**3. Statistical analysis**

In the following, we focus on geoadditive Gaussian models for continuous response variables to analyze the effects of metrical, categorical, and spatial covariates on stunting, wasting and underweight response variables in latent variable analyses. Furthermore, we use "nutritional status" as the indicator in the analysis of the latent variable models as mentioned.

### **3.1 Geoadditive gaussian model**

In this analysis, we apply a noval approach by exploring regional patterns of childhood malnutrition and possible nonlinear effects of the factor within latent model framework using geoadditive Bayesian gaussian model for continuous response variable.The model

Associations Between Nutritional Indicators Using Geoadditive

= ' ' = 1,..,3, *lin*

With geoadditive predicator, to have geoadditive model

*geo*

**3.2.1 Mesurment model** 

Gaussian errors <sup>2</sup> ~ (0, ) *ij N*

Where

 =var( )=1 

Fahrmeir 2009).

individual *i* ,

**variable models** to overcome the drawbacks of separate analysis.

covariates (indirect effects) (Fharmeir and Raach, 2007; Khatab, 2007)

**3.2 A bayesian geoadditive LVM (latent varaible models)** 

coefficients. The restriction to = ( )=1 *var*

(Fahrmeir L 2007; Khatab and Fahrmeir 2009).

**3.2.2 General geoadditive structural model** 

 

*<sup>j</sup>* is the ''factor loading'', and

Latent Variable Models with Application to Child Malnutrition in Nigeria 549

used for this investigation has been described else where.(Raach 2005; Khatab 2007) . Basicaly in the early stage of this study we used the geoadditive Bayesian gaussian model

= ( ) ( ) ( ) () ' 01 2 3

*ij <sup>j</sup> i i i spat ij j <sup>i</sup>*

where *w* includes the categorical covariates in effect coding. The function 1*f* , 2*f* and 3*f* are non-linear smooth effects of the metrical covariates (body mass index, child, and mother's age) which are modelled by Bayesian P-splines, and *spat f* is the effect of the spatial covariate 1;...; *is S* labeling the districts in Nigeria. Regression models with predictors are referred to as geoadditive models. However, in this work we have used g**eoadditive latent** 

A latent variable model with covariates consists of two main approaches: the measurment model for continuous response with covaraites influencing the indicators directly (direct effects); and the structural model explaining the modificatio of the latent variables by

= ' <sup>0</sup> , = 1,.., , = 1,.., , *ij j i j i ij y*

 *j* 

direct effects which affect the observed variables directly and *<sup>j</sup> a* is the vector of regression

*ii i i* = ( ) ... ( ) ( ) , *u* 1 1 *q geo i i*

**RESULTS.** We applied a geoadditive latent variable model, using the three types of undernutrition as indicators of latent nutritional status. The decision which covariates should be used in the measurement model, and which should be used in the structural equation, is based on the same criteria that was used in (Khatab 2007; Khatab and

Continuous variables are observed directly, hence the underlying variable is obsolete.

*<sup>i</sup>* represents the nutritional status with independent and identically distributed

*i* is the effect of

 *aw i n* 

. In this model,

with independent and identically distributed Gaussian errors ~ (0,1)

is necessary for identifiability reasons.

*xwj* (2)

*f Chage f BMI f Mageb f s w* (3)

 

*j p* (4)

*i* is the unobservable value of

is necessary for identifability reasons .

 *f x f x f s* (5)

for

*<sup>i</sup>* . In addition, *wi* are the

*<sup>i</sup> N* . The restriction to

for the separate analysis. In this model we replace the strictly linear predictor

*ij ij j ij j*

 


Table 1. Factors analyzed in malnutrition study

used for this investigation has been described else where.(Raach 2005; Khatab 2007) . Basicaly in the early stage of this study we used the geoadditive Bayesian gaussian model for the separate analysis. In this model we replace the strictly linear predictor

$$\eta\_{ij}^{\text{lin}} = \mathbf{x}\_{ij}^{\prime} \boldsymbol{\beta}\_{\rangle} + \mathbf{w}\_{ij}^{\prime} \boldsymbol{\gamma}\_{\rangle} \quad \mathbf{j} = \mathbf{1}, \ldots, \mathbf{3},\tag{2}$$

