nrm: Estimating a Nominal Response Model
In mcIRT: IRT models for multiple choice items (mcIRT)

Description Usage Arguments Details Value Author(s) References See Also Examples

This function fits a Nominal Response Model as proposed by Bock (1972). The model estimates two parameters for each response category: ζ and λ. The model ist best suited for a set of items with more than two nominal response categories.

P_{iqv} = \frac{exp(ζ_{ik} - λ_{ik} θ_j)}{∑_{v=1}^{m} exp(ζ_{iv} - λ_{iv} θ_j)}

nrm(reshOBJ,etastart=-0.1, ctrl=list())
## S3 method for class 'nrm'
summary(object, RETURN=FALSE, ...)
## S3 method for class 'nrm'
print(x, ...)
## S3 method for class 'nrm'
deviance(object, ...)
## S3 method for class 'nrm'
logLik(object, ...)

`reshOBJ`	A object of class reshNRM is expected. So the step before fitting the model is to reshape the data by means of the `reshMG` function.
`etastart`	A numerical vector. Starting values for the eta parameters can be change (but is not necessary in typical cases).
`ctrl`	A list of argumets to customize the computations.
`object`	An object of class `nrm`.
`x`	An object of class `nrm`.
`RETURN`	A logical vector of length 1. If TRUE all result tables are returned by the summary function.
`...`

The eta parameters in etastart denote the estimable parameters of the model. For example, for an item with 4 categories, 3 γ's (which substitute the 4 ζ's) and 3 ξ's (which substitute the 4 λ's) are constrained for normalization (parameter sets add up to zero).

The following arguments can be comitted within a list (ctrl argument):

nodes A numerical vector of length 1. Set the number of quadrature nodes/points for the a-priori distribution. The distribution is assumed to be normal. Default: 21
absrange A numerical vector of length 1. Denotes the absolute range of the a-priori distribution. The default value is 5, so the normal distribution ranges from [-5 ; 5].
verbose If TRUE, the estimation process is displayed in terms of the actual EMstep. Default: TRUE
sigmaest If TRUE, the variance of the latent person distribution is estimated. Otherwise it is set to 1 (for each group). Default: FALSE
exac A numerical vector of length 1. If the difference between two consecutive EM steps (sum of absolute difference of parameters) is not larger than 'exac' - the estimation stops. Default: 0.00001
EMmax A numerical vector of length 1. This argument sets the maximum number of EM steps. The default value is 500. Feel free to enlarge this number. Default: 500
NRmax A numerical vector of length 1. This argument sets the maximum number of Newton Raphson steps within the M-Step of the EM Algorithm. Default: 20
NRexac A numerical vector of length 1. If the difference between two consecutive NR steps is not larger than 'exac' - the estimation stops. Default: 0.01
Clist A list which contains informations about which parameters should be held constant during estimation. Each list element has to look similar to this expression: "eta\d* = -*\d*" (of course real digits instead of Regexes!). The term eta refers to a column in the Q matrix in the reshape object which actually represents the parameters. The right side of the equation is the constant the parameter should be set to. So an entry in the Clist could look like "eta2 = -1" which means that the second eta parameter will not be estimated and is set to the value of -1.
nonpar If TRUE, the prior distribution is nonparametric and is reestimated in each EM step by use of the expected number of examinees on each quadrature node. Default: FALSE. Computations are based on the EH (empirical histogram) estimation method of Woods (2007, 2011). It is possible to estimate EHs for more than one group. First experiences showed that a huge amount of EM steps are needed (> 5000) to approximate the latent ability distributions.
quads Supply specific quadrature nodes and weights for nonparametric estimation (npar must be TRUE). It has to be a list of length = number of groups. Inside each list element a list with two elements (a vector of nodes and a vector of weights) is expected. To get an idea what this looks like use mcIRT:::quadIT(nodes=15,ngr=2).

`etapar`	A numerical vector of eta-parameters
`last_estep`	A list of informations concerning the last e-step before convergence. This is nothing the typical user should care about.
`last_mstep`	Contains the log likelihood and the Hessian-Matrix of the last M-step of the EM-Algorithm.
`n_steps`	The number of passed EM steps which lead to the convergent result.
`erg_distr`	Estimates concerning the latent person distribution.
`QUAD`	Denotes a list of the quadrature nodes and weights which were used as a-priori distribution.
`starting_values`	A list with infos concerning the starting values. The first entry gives merely the structure of the starting values whereas `$ulstv` gives the used starting values for the first EM step
`EAPs`	The exact a-posteriori values for each person - which is something like a person parameter estimate.
`ZLpar`	The list of item parameter estimates for each group.
`SE`	The list of standard errors for the item parameter estimates.
`reshOBJ`	The committed reshape object (which includes the data).
`Catinf`	A list which contains 1) the information amount for each category/item/group for a sequence of ability values; 2) the sequence of ability values; 3) test information (sum above all items) for each of the ability values.
`call`	Shows the actual call of the nrm function.

Manuel Reif

Bock, R. D. (1972). Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika, 37, 29-51.

Baker, F. B., & Kim, S.-H. (2004). Item response theory. Parameter estimation techniques (2nd ed.). New York, NY: Marcel-Dekker.

Harwell, M. R., Baker, F. B. & Zwarts, M. (1988). Item parameter estimation via marginal maximum likelihood and an EM algorithm: A didactic. Journal of Educational Statistics, 13, 243-271.

Woods, C. M. (2007). Empirical Histograms in Item Response Theory With Ordinal Data. Education and Psychological Measurement, 67:1, 73-87.

Woods, C. M. (2011). DIF Testing With an Empirical-Histogram Approximation of the Latent Density for Each Group. Applied Measurement in Education, 24:3, 256-279.

reshMG

NRM.sim

nelm

# Simulating 3 Items within a loop
ParList <- lapply(1:3,function(x)
{
  Item1 <- c(c(-2,-1,1,2),c(-1.2,0.3,0.2,0.7))
  names(Item1) <- c(paste("zeta",1:4,sep=""),paste("lamb",1:4,sep=""))
  Item1
})

names(ParList) <- paste("item",1:3,sep="")

# simulate person parameters
perp1     <- rnorm(1100,0,1)

# simulate data.frame 
sim.nrm.1 <- NRM.sim(ParList,perp1)

# reshape
reshdat1 <- reshMG(sim.nrm.1,items=1:3,correct=rep(0,3))

# fit the nrm with a maximum number of 40 EM iterations, which is NOT recommanded 
# and is just applied here because estimating the model during example checks on cran
# took too long with default settings

res.nrm <- nrm(reshdat1)

summary(res.nrm)