Hypothesis testing in mixture regression models - Columbia University

4 H.-T. Zhu and H. Zhang
in model (1) are very important in genetic studies for assessing potential gene–environment
interactions.
Beyond this example, finite mixtures of Bernoulli distributions such as model (1) have received
much attention in the last five decades. See Teicher (1963) for an early example. More recently,
Wang and Puterman (1998) among others generalized binomial finite mixtures to mixture
logistic regression models, and Zhang et al. (2003) applied mixture cumulative logistic models
to analyse correlated ordinal responses.
1.2. Example 2: mixture of non-linear hierarchical models
Longitudinal and genetic studies commonly involve a continuous response {Y ij }, also referred
to as a quantitative trait. See, for example, Diggle et al. (2002), Haseman and Elston (1972) and
Risch and Zhang (1995). Pauler and Laird (2000) used general finite mixture non-linear hierarchical
models to analyse longitudinal data from heterogeneous subpopulations. Specifically,
when there are only two subgroups, the model is of the form
Y ij = g{x ij , β, U i z ij µ 1 + .1 − U i /z ij µ 2 } + " i,j ,
where the " i,j s are independent and identically distributed according to N.0, σ 2 / and g.·/ is
a prespecified function. Here, the known covariates x ij may contain observed time points to
reflect a time course in longitudinal data.
1.3. Example 3: a finite mixture of Poisson regression models
Poisson distribution and Poisson regression have been widely used to analyse count data
(McCullagh and Nelder, 1989), but observed count data often exhibit overdispersion relative
to this. Finite mixture Poisson regression models (Wang et al., 1996) provide a plausible explanation
for overdispersion. Specifically, conditionally on all U i s, the Y ij s are independent
and follow the Poisson regression model
p.Y ij = y ij |x ij , U i / = 1
y ij ! λy ij
ij exp.−λ ij/, .2/
where λ ij = exp{x ij β + U i z ij µ 1 + .1 − U i /z ij µ 2 }.
To summarize the models presented above, we consider a random sample of n independent
observations {y i , X i } n 1
with the density function
p i .y i , x i ; ω/ = {.1 − α/f i .y i , x i ; β, µ 1 / + α f i .y i , x i ; β, µ 2 /} g i .x i /, .3/
where g i .x i / is the distribution function of X i . Further, ω = .α, β, µ 1 , µ 2 / is the unknown parameter
vector, in which β (q 1 × 1) measures the strength of association that is contributed by
the covariate terms and the two q 2 × 1 vectors, µ 1 and µ 2 , represent the different contributions
from two different groups.
Equivalently, if we consider P.U i = 0/ = 1 − P.U i = 1/ = α, and assume that the conditional
density of y i given U i is p i .y i |U i / = f i {y i , x i ; β, µ 2 .1 − U i / + µ 1 U i }, then model (3) is the
marginal density of y i . In fact, McCullagh and Nelder (1989) considered a special case in which
f i is from an exponential family distribution, i.e.
∏
f i {y i , x i ; β, µ 2 .1 − U i / + µ 1 U i } = n i
exp[φ{y ij θ ij − a.θ ij /} + c.y ij , φ/], .4/
j=1
where θ ij = h{x ij , β, U i µ 1 + .1 − U i /µ 2 }, h.·/ is a link function and φ is a dispersion parameter.
This family of mixture regression models is very useful in practice.