Page 254 - Special Topic Session (STS) - Volume 2

STS489 Danielle J.R. et al.
thus Equation (1) is a semi-parametric model. The spatial effect of district in
ℎ
which the child resides, ∈ (1, . . . , 370), is given by ( ) which
ℎ
represents the effects of unobserved covariates that are not included in the
model and also accounts for spatial autocorrelation (Kandala and Madise,
2004). This spatial effect may be partitioned into a spatially correlated
(structured) and an uncorrelated (unstructured) effect as follows:

( ) = ( ) + ( ) (2)
ℎ

ℎ
ℎ
The structured spatial effect ( ) accounts for the assumption that

ℎ
districts close in proximity would have similar observations. However, the
unstructured spatial effect ( ) accounts for the spatial variation due to
ℎ
effects of unmeasured local factors that are not spatially related.
In this study, inference was fully Bayesian, hence all parameters and
functions were treated as random variables. The fixed effect parameters in β
were assigned vague Gaussian priors (0, 1000), where the precision = 0.001 =
1/variance. The Bayesian perspective of penalised splines (P-splines) was
adopted for the unknown smooth functions where second-order random

walk smoothness priors and third degrees splines were used (Lang and
Brezger, 2004). For the structured spatial effect, ( ), intrinsic Gaussian
ℎ

Markov random field (IGMRF) priors specified by Besag et al. (1991) were used.
The unstructured spatial effect ( ) was assigned i.i.d. Gaussian priors.
ℎ
The variance components of the random and spatial effects are unknown
precision parameters that require estimation. Therefore, hyper-priors were
assigned to them in a second stage of hierarchy. These hyper-priors are
defined on a logarithmic scale and thus a log-gamma (1,0.001) distribution
was used. A sum-to-zero constraint was imposed on the non-linear and spatial
effects to ensure model identifiability between the intercept and these effects.
Three types of models were fitted:
 Model 1: GLM model: Linear fixed effects of all variables, categorical and
continuous.
 Model 2: GAM model: Linear fixed effects of categorical variables and some
continuous variables, and non-linear effect of the child’s age in months.
 Model 3: Geoadditive Model: Model 2 with the inclusion of the spatial
effects.
The posterior distributions of the parameters in the models were estimated
using Integrated Nested Laplace Approximation, and thus the INLA package
in R was used (http://www.r-inla.org/) (Rue et al., 2009). The final geoadditive
model was selected using the Deviance Information Criteria (DIC) and the
effective number of parameters ( ). QGIS 3.4

(https://qgis.org/en/site/index.html) was used to create maps displaying the

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