AStA Advances in Statistical Analysis

, Volume 100, Issue 3, pp 289–311 | Cite as

Modeling body height in prehistory using a spatio-temporal Bayesian errors-in variables model

  • Marcus GroßEmail author
Original Paper


Body height is commonly employed as a proxy variable for living standards among human populations. In the following, the human standard of living in prehistory will be examined using body height as reconstructed through long bone lengths. The aim of this paper is to model the spatial dispersion of body height over the course of time for a large archeological long bone dataset. A major difficulty in the analysis is the fact that some variables in the data are measured with uncertainty, like the date, the sex and the individual age of the available skeletons. As the measurement error processes are known in this study, it is possible to correct this using so-called errors-in-variables models. Motivated by this dataset, a Bayesian additive mixed model with errors-in-variables is proposed, which fits a global spatio-temporal trend using a tensor product spline approach, a local random effect for the archeological sites and corrects for mismeasurement and misclassification of covariates. In application to the data, the model reveals long-term spatial trends in prehistoric living standards.


Errors-in-variables Measurement error Misclassification Additive mixed models Bayesian methods  Nonparametric regression Tensor product splines Prehistoric living standard 



This work originated from the LiVES project, which is funded by the Emmy-Noether-Program of the German Research Foundation (DFG). The author gratefully acknowledges Eva Rosenstock, the principal investigator of the LiVES project, for access to the data collection and giving her advice on archaeology. Moreover, thanks are due to Alisa Hujic for her counselation in anthropological issues. The author is also grateful to the editor and referees for helpful comments which led to an improved manuscript.


