Principal Coordinates, Genstat and Me

By Professor Richard Bateman

Prelude

Picture the scene. It is the autumn of 1979. A callow youth named Richard, recently hired as a laboratory technician in the Pedology Department of the then Rothamsted Experimental Station, is identifying and counting sand-sized grains of a wide range of detrital minerals for Quaternary palaeoenvironment expert Prof. John Catt. This task is one aspect of a wider study seeking to better understand Britain during the Younger Dryas – a geological period that began ca 13,000 years ago when the ocean currents of the North Atlantic Conveyor temporarily failed and Britain consequently re-entered the ice age for a brief but environmentally catastrophic ca 1,200-year period. The project has already been criticised as worryingly esoteric and likely unfundable, senior managers wrongly believing it inconceivable that this catastrophic cessation of flow might be repeated in the modern era due to climate change. Persuasive outcomes, backed by solid quantitative data, are badly needed to counteract the sceptics.

Birth of a Multipurpose Computational Routine

Viewed from today’s perspective, our search for a solution offers a nostalgic window into the challenges posed by late 1970s computing. I first compared notes with our then doctoral student, Andrew Moffat, who in turn consulted Pedology Department statistician James Rayner. Rayner argued that the large volume of complex data that has been accumulated can be explored most effectively through some form of multivariate ordination. Further consultation resulted in the choice of the Gower similarity coefficient [1] combined with principal coordinates [2] (both techniques previously devised within the fertile mind of Rothamsted’s statistical guru John Gower) as constituting the best solution to the challenges posed by the data. Gower generously prepared a script ostensibly capable of achieving (almost) everything they desired. The routine consisted of a mere two-dozen elegant lines of Genstat command language. This computerised statistical package, born at Rothamsted roughly a decade earlier under the watchful eye of John Nelder, was only then achieving the wider recognition that it deserved. 

I painstakingly typed our data into a cacophonous teleprinter terminal and preserved them on yellowing paper rolls of lavatorial quality, before they were prefixed with the elegant Genstat protocol and an attempt was made to actually run the analysis. I then forced myself to allow sufficient time for the analysis to work through the queued work packages before I set off on the nerve-wracking walk across the campus to the Statistics Department in order to collect the output produced by Rothamsted’s intimidating mainframe. The presence in the relevant pigeonhole of an inch-thick concertinaed wad of perforated paper signalled success, whereas failure was documented via the admonishment of a single sheet bearing nothing more than unsympathetic error messages.

The positioning of the sample points on the initial ordinations led immediately to insightful interpretations of the mineral data, but the inability of principal coordinates to mirror principal components by quantitatively determining the relative contributions of the original variables to the plots proved an initial handicap. A return visit to John Gower’s office provided the solution within 24 hours – a new function written for Genstat simply summarised as the ‘RELATE’ command, which generates pseudo-F statistics relating the original variables to the plots. Now we could finally learn not only how our samples were related but which variables dictated those relationships – a key step toward identifying the underlying geological process(es).

One Routine Fits All?

Thus was born a versatile and powerful analytical routine – a routine that I, happily no longer callow and sadly no longer youthful, continue to employ frequently today, albeit via a slimline laptop rather than a wardrobe-sized mainframe. Genstat in general, and my personalised “Gower package” in particular, have followed me faithfully throughout my rather peripatetic post-Rothamsted research career as I passed through London University, the Smithsonian Institution, Oxford University, followed by the Royal Botanic Gardens Edinburgh, the Natural History Museum and finally (?!) the Royal Botanic Gardens Kew. Collectively, these roles have caused me to explore an exceptional range of kinds of quantitative data, and in many cases, my treasured 1979 Genstat protocol has yielded crucial insights. Below is a small selection of examples designed to illustrate the first three of four overlapping phases that constituted my academic career, having variously operated as a sedimentologist, palaeontologist, plant morphologist and plant geneticist. 

The Case of the Disappearing Minerals

We’ll begin at the beginning, back at Rothamsted and still wrestling with the origin of the Shirdley Hill Sand of southwest Lancashire, as putatively deposited in the periglacial conditions of the Younger Dryas. Unconsolidated and rich in quartz, the deposit still provides the raw material of the region’s famous glass manufacturing industry, but where did the Sand come from, and how was it deposited? All possible source sediments were identified, sampled and analysed for their mineral content.

