1. SCIENZE CHIMICHE E SINDONE
Robust statistical analysis on the 14-C dating by the three official laboratories Marco Riani Director of the Interdepartmental Research Centre of Robust Statistics of the University of Parma
Joint work with A.C. Atkinson (LSE), F. Crosilla (Univ. of Udine), G. Fanti (Univ. of Padua)
SCIENZE CHIMICHE E SINDONE

2. Starting point: strip of material cut from one corner of the TS in 1988

3. Starting point: strip of material cut from one corner of the TS
Arizona 4 measurements (Only A1 or A1+A2)?
Oxford 3 measurements x2
Zurich 5 measurements x1
Notation
A1  Arizona dated only A1
A1+A2 Arizona dated both A1 and A2

4. The general model for observation j at site i is:
Estimated ages of the individual samples with calculated s.e. (source Nature)
The general model for observation j at site i is:

5. Question: are the µi all equal?
3 possibilities for the error structure
Unweighted analysis. Standard analysis of variance (ANOVA) with vij=1
Original weights. We weight the observations by 1/vij. (weighted ANOVA)
Modified weights for Arizona: keeping into account the fact that the S.E. for Arizona are roughly 2/3 of those for the other sites
Damon et al. (1988): “the errors include the statistical (counting) error, the scatter of results for standards and blanks, and the uncertainty in the Ó13C determination (Arizona includes the Ó13C error at a later stage, when combining subsample results)”.

6. Question: are the µi all equal?

7. Summary: are the µi all equal?
Unweighted analysis. Standard ANOVA
Original weights. Weighted ANOVA
Modified weights. Weighted ANOVA with correction for ARIZONA
Remark: ANOVA can correctly be applied only if the variances across the three sites are similar (homogeneous). Box test.

8. P-values of the results of the ANOVA test
Conclusions:
the variability is homogeneous among sites
the means are significantly different

9. In addition to the TS each laboratory dated 3 controls
Linen from Nubian tomb
An Egyptian mummy from Thebes
Threads from a cope from Var (France)
None of the datings of these samples was controversial
Question: the difference in means found for the TS can also be found for the 3 controls?

10. P-values of the results of the ANOVA test for the three controls
Conclusions:
the variability is homogeneous among sites
the means are not significantly different

11. Shroud sample and control samples
Homogeneity of variances
Heterogeneity of means
Control samples
Homogeneity of variances  No evidence of systematic differences between laboratories
Homogeneity of means  No evidence of heterogeneity in the means
Question: where does the source of egregious heterogeneity on the TS come from?

12. We now try to check if there is any trend in the age of the material
Spatial layout
We now try to check if there is any trend in the age of the material x1 x2 x2 x1
x1=horizontal (longitudinal) coordinate x2=vertical (lateral) coordinate

13. More formally we want to test whether the following regression model is true
y is 12×1 vector containing the raw dates obtained from the 3 laboratories. X is a 12 × 3 matrix containing respectively the intercept, the longitudinal (horizontal) coordinate (x1) and the lateral (vertical) coordinate (x2).

14. The structure of the regression model
In our case we have Y = β0 + β1 X1 + β2 X2 + ε Y = radiocarbon dating X1 = horizontal coordinate X2 = vertical coordinate ε = error term For example if β1 =0 we expect to find an estimated value of β1 which is centered around to zero and is not significant. In order to test the significance of X1 and X2 we use the t-statistics x1 x2

15. Recap about t-statistics
t-statistics are used to test the significance of corresponding regressor If the true value of the regression parameter β is equal to 0, then the sampling distribution of the t-statistic is the Student's t-distribution with 𝜈=(n − k) degrees of freedom, where n is the number of observations, and k is the number of regressors (including the intercept)

16. Problem: we do not know how the material was cut inside each laboratoryx1 x2

17. We consider 387072 regression models of the kind
Problem: we do not know how the material was cut inside each laboratory
Solution:
we consider all the possible ways in which the material could have been cut
We consider regression models of the kind
Y = β0 + β1 X1 + β2 X2 + ε

18. Arrangements investigated for the Oxford sample (24=3!×4)

19. Arrangements investigated for the Zurich sample (96=24×4)x2 x1 x2 x1 x2 x1

20. Arrangements investigated for Arizona (72=if Arizona only dated A1)x2 x1 x2 x1

21. Arrangements investigated for Arizona (96=if Arizona dated both A1 and A2)x2 x1 x2 x1

22. We consider 387072 possible bivariate regressions
For each regression we store the values of the t statistic of the horizontal (x1) and vertical coordinate (x2) How to interpret the numbers for the significance of x1? How to interpret the numbers for the significance of x2?
We have to produce a confidence band!

