A review of Resolving the Geochemical Provenance of Stonehenge Bluestones: A Sample-Size-Independent Multivariate Framework (G. W. Taylor, June 2026)
Taylor's paper, DOI:10.13140/RG.2.2.17039.96164 , sets out to demonstrate that the Stonehenge bluestones are geochemically identical to outcrops in the Mynydd Preseli, using rare earth element (REE) ratio data and three complementary techniques: PERMANOVA, SIMPER, and Euclidean nearest-neighbour distance mapping. It is generous with its data. Table 1 gives the full REE ratio matrix for all forty-two analyses, Table 2 the complete pairwise PERMANOVA output, and Table 3 the entire 20 × 22 distance matrix with nearest-neighbour assignments. That openness makes a substantive review possible, and everything below is derived from those three tables and from the published source of the underlying measurements.
The conclusion the paper reaches is, in outline, correct. The problem is that the method used to reach it cannot distinguish that conclusion from its opposite — and the paper's own data, read against their source, contain the demonstration.
1. What the paper argues
The argument runs in three stages.
A global PERMANOVA on the pooled data fails to reject the null hypothesis of no difference between Stonehenge and Welsh samples (p = 0.85, pseudo-F = 0.074). The author treats this as positive support for common origin. Because the pairwise PERMANOVA matrix contains a handful of rejections, and because archaeological sample sizes are small, those pairwise results are judged unreliable and set aside.
To bypass the sample-size problem, all Stonehenge analyses are pooled into one group and all Welsh analyses into another, and SIMPER is used to compare group means. The means agree closely across all twelve ratios — La/Lu at 15.2 against 15.3, La/Yb at 2.12 against 2.12 — which is presented as demonstrating homogeneity.
Finally, Euclidean distances are computed between every Stonehenge analysis and every Welsh analysis. Several minima are very small, and these are offered as sample-level confirmation, with the two smallest presented as headline results.
Stated at its strongest, the argument is: three methods at three different scales all fail to find a difference, so there is no difference to find.
2. The data are from Bevins, Pearce and Ixer (2021), uncited
The paper does not say where its measurements come from. They can be identified exactly.
Taylor's Table 1 consists of twelve ratios, La divided by each of the other REE. Taking sample CGD1 from Table 2 of Bevins et al. (2021) — La 3.36, Ce 9.05, Pr 1.42, Nd 7.50, Sm 2.42, Eu 0.97, Gd 2.75, Tb 0.50, Dy 3.30, Ho 0.65, Er 1.78, Yb 1.71, Lu 0.24 ppm — and performing those divisions reproduces Taylor's row to every reported digit:
| Ratio | Computed from Bevins et al. (2021) | Taylor Table 1 |
|---|---|---|
| La/Ce | 0.371271 | 0.371271 |
| La/Pr | 2.366197 | 2.366197 |
| La/Nd | 0.448000 | 0.448 |
| La/Sm | 1.388430 | 1.38843 |
| La/Eu | 3.463918 | 3.463918 |
| La/Gd | 1.221818 | 1.221818 |
| La/Tb | 6.720000 | 6.72 |
| La/Dy | 1.018182 | 1.018182 |
| La/Ho | 5.169231 | 5.169231 |
| La/Er | 1.887640 | 1.88764 |
| La/Yb | 1.964912 | 1.964912 |
| La/Lu | 14.000000 | 14 |
The same reconstruction succeeds for OU10, OU10 rpt, PCM7 and PCM7 rpt. Taylor's entire dataset is the REE table of Bevins et al. (2021), converted to La/X ratios.
That paper appears nowhere in Taylor's bibliography, which cites three methodological references (Clarke, 1993; Anderson, 2001; Gower, 1966) and no archaeological or geological source at all. The dataset represents years of laboratory work, sample access negotiated with the Natural History Museum and the Salisbury and South Wiltshire Museum, and analytical development at Aberystwyth. It should be cited.
Two errors follow from the missing attribution. Taylor's Table 1 caption describes the measurements as LA-ICP-MS data. Bevins et al. (2021) used solution nebulisation ICP-MS on acid-digested bulk powders — a different technique producing a different kind of measurement. And the ratios are described as "mineral element proportionality ratios"; they are ratios of whole-rock elemental concentrations, and carry no information about mineral proportions.
