sirt-package {sirt}

Supplementary Item Response Theory Models

Description

Supplementary functions for item response models aiming to complement existing R packages. The functionality includes among others multidimensional compensatory and noncompensatory IRT models (Reckase, 2009, <doi:10.1007/978-0-387-89976-3>), MCMC for hierarchical IRT models and testlet models (Fox, 2010, <doi:10.1007/978-1-4419-0742-4>), NOHARM (McDonald, 1982, <doi:10.1177/014662168200600402>), Rasch copula model (Braeken, 2011, <doi:10.1007/s11336-010-9190-4>; Schroeders, Robitzsch & Schipolowski, 2014, <doi:10.1111/jedm.12054>), faceted and hierarchical rater models (DeCarlo, Kim & Johnson, 2011, <doi:10.1111/j.1745-3984.2011.00143.x>), ordinal IRT model (ISOP; Scheiblechner, 1995, <doi:10.1007/BF02301417>), DETECT statistic (Stout, Habing, Douglas & Kim, 1996, <doi:10.1177/014662169602000403>), local structural equation modeling (LSEM; Hildebrandt, Luedtke, Robitzsch, Sommer & Wilhelm, 2016, <doi:10.1080/00273171.2016.1142856>).

Details

The sirt package enables the estimation of following models:

• Multidimensional marginal maximum likelihood estimation (MML) of generalized logistic Rasch type models using the generalized logistic link function (Stukel, 1988) can be conducted with rasch.mml2 and the argument itemtype="raschtype". This model also allows the estimation of the 4PL item response model (Loken & Rulison, 2010). Multiple group estimation, latent regression models and plausible value imputation are supported. In addition, pseudo-likelihood estimation for fractional item response data can be conducted.

• Multidimensional noncompensatory, compensatory and partially compensatory item response models for dichotomous item responses (Reckase, 2009) can be estimated with the smirt function and the options irtmodel="noncomp" , irtmodel="comp" and irtmodel="partcomp".

• The unidimensional quotient model (Ramsay, 1989) can be estimated using rasch.mml2 with itemtype="ramsay.qm".

• Unidimensional nonparametric item response models can be estimated employing MML estimation (Rossi, Wang & Ramsay, 2002) by making use of rasch.mml2 with itemtype="npirt". Kernel smoothing for item response function estimation (Ramsay, 1991) is implemented in np.dich.

• The multidimensional IRT copula model (Braeken, 2011) can be applied for handling local dependencies, see rasch.copula3.

• Unidimensional joint maximum likelihood estimation (JML) of the Rasch model is possible with the rasch.jml function. Bias correction methods for item parameters are included in rasch.jml.jackknife1 and rasch.jml.biascorr.

• The multidimensional latent class Rasch and 2PL model (Bartolucci, 2007) which employs a discrete trait distribution can be estimated with rasch.mirtlc.

• The unidimensional 2PL rater facets model (Lincare, 1994) can be estimated with rm.facets. A hierarchical rater model based on signal detection theory (DeCarlo, Kim & Johnson, 2011) can be conducted with rm.sdt. A simple latent class model for two exchangeable raters is implemented in lc.2raters. See Robitzsch and Steinfeld (2018) for more details.

• The discrete grade of membership model (Erosheva, Fienberg & Joutard, 2007) and the Rasch grade of membership model can be estimated by gom.em.

• Some hierarchical IRT models and random item models for dichotomous and normally distributed data (van den Noortgate, de Boeck & Meulders, 2003; Fox & Verhagen, 2010) can be estimated with mcmc.2pno.ml.

• Unidimensional pairwise conditional likelihood estimation (PCML; Zwinderman, 1995) is implemented in rasch.pairwise or rasch.pairwise.itemcluster.

• Unidimensional pairwise marginal likelihood estimation (PMML; Renard, Molenberghs & Geys, 2004) can be conducted using rasch.pml3. In this function local dependence can be handled by imposing residual error structure or omitting item pairs within a dependent item cluster from the estimation.
  The function rasch.evm.pcm estimates the multiple group partial credit model based on the pairwise eigenvector approach which avoids iterative estimation.