23. Ordered p-values for Vertical coordinate
Significance level of t-statistic from configurations and envelopes from 100 simulations of each configuration

24. Ordered p-values for horizontal coordinate
Significance level of t-statistic from configurations and envelopes from 100 simulations of each configuration

25. Histograms of the values of t-statistics from 387072 possible configurations for vertical coordinate
If the data were random the histogram of the t-statistics would be centered around 0

26. In all the 387072 we obtain a negative value of the t-statistic!!!
Histograms of the values of t-statistics from possible configurations for horizontal coordinate
In all the we obtain a negative value of the t-statistic!!!

27. An important point to remark
If there is no effect of the x1 coordinate we expect that roughly half of the values of the t-stat are positive and roughly half are negative
All the values of the t-stat ARE NEGATIVE

28. Conclusions and questions up to now
x2 is not significant but x1 is! Result is expected because the sample is long and (in x1) and thin (in x2) x1 x2
What gives rise to the two peaks for x1?

29. Projections of cases to consider
Whether Arizona analyzed A2 or not clearly has a large effect on the range of values of x1 and so on the t-statistic x1 x2
Reduction of cases:
For A1+A2 the original 96 cases become 52 For A1 the original 72 become 31
Just 83 possible configurations for Arizona.
The horizontal projection of the configurations gives possibilities

30. Arrangements investigated for Arizona (72=if Arizona only dated A1) Considering just the horizontal projection we have only 31 cases) x2 x1 x2 x1
C4,2=6 cases
1 case
4!=24 cases

31. Summary of the results
For each of the 83 possible horizontal configurations for Arizona there are 507 different ways to obtain another configuration for Oxford or Zurich We show the distribution of the t-statistics for regression only on x1 for each of these 83 configurations using boxplots

32. Recap on the boxplot
It is based on first quartile, median, third quartile and a rule for detecting the outliers

33. Recap on the boxplot
Example of histograms and boxplot and for normal data

34. Recap on the boxplot
Example of histograms and boxplots and for asymmetric data

35. Summary of the results
For each of the 83 possible horizontal configurations for Arizona there are 507 different ways to obtain another configuration for Oxford or Zurich In the plot of the next slide we present the boxplots of the t-statistics for regression only on x1 divided according to these 83 configurations

36. Boxplots of the distribution of the t-statistics for x1 for each of the 83 configurations of Arizona
On top of the boxplots for A1+A2 we put the value of y (radiocarbon dating) associated with x1=41 1 83
Index number

37. Question: which configurations are plausible?
Histograms of values of t-stat divided according to the value of y from A1+A2 associated with x1=41 x1 x2
Question: which configurations are plausible?

38. A1: Analysis of robust residuals for a representative configuration
LTS residuals

39. A1: Analysis of residuals for a representative configuration
Monitoring residuals plot

40. Monitoring residuals plot
A1+A2: Analysis of residuals for a representative configuration when x1=41 is associated with y=591
x1=41 and y=591 is a clear outlier
Monitoring residuals plot

41. A1+A2: Analysis of residuals for a representative configuration when x1=41 is associated with y=591
LTS robust residuals

42. Arizona dated only A1 or A1+A2?
The configurations with A1+A2 lead to detection of the presence of one or more outliers! Arizona is likely to have dated only A1!

43. Confirmation from Arizona researchers (two years after our analysis)
Arizona only dated A1 x1 x2

44. Overall conclusions
The 12 measurements of the age of the TS cannot be considered as repeated measurements from a single unknown quantity!
Evidence of a strong linear trend
Arizona only dated A1
Hetero-geneity of the data

45. Overall conclusions
The statement of Damon et al. (1988) that “The results provide conclusive evidence that the linen of the TS is mediaeval” must be reconsidered in the light of the strong evidence produced by our use of robust statistical techniques Any analysis of the 12 measurements which does not take into account the spatial trend ignores an important component!

46. More about the TS in «Statistics and Computing»

47. Fit and forecasts (with 95% confidence bands) for one of the plausible configurations
millimiters

48. Other datings
It is worthwhile also to mention the results from mechanical, chemical and numismatic dating (Fanti G., P. Malfi, Pan Stanford 2015; Fanti G. P. Malfi, F. Crosilla & Baraldi P., A. Tinti, MATEC Web of Conf, 36, 2015). While chemical dating (FT-IR and Raman) gives dates of 300B.C. ±400 and 200 B.C. ±400, mechanical one furnishes 400 A.D. .±400 and numismatic an age before 692 A.D. These datings are fully compatible with the age in which Jesus Christ lived in Palestine.