3. The groups are sampling categories, not geochemical ones
Taylor's group codes — PO1, SLF1, SODC1, SOF1, SLF2v, SLF2vi, PO2ii, PO2iii, PO2iv, PO3, SLF3 — have no stated derivation. They are Bevins et al.'s group numbers prefixed with their Sample source field:
| Taylor code | Bevins group | Bevins sample source | n |
|---|---|---|---|
| PO1 | Preseli Group 1 | Preseli outcrop | 3 |
| SLF1 | Stonehenge Group 1 | Stonehenge Landscape fragment | 7 |
| SODC1 | Stonehenge Group 1 | Stonehenge orthostat drill core | 4 |
| SOF1 | Stonehenge Group 1 | Stonehenge orthostat fragment | 1 |
Bevins et al.'s Stonehenge Group 1 is a single population of twelve analyses, argued on REE grounds to derive from one intrusive body and attributed to Carn Goedog. Taylor has divided it into three separate "geochemical groups" according to whether each sample was a drill core, a chip off an orthostat, or a surface find from the Stonehenge Landscape — and then run PERMANOVA between them.
This explains the two singleton groups. SOF1 contains one analysis because exactly one Group 1 sample happened to be an orthostat fragment (SH67); SLF2vi contains one for the same reason. These are not rare geological units. They are the intersection of a geochemical group with a collection method.
The scheme is also applied inconsistently: SLF2v contains a drill core (SH62), and SLF3 contains three orthostat fragments, which suggests labels carried down from the first row of each sorted block rather than a deliberate classification.
4. Twelve Welsh analyses and one Stonehenge analysis have been removed
Bevins et al. (2021) report 32 Preseli analyses and 23 Stonehenge analyses. Taylor uses 20 and 22 respectively. The omissions are unstated, and they are not random.
Dropped from the Welsh set: all four Group 2i samples (PCP12, PCS13, PCTF14, PCA15), together with PCM6, PMB9, PMB10, PCB16, PCB19, PCGF27, PCGF27 rpt and PCAW47.
Group 2i matters more than any other. Its distinctive concave-down, MREE-enriched patterns are what allow Bevins et al. to exclude Craig Talfynydd, Carn Sian, Carn Arthur and Carn Bica as sources for any Stonehenge dolerite. It is the clearest discriminating result in the source paper, and it is the one group removed in its entirety.
The twenty retained Welsh analyses span La from 3.04 to 4.26 ppm. All twelve dropped analyses fall outside that band — eight above it, four below. The retained set is, exactly, the central twenty of thirty-two by REE abundance.
Dropped from the Stonehenge set: SH42, at 5.50 ppm La the most REE-rich Stonehenge analysis, and the single sample that Bevins et al. identify as falling outside the envelope of the Preseli patterns in their Fig. 4.
No reason is given for any of this. Whatever the cause, the effect is that both tails were removed from the Welsh distribution and the one non-conforming Stonehenge analysis was removed from the other, before a test for homogeneity was run.
5. The conclusion is almost certainly right
It is worth being clear about this before going further. That the great majority of the Stonehenge bluestones derive from the Mynydd Preseli has been the settled position since Thomas (1923), and successive work by Bevins, Ixer, Pearce and colleagues has narrowed several lithologies to individual outcrops, with excavation at Carn Goedog recovering a Neolithic quarry (Parker Pearson et al., 2019). Nobody reading this review needs persuading of the Preseli connection.
That is precisely why the paper repays close reading. We already know the answer. A method applied to a case with a known answer, which cannot recover that answer reliably, has been shown not to work — and the demonstration is far cleaner than it would be on an open question. What follows is not a defence of some rival provenance. It is an argument that this framework would have produced the same confident result had the stones come from anywhere.
6. What a provenance method has to demonstrate
A geochemical fingerprint is only useful if it discriminates. Showing that a Stonehenge sample resembles a Welsh outcrop establishes nothing on its own; the question is always whether it resembles that outcrop more than it resembles the alternatives, and by a margin larger than the measurement error.
