mle.FactorTree {FactorCopula}

Maximum likelhood estimation of factor tree copula models

Description

We use a two-stage estimation approach toward the estimation of factor tree copula models for item response data.

Usage

mle1FactorTree(y, A, cop, gl, hessian, print.level) 
mle2FactorTree(y, A, cop, gl, hessian, print.level) 

Arguments

y

n \times d matrix with the ordinal reponse data, where n and d is the number of observations and ordinal variables, respectively.

A

d \times d vine array with 1,...,d on diagonal, note only the first row and diagnoal values are used for the 1-truncated vine model

cop

(2d-1)-vector with the names of bivariate copulas that link each of the oberved variabels with the 1st factor (1-factor part of the model), and conditional dependence of variables given the latent factor (1-truncated vine tree part of the model) . Choices are “bvn” for BVN, “bvt\nu” with \nu = \{1, \ldots, 9\} degrees of freedom for t-copula, “frk” for Frank, “gum” for Gumbel, “rgum” for reflected Gumbel, “1rgum” for 1-reflected Gumbel, “2rgum” for 2-reflected Gumbel.

gl

Gauss legendre quardrature nodes and weights.

hessian

If TRUE, the hessian of the negative log-likelihood is calculated during the minimization process.

print.level

Determines the level of printing which is done during the minimization process; same as in nlm.

Details

Estimation is achieved by maximizing the joint log-likelihood over the copula parameters with the univariate cutpoints fixed as estimated at the first step of the proposed two-step estimation approach.

Value

A list containing the following components:

cutpoints

The estimated univariate cutpoints.

loglik

The maximized joint log-likelihood.

taus

The estimated copula parameters in Kendall's tau scale.

SEs

The SEs of the Kendall's tau estimates.

Author(s)

Sayed H. Kadhem
Aristidis K. Nikoloulopoulos [e-mail]

References

Joe, H. (2014). Dependence Modelling with Copulas. Chapman & Hall, London.

Kadhem, S.H. and Nikoloulopoulos, A.K. (2022b) Factor tree copula models for item response data. Arxiv e-prints, <arXiv: 2201.00339>. https://arxiv.org/abs/2201.00339.

Examples

#------------------------------------------------
# Setting quadreture points
nq <- 5  
gl <- gauss.quad.prob(nq) 
#------------------------------------------------
#                    PTSD Data
#------------------             -----------------
data(PTSD)
ydat=PTSD
n=nrow(ydat)
d=ncol(ydat)
#------------------------------------------------
#                   Estimation
#------------------             -----------------
#selecting vine tree based on polychoric
rmat=polychoric0(ydat)$p
A.polychoric=selectFactorTrVine(y=ydat,rmat,alg=3)

#---------------- 1-factor tree  ----------------
# 1-factor tree copula model
copf1 <- rep("frk",d)
coptree <- rep("frk",d-1)
cop <- c(copf1,coptree)
est1factortree <- mle1FactorTree(y=ydat, A=A.polychoric$VineTreeA, cop, 
gl, hessian=FALSE, print.level=2) 

#---------------- 2-factor tree  ----------------
# 2-factor tree copula model
copf1 <- rep("frk",d)
copf2 <- rep("frk",d)
coptree <- rep("frk",d-1)
cop <- c(copf1,copf2,coptree)

est2factortree <- mle2FactorTree(y=ydat, A=A.polychoric$VineTreeA, 
cop, gl, hessian=FALSE, print.level=2)