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Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept. 25-26, 2007. 1 Bayesian Geoadditive Latent Variable Models.

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Presentation on theme: "Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept. 25-26, 2007. 1 Bayesian Geoadditive Latent Variable Models."— Presentation transcript:

1 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses Ludwig Fahrmeir Department of Statistics, University of Munich, Germany. 1. Applications: - Internet survey ‘Prospect Germany 1‘ - Child morbidity and malnutrition in Nigeria - Post war security in Cambodia 2. Geoadditive latent variable models 3. MCMC inference based on auxiliary variables 4. Applications: Results

2 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Applications: Internet Survey “Prospect Germany 1“ (1) Survey initiated by McKinsey & Co., stern.de and t-online (2001) General goal: To find out in which areas of life are people willing to bear the responsibility, and in which areas are they expecting the state to take the responsibility. Data from a subsample of 6804 individuals. Most variables are binary or (ordered) categorical. Continuous covariate: age of participant (in years). Spatial information: 402 administrative districts in Germany. Analysis with ten indicators and two latent variables: First latent variable reflects the participant‘s attitude when social coverage would be one‘s own responsibility, or if the state has to take care of social coverage. Second latent variable reflects the ambition of the person to achieve something in their job and in society.

3 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Applications: Internet Survey “Prospect Germany 1“ (2) Table 1: Variable names, variable types, questions/statements, and response categories of the ten indicators.

4 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Applications: Internet Survey “Prospect Germany 1“ (3) Table 2: Variable names, variable types and response categories of the four indirect covariates used in the analyses.

5 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Applications: Child morbidity and malnutrition in Nigeria (1) Data: about 5000 children from DHS for Nigeria (2003) Goal: Assess impact of personal, socioeconomic and public health factors as well as spatial location on the latent variables „morbidity“ and „malnutrition“ of children. Responses / indicators: child had diarrhoea (or not), child had cough (or not), child had fever (or not) within two weeks before the interview. Malnutrition status of child measured through Z-score for stunting, wasting and underweight.

6 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Applications: Child morbidity and malnutrition in Nigeria (2) Covariates: age of child (in months) age of mother (in years) body mass index of mother Categorical covariates characterize socio-economic and public health environment. Spatial information: district of Nigeria where mother and child live.

7 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Applications: Post war security in Cambodia (1) Conflict and violence data collected by the monitoring arm of the Government of Cambodia‘s decentralization program SEILA (the Khmer word for foundation stone). We use data for 2002, obtained from headmen and leaders of over villages and urban neighbourhoods. More details on the data as well as sociological and political background is given in Benini, Owen and Rue (2006). They used separate geoadditive count data models to analyze the impact of the legacy of war, poverty and resource competition, urbanity, and governance quality on the three dependent variables - number of serious crimes committed in community, - number of land conflicts in community, - number of households in community known to have domestic violence problems.

8 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Applications: Post war security in Cambodia (2) We apply a Poisson indicator LVM to these three indicators, focussing on the latent variable „disposition for violence“. Instead of the total numbers of counts per year, we use the monthly averages y 1, y 2 and y 3 of the three count variables as target variables. Because the yearly numbers are only estimates provided by local leaders, the effect of averaging can be neglected. It helps to make data analysis computationally feasible..

9 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Applications: Post war security in Cambodia (3)

10 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Geoadditive latent variable models (1) 2.1 Measurement models for observable responses

11 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Geoadditive latent variable models (2) 2.2 Structural models for latent variables (1)

12 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Geoadditive latent variable models (3) 2.2 Structural models for latent variables (2)

13 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Geoadditive latent variable models (4) 2.3 Other priors

14 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , MCMC inference based on auxiliary variables (1)

15 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , MCMC inference based on auxiliary variables (2)

16 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , MCMC inference based on auxiliary variables (3)

17 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , MCMC inference based on auxiliary variables (4)

18 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Results: Internet Survey „Prospect Germany 1“ (1)

19 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Results: Internet Survey „Prospect Germany 1“ (2) Estimates of factor loadings and parametric indirect effects.

20 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Results: Internet Survey „Prospect Germany 1“ (3) Estimates of the smooth functions modelled by P-splines priors. The mean values are connected by the solid line, 10%- and 90%-quantiles are connected by the dashed line.

21 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Results: Internet Survey „Prospect Germany 1“ (4) Estimated Spatial effects Spatial effects for the first latent variable Regions with - a significant negative effect (red) - a significant positive effect (green) - a non-significant effect (yellow)

22 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Results: Internet Survey „Prospect Germany 1“ (5) Estimated Spatial effects Spatial effects for the second latent variable Regions with - a significant negative effect (red) - a significant positive effect (green) - a non-significant effect (yellow)

23 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Results: Child morbidity and malnutrition in Nigeria (1)

24 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Results: Child morbidity and malnutrition in Nigeria (2)

25 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Results: Child morbidity and malnutrition in Nigeria (3)

26 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Nonlinear effects for the first latent variable for Nigeria 4. Results: Child morbidity and malnutrition in Nigeria (4)

27 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Nonlinear effects for the second latent variable for Nigeria 4. Results: Child morbidity and malnutrition in Nigeria (5)

28 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Spatial effects for the first and second latent variable for Nigeria 4. Results: Child morbidity and malnutrition in Nigeria (6)

29 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Results: Post war security in Cambodia (1)

30 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Results: Post war security in Cambodia (2)

31 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Commune rate for land conflicts, rated to population (light: below average, dark: above average) 4. Results: Post war security in Cambodia (3)

32 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Map of Cambodia with the estimated spatial effects for all 1628 communities 4. Results: Post war security in Cambodia (4)

33 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Estimated effects: f1: f2: 4. Results: Post war security in Cambodia (5)

34 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , References Benini, A., Owen, T. and Rue, H. (2006). A semi-parametric spatial regression approach to post-war human security: Cambodia Technical Report. Fahrmeir, L. and Raach, A. (2007). A Bayesian semiparametric latent variable model for mixed responses. Psychometrica, in press. Fahrmeir, L., Steinert, S. (2006). A geoadditive Bayesian latent variable model for Poisson indicators. Discussion Paper 508, Sonderforschungsbereich 386. Ludwig-Maximilians-Universität München. Frühwirth, S., Frühwirth, R., Held, L. and Rue, H. (2007). Improved auxiliary mixture sampling for hierarchical models of non-Gaussian data. IFAS Research Paper Frühwirth, S. and Wagner, H. (2006). Auxiliary mixture sampling for parameter-driven models of time series of counts with application to state space modelling. Biometrika 93(4),

35 Bayesian Geoadditive Latent Variable Models for Discrete and Continuous Responses, Zurich, Sept , Thanks to Alexander Raach Sven Steinert Khaled Khatab The BayesX Group


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