Bayesian Inference of Spatially Correlated Binary Data Using Skew-Normal Latent Variables with Application in Tooth Caries Analysis

Abstract

The analysis of spatially correlated binary data observed on lattices is an interesting topic that catches the attention of many scholars of different scientific fields like epidemiology, medicine, agriculture, biology, geology and geography. To overcome the encountered difficulties upon fitting the autologistic regression model to analyze such data via Bayesian and/or Markov chain Monte Carlo (MCMC) techniques, the Gaussian latent variable model has been enrolled in the methodology. Assuming a normal distribution for the latent random variable may not be realistic and wrong, normal assumptions might cause bias in parameter estimates and affect the accuracy of results and inferences. Thus, it entails more flexible prior distributions for the latent variable in the spatial models. A review of the recent literature in spatial statistics shows that there is an increasing tendency in presenting models that are involving skew distributions, especially skew-normal ones. In this study, a skew-normal latent variable modeling was developed in Bayesian analysis of the spatially correlated binary data that were acquired on uncorrelated lattices. The proposed methodology was applied in inspecting spatial dependency and related factors of tooth caries occurrences in a sample of students of Yasuj University of Medical Sciences, Yasuj, Iran. The results indicated that the skew-normal latent variable model had validity and it made a decent criterion that fitted caries data.

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Afroughi, S. (2015) Bayesian Inference of Spatially Correlated Binary Data Using Skew-Normal Latent Variables with Application in Tooth Caries Analysis. Open Journal of Statistics, 5, 127-139. doi: 10.4236/ojs.2015.52016.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Allard, D. and Naveau, P. (2007) A New Spatial Skew-Normal Random Field Model. Communications in Statistics— Theory and Methods, 36, 1821-1834.
http://dx.doi.org/10.1080/03610920601126290
[2] Allard, D. and Soubeyrand, S. (2012) Skew-Normality for Climatic Data and Dispersal Models for Plant Epidemiology: When Application Fields Drive Spatial Statistics. Spatial Statistics, 1, 50-64.
http://dx.doi.org/10.1016/j.spasta.2012.03.001
[3] Hosseini, F., Eidsvik, J. and Mohammadzadeh, M. (2011) Approximate Bayesian Inference in Spatial GLMM with Skew Normal Latent Variables. Computational Statistics and Data Analysis, 55, 1791-1806.
http://dx.doi.org/10.1016/j.csda.2010.11.011
[4] Hughes, J., Haran, M. and Caragea, P.C. (2011) Autologistic Models for Binary Data on a Lattice. Environmetrics, 22, 857-871.
http://dx.doi.org/10.1002/env.1102
[5] Karimi, O. and Mohammadzadeh, M. (2012) Bayesian Spatial Regression Models with Closed Skew Normal Correlated Errors and Missing Observations. Statistical Papers, 53, 205-218.
http://dx.doi.org/10.1007/s00362-010-0329-2
[6] Kim, H.M. and Mallick, B.K. (2004) A Bayesian Prediction Using the Skew Gaussian Distribution. Journal of Statitical Planning and Inference, 120, 85-101.
http://dx.doi.org/10.1016/S0378-3758(02)00501-3
[7] Khormi, H.M. and Kumar, L. (2012) The Importance of Appropriate Temporal and Spatial Scales for Dengue Fever Control and Management. Science of Total Environment, 430, 144-149.
http://dx.doi.org/10.1016/j.scitotenv.2012.05.001
[8] March, D., Alós, J., Cabanellas-Reboredo, M., Infantes, E., Jordi, A. and Palmer, M. (2013) A Bayesian Spatial Approach for Predicting Seagrass Occurrence. Estuarine, Coastal and Shelf Science, 131, 206-212.
http://dx.doi.org/10.1016/j.ecss.2013.08.009
[9] BO, Y.-C. Song, C., Wang, J.-F. and Li, X.-W. (2014) Using an Autologistic Regression Model to Identify Spatial Risk Factors and Spatial Risk Patterns of Hand Foot and Mouth Disease (HFMD) in Mainland China. BMC Public Health, 14, 358.
