Item Response Theory is a comprehensive statistical framework that is used widely in educational and psychological research to model an examinee’s individual response patterns on a test or other instrument by specifying an interaction between the underlying latent trait and item characteristics.

A number of commercial software packages are available for the estimation of IRT models, such as Bilog-MG (Zimowski, Muraki, Mislevy, & Bock, 2006), Multilog (Thissen, 1991), Parscale (Muraki & Bock, 1997), ConQuest (Adams, Wu, & Wilson, 2012), IRTPRO (Cai, du Toit, & Thissen, 2011), and FlexMIRT (Cai, 2012). In recent years, some free IRT packages have been developed in the R environment (R Development Core Team, 2018), such as ltm (Rizopoulos, 2006), mirt (Chalmers, 2012), TAM (Robitzsch, Kiefer, & Wu, 2019), and sirt (Robitzsch, 2019). Many of these tools use different parameterizations of the model, making direct comparisons of results a challenge.

In this blog, we first demonstrate how to obtain comparable item parameter estimates in PARSCALE, mirt, TAM, for the two-parameter IRT model. Second, we demonstrate how to specify item parameters in order to generate response data in lsasim (Matta, Rutkowski, Rutkowski, & Liaw, 2018).

 

Traditional IRT metric

In general, the logistic form of the two-parameter IRT model can be written as

\[ p(y_{ij} = 1 | \theta_{j}) = \frac{1} {1 + \text{exp} \left[ - Da_{i} (\theta_{j} - b_{i}) \right]} \]

where \(y_{ij}\) is the response to item \(i\) by respondent \(j\), \(\theta_{j}\) is the latent trait for respondent \(j\), \(D\) is a scaling constand (\(D\) = 1.7 to scale the logistic to the normal ogive metric; \(D\) = 1 to preserve the logistic metric), and \(b_{i}\) and \(a_{i}\) are the difficulty parameter and discrimination (slope) parameter, respectively, for item \(i\).

When models are estimated in the logistic metric, which means that there is no \(D\) = 1.7 scaling factor, \(a_{i}\) discrimination (slope) parameters will be approximately 1.7 times higher than they would be if reported in the normal ogive metric.

 

Install R pacakges.

[1] '1.30'
[1] '3.2.24'
[1] '2.0.0.9016'

Load response data.

  id V1 V2 V3 V4 V5
1  1  0  0  0  0  0
2  2  0  1  1  1  1
3  3  1  1  1  1  1
4  4  0  0  0  0  0
5  5  0  0  1  1  0
6  6  1  0  1  1  1

The PARSCALE version

In the PARSCALE parameterization, \(D\) can be set to either 1 or 1.7.

In the first command file, the scale constant is set to 1.

 

The output reported item parameters estimation in Phase 2, where \(D\) = 1.

 

In the second command file, the scale constant is set to 1.7 for slope parameters.

 

The output reported item parameters estimation in Phase 2, where \(D\) = 1.7.

 

 

When models are estimated in the logistic metric (\(D\) = 1), discrimination parameters are approximately 1.7 times higher than they reported in the normal ogive metric (\(D\) = 1.7).

[1] 1.700607 1.699561 1.699746 1.699758 1.699915

 

The mirt version

In the mirt parameterization, the functions are written with the logistic metric, i.e., \(a_{i}\theta_{j} + d_{i}\), where \(d_{i}\) denotes item easiness. For the unidimensional models, the \(d\) parameters can be converted into traditional IRT \(b\) parameters. When IRTpars = TRUE, \(b = -d/a\) while the \(a\) parameters will be identical under this parameterization.

         a1          d g u
V1 2.522040 -2.1444262 0 1
V2 2.325332 -2.6195831 0 1
V3 1.335560  2.5464438 0 1
V4 2.103789  1.2154576 0 1
V5 1.991937  0.3083698 0 1
          a          b g u
V1 2.522040  0.8502744 0 1
V2 2.325332  1.1265417 0 1
V3 1.335560 -1.9066488 0 1
V4 2.103789 -0.5777469 0 1
V5 1.991937 -0.1548090 0 1

 

The TAM version

In the TAM parameterization, the functions are written with the logistic metric in mind, i.e., \(B_{i} \theta_{j} - xsi_{i}\), where \(B\) represents item slopes and \(xsi\) denotes item difficulties.

The first column shows \(B\) item slopes and the second column shows \(xsi\) item difficulties. \(B\) are equivalent to traditional IRT \(a\) parameters.

       [,1]       [,2]
V1 2.523893  2.1453010
V2 2.323765  2.6181900
V3 1.335626 -2.5465196
V4 2.104199 -1.2156551
V5 1.991855 -0.3084357

In order to get traditional IRT \(b\) parameters, \(xsi\) has to be divided by \(B\).

       [,1]       [,2]
V1 2.523893  0.8499967
V2 2.323765  1.1267017
V3 1.335626 -1.9066110
V4 2.104199 -0.5777281
V5 1.991855 -0.1548485

 

The lsasim version

The functions of cognitive item responses generation are written with the logistic metric in the lsasim. \(a_{i}\) and \(b_{i}\) parameters in the traditional IRT metric are required when users want to specify item parameters.

Specify the number of subjects, the number of items, and the number of booklets.

Generate latent trait.

Specify item parameters.

Specify rotated booklet design.

Generate cognitive item response data.

  i001 i002 i003 i004 i005 subject
1    0    0    0    0    0       1
2    1    0    1    1    1       2
3    0    0    1    1    0       3
4    0    0    1    1    1       4
5    0    0    1    1    0       5
6    1    0    1    1    1       6