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IPS192 Hukum C. et al.
plug-in empirical predictor (EP) of the population count in area is obtained
̂
by replacing , by its predicted value ̂ = ( | ) under the model (1) as

ŷ = + ̂ = + ( − )̂ , (3)

̂

with ̂ = expit ( + ̂) for binary data, where = (0, … , 1, . . ,0) is 1×

vector with 1 in the -th position and ̂ = (̂ , … . , ̂ ) . In SAE problems, the
1

sample size is often negligible relative to the population size , then ̂ =

̂ . An estimate of the proportion in area is given by ̂ .

We now introduce a spatially non-linear extension of an area level GLMM.
We refer this model as spatially non-linear generalized linear mixed model
(SNLGLMM). We start by developing the nonparametric extension of the
GLMM, and then suggest a spatial extension of this model. Typically, the fixed
effect part of a GLMM is assumed to be linear. However, in reality the
functional form of this relationship may be unknown or it may have a
complicated functional form. Without loss of generality we restrict our
development to the case of a single covariate x and use nonparametric
regression modelling based on a P-spline approximation. The spatially non-
linear GLMM (SNLGLMM) is then of the form
( ) = ƞ = ( ) + (4)

2
where ~(0, ) is the area specific random effect and = ( | , ) =

ℎ{( ) + }. In particular, the spatially non-linear logistic-normal mixed

model and the spatially non-linear Poissonnormal mixed model for binary and
count data, respectively, are defined as ( ) = ƞ = ( ) + with =

{( ) + } and log( ) = ƞ = ( ) + with = {( )}. The

function ( ) in (4) is unknown, but can be approximated sufficiently well by

the P-spline approximation

( , , ) = + + ⋯ … . + + ∑ ( − ) (5)
0
+

1

=1

Here is the degree of the spline, () = if > 0 and its otherwise,
+

for = 1, ……, is a set of fixed constants called knots, = ( , , … , ) is
0

,
the coefficient vector of the parametric portion of the model and =

( , … . , ) is the vector of spline coefficients, is the number of spline knots,

1
2
and ~(0, ); = 1,……, . Provided that the knot locations are sufficiently

spread out over the range of and is sufficiently large, the class of functions
defined by (5) can approximate most smooth functions. Ruppert et al. (2003,
chapter 5) suggest the use of a knot for every four observations, up to a
maximum of about 40 knots for a univariate application. This is usually done
by placing these knots at equally spaced quantiles of the distribution of the
covariate.
Note that the P-spline approximation consists of a linear combination of
appropriately chosen basis functions. For simplicity, the approximating
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