[10] Thanapongthram, W., Linard, C., Pamaranon, N., Kawkalong, S., Noimoh, T., Chanachai, K., Parakgamawongsa, T. and Gilbert, M. (2014) Spatial Edidemiology of Porcine Reproductive and Respiratory Syndrome in Thailand. BMC Veterinary Research, 10, 174.
http://dx.doi.org/10.1186/s12917-014-0174-y
[11] Mattsson, B.J., Zipkin, E.F., Gardner, B., Blank, P.J., Sauer, J.R. and Role, J.A. (2013) Explaining Local-Scale Species Distributions: Relative Contributions of Spatial Autocorrelation and Landscape Heterogeneity for an Avian Assemblage. PLoS ONE, 8, e55097.
http://dx.doi.org/10.1371/journal.pone.0055097
[12] Afroughi, S., Faghihzadeh, S., Jafari Khaledi, M., Ghandehari Motlagh, M. and Hajizadeh, E. (2011) Analysis of Clustered Spatially Correlated Binary Data Using Autologistic Model and Bayesian Method with an Application to Dental Caries of 3 - 5-Year-Old Children. Journal of Applied Statistics, 38, 2763-2774.
http://dx.doi.org/10.1080/02664763.2011.570315
[13] Afroughi, S., Ghandehari Motlagh, M., Faghihzadeh, S. and Jafari Khaledi, M. (2013) A Model for Analyzing Spatially Correlated Binary Data Clustered in Uncorrelated Lattices. Statistical Methodology, 14, 1-14.
http://dx.doi.org/10.1016/j.stamet.2013.01.004
[14] Besag, J.E. (1972) Nearest-Neighbor Systems and the Auto-Logistic Model for Binary Data. Journal of the Royal Statistical Society. Series B, 34, 75-83.
[15] Besag, J.E. (1974) Spatial Interaction and the Statistical Analysis of Lattice Systems (with Discussion). Journal of the Royal Statistical Society. Series B, 36, 192-236.
[16] Genton, M.G., He, L. and Liu, X. (2001) Moments of Skew Normal Random Vectors and Their Quadratic Forms. Statistics & Probability Letters, 51, 319-325.
http://dx.doi.org/10.1016/S0167-7152(00)00164-4
[17] Gumpertz, M.L., Graham, J.M. and Ristaino J.B. (1997) Autologistic Model of Spatial Pattern of Phytophthora Epidemic in Bell Pepper: Effects of Soil Variables on Disease Presence. Journal of Agricultural, Biological and Environmental Statistics, 2, 131-156.
http://dx.doi.org/10.2307/1400400
[18] He, F.L., Zhou, J. and Zhu, H.T. (2003) Autologistic Regression Model for the Distribution of Vegetation. Journal of Agricultural, Biological and Environmental Statistics, 8, 205-222.
http://dx.doi.org/10.1198/1085711031508
[19] Huffer, F.W. and Wu, H.L. (1998) Markov Chain Monte Carlo for Autologistic Regression Models with Application to the Distribution of Plant Species. Biometrics, 54, 509-524.
http://dx.doi.org/10.2307/3109759
[20] Kaiser, M.S. and Caragea, P.C. (2009) Exploring Dependence with Data on Spatial Lattices. Biometrics, 65, 857-865.
http://dx.doi.org/10.1111/j.1541-0420.2008.01118.x
[21] Sherman, M., Apanasovich, T.V. and Carroll, R.J. (2006) On Estimation in Binary Autologistic Spatial Models. Journal of Statistical Computation and Simulation, 76, 167-179.
http://dx.doi.org/10.1080/00949650412331320873
[22] Caragea, P.C. and Kaiser, M.S. (2009) Autologistic Models with Interpretable Parameters. Journal of Agricultural, Biological, and Environmental Statistics, 14, 281-300.
http://dx.doi.org/10.1198/jabes.2009.07032
[23] Besag, J.E. (1975) Statistical Analysis of Non-Lattice Data. The Statistician, 24, 179-195.
http://dx.doi.org/10.2307/2987782
[24] Besag, J.E. (1993) Towards Bayesian Image Analysis. Journal of Applied Statistics, 20, 107-119.
http://dx.doi.org/10.1080/02664769300000061
[25] Besag, J.E., York, J. and Molli, A. (1991) Bayesian Image Restoration with Two Applications in Spatial Statistics. Annals of the Institute of Statistical Mathematics, 43, 1-20.
http://dx.doi.org/10.1007/BF00116466
[26] De Oliveira, V. (2000) Bayesian Prediction of Clipped Gaussian Random Fields. Computational Statistics & Data Analysis, 34, 299-314.
http://dx.doi.org/10.1016/S0167-9473(99)00103-6
[27] Flecher, C., Allard, D. and Naveau, P. (2010) Truncated Skew-Normal Distribution: Moments, Estimation by Weighted Moments and Application to Climatic Data. METRON, 68, 331-345.
http://dx.doi.org/10.1007/BF03263543
[28] Jara, A., Quintana, F. and Martin, E.S. (2008) Linear Mixed Models with Skew-Elliptical Distributions: A Bayesian Approach. Computational Statistics and Data Analysis, 52, 5033-5045.
http://dx.doi.org/10.1016/j.csda.2008.04.027
[29] Zhang, H. and El-Shaarawi, A. (2010) On Spatial Skew-Gaussian Processes and Applications. Environmetrics, 21, 33-47.