  1. Angel, J.: Health as a crucial factor in the changes from hunting to developed farming in the eastern mediterranean. In: Cohen, M., Armelagos, G. (eds.) Palaeopathology at the Origins of Agriculture. Academic Press, New York (1984)Google Scholar
  2. Bach, H.: Zur Berechnung der Körperhöhe aus den langen Gliedmaßenknochen weiblicher Skelette. Anthropol. Anz. 29, 12–21 (1965)Google Scholar
  3. Bennicke, P.: Palaeopathology of Danish Skeletons. A comparative study of demography, disease and injury. Ph.D. thesis, University Copenhagen (1985)Google Scholar
  4. Berkson, J.: Are there two regressions? J. Am. Stat. Assoc. 45(250), 164–180 (1950)CrossRefzbMATHGoogle Scholar
  5. Berry, S., Carroll, R., Ruppert, D.: Bayesian smoothing and regression splines for measurement error problems. J. Am. Stat. Assoc. 97(457), 160–169 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bishop, C.: Pattern Recognition and Machine Learning. Information Science and Statistics. Springer, New York (2006)Google Scholar
  7. Blackwell, M., Honaker, J., King, G.: A unified approach to measurement error and missing data: details and extensions. Technical report, Harvard University OpenScholar (2014)Google Scholar
  8. Breitinger, : Zur Berechnung der Körperhöhe aus den langen Gliedmaßenknochen. Anthropol. Anz. 14, 249–274 (1938)Google Scholar
  9. Buonaccorsi, J.: Measurement Error: Models, Methods, and Applications. Chapman & Hall/CRC Interdisciplinary Statistics. Taylor & Francis, Boca Raton, FL (2010)Google Scholar
  10. Burger, J., Kirchner, M., Bramanti, B., Haak, W., Thomas, M.G.: Absence of the lactase-persistence-associated allele in early neolithic europeans. Proc. Natl. Acad. Sci. 104(10), 3736–3741 (2007)CrossRefGoogle Scholar
  11. Carroll, R., Ruppert, D., Stefanski, L., Crainiceanu, C.: Measurement Error in Nonlinear Models: A Modern Perspective, 2nd edn. Chapman & Hall, London (2006)CrossRefzbMATHGoogle Scholar
  12. Carroll, R.J., Delaigle, A., Hall, P.: Non-parametric regression estimation from data contaminated by a mixture of berkson and classical errors. J. Royal Stat. Soc.: Ser. B (Stat. Methodol.) 69(5), 859–878 (2007)MathSciNetCrossRefGoogle Scholar
  13. Carroll, R.J., Maca, J.D., Ruppert, D.: Nonparametric regression in the presence of measurement error. Biometrika 86(3), 541–554 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Celeux, G., Forbes, F., Robert, C.P., Titterington, D.M., et al.: Deviance information criteria for missing data models. Bayesian Anal. 1(4), 651–673 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Cook, J.R., Stefanski, L.A.: Simulation-extrapolation estimation in parametric measurement error models. J. Am. Stat. Assoc. 89(428), 1314–1328 (1994)CrossRefzbMATHGoogle Scholar
  16. Cressie, N., Wikle, C.: Statistics for Spatio-Temporal Data. Wiley, Hoboken (2011)zbMATHGoogle Scholar
  17. Currie, I.D., Durban, M., Eilers, P.H.: Smoothing and forecasting mortality rates. Stat. Model. 4(4), 279–298 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Delaigle, A., Fan, J., Carroll, R.J.: A design-adaptive local polynomial estimator for the errors-in-variables problem. J. Am. Stat. Assoc. 104(485), 348–359 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Delaigle, A., Hall, P., Qiu, P.: Nonparametric methods for solving the berkson errors-in-variables problem. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 68(2), 201–220 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Delaigle, A., Meister, A.: Nonparametric regression estimation in the heteroscedastic errors-in-variables problem. J. Am. Stat. Assoc. 102(480), 1416–1426 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Eilers, P.H., Marx, B.D.: Flexible smoothing with b-splines and penalties. Stat. Sci. 11(2), 89–121 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Fahrmeir, L., Kneib, T., Lang, S., Marx, B.: Regression: Models, Methods and Applications. Springer, Heidelberg (2013)Google Scholar
  23. Fan, J., Truong, Y.K.: Nonparametric regression with errors in variables. Ann. Stat. 21(4), 1900–1925 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Floud, R., Fogel, R.W., Harris, B., Hong, S.C.: The Changing Body: Health, Nutrition, and Human Development in the Western World since 1700. Cambridge University Press, New York (2011)CrossRefGoogle Scholar