With geoadditive predicator, to have geoadditive model

$$\eta\_{ij}^{\text{geo}} = \beta\_{0j} + f\_1(\text{Change}\_i) + f\_2(\text{BMI}\_i) + f\_3(\text{Mage}b\_i) + f\_{\text{sput}\_j}(\text{s}) + w\_{ij}\text{'}\_j \tag{3}$$

where *w* includes the categorical covariates in effect coding. The function 1*f* , 2*f* and 3*f* are non-linear smooth effects of the metrical covariates (body mass index, child, and mother's age) which are modelled by Bayesian P-splines, and *spat f* is the effect of the spatial covariate 1;...; *is S* labeling the districts in Nigeria. Regression models with predictors are referred to as geoadditive models. However, in this work we have used g**eoadditive latent variable models** to overcome the drawbacks of separate analysis.

#### **3.2 A bayesian geoadditive LVM (latent varaible models)**

A latent variable model with covariates consists of two main approaches: the measurment model for continuous response with covaraites influencing the indicators directly (direct effects); and the structural model explaining the modificatio of the latent variables by covariates (indirect effects) (Fharmeir and Raach, 2007; Khatab, 2007)

#### **3.2.1 Mesurment model**

548 Current Topics in Tropical Medicine

Incomp.sec 4194(62.97%) 1

Higher 2467(37.04%) -1.ref **Pregnancy's treatment**  Yes 697(10.46%) 1 No 5964(89.54%) -1.ref **Drinking water**  Controlled 5374(80.68%) 1 Not controlled 1287(19.32%) -1.ref

**Had radio** 

Yes 5374(80.68%) 1 No 1559(19.32%) -1.ref **Has electricity**  Yes 6203(93.12%) 1 No 458(6.88%) -1.ref **Toilet facility** 

**Antenatal visit**  Yes 4181(63%) 1 No 2342(35%) -1.ref

Own flush toile facility 1768(28%) 1 Other and no toilet facility 4511(71.8%) -1.ref

No, Incomp.prim, Comp.prim,

Compl.sec,

Missing 1%

Missing 1%

Missing 2%

Table 1. Factors analyzed in malnutrition study

**Factor N(%) Coding effect Place of residence**  Urban 2237(33.58%) 1 Rural 4424(66.42%) -1.ref **Child's sex**  Male 3487(52.35%) 1 Female 3174(47.65%) -1.ref **Working**  Yes 1209(18.15%) 1 No 5452(81.85%) -1.ref **Mother's Education** 

$$\mathbf{x}\_{i} y\_{i\dot{\jmath}} = \boldsymbol{\lambda}\_{0} + \boldsymbol{a}\_{\dot{\jmath}} \mathbf{\dot{w}}\_{i} + \boldsymbol{\lambda}\_{\dot{\jmath}} \boldsymbol{\nu}\_{i} + \boldsymbol{\varepsilon}\_{i\dot{\jmath}}, \ \mathbf{i} = \mathbf{1}, \ldots, \mathbf{n}, \ \mathbf{j} = \mathbf{1}, \ldots, \mathbf{p}, \tag{4}$$

Where *<sup>i</sup>* represents the nutritional status with independent and identically distributed Gaussian errors <sup>2</sup> ~ (0, ) *ij N* . In this model, *i* is the unobservable value of for individual *i* , *<sup>j</sup>* is the ''factor loading'', and *j i* is the effect of *<sup>i</sup>* . In addition, *wi* are the direct effects which affect the observed variables directly and *<sup>j</sup> a* is the vector of regression coefficients. The restriction to = ( )=1 *var* is necessary for identifability reasons . (Fahrmeir L 2007; Khatab and Fahrmeir 2009).

Continuous variables are observed directly, hence the underlying variable is obsolete.
