The negative value of the sum of bivariate marginal log-likelihoods at
CL1 estimates.
r
The CL1 estimates of the latent correlations.
b
The CL1 estimates of the regression coefficients.
gam
The CL1 estimates of the cutpoints.
xdat
(x1,x2,…,xn)⊤, where the matrix xi,i=1,…,n for a given unit will
depend on the times of observation for that unit (ji) and will have
number of rows ji, each row corresponding to one of the ji elements
of yi and p columns where p is the number of covariates. This xdat matrix is
of dimension (N×p), where N=∑i=1nji is the total
number of observations from all units.
id
An index for individuals or clusters.
tvec
A vector with the time indicator of individuals or clusters.
WtScMat
A list containing the following components.
omega: The array with the Ωi,i=1,…,n matrices;
delta: The array with the Δi,i=1,…,n matrices;
X: The array with the Xi,i=1,…,n matrices.
corstr
Indicates the latent correlation structure of normal copula.
Choices are “exch”, “ar”, and “unstr” for exchangeable, ar(1)
and unstructured correlation structure, respectively.
link
The link function.
Choices are “logit” for
the logit link function, and “probit” for
the probit link function.
mvncmp
The method of computation of the MVN rectangle probabilities.
Choices are 1 for mvnapp (faster), and 2 for
pmvnorm (more accurate).
Details
First, consider the sum of univariate log-likelihoods
ϕ2(⋅;ρ) denotes the standard bivariate normal density with correlation
ρ.
Let a⊤=(β⊤,γ⊤) be the
column vector of all r=p+q univariate parameters. Differentiating L1 with respect to a leads to the independent estimating equations or univariate composite score functions:
g1=g1(a)=∂a∂L1=∑i=1nXi⊤si(1)(a)=0,
Differentiating L2 with respect to R=(ρjk,1≤j<k≤d)
leads to the bivariate composite score functions (Zhao and Joe, 2005):
where si(2)(a,R)=∂R∂∑j<kℓ2(νij,νik,γ,ρjk;yij,yik) and si,jk(2)(γ,ρjk)=∂ρjk∂ℓ2(νij,νik,γ,ρjk;yij,yik).
The CL1 estimates a and R of the discretized MVN model are obtained
by solving the above CL1 estimating functions.
The asymptotic covariance matrix for the estimator that solves them, also known as the inverse Godambe (Godambe, 1991) information matrix, is
Gao, X. and Song, P.X.K. (2011).
Composite likelihood EM algorithm with applications to
multivariate hidden Markov model.
Statistica Sinica21, 165–185.
Godambe, V. P. (1991)
Estimating Functions.
Oxford: Oxford University Press
Nikoloulopoulos, A.K., Joe, H. and Chaganty, N.R. (2011)
Weighted scores method for regression models with dependent data.
Biostatistics, 12, 653–665. doi: 10.1093/biostatistics/kxr005.
Nikoloulopoulos, A.K. (2016)
Correlation structure and variable selection in generalized estimating equations via composite likelihood information criteria.
Statistics in Medicine, 35, 2377–2390. doi: 10.1002/sim.6871.
Nikoloulopoulos, A.K. (2017)
Weighted scores method for longitudinal ordinal data.
Arxiv e-prints, <arXiv:1510.07376>. https://arxiv.org/abs/1510.07376.
Varin, C. and Vidoni, P. (2005).
A note on composite likelihood inference and model selection.
Biometrika92, 519–528.
Zhao, Y. and Joe, H. (2005)
Composite likelihood estimation in multivariate data analysis.
The Canadian Journal of Statistics, 33, 335–356.