# Regression analysis with partially labelled regressors: carbon dating of the Shroud of Turin

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## Abstract

The twelve results from the 1988 radio carbon dating of the Shroud of Turin show surprising heterogeneity. We try to explain this lack of homogeneity by regression on spatial coordinates. However, although the locations of the samples sent to the three laboratories involved are known, the locations of the 12 subsamples within these samples are not. We consider all 387,072 plausible spatial allocations and analyse the resulting distributions of statistics. Plots of robust regression residuals from the forward search indicate that some sets of allocations are implausible. We establish the existence of a trend in the results and suggest how better experimental design would have enabled stronger conclusions to have been drawn from this multi-centre experiment.

## Keywords

Computer-intensive method Forward search Robust statistics Simulation envelope## References

- Abraham, B., Box, G.E.P.: Linear models and spurious observations. Appl. Stat.
**27**, 131–138 (1978) MathSciNetMATHCrossRefGoogle Scholar - Atkinson, A.C., Riani, M.: Robust Diagnostic Regression Analysis. Springer, New York (2000) MATHCrossRefGoogle Scholar
- Atkinson, A.C., Riani, M., Cerioli, A.: The forward search: theory and data analysis (with discussion). J. Korean Stat. Soc.
**39**, 117–134 (2010). doi: 10.1016/j.jkss.2010.02.007 MathSciNetCrossRefGoogle Scholar - Bailey, R.A., Nelson, P.R.: Hadamard randomization: a valid restriction of random permuted blocks. Biom. J.
**45**, 554–560 (2003) MathSciNetCrossRefGoogle Scholar - Ballabio, G.: (2006). New statistical analysis of the radiocarbon dating of the Shroud of Turin. Unpublished manuscript. See http://www.shroud.com/pdfs/doclist.pdf
- Box, G.E.P.: Non-normality and tests on variances. Biometrika
**40**, 318–335 (1953) MathSciNetMATHGoogle Scholar - Box, G.E.P., Hunter, W.G., Hunter, J.S.: Statistics for Experimenters. Wiley, New York (1978) MATHGoogle Scholar
- Buck, C.E., Blackwell, P.G.: Formal statistical models for estimating radiocarbon calibration curves. Radiocarbon
**46**, 1093–1102 (2004) Google Scholar - Christen, J.A.: Summarizing a set of radiocarbon determinations: a robust approach. Appl. Stat.
**43**, 489–503 (1994) CrossRefGoogle Scholar - Christen, J.A., Sergio Perez, E.: A new robust statistical model for radiocarbon data. Radiocarbon
**51**, 1047–1059 (2009) Google Scholar - Damon, P.E., Donahue, D.J., Gore, B.H., et al.: Radio carbon dating of the Shroud of Turin. Nature
**337**, 611–615 (1989) CrossRefGoogle Scholar - Fanti, G., Botella, J.A., Di Lazzaro, P., Heimburger, T., Schneider, R., Svensson, N.: Microscopic and macroscopic characteristics of the Shroud of Turin image superficiality. J. Imaging Sci. Technol.
**54**, 040201 (2010) CrossRefGoogle Scholar - Freer-Waters, R.A., Jull, A.J.T.: Investigating a dated piece of the Shroud of Turin. Radiocarbon
**52**, 1521–1527 (2010) Google Scholar - Müller, W.G.: Collecting Spatial Data, 3rd edn. Springer, Berlin (2007) MATHGoogle Scholar
- Ramsey, C.B.: Dealing with outliers and offsets in radiocarbon data. Radiocarbon
**51**, 1023–1045 (2009) Google Scholar - Reimer, P.J., Baillie, M.G.L., Bard, E., et al.: INTCAL04 terrestrial radiocarbon age calibration. Radiocarbon
**46**, 1029–1058 (2004) Google Scholar - Rousseeuw, P.J.: Least median of squares regression. J. Am. Stat. Assoc.
**79**, 871–880 (1984) MathSciNetMATHCrossRefGoogle Scholar - Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Stat. Sci.
**4**, 409–435 (1989) MathSciNetMATHCrossRefGoogle Scholar - Walsh, B.: The 1988 Shroud of Turin radiocarbon tests reconsidered. In: Walsh, B. (ed.) Proceedings of the 1999 Shroud of Turin International Research Conference, Richmond, Virginia, USA, pp. 326–342. Magisterium Press, Glen Allen (1999) Google Scholar