personfit: Latent traits estimation
In LatentREGpp: Item Response Theory Implemented in R and Cpp

Description Usage Arguments Value Methods to estimate the Latent Traits Notation Examples

Estimates the latent traits by using either the Expected A Posteriori (EAP) or Mode A Posteriori (MAP) method. A Normal distribution with mean vector zero and covariance matrix the identity is assumed. Quasi-Monte Carlo quadrature is suggested when the data dimension is large (>3).

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personfit(data, dim, model = "2PL", zetas, quad_tech = "Gaussian",
  quad_points = NULL, init_traits = NULL, method = "MAP",
  by_individuals = TRUE, verbose = FALSE)

`data`	The matrix containing the answers of tested individuals
`dim`	The dimensionality of the test
`model`	"1PL", "2PL" or "3PL"
`zetas`	The item parameters. A matrix of dim (num of items * num of parameters from item that has the greater number of categories) where each row is a vector of the form: (\boldsymbol{a}_i,γ_{i1},γ_{i2},...,γ_{im_i}, NA,NA,...,NA,c_i) according to the notation in the section "Notation". The function "LatentREGpp::itemfit( )" returns the zetas with this structure.
`quad_tech`	A string with technique. "Gaussian" for Gaussian quadrature or "QMCEM" for Quasi-Monte Carlo quadrature
`quad_points`	Amount of quadrature points by dimention. Default NULL
`init_traits`	Initial values by pattern or by individual as the case may be. Default NULL.
`method`	"EAP" or "MAP". "MAP" by default.
`by_individuals`	if True, return latent trait by individual, otherwsie by response pattern. True by default.
`verbose`	True for get information about estimation process in runtime. False in otherwise.

depends on value of by_individuals argument.

: If by_individuals=TRUE. Returns a matrix with latent trait for each individual
: If by_individuals=FALE. Returns a list of length 3 with the latent traits for each pattern, the reponse patterns and the frecuencie of each pattern

EAP

In general the EAP Method is based on the next expression.

\frac{\int θ_lp(U_{l}=u_l /θ_l,ζ) p(θ_l/η)\partialθ_l}{\int p(U_{l}=u_l /θ_l,ζ) p(θ_l/η)\partialθ_l}

where:

: θ_l is the latent trait associated with pattern l.
: U_l refers to response pattern l
: ζ are items parameters
: η are the hiperparameters from the prior distribution to the traits.

MAP

The method consists of maximize the following expression regard to θ_l

\frac{p(U_{l}=u_l /θ_l,ζ)p(θ_l/η)} {\int p(U_{l}=u_l /θ_l,ζ) p(θ_l/η)\partialθ_l}

In the Polytomous Multidimensional case, the probability that an examinee with latent trait vector θ_l responses categorie k to item i is,

P(U_{li}=k \mid \boldsymbol{θ}_l, \boldsymbol{a}_i,γ_{ik}) =c_i+(1-c_i)\boldsymbol{Ψ}(η_{lik})

where

: η_{lik}=\boldsymbol{a}_i^t\boldsymbol{θ}_l + γ_{ik}

: \boldsymbol{a}_i=(\boldsymbol{a}_1,\boldsymbol{a}_2,...,\boldsymbol{a}_d)^t is a parameter associated with discrimination of item, d is the data dimention
: \boldsymbol{θ}_l=(\boldsymbol{θ}_{l1},\boldsymbol{θ}_{l2},...,\boldsymbol{θ}_{ld})^t

is the latent trait multidimensional associated with pattern l.

: γ_{ik} is a parameter associated with item and the categorie, i=1,2,..,p (number of items) k=1,2...,m_i (number of categories from item)
: c_i is the guessing parameter, make sense in the dichotomous case.

## Not run: 
#Example 1

#simulate 10 polyotmous items, the first 5 with 4 response categories and the
#others with 5 response categories, By default the number of individuals is 1000
#, model is "2PL" and the dimention is 1
dats=simulate_polytomous(ncatgs = c(rep(4,10),rep(5,10)),seed_data = 5000L)
#estimate the items parameters
est=itemfit(dats$data,dim = 1,model = "2PL")
#calculates the latent traits estimation.
personfit(dats$data,dim = 1,zetas = est$zetas,method = "MAP")

#Example 2

#simulate 10 dichotomous items, the trait dimention is 2
#other arguments by default
dats=simulate_dichotomous(dim.data = 2,size.cluster = c(5,5), seed_data = 5000L)
#estimate the items parameters
est=itemfit(dats$data,dim = 2,clusters = c(5,5),model = "2PL")
#calculates the latent traits estimation
personfit(dats$data,dim = 2,zetas = est$zetas,method = "MAP",by_individuals=F)

## End(Not run)