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In this section, a comparison is made between the results of two semiparametric models, and two parametric models which shall approximate the semiparametric models. For this purpose, we take the two semiparametric models M3b und M4a (see Table 7.5). The nonparametric function of the metric covariate Age of model M3b is approximated by a cubic, parametric predictor which yields

ηi = γ1·Sex2i+γ2·Inc2i+γ3·Inc3i+

+γ4·Inc4i+γ5·Agei+γ6·Age2i +γ7·Age3i .

For the effect of the spatial covariate Reg in model M4a, a classification of the regions into three areas is carried out: south, east, and northwest (see Figure 7.18). The reason for this segmentation stems from the hypothesis that people in the comparably wealthy south of Germany might have a different attitude towards state provisions than people in the northwest with a medium prosperity, and people in the east with a traditionally weaker economy. Consequently each observation is assigned one of those three geographical locations, and this new categorized variable is termed RegPar. Since the nonparametric spatial effect of M4a is centered around zero, an effect coding ofRegParhas to be employed which delivers the difference effect of a certain region to the mean of all functional values. The predictor is defined as

ηi = γ1·RegParSouthi+γ2·RegParEasti.

Let us start with the discussion of model M3b with the metric variable Age. Estimates of the factor loadings and the parametric effects Sex and Inc given in Table 7.20 are virtually identical to the estimates of the nonparametric model estimates in Table 7.10. The estimated function ofAgebased on the parameters{γ5, γ6, γ7}of the cubic polynom of

Ageis plotted in Figure 7.19. The same figure shows a comparison of the cubic parametric estimate with the original nonparametric P-splines estimate. From the ages 20 to 60, the

7.5 Approximation of smooth functions 143

−0.074 0 0.104

−0.021 0 0.022

Figure 7.17: Left: Estimates of nonparametric effect of spatial covariate Reg for the first (top) and second (bottom) latent variable. Minimum and maximum values are set to 2.5% and 97.5% quantiles of observed function estimates, respectively. Right: Plot of regions with a significant negative effect (red), significant positive effect (green), and non-significant effect (yellow) for the first (top) and second (bottom) latent variable.

Figure 7.18: Categorization of the regional landscape into three categories ”Northwest” (grey), ”South” (white), and ”East” (black) of the new variable RegPar.

cubic parametric estimate demonstrates a very close resemblance to the functional form of the nonparametric estimate; for ages higher than 60 however, the cubic estimate rises much to high up to the age of 70. Furthermore, the 80% credible region is particularly wide. The reason for this broad credible interval lies in the fact that the three cubic parameters {γ5, γ6, γ7} are not adjusted in each MCMC iteration in such a way that the resulting

function estimate is centered around zero. Due to this lack of centering, a very broad and not meaningful credible region is obtained. Hence there are two advantages of the nonparametric approach; firstly, the nonparametric estimate follows the actual function in a better way than the parameterized functions; secondly, the centering of the estimated function estimate in each MCMC iteration leads to a narrow credible interval.

Results of the parametric estimates of the approximated model of M4a with three spatial regions are summarized in Table 7.21. The factor loadings are very close to the results of the parametric spatial model in Table 7.11. The two effects for the regions ”South” and ”East” are noticable and significant. Since effect coding is employed where the regression coefficients add up to zero, the parameter for the reference category ”Northwest” is obtained byγN orthwest =−(γ1+γ2) = −0.019. The drawback of the categorized regions is that you

need a hypothesis which regions might have a similar effect on the latent construct, and thus have to be merged before the conduct of the analysis; it is not reasonable to estimate a parametric analysis including 400 regions due to very high confidence regions of the individual parameters. Furthermore, the variation of the latent scores within a merged region is not visible in a parametric model.

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Parameter Mean Std. 10% Mode 90%

dev. quantile quantile

Factor loadings 1. System λ11 0.804 0.017 0.782 0.804 0.826 2. Initiative λ21 -0.851 0.014 -0.870 -0.851 -0.833 3. Retirement λ31 1.311 0.025 1.279 1.311 1.342 4. Emergency λ41 0.646 0.012 0.631 0.646 0.661 5. Health λ51 0.911 0.015 0.892 0.911 0.930 Parametric indirect effects

Sex2 γ1 -0.354 0.022 -0.382 -0.354 -0.327 Inc2 γ2 0.283 0.030 0.244 0.283 0.322 Inc3 γ3 0.622 0.030 0.583 0.622 0.661 Inc4 γ4 1.121 0.034 1.076 1.120 1.165 Age γ₅ 0.102 0.027 0.068 0.102 0.137 Age2 γ6 −3.14·10−3 6.45·10−4 −3.97·10−3 −3.13·10−3 −2.32·10−3 Age3 γ7 2.73·10−5 4.94·10−6 2.10·10−5 2.72·10−5 3.37·10−5 Table 7.20: Results of the approximated model of M3b – estimates of factor loadings and parametric effects for the cubic factor Age.

Cubic parametric estimate

Age −0.5 0.0 0.5 20 30 40 50 60 70 P−splines estimate Age −0.2 0.0 0.1 0.2 20 30 40 50 60 70

Figure 7.19: Results of the approximated model of M3b. Left: Function of Age based on the cubic polynom of Age – the mean value is drawn as a straight line, and 10%- and 90%-quantiles are dashed lines. Right: The original nonparametric function estimate of Age from Figure 7.8 – the mean cubic parametric function estimate of Age (left picture) is drawn as a dotted line.

Parameter Mean Std. 10% Mode 90%

dev. quantile quantile

Factor loadings 1. System λ11 0.849 0.018 0.826 0.849 0.873 2. Initiative λ₂₁ -0.893 0.015 -0.913 -0.893 -0.873 3. Retirement λ31 1.457 0.028 1.422 1.456 1.494 4. Emergency λ41 0.690 0.012 0.674 0.690 0.706 5. Health λ51 0.970 0.017 0.949 0.969 0.991 Parametric indirect effects

RegPar-South γ1 0.134 0.013 0.117 0.134 0.150

RegPar-East γ₂ -0.115 0.015 -0.134 -0.115 -0.097

Table 7.21: Results of the approximated model of M4a – estimates of factor loadings and parametric effects for the parametric spatial analysis of covariate RegPar.