 \\
X.2 & Standard Error Mean & 0.17 & 0.95 \\
X.3 & Standard Error SD & 0.06 & 0.16 \\
X.4 & Outfit MSE Mean & 0.77 & 0.77 \\
X.5 & Outfit MSE SD & 0.35 & 0.80 \\
X.6 & Infit MSE Mean & 0.86 & 0.91 \\
X.7 & Infit MSE SD & 0.18 & 0.47 \\
X.8 & Std. Outfit Mean & -1.03 & 0.08 \\
X.9 & Std. Outfit SD & 1.83 & 0.64 \\
X.10 & Std. Infit Mean & 0.86 & -0.12 \\
X.11 & Std. Infit SD & 1.67 & 0.91 \\
item.separation.reliability & Reliability of Separation & 1.00 & 0.21 \\
\bottomrule
\end{longtable}

Table 2 includes detailed results for the 10 tasks included in the
Transitive Reasoning test. For each item, the proportion of correct
responses is presented, followed by the logit-scale location
(\(\delta\)), SE, and model-data fit statistics. Examination of these
results indicates that Task 9 was the most difficult
(\(Proportion Correct = 0.30\); \(\delta\) = 2.92 ; \(SE = 0.13\)),
followed by Task 10 (\(Proportion Correct = 0.52\); \(\delta\) = 1.84;
\(SE = 0.12\)). The easiest item was Task 6
(\(Proportion Correct = 0.97\); \(\delta\) = -2.18; \(SE = 0.29\)).

\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# Print Table 2:}
\NormalTok{knitr}\SpecialCharTok{::}\FunctionTok{kable}\NormalTok{(}
\NormalTok{  Table2, }\AttributeTok{booktabs =} \ConstantTok{TRUE}\NormalTok{,}
  \AttributeTok{caption =} \StringTok{\textquotesingle{}Item Calibration\textquotesingle{}}
\NormalTok{)}
\end{Highlighting}
\end{Shaded}

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\columnwidth - 16\tabcolsep) * \real{0.0792}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 16\tabcolsep) * \real{0.0792}}
  >{\raggedleft\arraybackslash}p{(\columnwidth - 16\tabcolsep) * \real{0.1881}}
  >{\raggedleft\arraybackslash}p{(\columnwidth - 16\tabcolsep) * \real{0.1386}}
  >{\raggedleft\arraybackslash}p{(\columnwidth - 16\tabcolsep) * \real{0.0792}}
  >{\raggedleft\arraybackslash}p{(\columnwidth - 16\tabcolsep) * \real{0.1089}}
  >{\raggedleft\arraybackslash}p{(\columnwidth - 16\tabcolsep) * \real{0.1188}}
  >{\raggedleft\arraybackslash}p{(\columnwidth - 16\tabcolsep) * \real{0.0990}}
  >{\raggedleft\arraybackslash}p{(\columnwidth - 16\tabcolsep) * \real{0.1089}}@{}}
\caption{Item Calibration}\tabularnewline
\toprule
\begin{minipage}[b]{\linewidth}\raggedright
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Task ID
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Proportion Correct
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Item Location
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Item SE
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Outfit MSE
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Std. Outfit
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Infit MSE
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Std. Infit
\end{minipage} \\
\midrule
\endfirsthead
\toprule
\begin{minipage}[b]{\linewidth}\raggedright
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Task ID
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Proportion Correct
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Item Location
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Item SE
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Outfit MSE
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Std. Outfit
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Infit MSE
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Std. Infit
\end{minipage} \\
\midrule
\endhead
task\_09 & task\_09 & 0.30 & 2.92 & 0.13 & 1.05 & 0.46 & 0.90 & -1.73 \\
task\_10 & task\_10 & 0.52 & 1.84 & 0.12 & 0.87 & -1.82 & 0.90 &
-2.41 \\
task\_04 & task\_04 & 0.78 & 0.44 & 0.13 & 1.28 & 2.44 & 1.16 & 2.27 \\
task\_05 & task\_05 & 0.80 & 0.31 & 0.13 & 0.93 & -0.62 & 0.99 &
-0.11 \\
task\_02 & task\_02 & 0.81 & 0.26 & 0.13 & 1.13 & 1.07 & 1.09 & 1.19 \\
task\_07 & task\_07 & 0.84 & -0.02 & 0.14 & 0.65 & -2.77 & 0.80 &
-2.47 \\
task\_03 & task\_03 & 0.88 & -0.42 & 0.16 & 0.58 & -2.74 & 0.75 &
-2.69 \\
task\_01 & task\_01 & 0.94 & -1.24 & 0.20 & 0.66 & -1.33 & 0.76 &
-1.62 \\
task\_08 & task\_08 & 0.97 & -1.91 & 0.26 & 0.34 & -2.21 & 0.69 &
-1.57 \\
task\_06 & task\_06 & 0.97 & -2.18 & 0.29 & 0.18 & -2.79 & 0.61 &
-1.78 \\
\bottomrule
\end{longtable}

