84 parameter and is estimated jointly with the regression parameters by the
maximum likelihood ML method. For discrete distributions negative binomial, Poisson, binomial and multinomial,
is estimated by Pearson chi- square as follows:
where N = N - p
x
for the restricted maximum pseudo-likelihood REPL method.
2. If a model only has G-side random effects, then the G matrix is user-specified
and R=
I. is estimated jointly with the covariance parameters in G for
continuous distributions and = 1 for discrete distributions.
3. If a model only has R-side residual effects, then G = 0 and the R matrix is user-specified. All covariance parameters in R are estimated using the REPL
method.
4. If a model has both G-side and R-side effects, all covariance parameters in G and R are jointly estimated using the REPL method.
Type 2 is appropriate to the model in this study. For ordinal multinomial distribution,
of equation 9 and R =
I which means that R-side effects are not supported for the multinomial distribution.
is set to 1.
4.2.2.2 Logistic Response Function
The probability that , conditional on random effects, under logit
formulation with the mixed-effects regression model for the underlying latent variable
, as shown by equation 3 in section 4.2.1, is given by
85 where
represents the random effects; and
· represents the logistic cumulative distribution function cdf. In the following
model development, the logit response function and the expansion of formula is based on Liu and Hedeker 1993.
Maximum Marginal Likelihood estimation
Let Y
sij
be the vector of ordinal responses from area s and subject i for all the si occasions with n
si
items at each occasion. Assuming independence of the responses conditional on the random effect, the conditional likelihood of any
pattern Y
sij
, given u
i
, is
where
Then the marginal likelihood of Y
s
in the population is expressed as the following integral of the conditional likelihood, L., weighted by the prior density
where represents the distribution of random effects in the population the
joint distribution of , a standard normal density. With assumption conditional
on the level-2 effect , the responses from n
i
occasions in subject i are independent, the marginal probability can be rewritten as
where
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