Two requirements follow:
- A comparison set. At least one candidate source outside the favoured region, so that a match can be shown to be selective rather than universal.
- A resolution limit. An estimate of how much two measurements of the same rock differ, so that "close" can be distinguished from "indistinguishable given the noise."
The paper meets neither. No non-Welsh source appears anywhere, so the specificity of the match is never tested. And no error estimate is offered — the framework is presented as a way of avoiding variance estimates rather than as a way of quantifying them.
The second gap can be closed from the source data, and doing so is the substance of the next section.
7. The replicates give the resolution limit directly
Bevins et al. (2021) include three analytical replicate pairs: OU10 and OU10 rpt, PCM7 and PCM7 rpt, PCGF27 and PCGF27 rpt. Each pair is one rock analysed twice. Two of the pairs survive into Taylor's dataset; the third was among the samples dropped.
Two measurements of one rock should, if the method has any resolving power, be closer to each other than either is to a genuinely different rock. The distance between replicates is therefore an estimate of the method's noise floor. Computing Taylor's metric on the Bevins concentrations gives it exactly:
| Replicate pair | Euclidean distance | In Taylor's dataset? |
|---|---|---|
| PCGF27 / PCGF27 rpt | 0.1475 | dropped |
| OU10 / OU10 rpt | 0.1909 | retained |
| PCM7 / PCM7 rpt | 0.2952 | retained |
Now set those against the paper's results.
The headline match is smaller than the noise floor. The abstract, methodology and conclusion all cite OU11 to CGD2 at d = 0.143 as the exemplary near-zero pairing. Every one of the three replicate distances exceeds it. A sample in this dataset sits further from itself than the flagship match sits from its claimed source.
Most of the reported matches fall inside the noise floor. Of the twenty-two nearest-neighbour distances in Taylor's "Closest Euclidean" row, thirteen are below 0.295 and four are below 0.191. For these the assignment carries no information.
The replicates disagree with each other about provenance. OU10 and OU10 rpt are one stone:
| Analysis | Nearest Welsh outcrop | Group | d | Runner-up | Group | d |
|---|---|---|---|---|---|---|
| OU10 | CGD2 | PO1 | 0.203463 | PCM30 | PO3 | 0.317994 |
| OU10 rpt | PCM30 | PO3 | 0.222013 | CGD2 | PO1 | 0.345259 |
The same physical stone is assigned to Carn Goedog on one analytical run and to a Group 3 locality on the other. On Taylor's Fig. 1 these are different red boxes. The ranking is reversed by the difference between two analyses of one rock.
The Welsh-side replicate behaves the same way. Against SH33, PCM7 rpt ranks first at 0.196369 while PCM7 — the same rock — ranks fifth at 0.422311, behind PCM31, PCGF29 and PCM32.
It is worth being precise about the cause. OU10 and OU10 rpt differ in La by 3.63 against 3.67 ppm, about one percent, and both report Lu as 0.24. That one percent alone shifts La/Lu by 0.167, which supplies 76% of the squared distance between them. The noise floor here is substantially an artefact of two-significant-figure reporting of the heavy REE, amplified by the choice to divide by them.
The margins between competing sources are far smaller than the noise. SH62 is 0.550687 from PCDL25 and 0.561924 from PCAW49 — a separation of 0.011, a twentieth of the OU10 replicate distance. SH61 sits 0.023 from a decision between PCC11 and PCAW49. SH37 has four candidates within 0.08 of one another, spanning two different Welsh groups. Every assignment in Table 3 is a coin toss.
8. The assignments against the published attributions
Reading the bottom rows of Taylor's Table 3 produces the paper's actual provenance result: each Stonehenge analysis matched to a Welsh source. Compared with Bevins et al. (2021):
Carn Goedog. Bevins et al. assign twelve Stonehenge analyses to Group 1, sourced to Carn Goedog and corroborated by excavation of a Neolithic quarry there. Taylor's nearest-neighbour method sends four of the twelve to PO1. The other eight go to Carn Ddafad-las, the ground between Cerrigmarchogion and Mynydd Bach, and Group 3 outcrops.
