ltm (version 0.8-9)

rasch: Rasch Model

Description

Fit the Rasch model under the Item Response Theory approach.

Usage

rasch(data, constraint = NULL, IRT.param = TRUE, start.val = NULL, 
    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.

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 discri

IRT.param

logical; if TRUE then the coefficients' estimates are reported under the usual IRT parameterization. See Details for more info.

start.val

the character string "random" or a numeric vector of $p+1$ starting values, where the first $p$ values correspond to the easiness parameters while the last value corresponds to the discrimination parameter. If "random", random starting

na.action

the na.action to be used on data. 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

control

a list of control values, [object Object],[object Object],[object Object],[object Object]

Value

• An object of class rasch with components,
• coefficients
  a matrix with 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 used 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 using rasch(..., start.val = "random").

Details

The Rasch model is a 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 mainly used in educational testing where the aim is to study the abilities of a particular set of individuals. The model is defined as follows $$\log\left(\frac{\pi_i}{1-\pi_i}\right) = \beta_{i} + \beta z,$$ where $\pi_i$ denotes the conditional probability of responding correctly to the $i$th item given $z$, $\beta_{i}$ is the easiness parameter for the $i$th item, $\beta$ 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., $$\log\left(\frac{\pi_i}{1-\pi_i}\right) = \beta (z - \beta_i^*).$$ 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.

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. Rizopoulos, D. (2006) ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17(5), 1--25. URL http://www.jstatsoft.org/v17/i05/

See Also

coef.rasch, fitted.rasch, summary.rasch, anova.rasch, plot.rasch, vcov.rasch, GoF.rasch, item.fit, person.fit, margins, factor.scores

Examples

## The common form of the Rasch model for the 
## LSAT data, assuming that the discrimination
## parameter equals 1
rasch(LSAT, constraint = cbind(ncol(LSAT) + 1, 1))


## The Rasch model for the LSAT data under the 
## normal ogive; to do that fix the discrimination
## parameter to 1.702
rasch(LSAT, constraint = cbind(ncol(LSAT) + 1, 1.702))

## The Rasch model for the LSAT data with
## unconstraint discrimination parameter
rasch(LSAT)

## The Rasch model with (artificially created) 
## missing data
data <- LSAT
data[] <- lapply(data, function(x){
    x[sample(1:length(x), sample(15, 1))] <- NA
    x
})
rasch(data)