Agriculture Reference
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11.3.3 Generalized Linear Mixed Models
The two previous models assume continuous response variables. Now suppose that
y dk is binary, taking the value 1 if the phenomenon under investigation is present
and 0 otherwise. In this situation, the SA quantities of interest are usually pro-
portions or counts (for example, the proportion or total of unemployed persons in
the area or agricultural sector). In such cases, the mixed linear models described
above are no longer applicable.
MacGibbon and Tomberlin ( 1989 ) defined a generalized linear mixed model
(GLMM) for SAE that is widely used for this kind of problem. Suppose that y dk
is binary, and the variables of interest are the SA proportions P d ¼ X
k
y dk =
N d ,
d ¼ 1, 2,
MacGibbon and Tomberlin ( 1989 ) used a logistic regression
model to estimate P d as
...
, D
:
p dk
1 p dk
¼ x dk βþ˅ d ;
logit p d ðÞ ¼ log
ð 11
:
32 Þ
, x dk are unit-
specific covariates, and the outcomes y dk are assumed to be Bernoulli independent.
Model ( 11.32 ) has been analyzed more recently by Jiang and Lahiri ( 1998 ). For
another definition of a logistic regression model with random regression coeffi-
cients, see Malec et al. ( 1997 ).
Ghosh et al. ( 1998 ) proposed a general methodology for inference using the
generalized linear model (GLM) with random area effects. The sample values y dk s
are assumed to be independent, conditionally on the
iid N 0
where Pr( y dk ¼ 1| p dk ) ¼ p dk , Pr( y dk ¼ 0| p dk ) ¼ 1 p dk ,
˅ d
; ˃
2
˅
e
ʸ dk s, with probability density
functions belonging to exponential family defined as (Ghosh et al. 1998 )
;
1
fy dk ʸ dk
ð
Þ exp
˕
dk y dk ʸ dk a
ð
ʸ d ðÞ
Þ þby dk ; ˕ dk
ð
Þ
ð 11
:
33 Þ
j
where ʸ dk are the canonical parameters, the scale parameters ˕ dk > 0 should be
known, and the functions a (.) and b (.) are known. The
ʸ dk s are modeled as
ʸ d ðÞ ¼x dk βþ˅ d þ e dk ;
h
ð 11
:
34 Þ
are
mutually independent. The exponential family covers the well-known probability
distributions such as the Normal, Binomial, Bernoulli, and Poisson distributions
(see McCullagh and Nelder ( 1989 ) for more details about GLM).
The approach outlined in this section is particularly suitable when using point
frame sampling, and surveying a qualitative variable. For example, if the observed
variable is a land cover/land use code (i.e., a categorical variable), a multinomial
and e dk
iid N 0
iid N 0
where h is a strictly increasing function,
˅ d
2
˅
e
; ˃
; ˃
e
e
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