GENERALIZED LINEAR MODELS 19
tudinal data, unobserved responses will be imputed from observed responses
for the same individual; the assumed correlation structure will usually dictate
(at least in part) the functional form of the imputation. This theme recurs
throughout the book, and therefore our review pays particular attention to
aspects of variance-covariance specification.
2.2.4 Additional notation
Random variables and their realizations are denoted by Roman letters (e.g.,
X, x), and parameters are representedbyGreek letters (e.g., α, θ). Vector-
and matrix-valued random variables and parameters are represented using
boldface (e.g., x, Y , β, Σ). For any matrix or vector A,weuseA
T
to denote
transpose. If A is invertible, then A
−1
is its inverse, |A| is its determinant,
and L = A
1/2
is the lower triangular matrix square root (Cholesky factor)
such that LL
T
= A.Aq-dimensional identity matrix is denoted I
q
and a
diagonal matrix by diag(a), where a is thevector of diagonal elements. The
parameterizations of specific probability distributions used in the text can be
found in the Appendix.
2.3 Generalized linear models for cross-sectional data
The generalized linear model (GLM) forms the foundation for many ap-
proaches to regression with multivariate responses, such as longitudinal or
clustered data. Models such as random effects or mixed effects models, latent
variable and latent class models, and regression splines, all highly flexible and
general, are based on the GLM framework. Moment-based methods such as
generalized estimating equations (GEE) also follow directly from the GLM
forcross-sectional data (Liang and Zeger, 1986).
The GLM is a regression modelforadependent variable Y arising from
the exponential family of distributions
p(y | θ, ψ)=exp{(yθ − b(θ)) /a(ψ)+c(y, ψ)} ,
where a, b,andc are known functions, θ is the canonical parameter,andψ
is a scale parameter.Theexponential family includes several commonly used
distributions, such as normal, Poisson, binomial, and gamma. It can be readily
shown that
E(Y )=b
(θ)
var(Y )=a(ψ)b
(θ),
where b
(θ)andb
(θ)arefirstandsecond derivatives of b(θ)withrespect to
θ (see McCullagh and Nelder, 1989, Section 2.2.2 for details).
The effect of covariates x
i
=(x
i1
,...,x
ip
)canbemodeledbyintroducing