• Some item response models in sirt can be estimated via Markov Chain Monte Carlo (MCMC) methods. In mcmc.2pno the two-parameter normal ogive model can be estimated. A hierarchical version of this model (Janssen, Tuerlinckx, Meulders & de Boeck, 2000) is implemented in mcmc.2pnoh. The 3PNO testlet model (Wainer, Bradlow & Wang, 2007; Glas, 2012) can be estimated with mcmc.3pno.testlet. Some hierarchical IRT models and random item models (van den Noortgate, de Boeck & Meulders, 2003) can be estimated with mcmc.2pno.ml.

• For dichotomous response data, the free NOHARM software (McDonald, 1982, 1997) estimates the multidimensional compensatory 3PL model and the function R2noharm runs NOHARM from within R. Note that NOHARM must be downloaded from http://noharm.niagararesearch.ca/nh4cldl.html at first. A pure R implementation of the NOHARM model with some extensions can be found in noharm.sirt.

• The measurement theoretic founded nonparametric item response models of Scheiblechner (1995, 1999) – the ISOP and the ADISOP model – can be estimated with isop.dich or isop.poly. Item scoring within this theory can be conducted with isop.scoring.

• The functional unidimensional item response model (Ip et al., 2013) can be estimated with f1d.irt.

• The Rasch model can be estimated by variational approximation (Rijmen & Vomlel, 2008) using rasch.va.

• The unidimensional probabilistic Guttman model (Proctor, 1970) can be specified with prob.guttman.

• A jackknife method for the estimation of standard errors of the weighted likelihood trait estimate (Warm, 1989) is available in wle.rasch.jackknife.

• Model based reliability for dichotomous data can be calculated by the method of Green and Yang (2009) with greenyang.reliability and the marginal true score method of Dimitrov (2003) using the function marginal.truescore.reliability.

• Essential unidimensionality can be assessed by the DETECT index (Stout, Habing, Douglas & Kim, 1996), see the function conf.detect.

• Item parameters from several studies can be linked using the Haberman method (Haberman, 2009) in linking.haberman. See also equating.rasch and linking.robust. The alignment procedure (Asparouhov & Muthen, 2013) invariance.alignment is originally for comfirmatory factor analysis and aims at obtaining approximate invariance.

• Some person fit statistics in the Rasch model (Meijer & Sijtsma, 2001) are included in personfit.stat.

• An alternative to the linear logistic test model (LLTM), the so called least squares distance model for cognitive diagnosis (LSDM; Dimitrov, 2007), can be estimated with the function lsdm.

• Local structural equation models (LSEM) can be estimated with the lsem.estimate function (Hildebrandt et al., 2016).

Author(s)

Alexander Robitzsch [aut,cre] (<https://orcid.org/0000-0002-8226-3132>)

Maintainer: Alexander Robitzsch <[e-mail]>

References

Asparouhov, T., & Muthen, B. (2014). Multiple-group factor analysis alignment. Structural Equation Modeling, 21(4), 1-14. doi:10.1080/10705511.2014.919210

Bartolucci, F. (2007). A class of multidimensional IRT models for testing unidimensionality and clustering items. Psychometrika, 72, 141-157.

Braeken, J. (2011). A boundary mixture approach to violations of conditional independence. Psychometrika, 76(1), 57-76. doi:10.1007/s11336-010-9190-4

DeCarlo, T., Kim, Y., & Johnson, M. S. (2011). A hierarchical rater model for constructed responses, with a signal detection rater model. Journal of Educational Measurement, 48(3), 333-356. doi:10.1111/j.1745-3984.2011.00143.x

Dimitrov, D. (2003). Marginal true-score measures and reliability for binary items as a function of their IRT parameters. Applied Psychological Measurement, 27, 440-458.

