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Bayesian factor analysis for spatially correlated data: application to cancer incidence data in Scotland

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Abstract

A hierarchical Bayesian factor model for multivariate spatially correlated data is proposed. Multiple cancer incidence data in Scotland are jointly analyzed, looking for common components, able to detect etiological factors of diseases hidden behind the data. The proposed method searches factor scores incorporating a dependence within observations due to a geographical structure. The great flexibility of the Bayesian approach allows the inclusion of prior opinions about adjacent regions having highly correlated observable and latent variables. The proposed model is an extension of a model proposed by Rowe (2003a) and starts from the introduction of separable covariance matrix for the observations. A Gibbs sampling algorithm is implemented to sample from the posterior distributions.

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References

  • Aguilar O, West M (2000) Bayesian dynamic factor models and portfolio allocation. J Bus Econ Stat 18: 338–357

    Article  Google Scholar 

  • Anselin L (2001) Rao’s score test in spatial econometrics. J Stat Plan Inference 97: 113–139

    Article  MathSciNet  MATH  Google Scholar 

  • Anselin L (2007) Spatial econometrics in RSUE: retrospect and prospect. Reg Sci Urban Econ 37: 450–456

    Article  Google Scholar 

  • Arminger G, Muthén BO (1998) A Bayesian approach to nonlinear latent variable models using the Gibbs sampler and the metropolis-hastings algorithm. Psychometrika 63: 271–300

    Article  Google Scholar 

  • Besag J, York J, Mollie A (1991) Bayesian image restoration, with two applications in spatial statistics. Ann Inst Stat Math 43: 1–20

    Article  MathSciNet  MATH  Google Scholar 

  • Best N, Richardson S, Thomson A (2005) A comparison of Bayesian spatial models for disease mapping. Stat Methods Med Res 14: 35–59

    Article  MathSciNet  MATH  Google Scholar 

  • Breslow NE, Clayton DG (1993) Approximate inference in generalized linear mixed models. J Am Stat Assoc 88: 9–25

    Article  MATH  Google Scholar 

  • Brooks S, Gelman A (1998) Alternative methods for monitoring convergence of iterative simulations. J Comput Graph Stat 7: 434–455

    Article  MathSciNet  Google Scholar 

  • Burkitt DP (1969) Related disease-related cause?. Lancet 2: 1229–1231

    Article  Google Scholar 

  • Burkitt DP (1970) Relationship as a clue to causation. Lancet 2: 1237–1240

    Article  Google Scholar 

  • Cancer in Scotland (October 2010) Information Services Division, NHS, National Services Scotland

  • Christensen WF, Amemiya Y (2003) Modeling and prediction for multivariate spatial factor analysis. J Stat Plan Inference 115: 543–564

    Article  MathSciNet  MATH  Google Scholar 

  • Clayton D, Kaldor J (1987) Empirical bayes estimates of age- standardized relative risks for use in disease mapping. Biometrics 43: 671–681

    Article  Google Scholar 

  • Cressie N (1993a) Regional mapping of incidence rates using spatial Bayesian models. Med Care 31: 60–65

    Article  Google Scholar 

  • Cressie N (1993b) Statistics for spatial data. Wiley, New York

    Google Scholar 

  • Cressie N, Wikle CK (2001) Statistics for spatio-temporal data. Wiley, New York

    Google Scholar 

  • Cressie N, Calder K, Clark J, VerHoef J, Wikle CK (2009) Accounting for uncertainty in ecological analysis: the strengths and limitations of hierarchical statistical modeling. Ecol Appl 19: 553–557

    Article  Google Scholar 

  • Diggle PG, Moyeed RA, Tawn JA (1992) Model-based geo-statistics. Appl Stat 47: 299–350

    MathSciNet  Google Scholar 

  • Downing A, Forman D, Gilthorpe MS, Edwards KL, Manda SO (2008) Joint disease mapping using six cancers in the Yorkshire region of England. Int J Health Geogr 28: 7–41

    Google Scholar 

  • Fathalla MF (1971) Incessant ovulation-a factor in ovarian neoplasia?. Lancet 2(7716): 163

    Article  Google Scholar 

  • Fritschi L, Glassm DC, Tabrizi JS, Leavy JE, Ambrosini GL (2007) Occupational risk factors for prostate cancer and benign prostatic hyperplasia: a case-control study in Western Australia. Occup Environ Med 64(1): 60–65

    Article  Google Scholar 

  • Gilks WR, Best NG, Tan KKC (1995) Adaptive Rejection Metropolis Sampling within Gibbs Sampling. Appl Stat 44: 455–472

    Article  MATH  Google Scholar 

  • Grönberg H (2003) Prostate cancer epidemiology. Lancet 361: 859–864

    Article  Google Scholar 

  • Haining R, Grith D, Bennett R (1989) Maximum likelihood estimation with missing spatial data and with an application to remotely sensed data. Commun Stat Theory Methods 1875–1894

  • Hayashi K, Sen PK (2001) Bias-corrected estimator of factor loadings in Bayesian factor analysis. Educ Psychol Meas 62(6): 944–959

    Article  MathSciNet  Google Scholar 

  • Hogan JW, Tchernis R (2004) Bayesian factor analysis for spatially correlated data, with application to summarizing area-level material deprivation from census data. J Am Stat Assoc 99(466): 314–324

    Article  MathSciNet  MATH  Google Scholar 

  • Journel AG (1983) Geostatistics. Encyclopedia of statistical sciences 3: 424–431

