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ProceedingsoftheInternationalWorkshopontheScientificapproachto

theAcheiropoietosImages,ENEAFrascati,Italy,4‐6May2010

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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ProceedingsoftheInternationalWorkshopontheScientificapproachto

theAcheiropoietosImages,ENEAFrascati,Italy,4‐6May2010

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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ProceedingsoftheInternationalWorkshopontheScientificapproachto

theAcheiropoietosImages,ENEAFrascati,Italy,4‐6May2010

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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ProceedingsoftheInternationalWorkshopontheScientificapproachto

theAcheiropoietosImages,ENEAFrascati,Italy,4‐6May2010

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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ProceedingsoftheInternationalWorkshopontheScientificapproachto

theAcheiropoietosImages,ENEAFrascati,Italy,4‐6May2010

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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