Dimitrov, D. M. (2007). Least squares distance method of cognitive validation and analysis for binary items using their item response theory parameters. Applied Psychological Measurement, 31, 367-387.

Erosheva, E. A., Fienberg, S. E., & Joutard, C. (2007). Describing disability through individual-level mixture models for multivariate binary data. Annals of Applied Statistics, 1, 502-537.

Fox, J.-P. (2010). Bayesian item response modeling. New York: Springer. doi:10.1007/978-1-4419-0742-4

Fox, J.-P., & Verhagen, A.-J. (2010). Random item effects modeling for cross-national survey data. In E. Davidov, P. Schmidt, & J. Billiet (Eds.), Cross-cultural Analysis: Methods and Applications (pp. 467-488), London: Routledge Academic.

Fraser, C., & McDonald, R. P. (1988). NOHARM: Least squares item factor analysis. Multivariate Behavioral Research, 23, 267-269.

Glas, C. A. W. (2012). Estimating and testing the extended testlet model. LSAC Research Report Series, RR 12-03.

Green, S.B., & Yang, Y. (2009). Reliability of summed item scores using structural equation modeling: An alternative to coefficient alpha. Psychometrika, 74, 155-167.

Haberman, S. J. (2009). Linking parameter estimates derived from an item response model through separate calibrations. ETS Research Report ETS RR-09-40. Princeton, ETS. doi:10.1002/j.2333-8504.2009.tb02197.x

Hildebrandt, A., Luedtke, O., Robitzsch, A., Sommer, C., & Wilhelm, O. (2016). Exploring factor model parameters across continuous variables with local structural equation models. Multivariate Behavioral Research, 51(2-3), 257-278. doi:10.1080/00273171.2016.1142856

Ip, E. H., Molenberghs, G., Chen, S. H., Goegebeur, Y., & De Boeck, P. (2013). Functionally unidimensional item response models for multivariate binary data. Multivariate Behavioral Research, 48, 534-562.

Janssen, R., Tuerlinckx, F., Meulders, M., & de Boeck, P. (2000). A hierarchical IRT model for criterion-referenced measurement. Journal of Educational and Behavioral Statistics, 25, 285-306.

Jeon, M., & Rijmen, F. (2016). A modular approach for item response theory modeling with the R package flirt. Behavior Research Methods, 48(2), 742-755. doi:10.3758/s13428-015-0606-z

Linacre, J. M. (1994). Many-Facet Rasch Measurement. Chicago: MESA Press.

Loken, E. & Rulison, K. L. (2010). Estimation of a four-parameter item response theory model. British Journal of Mathematical and Statistical Psychology, 63, 509-525.

McDonald, R. P. (1982). Linear versus nonlinear models in item response theory. Applied Psychological Measurement, 6(4), 379-396. doi:10.1177/014662168200600402

McDonald, R. P. (1997). Normal-ogive multidimensional model. In W. van der Linden & R. K. Hambleton (1997): Handbook of modern item response theory (pp. 257-269). New York: Springer. doi:10.1007/978-1-4757-2691-6_15

Meijer, R. R., & Sijtsma, K. (2001). Methodology review: Evaluating person fit. Applied Psychological Measurement, 25, 107-135.

Proctor, C. H. (1970). A probabilistic formulation and statistical analysis for Guttman scaling. Psychometrika, 35, 73-78.

Ramsay, J. O. (1989). A comparison of three simple test theory models. Psychometrika, 54, 487-499.

Ramsay, J. O. (1991). Kernel smoothing approaches to nonparametric item characteristic curve estimation. Psychometrika, 56, 611-630.

Reckase, M. (2009). Multidimensional item response theory. New York: Springer. doi:10.1007/978-0-387-89976-3

Renard, D., Molenberghs, G., & Geys, H. (2004). A pairwise likelihood approach to estimation in multilevel probit models. Computational Statistics & Data Analysis, 44, 649-667.