[30] Arellano-Valle, R.B., Ozan, S., Bolfarine, H. and Lachos, V.H. (2005) Skew Normal Measurement Error Models. Journal of Multivariate Analysis, 96, 265-281.
http://dx.doi.org/10.1016/j.jmva.2004.11.002
[31] Mohammadzadeh, M. and Hosseini, F. (2011) Maximum Likelihood Estimation for Spatial GLM Models. Procedia Environmental Sciences, 3, 63-68.
http://dx.doi.org/10.1016/j.proenv.2011.02.012
[32] Arellano-Valle, R.B., Bolfarine, H. and Lochos, V.H. (2005) Skew-Normal Linear Mixed Models. Journal of Data Science, 3, 415-438.
[33] Kim, H.M., Ha, E. and Mallick, B.K. (2004) Spatial Prediction of Rainfall Using Skew-Normal Processes. In: Genton, M.G., Ed., Skew-Elliptical Distributions and Their Applications: A Journey beyond Normality, Chapter 16, Chapman & Hall/CRC, London, 279-289.
http://dx.doi.org/10.1201/9780203492000.ch16
[34] Kim, H.M. and Mallick, B.K. (2002) Analyzing Spatial Data Using Skew-Gaussian Processes. In: Lawson, A. and Deninson, D., Eds., Spatial Cluster Modelling, Chapman & Hall/CRC, London, 164.
http://dx.doi.org/10.1201/9781420035414.ch9
[35] Lin, T.I. (2009) Maximum Likelihood Estimation for Multivariate Skew-Normal Mixture Models. Journal of Multivariate Analysis, 100, 257-265.
http://dx.doi.org/10.1016/j.jmva.2008.04.010
[36] Asili, S., Rezaei, S. and Najjar, L. (2014) Using Skew-Logistic Probability Density Function as a Model for Age-Spe- cific Fertility Rate Pattern. BioMed Research International, 2014, Article ID: 790294.
http://dx.doi.org/10.1155/2014/790294
[37] Azzalini, A. (1985) A Class of Distributions Which Includes the Normal Ones. Scandinavian Journal of Statistics, 12, 171-178.
[38] Azzalini, A. (1986) Further Results on a Class of Distributions Which Includes the Normal Ones. Statistica, 46, 199-208.
[39] Flecher, C., Naveau, P. and Allard, D. (2009) Estimating the Closed Skew-Normal Distributions Parameters Using Weighted Moments. Statistics & Probability Letters, 79, 1977-1984.
http://dx.doi.org/10.1016/j.spl.2009.06.004
[40] Gupta, A.K. and Chen, J.T. (2004) A Class of Multivariate Skew-Normal Models. Annals of the Institute of Statistical Mathematics, 56, 305-315.
http://dx.doi.org/10.1007/BF02530547
[41] Gupta, A.K., Nguyen, T.T. and Sanqui, J.A.T. (2004) Characterization of the Skew-Normal Distribution. Annals of the Institute of Statistical Mathematics, 56, 351-360.
http://dx.doi.org/10.1007/BF02530549
[42] Lin, T.I., Lee, J.C. and Yen, S.Y. (2007) Finite Mixture Modeling Using the Skew-Normal Distribution. Statistica Sinica, 17, 909-927.
[43] Azzalini, A. and Dalla Valle, A. (1996) The Multivariate Skew-Normal Distribution. Biometrika, 83, 715-726.
http://dx.doi.org/10.1093/biomet/83.4.715
[44] Azzalini, A. and Capitanio, A. (1999) Statistical Applications of the Multivariate Skew Normal Distribution. Journal of the Royal Statistical Society: Series B, 61, 579-602.
http://dx.doi.org/10.1111/1467-9868.00194
[45] Azzalini, A. (2005) The Skew-Normal Distribution and Related Multivariate Families. Scandinavian Journal of Statistics, 32, 159-188.
http://dx.doi.org/10.1111/j.1467-9469.2005.00426.x
[46] Liseo, B. and Loperfido, N. (2003) A Bayesian Interpretation of the Multivariate Skew-Normal Distribution. Statistics & Probability Letters, 61, 395-401.
http://dx.doi.org/10.1016/S0167-7152(02)00398-X
[47] Minozzo, M. and Ferracuti, L. (2012) On the Existence of Some Skew-Normal Stationary Processes. Chilean Journal of Statistics, 3, 157-170.
