86
For estimation of the p covariate coefficients , r item discrimination parameters u,
and K –1 threshold values
k = 1, …, K-1, the marginal log likelihood for the
patterns from the n
s
level-2 subjects is differentiated,
Let
θ is an arbitrary parameter vector, then we obtain
It is tractable for probit formulation and as long as the number of level-2 random effects is no greater than three or four,
a condition which is typically satisfied for longitudinal or clustered studies Liu and Hedeker 2006. In this study, cumulative
logit is used, which is not tractable Vasdekis et.al. 2010 or has no closed form solution Hardin and Hilbe 2003. To handle this problem, Wolfinger and
O’Connell gave a solution using Linear Mixed Pseudo model with first-order Taylor series approximation that will be discussed at the following sub section.
4.2.2.3 Wolfinger and O’Connell Approach
Procedure of pseudo-likelihood estimation by
Wolfinger and O’Connell is described as follows. For the generalized linear mixed model GLMM consider a
data vector y of length n satisfying
and a differentiable monotonic link function
such that
where
is a vector of unknown fixed effects with known mode1 matrix X of rank p, and u is a vector of unknown random effects with known model matrix Z.
Assume Eu = 0 and covu = D, where D is unknown.
87
Also, e is a vector of unobserved errors with
and
Here
,
is a diagonal matrix contain evaluations at of a known variance
function for the generalized linear model under consideration and R is unknown.
The next step is to build PL and REPL methods for fitting the GLMM by using three approximations: two analytic and one probabilistic
For the first analytic approximation, let
and be known estimates of and u, and define
which is a vector consisting of evaluations of at each component of
. Now let
10 where
is a diagonal matrix with elements consisting of evaluations of the first derivative of
. Note that
in equation 10 is a Taylor series approximation to
expanding about and . Next, for the probabilistic approximation, the conditional distribution of
given
and u with a Gaussian distribution having the same first two moments as e
|
, u which we assume corresponds to e | . In particular, assumed that |, u is
Gaussian with mean and variance
. The second and final analytic approximation is substituting
for in the
variance matrix. Then, since
for each component
i,
where is a diagonal matrix with elements constructed as above. Defined
then equivalently it can be specified For ordinal multinomial response,
88
and error terms with
and
and block diagonal weight matrix is
The Gaussian log pseudo-likelihood PL and restricted log pseudo- likelihood REPL, which are expressed as the functions of covariance parameters
in , corresponding to the linear mixed model for v are the following:
l l
R
where ,
,
89 N denotes the effective sample size, and
denotes the total number of non-
redundant parameters for B.
The parameter can be estimated by linear mixed model using the objection
function -2
l
θ; v or -2
l
R
θ; v, B and u are best linear unbiased prediction
BLUP Robinson 1991 and computed as
Iterative process The estimation of
θ uses the doubly iterative according to Wolfinger and
O’Connell and SPSS algorithm. The steps are as follows: 1. Obtaining an initial estimate of
, . Let
. Also set
the outer iteration index m = 0, M = maximum iterations.
2. Based on
, compute
and Fit a weighted linear mixed model with pseudo target v, fixed effects
design matrix X, random effects design matrix Z, and diagonal weight
matrix . The fitting procedure, which is called the inner iteration, yields
the estimates of
θ, and is denoted as θ
m
. If m = 0, go to step 4; otherwise go to the next step.
3. Check if the following criterion with tolerance level is satisfied:
. If it is met or maximum number of outer iterations is reached, stop.
Otherwise, go to the next step.
4. Compute
