MANEJO DE ERRORES Y ADMINISTRACIÓN DE EXCEPCIONES

Figures 4 and 5 show results for the spatial random effects modeled through the level- 2 and level-3 equations. The kernel density estimates of Figure 4 provide insight into the strength and importance of the various random effects. We first note that the interaction random effects are much weaker than the main random effects. Moreover, the district smooth effects and the uncorrelated district random effects are roughly of equal size and dominate the state random effects which are almost negligible. Figure 5 shows maps of the spatial heterogeneity not explained by covariates for males and females, respectively. Unexplained spatial heterogeneity is additively composed of the district smooth and uncorrelated random effect and the state random effect. Overall, unexplained heterogeneity is higher for females (see also in Figure 4 the right bottom panel). Moreover, females exhibit a more pronounced spatial pattern with higher probabilities of stunting in the north-west and lower probabilities in the south and the north-east. For males we observe a similar pattern although the north-south patterns are less distinct.

5.4

Model choice

Some final remarks regarding model choice are in order. General tools for model choice are pointwise and simultaneous credible intervals for the nonlinear effects as well as Bayesian goodness of fit criteria, particularly the DIC. Also beneficial for model choice is the detailed hierarchical modeling of spatial heterogeneity. For instance, the kernel densities of Figure 4 suggest that the interaction random effect can be restricted to a level-2 equation with a smooth and/or uncorrelated district effect. The spatial main effect could possibly be restricted to the level-2 equation omitting the level-3 states equation. To reduce the complexity of the full interaction model (6) we could in a first step exclude the smooth nonsignificant interactions (in terms of 95% pointwise credible intervals) which slightly reduces the DIC by approximately 15 points. A further reduction of the DIC is obtained by more parsimonious random effects. The best model (in terms of the DIC) is given by a full main effects spatial random effect

including the level-3 equation and a reduced spatial interaction with a simple i.i.d. Gaussian district random effect. In this model the DIC reduces by approximately 25 points compared to the full interaction model. Further reduction of the main effects spatial random effect to a level-2 equation shows almost identical DIC.

6

Conclusion

This paper proposes a multilevel version of STAR models by assuming that the re- gression coefficients of a particular nonlinear term obey another regression model with structured additive predictor. The proposed model may be regarded as an extended version of a multilevel model with nonlinear covariate terms in every level of the hier- archy. Our model framework also comprises proposals for generalizations of random slopes by assuming a common functional form that is scaled by cluster specific scaling factors. We have developed highly efficient MCMC schemes for simulation-based in- ference. The algorithms utilize the hierarchical structure of the models and rigorously exploit the sparsity of design matrices, cross products and penalty matrices. Thereby a considerable gain in numerical efficiency, reduction in computing time and mixing of Markov chains is achieved compared to non-hierarchical versions of the models. The methodology of this paper is the basis for a number of extensions that we plan for future research:

• First of all, we plan to extend multilevel STAR models to multivariate responses, in particular multicategorical regression and seemingly unrelated regression. • We also plan to model other parameters than the mean of the distribution in the

spirit of generalized additive models for location, scale and skewness (GAMLSS, Rigby and Stasinopoulos 2005).

• Another interesting (albeit rather challenging) field is to model hyperparameters in dependence of covariates, e.g. the variance parameter τ2 _{in the general prior}

the specification of a full STAR model should be possible for these hyperparam- eters. This allows for modeling locally adaptive functions or complex covariate driven spatial neighborhood definitions.

• We finally want to develop methodology for automatic model choice and variable selection in the spirit of Belitz and Lang (2008) in a frequentist setting and Scheipl, Fahrmeir, and Kneib (2011) in a Bayesian approach via spike and slab priors.

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Figures

0 .2 .4 .6 .8 1 max ACF 0 10 20 30 40 50 lag hierarchical non−hierarchical Effect of mean education years within districts

0 .2 .4 .6 .8 1 max ACF 0 10 20 30 40 50 lag hierarchical non−hierarchical Markov random field of district

0 .2 .4 .6 .8 1 max ACF 0 10 20 30 40 50 lag hierarchical non−hierarchical random effect of district

0 .2 .4 .6 .8 1 max ACF 0 10 20 30 40 50 lag hierarchical non−hierarchical random effect of region

−2 −1.5 −1 −.5 0 .5 1 effect 0 4 8 12 16 20 24 28 32 36 age in month

Average effect of child’s age

−.75 −.25 .25 .75 effect 15 20 25 30 35 age at first birth

Effect of mother’s age at first birth

−.2 −.1 0 .1 .2 effect 0 4 8 12 16 20 24 28 32 36 age in months

Varying effect of child’s age

−.5 0 .5 1 1.5 effect 15 20 25 30 35 age at first birth

Varying effect of mother’s age at first birth

−2 −1 0 1 effect 0 4 8 12 16 20 24 28 32 36 age in months male female Effect of child’s age by gender

−.4 −.2 0 .2 .4 .6 effect 15 20 25 30 35 age at first birth

male female Effect of mother’s age at first birth by gender

Figure 2: Effect of child’s age and mother’s age at first birth by gender. Shown is the posterior mean together with 95% pointwise and simultaneous credible intervals.

−1.2 −1 −.8 −.6 −.4 −.2 0 .2 .4 .6 effect −1.2 −.8 −.4 0 .4 .8 1.2 mean wealth index within districts

Effect of mean wealth index within districts

−.6 −.4 −.2 0 .2 .4 .6 effect .5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 mean education years within districts

Effect of mean education years within districts

−1 −.5 0 .5 1 effect −3 −2 −1 0 1 2 3 Difference of household’s wealth index to district mean

Effect of household’s wealth index

−.8 −.5 −.2 .1 .4 effect −4 0 4 8 12 16 20 Difference of mother’s education years to district mean

Effect of mother’s education

Figure 3: Nonlinear effects of the wealth index and the education years. Shown is the posterior mean together with 95% pointwise and simultaneous credible intervals.

0 2 4 6 density −.5 −.25 0 .25 .5 main effect interaction effect

kernel density estimators for random district effect

0 5 10 15 density −.5 −.25 0 .25 .5 main effect interaction effect

kernel density estimators for smooth district effect

0 10 20 30 40 density −.5 −.25 0 .25 .5 main effect interaction effect kernel density estimators for random region effect

0 .5 1 1.5 2 density −1.25 −.75 −.25 .25 .75 female male

kernel density estimators for unexplained heterogenity

Figure 4: Kernel densities of the spatial random effects.

−0.88 0 0.88 −1.1 0 1.1

Figure 5: Spatial heterogeneity not explained by covariates for males (left panel) and females(right panel).

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