  25. Formicola, V.: Stature reconstruction from long bones in ancient population samples: an approach to the problem of its reliability. Am. J. Phys. Anthropol. 90(3), 351–358 (1993)CrossRefGoogle Scholar
  26. Frost, C., Thompson, S.G.: Correcting for regression dilution bias: comparison of methods for a single predictor variable. J. R. Stat. Soc.: Ser. A (Stat. Soc.) 163(2), 173–189 (2000)CrossRefGoogle Scholar
  27. Fuller, W.A.: Measurement Error Models. Wiley, New York (1987)Google Scholar
  28. Gelfand, A., Diggle, P., Guttorp, P., Fuentes, M.: Handbook of Spatial Statistics. Chapman & Hall/CRC Handbooks of Modern Statistical Methods. Taylor & Francis, Boca Raton, FL (2010)Google Scholar
  29. Gelman, A., Rubin, D.B.: Inference from iterative simulation using multiple sequences. Stat. Sci. 7(4), 457–472 (1992)CrossRefGoogle Scholar
  30. Gustafson, P.: Measurement Error and Misclassification in Statistics and Epidemiology: Impacts and Bayesian Adjustments. CRC Press, New York (2003)Google Scholar
  31. Gustafson, P.: On model expansion, model contraction, identifiability and prior information: two illustrative scenarios involving mismeasured variables. Stat. Sci. 2, 111–137 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  32. Higdon, R., Schafer, D.W.: Maximum likelihood computations for regression with measurement error. Comput. Stat. Data Anal. 35(3), 283–299 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  33. Jaeger, U., Bruchhaus, H., Finke, L., Kromeyer-Hauschild, K., Zellner, K.: Säkularer Trend bei der Körperhöhe seit dem Neolithikum. Anthropol. Anz. 56(2), 117–130 (1998)Google Scholar
  34. Kass, R.E., Wasserman, L.: The selection of prior distributions by formal rules. J. Am. Stat. Assoc. 91(435), 1343–1370 (1996)CrossRefzbMATHGoogle Scholar
  35. Kemkes-Grottenthaler, A.: Das Frauendefizit archäologischer Serien—ein paläodemographisches Paradoxon? Anthropol. Anz. 55(3/4), 265–280 (1997)Google Scholar
  36. Koepke, N., Baten, J.: The biological standard of living in Europe during the last two millennia. Eur. Rev. Econ. History 9(1), 61–95 (2005)CrossRefGoogle Scholar
  37. Koepke, N., Baten, J.: Agricultural specialization and height in ancient and medieval Europe. Explor. Econ. History 45(2), 127–146 (2008)CrossRefGoogle Scholar
  38. Komlos, J.: Nutrition and Economic Development in the Eighteenth-Century Habsburg Monarchy: an Anthropometric History. Princeton University Press, Princeton (1989)CrossRefGoogle Scholar
  39. Komlos, J.: The Biological Standard of Living in Europe and America, 1700–1900: Studies in Anthropometric History. Variorum Press, Aldershot, England (1995)Google Scholar
  40. Krüttli, A., Bouwman, A., Akgül, G., Della Casa, P., Rühli, F., Warinner, C.: Ancient dna analysis reveals high frequency of european lactase persistence allele (t-13910) in medieval central europe. PloS One 9(1), e86251 (2014)CrossRefGoogle Scholar
  41. Lang, S., Brezger, A.: Bayesian p-splines. J. Comput. Graph. Stat. 13(1), 183–212 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  42. Lang, S., Sunder, M.: Non-parametric regression with BayesX: a flexible estimation of trends in human physical stature in 19th century America. Econ. Hum. Biol. 1(1), 77–89 (2003)CrossRefGoogle Scholar
  43. Larsen, C.S.: Biological changes in human populations with agriculture. Annu. Rev. Anthropol. 24, 185–213 (1995)CrossRefGoogle Scholar
  44. Lee, D.-J., Durbán, M.: P-spline ANOVA-type interaction models for spatio-temporal smoothing. Stat. Model. 11(1), 49–69 (2011)MathSciNetCrossRefGoogle Scholar
  45. Muff, S., Riebler, A., Held, L., Rue, H., Saner, P.: Bayesian analysis of measurement error models using integrated nested laplace approximations. J. R. Stat. Soc.: Ser. C (Appl. Stat.) 64(2), 231–252 (2015)MathSciNetCrossRefGoogle Scholar
  46. Mummert, A., Esche, E., Robinson, J., Armelagos, G.J.: Stature and robusticity during the agricultural transition: Evidence from the bioarchaeological record. Econ. Hum. Biol. 9(3), 284–301 (2011)CrossRefGoogle Scholar
  47. Pearson, K.: Mathematical contributions to the theory of evolution. V. On the reconstruction of the stature of prehistoric races. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character 192, 169–244 (1899)Google Scholar
  48. Perscheid, M.: Das Mainzer Lochkartenarchiv fur postkraniales Skelettmaterial prähistorischer Populationen. Homo 25(2), 121–124 (1974)Google Scholar
