pcIRT (version 0.2.4)

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. The functions allows setting linear restrictions on item category parameters \(\beta\).

Usage

MPRM(data, desmat, ldes, lp, start, control)
# S3 method for MPRM
print(x, ...)
# S3 method for MPRM
summary(object, ...)

Arguments

data

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

desmat

Design matrix

ldes

a numeric vector of the same length as the number of item category parameters indicating which parameters are set linear dependent of which other parameters (see details)

lp

a numeric vector with length equal to the number of item parameters set linear dependent. The vector indicates the number of scoring parameters (see details)

start

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

control

list with control parameters for the estimation process e.g. the convergence criterion. For details please see the help pages to the R built-in function optim

x

object of class MPRM

…

…

object

object of class MPRM

Value

data

data matrix according to the input

design

design matrix according to the input

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

linpar

estimated scoring parameters

linpar_se

estimated standard errors for scoring 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

Parameter estimations is done by CML method.

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. Furthermore linear restrictions can be set on the multidimensional polytomous Rasch model. Item category parameters can be set as being linear dependent to other item category parameters and the scoring parameter (as the multiple of the linear dependen parameters) is estimated. The restrictions are set by defining the arguments ldes and lp. ldes is a numerical vector of the same length as item category parameters in the general MPRM. A 0 in this vector indicates that no restriction is set. Putting in another number sets the item category parameter according to the vector position as linear dependent to that item category parameter with the position of the number included. For example, if item category parameter of item 1 and category 2 (that is position 2 in the vector ldes) should be linear dependent to the item category parameter of item 1 and category 1 (that is position 1 in the vector ldes), than the number 1 has to be on the second element of vector ldes. With the vector lp it is set, how many different scoring parameters have to be estimated and (if there are more than two) which of them should be equal. For example if 5 item category parameters are set linear dependent (by ldes) and according to the ldes vector the first, third and fourth have the same scoring parameters and the second and fifth have another scoring parameter, than lp must be a vector lp = c(1,2,1,1,2).

It is necessary that the design matrix is specified in accordance with the restrictions in ldes and lp.

References

Andersen, E. B. (1974). Das mehrkategorielle logistische Testmodell [The polytomous logistic test model] In. W. F. Kempf (Ed.), Probabilistische Modelle in der Sozialpsychologie [Probabilistic model in social psychology]. Bern: Huber.

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.

See Also

MPRM

Examples

# NOT RUN {
#simulate data set according to the general MPRM
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 the MPRM without any restrictions
res_mprm <- MPRM(simdat$datmat)

#estimate a MPRM with linear restrictions;
#for item 1 and 2 the second category is set linear dependent to the first
#category
ldes1 <- rep(0,length(res_mprm$itempar))
ldes1[c(2,5)] <- c(1,4)
lp1 <- rep(1,2)
#take the design matrix from the general MPRM and modify it according to the
#linear restriction
design1 <- res_mprm$design
design1[2,1] <- 1
design1[5,3] <- 1
design1[11,c(1,3)] <- -1
design1 <- design1[,-c(2,4)]

res_mprm2 <- MPRM(simdat$datmat, desmat=design1, ldes=ldes1, lp=lp1)

summary(res_mprm2)

# }