A robust statistical analysis of the 1988 Turin Shroud radiocarbon dating results

Abstract

Using the 12 published results from the 1988 radiocarbon dating of the TS (Turin Shroud), a robust statistical analysis has been performed in order to test the conclusion by Damon et al. (1998) that the TS is mediaeval. The 12 datings, furnished by the three laboratories, show a lack of homogeneity. We used the partial information about the location of the single measurements to check whether they contain a systematic spatial effect. This paper summarizes the results obtained by Riani et al. (2010), showing that robust methods of statistical analysis can throw new light on the dating of the TS.

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ProceedingsoftheInternationalWorkshopontheScientificapproachto
theAcheiropoietosImages,ENEAFrascati,Italy,4‐6May2010
A robust statistical analysis
of the 1988 Turin Shroud
radiocarbon dating results
G. Fanti1, F. Crosilla2, M. Riani3, A.C. Atkinson4
1Department of Mechanical Engineering University of Padua, Italy, giulio.fanti@unipd.it.
2Department of Geo-Resources and Territory, University of Udine, Italy fabio.crosilla@uniud.it.
3Department of Economics, University of Parma, Italy, mriani@unipr.it
4Department of Statistics, London School of Economics, London WC2A 2AE, UK,
a.c.atkinson@lse.ac.uk.
Abstract
Using the 12 published results from the 1988 radiocarbon dating of the TS (Turin Shroud), a robust statistical analysis has
been performed in order to test the conclusion by Damon et al. (1998) that the TS is mediaeval. The 12 datings, furnished by
the three laboratories, show a lack of homogeneity. We used the partial information about the location of the single
measurements to check whether they contain a systematic spatial effect. This paper summarizes the results obtained by Riani
et al. (2010), showing that robust methods of statistical analysis can throw new light on the dating of the TS.
Keyword: ANOVA, Forward Search, Robust methods, t-statistics, Turin Shroud.
1. INTRODUCTION
The results of the 1988 radiocarbon dating [1] of the TS
were published as providing conclusive evidence that the
linen fabric dates from between 1262 and 1384 AD, with
a confidence level of 95%.
However, after publication of the result, many speculated
that the sample had been contaminated due to the fire of
1532 which seriously damaged the TS, or to the sweat of
hands impregnating the linen during exhibitions, others
that the date was not correct due to the presence of
medieval mending and so on. We give references to some
of these concerns in Section 7.
The purpose of this paper is to summarize the results
obtained in Ref. 2 which show how robust methods of
statistical analysis, in particular the combination of
regression analysis and the forward search [3] combined
with computer power and a liberal use of graphics, can
help to shed new light on results that are a source of
scientific controversy. Throughout we analyse only
numbers from the data given in Ref. 1.
2. DESCRIPTION OF THE DATA
The samples for radio carbon dating were taken from a
strip of material cut from one corner of the TS. The strip
was divided into five parts; the three parts on the right of
Figure 1 were sent to laboratories in Arizona, Oxford and
Zurich. Arizona also received the fourth, smaller, part on
the left. A larger part on the left of Figure 1 was taken by
the Arcidiocesi of Turin as a “Riserva”.
Figure 2 indicates the cutting of the strip in question.
These samples were divided into a total of 12 sub-
samples for which datings were made. The resulting dates
ranged from 591 BP for a reading from Arizona, to 795
BP from Oxford.
3. HETEROGENEITY ANALYSIS
Damon et al. [1] noticed that the data show some
heterogeneity, which they assessed using a chi-squared
test. In this section we instead use the analysis of variance
to test whether these 12 observations can be considered as
homogeneous, i.e. as 12 repeated measurements coming
from a single unknown quantity.
More formally, a general model for observation j at site i
is
yij = µi + σvij εij (i = 1, 2, 3; j = 1, ..., ni), (1)
where the errors εij have a standard normal distribution.
Our central concern is the structure of the µi; at this point
whether they are all equal. However, before proceeding to
the test this hypothesis we need to establish the error
structure. Riani et al. [2] suggest the three following
possibilities
1. Unweighted Analysis. Standard analysis of variance:
all vij = 1
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ProceedingsoftheInternationalWorkshopontheScientificapproachto
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Figure 1. Diagram showing the piece removed from the TS and how it was partitioned. T: trimmed strip. R: retained part called
“Riserva”. O, Z, A1, A2: subsamples given to Oxford, Zurich, and Arizona (two parts) respectively.
Figure 2. Cutting of the linen strip from the TS for the 1988 radiocarbon dating. (G. Riggi di Numana, Fototeca 3M).