Table 3 includes detailed results for first 10 students who participated
in the Transitive Reasoning Test. For each student, the proportion of
correct responses is presented, followed by their logit-scale location
estimate (\(\theta\)), \(SE\), and model-data fit statistics.

\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# Print the first 10 rows of Table 3:}
\NormalTok{knitr}\SpecialCharTok{::}\FunctionTok{kable}\NormalTok{(}
  \FunctionTok{head}\NormalTok{(Table3,}\DecValTok{10}\NormalTok{), }\AttributeTok{booktabs =} \ConstantTok{TRUE}\NormalTok{,}
  \AttributeTok{caption =} \StringTok{\textquotesingle{}Person Calibration\textquotesingle{}}
\NormalTok{)}
\end{Highlighting}
\end{Shaded}

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\columnwidth - 16\tabcolsep) * \real{0.0294}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 16\tabcolsep) * \real{0.0980}}
  >{\raggedleft\arraybackslash}p{(\columnwidth - 16\tabcolsep) * \real{0.1863}}
  >{\raggedleft\arraybackslash}p{(\columnwidth - 16\tabcolsep) * \real{0.1569}}
  >{\raggedleft\arraybackslash}p{(\columnwidth - 16\tabcolsep) * \real{0.0980}}
  >{\raggedleft\arraybackslash}p{(\columnwidth - 16\tabcolsep) * \real{0.1078}}
  >{\raggedleft\arraybackslash}p{(\columnwidth - 16\tabcolsep) * \real{0.1176}}
  >{\raggedleft\arraybackslash}p{(\columnwidth - 16\tabcolsep) * \real{0.0980}}
  >{\raggedleft\arraybackslash}p{(\columnwidth - 16\tabcolsep) * \real{0.1078}}@{}}
\caption{Person Calibration}\tabularnewline
\toprule
\begin{minipage}[b]{\linewidth}\raggedright
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Person ID
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Proportion Correct
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Person Location
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Person SE
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Outfit MSE
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Std. Outfit
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Infit MSE
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Std. Infit
\end{minipage} \\
\midrule
\endfirsthead
\toprule
\begin{minipage}[b]{\linewidth}\raggedright
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Person ID
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Proportion Correct
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Person Location
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Person SE
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Outfit MSE
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Std. Outfit
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Infit MSE
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Std. Infit
\end{minipage} \\
\midrule
\endhead
1 & 1 & 0.8 & 1.93 & 0.94 & 0.24 & -0.52 & 0.41 & -1.19 \\
2 & 10 & 0.6 & 1.16 & 0.83 & 1.10 & 0.36 & 1.13 & 0.47 \\
3 & 100 & 0.4 & 1.16 & 0.83 & 5.62 & 3.50 & 2.10 & 2.55 \\
5 & 101 & 0.7 & 3.03 & 1.19 & 0.56 & -0.37 & 0.71 & -0.57 \\
6 & 102 & 0.9 & 0.52 & 0.77 & 0.17 & 0.12 & 0.49 & -0.61 \\
8 & 103 & 0.7 & 3.03 & 1.19 & 0.73 & -0.09 & 0.80 & -0.34 \\
9 & 104 & 0.9 & 1.93 & 0.94 & 0.17 & 0.12 & 0.49 & -0.61 \\
10 & 105 & 0.7 & 1.16 & 0.83 & 0.43 & -0.64 & 0.59 & -0.96 \\
11 & 106 & 0.9 & 1.93 & 0.94 & 0.17 & 0.12 & 0.49 & -0.61 \\
12 & 107 & 0.9 & 1.16 & 0.83 & 1.47 & 0.92 & 1.64 & 0.97 \\
\bottomrule
\end{longtable}

