Correction of the MDS paper

corrections

Section 2 of the paper posits that it is desirable for dissimilarity measures in detrital geochronology to exhibit the following four properties:

1. δ_i,j is independent of sample size N
2. δ_i,j = 0 if i = j and δ_i,j > 0 otherwise	(non-negativity)
3. δ_i,j = δ_j,i	(symmetry)
4. δ_i,k ≤ δ_i,j + δ_j,k	(triangle inequality)

Although this may appear reasonable at first glance, it turns out that the first two requirements are mutually incompatible. The paper’s assertion that the effect sizes of statistical tests such as Chi-square and Kolmogorov-Smirnov (K-S) fulfill all four requirements is incorrect on three levels. First, the effect size of the Chi-square test fulfils requirement 1 but not 2. Second, the K-S test does not have a calculable effect size. And third, the K-S statistic fulfils requirement 2 but not 1. The reason for the incompatibility of requirements 1 and 2 is simple. To guarantee non-negativity, the definition of dissimilarity measures commonly includes quadratic terms or absolute values, which inevitably introduce positive bias. Note that this is not only the case for the K-S statistic, but also for Euclidean distances such as the Aitchison dissimilarity. Interestingly, the removal of requirement 1 is good news for the applicability of multidimensional scaling to provenance research, because doing so greatly increases the number of suitable dissimilarity measures. With requirement 4 also being optional (in non-metric scaling), MDS turns out to be even more flexible and widely applicable than proposed in the original paper.

The observant reader may also note a slight mismatch between Equation 13 and the plot coordinates of Figure 3a. This is because Equation 13 was calculated by classical MDS whereas Figure 3a was calculated by metric MDS. Equation 13 further implies that two-dimensional MDS configurations are given by the first two dimensions of any higher dimensional case. Although this is true for classical MDS, it is important to note that it is not necessarily the case for (non)metric MDS. Fortunately, neither the examples nor the computer code provided with the paper are affected by this inaccuracy.