  49. Pham, T.H., Ormerod, J.T., Wand, M.: Mean field variational bayesian inference for nonparametric regression with measurement error. Comput. Stat. Data Anal. 68, 375–387 (2013)MathSciNetCrossRefGoogle Scholar
  50. Development Core Team, R.: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2008)Google Scholar
  51. Richardson, S., Gilks, W.R.: A bayesian approach to measurement error problems in epidemiology using conditional independence models. Am. J. Epidemiol. 138(6), 430–442 (1993)Google Scholar
  52. Rosenstock, E.: Eiweißversorgung und Körperhöhe: zur Übertragbarkeit anthropometrischer Ansätze auf die Archäologie. In: Schier, W., Meyer, M. (eds.) Vom Nil bis an die Elbe. Forschungen aus fünf Jahrzehnten am Institut für Prähistorische Archäologie der Freien Universität Berlin.Internationale Archäologie, Studia Honoraria 36. Rahden/Westfalen, (2014)Google Scholar
  53. Rosenstock, E., Groß, M., Hujic, A., and Scheibner, A.: Back to good shape: biological standard of living in the copper and bronze ages and the possible role of food. In: Kneisel, J., Kierleis, W., Taylor, N., dal Corso, M., Tiedtke, V. (eds.), Setting the Bronze Age Table: Production, Subsistence, Diet and Their Implications for European Landscapes. Proceedings of the International Workshop “Socio-environmental dynamics over the last 12.000 years: the creation of landscapes III (5th - 18th April 2013)”, Kiel. Habelt: Bonn, in press (2015)Google Scholar
  54. Rue, H., Held, L.: Gaussian Markov Random Fields: Theory and Applications. Chapman & Hall/CRC Monographs on Statistics & Applied Probability. CRC Press (2005)Google Scholar
  55. Ruppert, D., Wand, P., Carroll, R.: Semiparametric Regression. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, London (2003)CrossRefGoogle Scholar
  56. Schafer, D.W.: Semiparametric maximum likelihood for measurement error model regression. Biometrics 57(1), 53–61 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  57. Scheibner, A.: Prähistorische Ernährung in Vorderasien und Europa: eine Synopse der Quellen. Ph.D. thesis, Freie Universität Berlin (2015)Google Scholar
  58. Schier, W.: Extensiver Brandfeldbau und die Ausbreitung der neolithischen Wirtschaftsweise in Mitteleuropa und Südskandinavien am Ende des 5. Jahrtausends v. Chr. Praehistorische Z. 84(1), 15–43 (2009)Google Scholar
  59. Siegmund, F.: Die Körpergrösse der Menschen in der Ur- und Frühgeschichte Mitteleuropas und ein Vergleich ihrer anthropologischen Schätzmethoden. Books on Demand, Norderstedt 2010 (2011)Google Scholar
  60. Silventoinen, K.: Determinants of variation in adultbody height. J. Biosoc. Sci. 35, 263–285 (2003)CrossRefGoogle Scholar
  61. Spiegelhalter, D.J., Best, N.G., Carlin, B.P., van der Linde, A.: Bayesian measures of model complexity and fit. J. Royal Stat. Soc.: Ser. B (Stat. Methodol.) 64(4), 583–639 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  62. Spiegelman, D., McDermott, A., Rosner, B.: Regression calibration method for correcting measurement-error bias in nutritional epidemiology. Am. J. Clin. Nutr. 65(4), 1179–1186 (1997)Google Scholar
  63. Steckel, R.H.: Stature and the standard of living. J. Econ. Literature 33(4), 1903–1940 (1995)Google Scholar
  64. Steckel, R.H.: Research project: A history of health in Europe from the late paleolithic era to the present. Econ. Hum. Biol. 1(1), 139–142 (2003)CrossRefGoogle Scholar
  65. Taupin, M.-L.: Semi-parametric estimation in the nonlinear structural errors-in-variables model. Annal. Stat. 29(1), 66–93 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  66. Trotter, M., Gleser, G.: Estimation of stature from long bones of American Whites and Negroes. Am. J. Phys. Anthropol. 10(4), 463 (1952)CrossRefGoogle Scholar
  67. Weedon, M.N., Frayling, T.M.: Reaching new heights: insights into the genetics of human stature. Trends Genetics 24(12), 595–603 (2008)CrossRefGoogle Scholar
  68. Wood, S.N.: Thin plate regression splines. J. R. Stat. Soc.: Ser. B (Stat. Methodol.) 65(1), 95–114 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  69. Wood, S.N.: Low-rank scale-invariant tensor product smooths for generalized additive mixed models. Biometrics 62(4), 1025–1036 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  70. Wood, S.N.: Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. J. R. Stat. Soc. (B) 73(1), 3–36 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Freie Universität BerlinBerlinGermany

Personalised recommendations