SH45. The strongest positive attribution in Bevins et al. (2021) is SH45 to Preseli Group 2iii, the Cerrigmarchogion samples, described as a near-identical REE composition. Taylor's matrix assigns SH45 to PCC11 at d = 0.5404, with PCM2 — the nearest Group 2iii sample retained — next at 0.5735. The margin is 0.033, a ninth of the OU10 replicate distance. The method gets the source paper's cleanest result wrong, by a margin well inside its own noise.
Group 2iv. Bevins et al. explicitly exclude the outcrops between Cerrigmarchogion and Mynydd Bach as a source for any Stonehenge Group 2 dolerite. Two of Taylor's tightest matches — SH33 at 0.1964 and OU19A at 0.2328 — are both to PCM7 rpt, which belongs to that group.
Group 2v. SH62 and OU6 form one Stonehenge group in Bevins et al., linked by a shared marked positive Eu anomaly and tentatively associated with Carn Ddafad-las and Garn Ddu Fach. Taylor sends SH62 to PCDL25 at Carn Ddafad-las — agreeing with Bevins, but by a margin of 0.011 over the runner-up — and sends OU6 to PCM3 at Cerrigmarchogion, some eight kilometres west.
The spotted / non-spotted test. Bevins et al.'s Groups 1 and 3 are spotted dolerite; Group 2 is non-spotted. The distinction is visible without instruments. Seven of Taylor's twenty-two assignments cross it: six spotted Stonehenge analyses (OU8, OU12, OU14, OU19A, SH33, SH67) are sent to non-spotted Preseli sources, and one non-spotted stone (SH45) to a spotted one. No statistics are needed to see that these attributions cannot be right.
Reading Taylor's Table 3 alongside the published attributions is the most direct test available of whether the framework works, and it fails it in every case where the two make comparable claims.
9. What the distances are actually measuring
The methodology states that the distances are computed on normalised REE ratios. They are not. The calculation reproduces exactly from the untransformed values: taking OU11 and CGD2 and summing squared differences across all twelve ratios gives 0.020460, whose square root is 0.14304 — the reported 0.143. Full working is in the appendix.
This matters because the twelve variables are on wildly different scales. Across the dataset La/Lu ranges from about 13.8 to 16.8, a spread of 3.1, while La/Ce ranges from 0.355 to 0.407, a spread of 0.05. In an unstandardised Euclidean distance each variable contributes as the square of its difference, so La/Ce can contribute at most about 0.0025 to a squared distance while La/Lu can contribute over 9.
Decomposing the largest distance in the matrix, SH49 to PCDL26 at 3.7325: La/Lu supplies 75.5% of the squared distance, La/Tb a further 13.0%, La/Ho 6.1%. Three of the twelve variables account for 94.6% of the result. The remaining nine are, for practical purposes, absent. The twelve-variable multivariate distance is largely a single-variable comparison of La/Lu, dressed in twelve dimensions.
Three further problems attach to the variable set.
No chondrite normalisation. Bevins et al. work throughout with chondrite-normalised patterns, which removes the Oddo–Harkins alternation whereby even-atomic-number REE are roughly ten times more abundant than their odd-numbered neighbours. What remains after normalisation is the shape of the pattern, which is where the petrogenetic information lies. Taylor's raw La/X ratios retain the alternation, so a substantial part of what the distances measure is the periodic-table artefact that normalisation exists to remove.
No sensitivity to the Eu anomaly. Eu/Eu* is a local deviation from the value interpolated between Sm and Gd, and it is the single most discriminating parameter in Bevins et al.'s Group 2 analysis — it is what links SH62 and OU6 to Carn Ddafad-las. La/Eu conflates that local deviation with the overall LREE-to-HREE slope. Taylor's variable set cannot see a Eu anomaly at all.
Induced correlation. All twelve variables share La as numerator. Dividing one quantity by twelve others induces strong correlation among the results by arithmetic alone (Chayes, 1949). The PCA reporting 87.47% of variance on the first component is not evidence that the projection is dependable; it is largely a measurement of that induced correlation. There is also an unresolved inconsistency: an unstandardised PCA on these ratios would place considerably more than 87% on PC1, so the PCA appears to have been run on the correlation matrix while the distances were not standardised at all. The two analyses are not on the same footing and cannot corroborate one another.
