rasch  R Documentation 
Fit the Rasch model under the Item Response Theory approach.
rasch(data, constraint = NULL, IRT.param = TRUE, start.val = NULL, na.action = NULL, control = list(), Hessian = TRUE)
data 
a 
constraint 
a twocolumn numeric matrix with at most p rows (where p is the number of items), specifying fixedvalue 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 
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 values are used. If 
na.action 
the 
control 
a list of control values,

Hessian 
logical; if 
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
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 GaussHermite quadrature rule for the approximation of the required integrals.
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 
log.Lik 
the loglikelihood value at convergence. 
convergence 
the convergence identifier returned by 
hessian 
the approximate Hessian matrix at convergence returned by 
counts 
the number of function and gradient evaluations used by the quasiNewton algorithm. 
patterns 
a list with two components: (i) 
GH 
a list with two components used in the GaussHermite rule: (i) 
max.sc 
the maximum absolute value of the score vector at convergence. 
constraint 
the value of the 
IRT.param 
the value of the 
X 
a copy of the response data matrix. 
control 
the values used in the 
na.action 
the value of the 
call 
the matched call. 
In case the Hessian matrix at convergence is not positive definite, try to refit the model using
rasch(..., start.val = "random")
.
Although the common formulation of the Rasch model assumes that the discrimination parameter is fixed to 1,
rasch()
estimates it. If you wish to fit the constrained version of the model, use the constraint
argument accordingly. See Examples for more info.
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.
Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl
Baker, F. and Kim, SH. (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 doi: 10.18637/jss.v017.i05
coef.rasch
,
fitted.rasch
,
summary.rasch
,
anova.rasch
,
plot.rasch
,
vcov.rasch
,
GoF.rasch
,
item.fit
,
person.fit
,
margins
,
factor.scores
## 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)
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.