rasch {ltm}R Documentation

Rasch Model

Description

Fit the Rasch model under the Item Response Theory approach.

Usage

rasch(dat, start.val, na.action = NULL, control = list())

Arguments

dat a data.frame (that will be converted to a numeric matrix using data.matrix()) or a numeric matrix of manifest variables. The binary responses must be in 0/1 format.
start.val a numeric vector of p+1 starting values for the algorithm. The first p values correspond to the difficulty parameters while the last value corresponds to the discrimination parameter. If it is not supplied randomly chosen starting values are used instead.
na.action the na.action to be used on dat. In case of missing data, if na.action=NULL the model uses the available cases, i.e., it takes into account the observed part of sample units with missing values (valid under MAR mechanisms if the model is correctly specified). If you want to apply a complete case analysis then use na.action=na.exclude.
control a list of control values,
iter.qN
the number of quasi-Newton iterations. Default 150.
GHk
the number of Gauss-Hermite quadrature points. Default 20.
method
the optimization method to be used in optim. Default "BFGS".
verbose
logical; if TRUE info about the optimization procedure are printed.

Details

The Rasch model is special case of the unidimensional latent trait model when all the discrimination parameters are equal. This model was first discussed by Rasch (1960) and it is used mainly in educational testing where the aim is to study the abilities of a particular set of individuals.

The model is defined as follows

logit (π_i) = beta_{i0} + beta z,

where π_i denotes the probability of responding correctly to the ith item, beta_{i0} denotes the difficulty parameter for the ith item, β is the discrimination parameter (the same for all the items) and z denotes the latent ability.

The optimization algorithm works under the constraint that the discrimination parameter is always positive.

Value

An object of class rasch with components,

coefficients the loadings' values at convergence.
log.Lik the log-likelihood value at convergence.
convergence the convergence identifier returned by optim.
hessian the Hessian matrix at convergence returned by optim.
patterns a list with two components: (i) mat a numeric matrix that contains the observed response patterns. (ii) dat a data.frame that contains the observed and expected frequencies for each observed response pattern.
GH a list with two components used in the Gauss-Hermite rule: (i) Z a numeric matrix that contains the quadrature points. (ii) GHw a numeric vector that contains the corresponding weights.
max.sc the maximum absolute value of the score vector at convergence.
X the responses data matrix.
control the values used in the control argument.
call the matched call.

Warning

In case the Hessian matrix at convergence is not positive definite, try to re-fit the model. rasch will use new random starting values.

References

Baker, F. and Kim, S-H. (2004) Item Response Theory, 2nd ed. New York: Marcel Dekker.

Rasch, G. (1960) Probabilistic Models for Some Intelligence and Attainment Tests. Copenhagen: Paedagogiske Institute.

See Also

coef.rasch, summary.rasch, anova.rasch, plot.rasch, margins, factor.scores

Examples


## The Rasch model for the Wirs data:
rasch(Wirs)

## The Rasch model for the Lsat data:
rasch(Lsat)

## The Rasch model for the Abortion data:
rasch(Abortion)


[Package ltm version 0.1-1 Index]