pcIRT (version 0.1)

MPRM: Estimation of Multidimensional Polytomous Rasch model (Rasch, 1961)

Description

This function estimates the multidimensional polytomous Rasch model by Rasch (1961). The model estimates item category parameters $\beta$ for each item and each category and takes each category of data as another dimension.

$$P_{vih} = \frac{exp(\theta_{vh} + \beta_{ih})}{\sum_{j=0}^{m} exp(\theta_{vj} + \beta_{ij})}$$

Parameters are estimated by CML method.

Usage

MPRM(data, desmat, start)
## S3 method for class 'MPRM':
summary(object, \dots)
## S3 method for class 'MPRM':
print(x, \dots)

Arguments

data

Data matrix or data frame; rows represent observations (persons), columns represent the items

desmat

Design matrix; if missing, the design matrix will be created automatically.

start

Starting values for parameter estimation. If missing, a vector of 0 is used as starting values.

object

object of class MPRM

x

{object of class MPRM}

...

...

Value

• data
  data matrix according to the input
• design
  design matrix either according to the input or according to the automatically generated matrix
• logLikelihood
  conditional log-likelihood
• estpar
  estimated basic item category parameters
• estpar_se
  estimated standard errors for basic item category parameters
• itempar
  estimated item category parameters
• itempar_se
  estimated standard errors for item category parameters
• hessian
  Hessian matrix
• convergence
  convergence of solution (see help files in optim)
• fun_calls
  number of function calls (see help files in optim)

Details

The parameters of the multidimensional polytomous Rasch model (Rasch, 1961) are estimated by CML estimation. For the CML estimation no assumption on the person parameter distribution is necessary.

References

Andersen, E. B. (1995). Polytomous Rasch models and their estimation. In G. H. Fischer and I. Molenaar (Eds.). Rasch Models - Foundations, Recent Developements, and Applications. Springer.

Fischer, G. H. (1974). Einfuehrung in die Theorie psychologischer Tests [Introduction to test theory]. Bern: Huber.

Rasch, G. (1961). On general laws and the meaning of measurement in psychology, Proceedings Fourth Berekely Symposium on Mathematical Statistiscs and Probability 5, 321-333.

Examples

#simulate data set
simdat <- simMPRM(rbind(matrix(c(-1.5,0.5,0.5,1,0.8,-0.3, 0.2,-1.2), ncol=4),0), 500)

#estimate MPRM item parameters
res_mprm <- MPRM(simdat$datmat)

summary(res_mprm)