2. Original weights. We weight all observations by 1/vij,
where the vij are the standard errors published by Damon
et al. [1], that is, we perform an analysis of variance using
responses:
zij = yij/vij . (2)
3. Modified weights for Arizona. This last formulation
takes into account the fact that according to Damon et al.
the standard errors for Arizona, unlike the two other
laboratories, include only two of the three sources of error.
Reference 2 shows that irrespective of the kind of
ANOVA which is used, while the test for homogeneity of
the variances among the 3 laboratories never turns out to
be significant (the minimum p-value is greater than 0.3),
the test for homogeneity of means is always significant at
the 5% level.
Christen [4] used these data as an example of Bayesian
outlier detection with a mean shift outlier model
(Abraham and Box [5]) in which the null model was that
the data were a homogeneous sample from a single
normal population. He found that the two extreme
observations, 591 and 795 were indicated as outlying.
When these two observations were removed, the data
appeared homogeneous, with a posterior distribution of
age that agreed with the conclusion of Damon et al. [1].
4. SPATIAL HETEROGENEITY
We have appreciable, but only partial, knowledge of the
spatial layout of the samples from Damon et al. [1]. Three
pieces were dated by Oxford, four by Arizona and five by
Zurich. However it is not known how the samples in
Figure 1 were divided within the laboratories, nor is it
known whether the four readings from Arizona came only
from A1 or from A1 and A2.
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Figure 3. Arrangements investigated for the Arizona sample. The image on top assumes that Arizona dated both pieces (A1 and A2).
The image at the bottom assumes that Arizona only dated piece A1. Total number of cases considered is 168 = 96+72.
On the assumption that the four readings from Arizona
all came from A1, Walsh [6] showed evidence for the
regression of age on the known centre points of the pieces
of fabric. Ballabio [7], as well reviewing earlier work,
introduced a second spatial variable into the analysis, the
values of both variables depending on how the division
into subsamples was assumed to have been made. He was
defeated by the number of possibilities.
The possible configurations for the subsamples from
Arizona are shown in Figure 3. If we also consider all
possible plausible ways in which cuts could have been
made by the laboratories of Oxford and Zurich, we end up
with 96 and 23 configurations. In summary there are
387,072 possible cases to analyse.
5. MULTIPLE REGRESSION
To try to detect any trend in the age of the material we fit
a linear regression model in x1 (longitudinal) and x2
(transverse) distances. The analysis is not standard. Riani
et al. [2] permute the values of x1 and x2 and perform all
387,072 analyses.
The question is how to interpret this quantity of numbers.
Without any trend in the longitudinal and transverse
directions we expect to obtain a distribution of t-statistics
for the regression coefficients which is centred around
zero and we approximately expect to obtain half of the
387072 configurations with a positive value of the t-stat
and the other half with negative values. The top panel of
Figure 4 (taken from Ref. 2) shows the distribution of the
t-statistic for x2. This has a t like shape centred around 0.5.
The bottom panel of Figure 4, the t-statistic for x1, is
however quite different, showing two peaks. The larger
peak is centred around −2.9 whereas the thinner peak is
centred around −1. It is also interesting to notice that for
each of the 387,072 configurations we obtain a negative
value of the t-statistic for the longitudinal coordinate.
As we have shown that x2 is not significant (even if it is
surprisingly not centered around 0), we continue our
analysis with a focus on x1. In particular, we want to
discover what feature of the data leads to the bimodal
distribution in Figure 4. If we consider the longitudinal
projections of the 387,072 configurations we obtain
42,081 possibilities.
Summarising the results in Ref 2 which performs a
detailed analysis of all these longitudinal configurations, it
comes out that inference about the slope of the
relationship depends critically on whether configuration
A2 (see Figure 1) was analysed. More precisely, the only
configurations which give rise to non-significant values of
the t-statistic are those associated with:
1) configuration A2 (that are based on the assumption
that Arizona dated both A1 and A2), see Figure 1.
2) the response at the longitudinal coordinate x1 = 41
is y=591 or y=690.
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We now analyse the data structure, taking typical
members inside the configurations 41-591 and of 41-690
and look at some simple diagnostic plots.
To determine whether the proposed data configuration
41-591 is plausible we look at residuals from the fitted
regression model. In order to overcome the potential
problem of masking (when one outlier can cause another
to be hidden) we use a forward search [3] in which
subsets of m carefully chosen observations are used to fit
the regression model and see what happens as m increases
from 2 to 12. Figure 6 shows a forward plot of the
residuals of all observations, scaled by the estimate of
sigma at the end of the search, that is when all 12
observations are used in fitting. The plot shows the pattern
typical of a single outlier, here 41-591 which is distant
from all the other observations until m = n, when it affects
the fitted model.