Figure 1.1 illustrates the calibrations of the Participants and Items on
the logit scale that represents the latent variable. The calibrations
shown in this figure correspond to the results presented in Table 2 and
Table 3 for tasks and students, respectively. Starting at the bottom of
the figure, the horizontal axis (labeled \emph{Latent Dimension}) is the
logit scale that represents the latent variable. In this analysis, lower
numbers indicate less transitive reasoning ability, and higher numbers
indicate more transitive reasoning ability. The central panel of the
figure shows item difficulty locations on the logit scale for the 10
transitive reasoning tasks that were included in the analysis; the
y-axis for this panel shows the item labels. By default in eRm, the
items are ordered according to their original order in the response
matrix. The upper panel of the figure shows a histogram of person (in
this case, student) location estimates on the logit scale. Small
vertical lines on the x-axis of this histogram show the points on the
logit scale at which information (variance) is maximized for the sample
of persons and items in the analysis.

On average, the persons (students) are located higher on the logit scale
compared to the average item (task) locations. In addition, there
appears to be a relatively wide spread of person and item locations on
the logit scale, such that the transitive reasoning test appears to be a
useful tool for identifying differences in person locations and item
locations on the latent variable.

\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# Figure 1.1}
\FunctionTok{plotPImap}\NormalTok{(dichot.transreas, }\AttributeTok{main =} \StringTok{"Liking for Science Rating Scale Model Wright Map"}\NormalTok{)}
\end{Highlighting}
\end{Shaded}

\includegraphics{02-DichotomousRaschModel_files/figure-latex/unnamed-chunk-53-1.pdf}

\hypertarget{exercise}{%
\subsection{Exercise}\label{exercise}}

Estimate item and person locations with the dichotomous Rasch model for
the Chapter 2 Exercise data using any of the estimation procedures
included in this chapter. The Exercise 2 data include responses from 500
students to a mathematics achievement test that includes 35 items scored
as incorrect (\(x=0\)) or correct (\(x=1\)). Then, try writing a results
section similar to the example in this chapter to report your findings.

\hypertarget{supplementary-learning-materials}{%
\subsection{Supplementary Learning
Materials}\label{supplementary-learning-materials}}

\href{https://www.rasch.org/moulton.xls}{Rasch Estimation Demonstration
Spreadsheet}

\href{https://link.springer.com/content/pdf/10.3758/BF03192809.pdf}{Li,
Y. Using the open-source statistical language R to analyze the
dichotomous Rasch model. Behavior Research Methods 38, 532--541 (2006).
https://doi.org/10.3758/BF03192809}

\href{https://eric.ed.gov/?id=ED419814}{Rasch, G. (1960/1980).
Probabilistic models for some intelligence and attainment
tests.(Copenhagen, Danish Institute for Educational Research), expanded
edition (1980) with foreword and afterword by B.D. Wright. Chicago: The
University of Chicago Press.}

\href{https://pdfs.semanticscholar.org/8083/5035228bc338840ed6c67e879b4bcef11e07.pdf?_ga=2.216101590.1797749273.1596845271-1703835138.1596845271}{Wright,
B. D., \& Masters, G. N. (1982). Rating Scale Analysis: Rasch
Measurement. Chicago, IL: MESA Press.}

\end{document}