Rijmen, F., & Vomlel, J. (2008). Assessing the performance of variational methods for mixed logistic regression models. Journal of Statistical Computation and Simulation, 78, 765-779.

Robitzsch, A., & Steinfeld, J. (2018). Item response models for human ratings: Overview, estimation methods, and implementation in R. Psychological Test and Assessment Modeling, 60(1), 101-139.

Rossi, N., Wang, X. & Ramsay, J. O. (2002). Nonparametric item response function estimates with the EM algorithm. Journal of Educational and Behavioral Statistics, 27, 291-317.

Rusch, T., Mair, P., & Hatzinger, R. (2013). Psychometrics with R: A Review of CRAN Packages for Item Response Theory. http://epub.wu.ac.at/4010/1/resrepIRThandbook.pdf

Scheiblechner, H. (1995). Isotonic ordinal probabilistic models (ISOP). Psychometrika, 60(2), 281-304. doi:10.1007/BF02301417

Scheiblechner, H. (1999). Additive conjoint isotonic probabilistic models (ADISOP). Psychometrika, 64, 295-316.

Schroeders, U., Robitzsch, A., & Schipolowski, S. (2014). A comparison of different psychometric approaches to modeling testlet structures: An example with C-tests. Journal of Educational Measurement, 51(4), 400-418. doi:10.1111/jedm.12054

Stout, W., Habing, B., Douglas, J., & Kim, H. R. (1996). Conditional covariance-based nonparametric multidimensionality assessment. Applied Psychological Measurement, 20(4), 331-354. doi:10.1177/014662169602000403

Stukel, T. A. (1988). Generalized logistic models. Journal of the American Statistical Association, 83(402), 426-431. doi:10.1080/01621459.1988.10478613

Uenlue, A., & Yanagida, T. (2011). R you ready for R?: The CRAN psychometrics task view. British Journal of Mathematical and Statistical Psychology, 64(1), 182-186. doi:10.1348/000711010X519320

van den Noortgate, W., De Boeck, P., & Meulders, M. (2003). Cross-classification multilevel logistic models in psychometrics. Journal of Educational and Behavioral Statistics, 28, 369-386.

Warm, T. A. (1989). Weighted likelihood estimation of ability in item response theory. Psychometrika, 54, 427-450.

Wainer, H., Bradlow, E. T., & Wang, X. (2007). Testlet response theory and its applications. Cambridge: Cambridge University Press.

Zwinderman, A. H. (1995). Pairwise parameter estimation in Rasch models. Applied Psychological Measurement, 19, 369-375.

See Also

For estimating multidimensional models for polytomous item responses see the mirt, flirt (Jeon & Rijmen, 2016) and TAM packages.

For conditional maximum likelihood estimation see the eRm package.

For pairwise estimation likelihood methods (also known as composite likelihood methods) see pln or lavaan.

The estimation of cognitive diagnostic models is possible using the CDM package.

For the multidimensional latent class IRT model see the MultiLCIRT package which also allows the estimation IRT models with polytomous item responses.

Latent class analysis can be carried out with covLCA, poLCA, BayesLCA, randomLCA or lcmm packages.

Markov Chain Monte Carlo estimation for item response models can also be found in the MCMCpack package (see the MCMCirt functions therein).

See Rusch, Mair and Hatzinger (2013) and Uenlue and Yanagida (2011) for reviews of psychometrics packages in R.

Examples

##
##   |-----------------------------------------------------------------|
##   | sirt 0.40-4 (2013-11-26)                                        |
##   | Supplementary Item Response Theory                              |
##   | Maintainer: Alexander Robitzsch <a.robitzsch at bifie.at >      |
##   | https://sites.google.com/site/alexanderrobitzsch/software       |
##   |-----------------------------------------------------------------|
##
##                       _/              _/
##              _/_/_/      _/  _/_/  _/_/_/_/
##           _/_/      _/  _/_/        _/
##              _/_/  _/  _/          _/
##         _/_/_/    _/  _/            _/_/
##