DIAGNÓSTICO POR VARIABLES 1 Identidad cultural

Forfunstr, we may setK=I .

Furthermore, the inverse Gamma priors are assumed for τ_str2 [IG(astr, bstr)]

and τ2

unstr[IG(aunstr, bunstr)].

2.4

MCMC Inference

We use Markov Chain Monte Carlo (MCMC) simulations to draw samples from the posterior. Statistical inference is done by means of Markov chain Monte Carlo techniques in a full Bayesian setting. We restrict the presen- tation to models with predictor 2.4. Full Bayesian inference is based on the entire posterior distribution.

p(β, τ2, γ|y)∝p(y|β, τ2, γ)p(β, τ2, γ), (2.15) whereβ = (β1, .., βp) and τ2 =τ12, .., τp2 denote parameter vectors for func-

tion evaluations and variance. Then, under usual conditional independence assumptions, the posterior is given by:

p(β, τ2, γ|y)∝ n Y i=1 Li(yi;ηi) p Y j=1 © p(βj|τj2)p(τj2) ªYr k=1 p(γ_k)p(σ2) (2.16) Only for Gaussian responses, the full conditional distributions for unknown functionsβj, j= 1, .., p, and fixed effects parameters γ are Gaussian and for

variance components τj, j = 1, .., p and σ2 the full conditionals are inverse

gama distributions.

p(β|.)∝Qn_i=1Li(yi;ηi)p(βj|τj)

p(γ|.)Qn_i=1Li(yi;p(ηi)p(γ)

p(τ2_{|.) =p(f|τ}2_)p(τ2₎

p(σ2|.) =Qr_k=1p(γk)

Bayesian inference via MCMC is based on updating full conditionals of sin- gle parameters or blocks of parameters, given the rest and the data. For

Gaussian models, Gibbs sampling with so-called multimove steps can be applied. For non-Gaussian responses Gibbs sampling is no longer feasible and Metropolis Hastings algorithms are needed. More details can be found in Rue (2001) or Fahrmeir and Lang (2001a). For the predictor 2.4, let α

denote the vector of all unknown parameters in the model. Then, under usual conditional assumptions, the predictor is given by

p(α|y)∝Πn_i=1Li(yi, ηi)Πp_j=1{p(βj|τj2)p(τj2)}p(fstr|τstr2 )p(funstr|τunstr2 )Πrj=1p(γj)p(σ2),

whereβj, j= 1, .., p,are the vectors of regression coefficients corresponding

to the functionsf_j. The full conditionalsf_str, f_unstr and fixed effects param- etersγ are multivariate Gaussian in the case for Gaussian response variables. While the full conditionals for the variance components τ2_{, j = 1, .., p, str,}

unstr and σ2 are inverse gamma distributions. More details can be found in Rue (2001), Fahrmeir and Lang (2001b), Lang and Brezger (2000a), and Kandala, et.al.(2001b). The estimation of models in this thesis is based on different sampling schemes depending on the distribution of the response. Two types of responses are included in this thesis, namely binary responses and Gaussian responses.

Gaussian Response

For the Gaussian response variable, the full conditionals for fixed effects and non-linear effects are multivariate Gaussian. For the variance parameters, all full conditionals are inverse Gamma distribution. Straight forward calcu- lations show that precision matrices for nonlinear effects are band matrices. For a one dimensional P-spline the bandwidth of precision matrix is the maximum between the degree of the spline and the order of the random walk. The cholesky decomposition is mostly used for fast efficient matrix operation of band matrices. More details and description on the sampling scheme for Gaussian responses can be found in Lang and Brezger, 2001 and Rue, 2001.

2.4. MCMC INFERENCE 37 Non-Gaussian Responses

Here, we now turn the attention to general responses from an exponential family. In this case the full conditionals are no longer Gaussian. For fixed effects and i.i.d. random effects we use a slightly modified version of the iteratively weighted least squares proposal suggested by Gamerman (1997), see also Brezger and Lang (2006), CSDA. In addition, Fahrmeir and Lang (2001a) propose a MH-algorithm for updating unknown regression parame- ters based on conditional prior proposals. For updating, only likelihood is required but no approximations of characteristics of the posterior (e.g. the mode).

Chapter 3

Modelling of Child Diseases

in Egypt and Nigeria

Abstract

Our case study is based on the 2003 Demographic and Health Survey for Egypt (EDHS) and Nigeria (NDHS). It provided data on the prevalence and treatment of common childhood diseases such as diarrhea, cough and fever, which are seen as symptoms or indicators of children’s health status, causing increased morbidity and mortality. The causes of childhood illnesses are multiple. Theses causes are associated with a number of risk factors, including inadequate antenatal care, lack of or inadequate vaccination, high birth order, and malnutrition. The main focus of this chapter is to analyze the effects of these different types of covariates on the response variables diarrhea, fever, and cough, using data from the 2003 DHS Demographic and Health surveys (DHS) from Egypt and Nigeria. We started our analysis using a large number of factors which could affect the health of children in both countries as a first step. Based on the results of the first step, we then excluded some factors which have slight effects on the childhood diseases as a second step and compare the results. A Bayesian geoadditive model for binary response variable is used in this application based on Fahrmeir and Lang (2001).

3.1

Introduction

In this application, we concentrate on flexible modelling of effects of metri- cal covariates, categorical covariates, and spatial covariates on the response variables (diarrhea, fever, and cough). The analyses for the childhood dis- ease in Egypt and Nigeria are based on the data from the 2003 Demo- graphic and Health survey (DHS). One of the main objectives of DHS is to provide an up-to-date information on childhood disease. This intends to assist policy makers and administrators in evaluating and designing pro- grams and improve planning for future interventions in these areas, which in turn should reduce childhood morbidity and childhood mortality as well. We use the geoadditive logit models for the binary response variables (had diseases/no) in this chapter. Accordingly, we began the investigation with a large number of covariates including a large set of bio-demographic and socio-economic variables, including covariates such as preceding birth inter- val, current working status of mother, place of delivery, mother’s educational attainment, whether the mother received injections during pregnancy or not and whether the mother attended antenatal clinic or not. Other relevant fac- tors included such as mother’s age at birth, availability of any toilet facility, source of drinkable water, locality of residence and region of residence. At the end, it turned out that many of them were not significant. The categor- ical covariates were transformed into effect coding. The metrical covariates are modelled by second order random walk priors. All computations have been carried out with BayesX-version 1.4 (Brezger, Kneib and Lang 2005).