Ratios of compositional parts are in any case not amenable to ordinary Euclidean geometry; centred log-ratio transformation exists for exactly this case (Aitchison, 1986). Taylor's appendix presents the absence of transformation as a methodological virtue.
10. The PERMANOVA worked. Its results were discarded
This is the section a reviewer is most likely to get wrong, and the arithmetic repays care.
The pairwise matrix contains five p-values below 0.05. All five are discarded as artefacts of small sample sizes. Checking each against the group assignments of Bevins et al. (2021):
| Rejection | p | Groups compared | Genuinely different? |
|---|---|---|---|
| SODC1 vs PO2ii | 0.029 | Stonehenge Gp 1 vs Preseli Gp 2ii | Yes |
| SODC1 vs SLF3 | 0.035 | Stonehenge Gp 1 vs Stonehenge Gp 3 | Yes |
| PO2ii vs PO3 | 0.020 | Preseli Gp 2ii vs Preseli Gp 3 | Yes |
| PO2ii vs SLF3 | 0.009 | Preseli Gp 2ii vs Stonehenge Gp 3 | Yes |
| PO2iv vs SLF3 | 0.009 | Preseli Gp 2iv vs Stonehenge Gp 3 | Yes |
Five out of five. Not one false positive.
The non-rejections that matter also come out right. Preseli Group 1 against Stonehenge Group 1 — the Carn Goedog attribution — gives p = 0.972. Preseli Group 3 against Stonehenge Group 3 gives p = 0.178. Stonehenge Group 1 drill cores against Stonehenge Group 1 landscape fragments, which are the same population divided by collection method, give p = 0.184.
The test is underpowered and misses real differences: Preseli Group 1 against Preseli Group 3 fails to reject at p = 0.595 when it should not. But every difference it detected is real, and it recovered the published group structure wherever it had the samples to do so. Taylor deleted precisely the entries carrying the signal.
The showcase example the paper offers for discarding them is worth examining. It cites SOF1 against SODC1, where a pseudo-F of 5.317 accompanies a non-significant p of 0.197, as evidence that the pairwise results are unreliable. But SOF1 and SODC1 are both Stonehenge Group 1 — the same geochemical population, separated only by whether the sample was an orthostat fragment or a drill core. Failing to reject is the correct answer, and it says nothing about provenance. It is also an unavoidable answer: with group sizes of 1 and 4 there are only five distinct partitions of the data, so the smallest attainable p-value is 0.200, and 0.197 is the floor of the test.
That floor effect runs through every comparison involving the singleton groups. In a permutation test with group sizes n₁ and n₂ there are C(n₁+n₂, n₁) distinct partitions, and no p-value below 1/C(n₁+n₂, n₁) can be returned however large the true difference:
| Comparison | Group sizes | Partitions | Minimum possible p | Reported p | Reported F |
|---|---|---|---|---|---|
| SOF1 vs SODC1 | 1, 4 | 5 | 0.200 | 0.197 | 5.317 |
| SOF1 vs PO2iii | 1, 3 | 4 | 0.250 | 0.245 | 37.24 |
| SOF1 vs PO2iv | 1, 3 | 4 | 0.250 | 0.251 | 92.41 |
| SODC1 vs PO2ii | 4, 3 | 35 | 0.029 | 0.029 | 16.3 |
| SLF3 vs PO2ii | 7, 3 | 120 | 0.008 | 0.009 | 9.532 |
The rows involving SOF1 and SLF2vi should be deleted rather than interpreted: PERMANOVA on a group of one has no within-group variance to estimate, and those groups exist only because of the sampling-category split described in section 3. The rows at the lower floor are the opposite case — the most extreme outcome the design permits, and correct.
The paper's diagnosis of the problem also needs correcting. It argues that a single critical F value of 2.1532 cannot apply to pairs of differing sizes because the degrees of freedom differ. The deeper point is that the pseudo-F in PERMANOVA does not follow an F distribution at all — that is the reason for permuting (Anderson, 2001). There is no critical F for any pair, at any sample size. The recommendation to work from permutation p-values is right; the reasoning offered for it is not.
