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

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.