    Google Scholar 

  • Knorr-Held L, Best NG (2001) A shared component model for detecting joint and selective clustering of two diseases. J Roy Stat Soc Ser A 164: 73–85

    Article  MathSciNet  MATH  Google Scholar 

  • Lawley DN (1940) The estimation of factor loadings by the method of maximum likelihood. Proc Roy Soc Edinb 60: 82–84

    MathSciNet  Google Scholar 

  • Lawson AB (2001) Statistical methods in spatial epidemiology. Wiley, New York

    MATH  Google Scholar 

  • Le N, Sun W, Zidek J (1997) Bayesian multivariate spatial interpolater with data missing by design. J Roy Stat Soc Ser B 59: 501–510

    Article  MathSciNet  MATH  Google Scholar 

  • Lee SE, Press SJ (1998) Robustness of Bayesian factor analys estimates. Commun Stat Theory Methods 27: 1871–1893

    Article  MathSciNet  MATH  Google Scholar 

  • Leorato S, Mezzetti M (2011) Bayesian spatial panel data. Tecnical Report Università “Tor Vergata”, Rome

  • Lopes H, West M (2004) Bayesian model assessment in factor analysis. Stat Sinica 14: 41–67

    MathSciNet  MATH  Google Scholar 

  • Martin JL, McDonald RP (1975) Bayesian estimation in unrestricted factor analysis. A treatment for Heywood cases. Psychometrika 40: 505–517

    Article  MATH  Google Scholar 

  • Mezzetti M, Billari FC (2005) Bayesian correlated factor analysis of socio-demographic indicators. Stat Methods Appl 14: 223–241

    Article  MathSciNet  MATH  Google Scholar 

  • Polasek W (1997) Factor analysis and outliers: a Bayesian approach. Discussion paper, University of Basel

  • Press SJ, Shigemasu K (1989) Bayesian inference in factor analysis In: Gleser L, Perlman M, Press SJ, Sampson A (eds) Contributions to probability and statistics: essays in honor of Ingram Olkin (Chap. 15). Springer, New York

  • Press SJ, Shigemasu K (1997) Bayesian inference in factor analysis-revised, with an appendix by Rowe, D.B. Technical report No. 243, Department of Statistics, University of California, Riverside

  • Rowe DB (2000) Factorization of separable and patterned covariance matrices for gibbs sampling. Monte Carlo Methods Appl 6(3): 205–210

    Article  MathSciNet  MATH  Google Scholar 

  • Rowe DB (2002) Jointly distributed mean and mixing coefficients for Bayesian source separation using MCMC and ICM. Monte Carlo Methods Appl 8(4): 395–403

    Article  MathSciNet  MATH  Google Scholar 

  • Rowe DB (2003a) Multivariate Bayesian statistics: models for source separation and signal unmixing. CRC Press, Boca Raton

    MATH  Google Scholar 

  • Rowe DB (2003b) On using the sample mean in Bayesian factor analysis. J Interdiscip Math 6(3): 319–329

    MathSciNet  MATH  Google Scholar 

  • Rowe DB, Press SJ (1998) Gibbs sampling and hill climbing in Bayesian factor analysis. Technical Report No. 255, Department of Statistics, University of California, Riverside

  • Scheiner SM, Gurevitch J (2001) Design and analysis of ecological experiments. 2. Oxford University Press, Oxford

    Google Scholar 

  • Spadea T, Zengarini N, Kunst A, Zanetti R, Rosso S, Costa G (2010) Cancer risk in relationship to different indicators of adult socioeconomic position in Turin, Italy. Cancer Causes Control 21(7):1117–1130

    Article  Google Scholar 

  • Srivastava MS, von Rosen T, von Rosen D (2008) Models with a Kronecker product covariance structure. Estim Test Math Methods Stat 17(4): 357–370

    Article  MATH  Google Scholar 

  • Tzala E, Best N (2008) Bayesian latent variable modelling of multivariate spatio-temporal variation in cancer mortality. Stat Methods Med Res 17: 97–118

    Article  MathSciNet  MATH  Google Scholar 

  • Waller LA, Gotway CA (2004) Applied spatial statistics for public health data. Wiley-InterScience, New York

    Book  MATH  Google Scholar 

  • Waller LA, Carlin BP, Xia H, Gelfand AE (1997) Hierarchical spatio-temporal mapping of disease rates. J Am Stat Assoc 92: 607–617

    Article  MATH  Google Scholar 

  • Wang F, Wall MM (2003) Generalized common spatial factor model. Biostatistics 4: 569–582

    Article  MATH  Google Scholar 

  • Webster R, Oliver MA (2001) Geostatistics for environmental scientists. Wiley, New York

    MATH  Google Scholar 

  • Wikle CK (2003) Hierarchical models in environmental science. Int Stat Rev 71: 181–199

    Article  MATH  Google Scholar 

  • Yanai H, Inaba Y, Takagi H, Toyokawa H, Yamamoto S (1978) An epidemiological study on mortality rates of various cancer sites during 1958–1971 by means of factor analysis. Behaviormetrica 5: 55–74

    Article  Google Scholar 

  • Yasui Y, Lele S (1997) A regression method for spatial disease rates: an estimating function approach. J Am Stat Assoc 92: 21–32

    Article  MATH  Google Scholar 

Download references

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Correspondence to Maura Mezzetti.

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Mezzetti, M. Bayesian factor analysis for spatially correlated data: application to cancer incidence data in Scotland. Stat Methods Appl 21, 49–74 (2012). https://doi.org/10.1007/s10260-011-0177-9

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  • DOI: https://doi.org/10.1007/s10260-011-0177-9

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