11. Pooling for SIMPER
The SIMPER analysis pools all Stonehenge analyses into one group and all Welsh analyses into another, on the grounds that this removes small-sample noise. The paper's own Table 2 records Stonehenge Group 1 differing from Stonehenge Group 3, and Preseli Group 2ii differing from Preseli Group 3. Neither pool is internally homogeneous, by the paper's own test. Averaging each into a single mean profile produces two composite figures corresponding to no actual rock, whose agreement is guaranteed by the mixing rather than by shared origin.
The appendix result confirms it: a pooled pseudo-F of 0.074 means that variation between the two groups is about 7% of the variation within them. That is not a signature of common origin. It is a statement that the grouping explains essentially nothing, which is what happens when heterogeneous populations are pooled.
SIMPER (Clarke, 1993) is also built on Bray–Curtis dissimilarity, defined for abundance data; applied to element ratios it has no clear interpretation. The paper's own observation that the high-magnitude ratios dominate the dissimilarity profile is a symptom of this rather than a finding.
Underlying all of it is the load-bearing assumption that failing to reject H₀ is evidence for H₀. It is not, and the stated justification — that sample sizes are small and variances tight — describes the conditions under which a non-significant result is least informative.
12. Errors of fact
The text states that SH65 matches CGD1 at d = 0.1585. In Table 3 the distance from SH65 to CGD1 is 1.3004. The figure 0.1585 belongs to PCM30 — a different outcrop in a different group, and more than eight times nearer than CGD1. One of the paper's two headline pairings names the wrong source.
The note beneath Table 3 states that only two or three pairings reject the null hypothesis. Table 2 contains five p-values below 0.05, two of them within-side comparisons.
Figure 1 is an annotated reproduction of a published geological map of southwest Wales — its own legend refers to sources proposed by Thomas — presented without attribution. It is not the map from Bevins et al. (2021); its origin should be established and credited.
Separately, the submitted document contains material that reads as unedited drafting: passages addressed in the second person within a first-person paper, unrendered LaTeX in the body and appendix, two alternative titles both retained, and one sentence in the appendix describing an intention to draw the reader's attention away from a discrepancy in the author's own figure. That sentence cannot be what the author meant to publish, and should be removed.
13. What a working version would look like
The instinct behind the sample-level analysis is sound. Nearest-neighbour matching in geochemical space is a legitimate provenance technique, and preferring it to group-level tests when groups are small is a reasonable judgement. The execution is what fails. A version that would carry weight would need:
- Full attribution of the dataset to Bevins et al. (2021), with the analytical method correctly described.
- The complete dataset, including Group 2i and SH42. The samples that don't fit are the ones that establish resolution.
- Chondrite-normalised concentrations, centred-log-ratio transformed, rather than twelve raw ratios sharing a numerator.
- Standardisation before any distance is computed, so that all variables contribute and the result is not a proxy for La/Lu.
- Shape-sensitive parameters — Lan/Smn, Gdn/Ybn, Eu/Eu*, and the λ coefficients of O'Neill (2016) — rather than a set blind to the Eu anomaly.
- The published geochemical groups, not sampling categories. The pairwise structure is the provenance signal, and the rejections are the informative entries, because exclusion is what geochemistry can establish.
- Candidate sources outside Preseli, so that specificity can be demonstrated rather than assumed.
- A permutation null for the nearest-neighbour distances, answering whether the observed minima are smaller than would arise by chance from twenty candidate outcrops.
- The replicate distance reported as the resolution limit, with any assignment whose margin over the runner-up falls below it declared undetermined.
Applied honestly, that last step alone would leave most of the assignments in Table 3 unresolved. That is not a failure of the study; it is the correct result for this dataset, and stating it would be a genuine contribution.
14. Why this matters beyond one paper
Bevins et al.'s Fig. 4 shows the Stonehenge Group 1 and 3 REE patterns falling inside the envelope of the Preseli patterns, with one exception — SH42. They present this as consistency with a Preseli origin and as a limit on what REE alone can resolve, which is why the group definitions rest on compatible-element geochemistry (Bevins et al., 2014) and why the rhyolite work required zircon chemistry. Their section on sample sizes and analytical homogeneity states in advance that differences of ten percent to a factor of two are to be expected between Preseli field samples and Stonehenge drill cores, and that exact matching should not be looked for.