[48] Kim, H.M. and Mallick, B.K. (2005) A Bayesian Prediction Using the Elliptical and the Skew-Gaussian Processes. Technical Report.
http://citeseerx.ist.psu.edu/viewdoc/download?
doi=10.1.1.36.5432&rep=rep1&type=ps
[49] Bandyopadhyay, D., Lochos, V.H., Abanto-Valle, C.A. and Ghosh, P. (2010) Linear Mixed Models for Skew-Normal/ Independent Bivariate Responses with an Application to Periodontal Disease. Statistics in Medicine, 29, 2643-2655.
http://dx.doi.org/10.1002/sim.4031
[50] Lu, J., Gong, D., Shen, Y., Liu, M. and Chen, D. (2013) An Inversed Bayesian Modeling Approach for Estimating Nitrogen Export Coefficients and Uncertainty Assessment in an Agricultural Watershed in Eastern China. Agricultural Water Management, 116, 79-88.
http://dx.doi.org/10.1016/j.agwat.2012.10.015
[51] Wang, Z. and Zheng, Y. (2013) Analysis of Binary Data via a Centered Spatial-Temporal Autologistic Regression Model. Environmental and Ecological Statistics, 20, 37-57.
http://dx.doi.org/10.1007/s10651-012-0206-3
[52] Cressie, N. (1993) Statistics for Spatial Data. Revised Edition, Wiley, New York.
[53] Haining, R. (2003) Spatial Data Analysis: Theory and Practice. Cambridge University Press, London.
http://dx.doi.org/10.1017/CBO9780511754944
[54] Afroughi, S., Faghihzadeh, S., Jafari Khaledi, M. and Ghandehari Motlagh, M. (2010) Dental Caries Analysis in 3 - 5 Years Old Children: A Spatial Modeling. Archives of Oral Biology, 55, 374-378.
http://dx.doi.org/10.1016/j.archoralbio.2010.03.008
[55] Reich, B.J. and Bandyopadhyay, D. (2010) A Latent Factor Model for Spatial Data with Informative Missingness. Annals of Applied Statistics, 4, 439-459.
http://dx.doi.org/10.1214/09-AOAS278
[56] Ashour, S.K. and Abdel-Hameed, M.A. (2010) Approximate Skew-Normal Distribution. Journal of Advanced Research, 1, 341-350.
http://dx.doi.org/10.1016/j.jare.2010.06.004
[57] Liseo, B. and Parisi, A. (2013) Bayesian Inference for the Multivariate Skew-Normal Model: A Population Monte Carlo Approach. Computational Statistics & Data Analysis, 63, 125-138.
http://dx.doi.org/10.1016/j.csda.2013.02.007
[58] Figueiredo, F. and Gomes, I. (2013) The Skew-Normal Distribution in SPC. Statistical Journal, 11, 83-104.
[59] Robert, C.P. and Casella, G. (2004) Monte Carlo Statistical Methods. Springer, New York.
http://dx.doi.org/10.1007/978-1-4757-4145-2
[60] Horrace, W.C. (2005) Some Results on the Multivariate Truncated Normal Distribution. Journal of Multivariate Analysis, 94, 209-221.
http://dx.doi.org/10.1016/j.jmva.2004.10.007
[61] Robertson, H.T. and Allison, D.B. (2012) A Novel Generalized Normal Distribution for Human Longevity and Other Negatively Skewed Data. PLoS ONE, 7, e37025.
http://dx.doi.org/10.1371/journal.pone.0037025
[62] Mcdonald, R.E., Avery, D.R. and Dean, J.A. (2004) Dentistry for the Child and Adolescent. 8th Edition, Mosby, New York.
[63] Albert, J. (2007) Bayesian Computation with R. Springer, New York.
http://dx.doi.org/10.1007/978-0-387-71385-4
[64] Ward, E.J. (2008) A Review and Comparison of Four Commonly Used Bayesian and Maximum Likelihood Model Selection. Ecological Modelling, 211, 1-10.
http://dx.doi.org/10.1016/j.ecolmodel.2007.10.030
[65] Yiengprugsawan, V., Somkotra, T., Seubsman, S. and Sieigh, A.C. (2013) Longitudinal Associations between Oral Health Impacts and Quality of Life among a National Cohort of Thai Adults. Health and Quality of Life Outcomes, 11, 172.
http://dx.doi.org/10.1186/1477-7525-11-172

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