The conclusion from this analysis is that whether one of
the lower y values, 591 or 606, or one of the higher y
values, 690 or 701, from Arizona is assigned to x1 = 41,
an outlier is generated, indicating an implausible data set.
The comparable plots when it is assumed that Arizona
only analysed A1 are quite different in structure. There is
a stable scatter of residuals in the left-hand panel as the
forward search progresses, with no especially remote
observation. We conclude, that there is statistical evidence
that Arizona only analysed A1 and that there is a
significant trend in the longitudinal coordinates.
6. CONCLUSIONS
The Shroud data relative to the 1988 radiocarbon dating
show surprising heterogeneity. This leads us to conclude
that the twelve measurements of the age of the TS cannot
be considered as repeated measurements of a single
unknown quantity.
The presence of a linear trend explains the difference in
means that was found using the ANOVA test.
The evidence of the heterogeneity together with the
evidence of a strong linear trend lead us to conclude that
the statement of Damon et al.: “The results provide
conclusive evidence that the linen of the Shroud of Turin is
mediaeval” [1] needs to be reconsidered in the light of the
evidence produced by our use of robust statistical
techniques.
Figure 4. Two variable regression. Histograms of values of t-statistics from 387,072 possible configurations. Upper panel x2 (transverse
coordinate), lower panel x1 (longitudinal coordinate).
Transverse coordinate
Longitudinal coordinate
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ProceedingsoftheInternationalWorkshopontheScientificapproachto
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Figure 5. Analysis of residuals for one typical configuration when x1=41, y1=591. Forward plot of scaled residuals showing that this
assignment produces an outlier.
7. DISCUSSION
The arguments in favour of the authenticity of the TS are
rehearsed in other papers in this volume. For example, the
formation mechanism of the body images has not yet been
scientifically explained. One so far unexplained feature is
that the body image is extremely superficial in the sense
that only the external layer of the topmost linen fibre is
coloured [8]. See also [9] and [10].
At a more mundane level, we note that the weights used
in Section 3, taken from Ref. 1, were obtained from up to
8 repeat determinations. Burr et al. [11] describe the
process of analysis used at Arizona. As always, in any
data analysis, it is a help in understanding and modeling
the truth of a situation to work with the original data,
rather than data which have already been summarized,
even if only lightly.
REFERENCES
1. Damon P.E., Donahue D.J., Gore B.H., Hatheway A.L.,
Jull A.J.T., Linick T.W., Sercel P.J., Toolin L.J., Bronk
C.R., Hall E.T., Hedges R.E.M., Housley R., Law I.A.,
Perry C., Bonani G., Trumbore S., Wölfli W., Ambers J.C.,
Bowman S.G.E., Leese M.N., Tite M.S.: Nature, 337, 611-
615 (1989).
2. Riani M., Atkinson A.C., Fanti G., Crosilla F.: “Carbon
Dating of the Shroud of Turin: Partially Labelled
Regressors and the Design of Experiments” see:
www.lse.ac.uk/collections/statistics/research/RAFC04Ma
y2010.pdf
3. Atkinson, A.C. and M. Riani: Robust Diagnostic
Regression Analysis, (New York: Springer–Verlag) 2000.
4. Christen, A.: Applied Statistics, 43, 489-503 (1994).
5. Abraham, B. and Box, G.E.P. Applied Statistics, 27,
131-138 (1978).
6. Wal s h B . The 1988 Shroud of Turin radiocarbon tests
reconsidered. Proceedings of the 1999 Shroud of Turin
International Research Conference Richmond, Virginia
USA, pp. 326–342. B. Walsh Ed., Glen Allen VA:
Magisterium Press (1999).
7. Ballabio G. New statistical analysis of the radiocarbon
dating of the Shroud of Turin (unpublished manuscript).
See http://www.shroud.com/pdfs/doclist.pdf.
8. Fanti G., Botella J.A. Di Lazzaro P., Heimburger T.,
Schneider R., Svensson N. “Journal of Imaging Science
and Technology 54, 040201-(8) (2010).
9. Fanti, G. Basso R., “The Turin Shroud, Optical
Research in the Past Present and Future”, Publisher Nova
Science Pub Inc., 2008.
10. Fanti G.: “La Sindone, una sfida alla scienza
moderna”, Aracne ed., Roma, Italy, 2008.
11. Burr G.S., Donahue D.J., Tang Y., Beck W.J.,
McHargue L., Biddulph D., Cruz R. and Jull A.J.T.: Nucl.
Instr. and Methods in Physics B 259, 149-153 (2007).
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