Taylor's headline result is that same overlap, with SH42 and Group 2i removed, restated as proof of identity and presented as a novel framework. The source paper's stated conclusion — that these data are consistent with Preseli origin but cannot on their own resolve outcrops — has been converted into its opposite by removing the uncertainty rather than by adding information.
The pattern is not unusual. "No statistically significant difference" is quietly doing the work of a positive finding across a good deal of provenance literature, and the smaller the sample the more confident the claim tends to become. The framework in this paper is an unusually explicit version of a common move: replacing tests that can fail with descriptive statistics that cannot, and describing the absence of an error estimate as independence from sample size.
The replicate check offers a cheap and general guard against it. Most analytical programmes run duplicates already — Bevins et al. ran three. Computing the distance between two analyses of the same sample, and refusing to report any assignment whose margin is smaller than that distance, costs nothing and would prevent a great deal of overclaiming. It is the one thing this paper's data do establish, and the source data supplied the means to establish it.
Appendix: worked arithmetic
A1. Reproducing d(OU11, CGD2) from untransformed ratios.
| Ratio | OU11 | CGD2 | Difference | Squared |
|---|---|---|---|---|
| La/Ce | 0.382173 | 0.373181 | 0.008992 | 0.0000809 |
| La/Pr | 2.464567 | 2.409396 | 0.055171 | 0.0030438 |
| La/Nd | 0.473525 | 0.460256 | 0.013269 | 0.0001761 |
| La/Sm | 1.462617 | 1.436000 | 0.026617 | 0.0007085 |
| La/Eu | 3.556818 | 3.626263 | −0.069445 | 0.0048226 |
| La/Gd | 1.298755 | 1.291367 | 0.007388 | 0.0000546 |
| La/Tb | 7.113636 | 7.039216 | 0.074420 | 0.0055383 |
| La/Dy | 1.046823 | 1.052786 | −0.005963 | 0.0000356 |
| La/Ho | 5.396552 | 5.439394 | −0.042842 | 0.0018354 |
| La/Er | 1.956250 | 1.983425 | −0.027175 | 0.0007385 |
| La/Yb | 2.100671 | 2.124260 | −0.023589 | 0.0005564 |
| La/Lu | 14.904760 | 14.958330 | −0.053570 | 0.0028698 |
| ÎŁ | 0.0204605 |
√0.0204605 = 0.14304, matching the reported 0.143. The distances are computed on untransformed ratios, not normalised values.
A2. Decomposition of the largest distance, d(SH49, PCDL26) = 3.7325.
| Ratio | Difference | Squared | % of total |
|---|---|---|---|
| La/Lu | 3.24276 | 10.5155 | 75.5 |
| La/Tb | 1.34784 | 1.8167 | 13.0 |
| La/Ho | 0.92424 | 0.8542 | 6.1 |
| La/Eu | 0.51740 | 0.2677 | 1.9 |
| La/Yb | 0.38592 | 0.1489 | 1.1 |
| La/Er | 0.37596 | 0.1413 | 1.0 |
| remaining six | 0.1870 | 1.3 | |
| ÎŁ | 13.9313 |
√13.9313 = 3.7325. Three variables account for 94.6% of the result.
A3. Replicate distances, computed from Bevins et al. (2021) Table 2.
Ratios were formed as in Taylor's Table 1 and the same untransformed Euclidean distance applied.
| Pair | La (ppm) | Lu (ppm) | Δ(La/Lu) | Σ of squares | d |
|---|---|---|---|---|---|
| OU10 / OU10 rpt | 3.63 / 3.67 | 0.24 / 0.24 | 0.16667 | 0.036440 | 0.1909 |
| PCM7 / PCM7 rpt | 3.04 / 3.10 | 0.21 / 0.21 | 0.28572 | 0.087112 | 0.2952 |
| PCGF27 / PCGF27 rpt | 5.16 / 5.20 | 0.32 / 0.32 | 0.12500 | 0.021750 | 0.1475 |
In each case the La/Lu term supplies the large majority of the total: 76%, 94% and 72% respectively. All three exceed the paper's headline match of d = 0.143.
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