Page 163 - Elements of Distribution Theory

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5.4 Hierarchical Models 149

Models for heterogeneity and dependence
The advantage of a hierarchical representation of a model is that in many applications
it is natural to construct models in this way. For instance, hierarchical models are useful
for incorporating additional heterogeniety into the model. Specifically, suppose that we
are willing to tentatively assume that random variables X 1 ,..., X n are independent and
identically distributed, each with a distribution depending on a parameter λ.However,we
may believe that there is more heterogeniety in the data than this model suggests. One
possible explanation for this is that the assumption that the same value of λ applies to each
X j maybetoostrong.Hence,wemightassumethatthedistributionon X j dependsonλ j , j =
1,..., n.However, taking λ 1 ,...,λ n as arbitrary parameter values may potentially allow
too much heterogeniety in the model. An alternative approach is to assume that λ 1 ,...,λ n
is a random sample from some distribution. The resulting distribution thus contains two
sources of variation in the X j : the variation inherent in the conditional distribution of X j
given λ, and the variation in the values in the sample λ 1 ,...,λ n .

Example 5.18 (Negative binomial distribution). Suppose that X has a negative binomial
distribution, as described in Example 5.17. Then
α α β + 1
E(X; θ) = and Var(X; θ) =
β β β
T
2
where θ = (α, β) ∈ R .In Example 5.17 it was shown that the distribution of X may
be viewed as a Poisson distribution with mean λ, where λ has a gamma distribution with
parameters α and β.
If X has a Poisson distribution, then
Var(X)
= 1;
E(X)
for the negative binomial distribution considered here,
Var(X; θ) 1
= 1 + .
E(X; θ) β
Hence, β measures the overdispersion of X relative to that of the Poisson distribution.

Hierarchical models are also useful for modeling dependence. For instance, as above,
suppose that X 1 ,..., X n are independent random variables, each with the same distribu-
tion, but with parameter values λ 1 ,...,λ n , respectively, where the λ j are random variables.
However, instead of assuming that λ 1 ,...,λ n are independent, identically distributed ran-
dom variables, we might assume that some of the λ j are equal; this might be appropriate if
there are certain conditions which affect more than one of X 1 ,..., X n . These relationships
among λ 1 ,...,λ n will induce dependence between X 1 ,..., X n . This idea is illustrated in
the following example.

Example 5.19 (Normal theory random effects model). Let X 1 and X 2 denote real-valued
random variables. Suppose that, given λ, X 1 and X 2 are independent, identically distributed
random variables, each with a normal distribution with mean λ and standard deviation σ.
Note that the same value of λ is assumed to hold for both X 1 and X 2 .