RSM: Estimation of rating scale models
In eRm: Extended Rasch Modeling

Description Usage Arguments Details Value Author(s) References See Also Examples

This function computes the parameter estimates of a rating scale model for polytomous item responses by using CML estimation.

1	RSM(X, W, se = TRUE, sum0 = TRUE, etaStart)

`X`	Input data matrix or data frame with item responses (starting from 0); rows represent individuals, columns represent items. Missing values are inserted as `NA`.
`W`	Design matrix for the RSM. If omitted, the function will compute W automatically.
`se`	If `TRUE`, the standard errors are computed.
`sum0`	If `TRUE`, the parameters are normed to sum-0 by specifying an appropriate `W`. If `FALSE`, the first parameter is restricted to 0.
`etaStart`	A vector of starting values for the eta parameters can be specified. If missing, the 0-vector is used.

The design matrix approach transforms the RSM into a partial credit model and estimates the corresponding basic parameters by using CML. Available methods for RSM-objects are print, coef, model.matrix, vcov, summary, logLik, person.parameters, plotICC, LRtest.

Returns an object of class 'Rm', 'eRm' and contains the log-likelihood value, the parameter estimates and their standard errors.

`loglik`	Conditional log-likelihood.
`iter`	Number of iterations.
`npar`	Number of parameters.
`convergence`	See `code` output in `nlm`.
`etapar`	Estimated basic item difficulty parameters (item and category parameters).
`se.eta`	Standard errors of the estimated basic item parameters.
`betapar`	Estimated item-category (easiness) parameters.
`se.beta`	Standard errors of item parameters.
`hessian`	Hessian matrix if `se = TRUE`.
`W`	Design matrix.
`X`	Data matrix.
`X01`	Dichotomized data matrix.
`call`	The matched call.

Patrick Mair, Reinhold Hatzinger

Fischer, G. H., and Molenaar, I. (1995). Rasch Models - Foundations, Recent Developements, and Applications. Springer.

Mair, P., and Hatzinger, R. (2007). Extended Rasch modeling: The eRm package for the application of IRT models in R. Journal of Statistical Software, 20(9), 1-20.

Mair, P., and Hatzinger, R. (2007). CML based estimation of extended Rasch models with the eRm package in R. Psychology Science, 49, 26-43.

RM,PCM,LRtest

##RSM with 10 subjects, 3 items
res <- RSM(rsmdat)
res
summary(res)                            #eta and beta parameters with CI
thresholds(res)                         #threshold parameters

Results of RSM estimation: 

Call:  RSM(X = rsmdat) 

Conditional log-likelihood: -107.5618 
Number of iterations: 11 
Number of parameters: 7 

Item (Category) Difficulty Parameters (eta):
                I2        I3         I4         I5         I6      Cat 2
Estimate 0.3051523 0.2558638 0.06730786 -0.1601342 -0.4442440 -0.1673936
Std.Err  0.2053696 0.2028161 0.19650733  0.1965178  0.2084872  0.4596567
             Cat 3
Estimate 0.1379066
Std.Err  0.7324949



Results of RSM estimation: 

Call:  RSM(X = rsmdat) 

Conditional log-likelihood: -107.5618 
Number of iterations: 11 
Number of parameters: 7 

Item (Category) Difficulty Parameters (eta): with 0.95 CI:
      Estimate Std. Error lower CI upper CI
I2       0.305      0.205   -0.097    0.708
I3       0.256      0.203   -0.142    0.653
I4       0.067      0.197   -0.318    0.452
I5      -0.160      0.197   -0.545    0.225
I6      -0.444      0.208   -0.853   -0.036
Cat 2   -0.167      0.460   -1.068    0.734
Cat 3    0.138      0.732   -1.298    1.574

Item Easiness Parameters (beta) with 0.95 CI:
           Estimate Std. Error lower CI upper CI
beta I1.c1    0.024      0.195   -0.359    0.407
beta I1.c2    0.215      0.600   -0.961    1.391
beta I1.c3   -0.066      0.932   -1.894    1.762
beta I2.c1   -0.305      0.205   -0.708    0.097
beta I2.c2   -0.443      0.636   -1.689    0.803
beta I2.c3   -1.053      1.011   -3.035    0.928
beta I3.c1   -0.256      0.203   -0.653    0.142
beta I3.c2   -0.344      0.629   -1.577    0.888
beta I3.c3   -0.905      0.997   -2.860    1.049
beta I4.c1   -0.067      0.197   -0.452    0.318
beta I4.c2    0.033      0.607   -1.157    1.223
beta I4.c3   -0.340      0.951   -2.204    1.524
beta I5.c1    0.160      0.197   -0.225    0.545
beta I5.c2    0.488      0.593   -0.675    1.651
beta I5.c3    0.342      0.910   -1.440    2.125
beta I6.c1    0.444      0.208    0.036    0.853
beta I6.c2    1.056      0.597   -0.114    2.226
beta I6.c3    1.195      0.884   -0.537    2.927


Design Matrix Block 1:
   Location Threshold 1 Threshold 2 Threshold 3
I1  0.02202    -0.02395    -0.19134     0.28135
I2  0.35112     0.30515     0.13776     0.61045
I3  0.30183     0.25586     0.08847     0.56116
I4  0.11328     0.06731    -0.10009     0.37261
I5 -0.11417    -0.16013    -0.32753     0.14517
I6 -0.39828    -0.44424    -0.61164    -0.13894