rasch {ltm}R Documentation

Rasch Model

Description

Fit the Rasch model under the Item Response Theory approach.

Usage

rasch(data, constraint = NULL, IRT.param = FALSE, start.val, 
            na.action = NULL, control = list())

Arguments

data 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.
constraint a two-column numeric matrix with at most p rows (where p is the number of items), specifying fixed-value constraints. The first column represents the item (i.e., 1 denotes the first item, 2 the second, etc., and p+1 the discrimination parameter) and the second column the value at which the corresponding parameter should be fixed. See Examples for more info.
IRT.param logical; if TRUE then the coefficients' estimates are reported under the usual IRT parameterization. See Details for more info.
start.val a numeric vector of p+1 starting values for the algorithm. The first p values correspond to the easiness 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 21.
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_{i} + beta z,

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

If IRT.param = TRUE, then the parameters estimates are reported under the usual IRT parameterization, i.e.,

logit (π_i) = beta (z - beta_i^*).

The fit of the model is based on approximate marginal Maximum Likelihood, using the Gauss-Hermite quadrature rule for the approximation of the required integrals.

Value

An object of class rasch with components,

coefficients the parameter values at convergence. These are always the estimates of beta_i, beta parameters, even if IRT.param = TRUE.
log.Lik the log-likelihood value at convergence.
convergence the convergence identifier returned by optim.
hessian the approximate Hessian matrix at convergence returned by optim.
counts the number of function and gradient evaluations required by the quasi-Newton algorithm.
patterns a list with two components: (i) X: a numeric matrix that contains the observed response patterns, and (ii) obs: a numeric vector that contains the observed 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 abscissas, and (ii) GHw: a numeric vector that contains the corresponding weights.
max.sc the maximum absolute value of the score vector at convergence.
constraint the value of the constraint argument.
IRT.param the value of the IRT.param argument.
X a copy of the response data matrix.
control the values used in the control argument.
na.action the value of the na.action 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.

If IRT.param = TRUE, calculation of the standard errors for hat{beta}_i^* requires that the discrimination parameter beta is not fixed.

Note

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

When the coefficients' estimates are reported under the usual IRT parameterization (i.e., IRT.param = TRUE), their standard errors are calculated using the Delta method.

Author(s)

Dimitris Rizopoulos dimitris.rizopoulos@med.kuleuven.be

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, fitted.rasch, summary.rasch, anova.rasch, plot.rasch, vcov.rasch, margins, factor.scores

Examples


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

## The Rasch model for the Wirs data, under the 
## constraint that: (i) the easiness parameter 
## for the first item equals 0.5, (ii) the easiness 
## parameter for the second item equals 1.2, and 
## (iii) the discrimination parameter equals 1
(constr <- rbind(c(1, 0.5), c(2, 1.2), c(7, 1)))
rasch(Wirs, constraint = constr)

## The Rasch model for the Lsat data;
## report parameter estimates under the
## usual IRT parameterization:
rasch(Lsat, IRT = TRUE)

## The Rasch model for the Lsat data, under the 
## constraint that the discrimination parameter
## equals 1
rasch(Lsat, constraint = rbind(c(6, 1)))

## The Rasch model for the Abortion data:
## report parameter estimates under the
## usual IRT parameterization:
rasch(Abortion, IRT = TRUE)


[Package ltm version 0.3-0 Index]