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IPS192 Hukum C. et al.
developing countries. In such situations, an area level version of the GLMM
can be used for SAE. In particular, when only area level data are available, an
area level version of the GLMM is fitted to obtain the plug-in empirical
predictor for the small areas, see for example, Chandra et al. (2011). In
economic, environmental and epidemiological applications, estimates for
areas that are spatially close may be more alike than estimates for areas that
are further apart. It is therefore reasonable to assume that the effects of
neighbouring areas, defined via a contiguity criterion, are correlated. Chandra
and Salvati (2018) describe an extension of the area level version of GLMM
that allows for spatially correlated random effects using a SAR model (SGLMM)
and define a plug-in empirical predictor (SEP) for the small area proportion
under this model. This model allows for spatial correlation in the error
structure, while keeping the fixed effects parameters spatially invariant.
Chandra et al. (2017) introduce a spatially nonstationary extension of the area
level version of GLMM, using an adaptation of the geographical weighted
regression (GWR) concept to extend the GLMM to incorporate spatial
nonstationarity (NSGLMM), which they then apply to the SAE problem to
define a plug-in empirical predictor (NSEP) for small areas. Non-stationary
spatial effects can be also modelled using a spatially non-linear extension of
the GLMM. In the GLMM, the relationship between the link function and the
covariates is often assumed to be linear. However, when the functional form
of the relationship between the link function and the covariates is unknown or
has a complicated functional form, an approach based on the use of a non-
linear regression model can offer significant advantages compared with one
based on a linear model. When geographically referenced area-level
responses play a central role in the analysis and need to be converted to maps,
we can use bivariate smoothing to fit a spatially heterogeneous GLMM. In
particular, we use P-splines that rely on a set of bivariate basis functions to
handle the spatial structures in the data, while at the same time including small
area random effects in the model. We denote this nonparametric P-spline-
based extension of the usual GLMM by SNLGLMM. See Opsomer et al. (2008)
and Ruppert et al. (2003). We then describe a non-linear version of the plug-
in empirical predictor for small areas (SNLEP) under an area level version of
SNLGLMM. We also develop mean squared error estimation for SNLEP using
the approach discussed in Chandra et al. (2011) and Opsomer et al. (2008).

2. SAE under a Spatially non-linear generalized mixed model
Consider a finite population U of size N, and assume that a sample s of
size n is drawn from this population according to a given sampling design,
with the subscripts s and r used to denote quantities related to the sampled
and non-sampled parts of the population. We assume that population is made
up of m small domains or small areas (or domains or areas) (=1,...,), where

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