The resulting principal coordinates plot [3] rejected the coastal aeolian sands as the source because they are too rich in staurolite, kyanite, olivine and epidote, but the plot appeared unable to determine whether glacial deposits (top left) or older consolidated sandstones (right) were the source – until we realised that the Shirdley Hill Sand samples are spread widely across the plot as a result of differential weathering among them. Specifically, the first coordinate shows relative concentration of the resistant mineral tourmaline as a result of the gradual dissolution of the more weatherable minerals augite and hornblende. The least weathered sand samples, at the left-hand end of the “axis of weathering”, signal the correct origin and mode of deposition of the Shirdley Hill Sand – glacial outwash sorted and redistributed by periglacial winds. This single diagram not only answered the original questions but also initiated a new line of research for the lab to actively pursue.

The Case of the Perplexing Pebbles

Next up is my doctoral project, pursued in various colleges of London University. The aim was to conceptually reconstruct entire fossil plants from their disarticulated organs, found scattered through a 352 million-year-old succession of volcanigenic deposits located along the Scottish coastline. Because the fossils had been captured in life-like three-dimensional cellular detail by the volcanically mineralised groundwaters, it became desirable to understand the environments in which these early land-plants had grown and soon after been so beautifully preserved. In addition, I needed to somehow correlate plant-rich water-worn stones collected on the modern beach with the horizon in the cliff from which each had recently been eroded by the sea.

The answers to both questions lay in a principal coordinates plot of rock geochemistry [4], in which the first coordinate reflected the amount of original ash relative to subsequent infilling with carbonates to form impure limestones (i.e. less aluminium, silicon, potassium, titanium relative to more calcium, manganese and volatiles). The second coordinate reflected whether the plants were deposited in waters permeated with calcium-magnesium carbonate, typical of the ambient geochemistry, or with purer calcium carbonate, which had been selectively precipitated out of the lake waters by microscopic algae. Moreover, the number of elemental variables proved sufficient to place within the complex stratigraphy three crucial plant-rich pebbles collected on the foreshore, two of them with high confidence – a novel use of geochemical data in fossil botany. 

The Case of the Mid-Atlantic Orchids

I first used Genstat for morphometric studies of living plants in the early 1980s [5], soon after my principal coordinates routine was created, but at that time my interests in systematic botany were merely those of an enthusiastic hobbyist. However, during the last 30 years systematic botany increasingly became my core profession, as I worked my way (equally systematically!) through Britain’s three main botanical systematics institutes: Royal Botanic Garden Edinburgh, Natural History Museum and Royal Botanic Gardens Kew.

Among my many published studies attempting to circumscribe morphometrically species in groups of temperate orchids, my favourite concerns the Butterfly Orchids of the remote Azorean archipelago. Taxonomists had long debated whether the islands were inhabited by one species of Butterfly Orchid or two. Intensive field sampling of orchids on the Azores and the British Isles, along with less intensive field sampling across mainland Europe, eventually yielded a matrix of 370 plants (representing 63 populations) × 38 quantified morphological characters. But such matrices are highly heterogeneous, mixing metric, meristic, scalar and bistate characters and, because plants in their natural habitat are never perfect, the matrices are further compromised by incurring a minority of empty cells. Happily, Gower similarity and principal coordinates cope well with such technical complications.

The first two principal coordinates resolved the analysed species unusually well [6], the majority of the putative species being separated by morphological discontinuities. The initial plots (A) suggested the presence of only two species on the Azores. However, reanalysing Azorean plants in the absence of their European brethren (B) revealed that three endemic species are present on the islands, rather than two as previously believed. One of the Azorean species, Platanthera azorica, closely resembles the eastern Mediterranean species P. holmboei, suggesting a potential ancestor and area of origin for the Azorean Butterfly Orchid lineage, which subsequently speciated repeatedly on the islands. This interpretation was further strengthened by integrating morphometric information with data on the genetics of both the plants and their associated mycorrhizal fungi, along with observations on their respective habitats. [7]

This classic study of speciation on oceanic islands also subsequently enabled precise conservation prioritisation of the three species under IUCN criteria. One of the endemic species grows in only two locations and constitutes Europe’s rarest orchid, formally categorised as Critically Endangered. Another species, categorised as Endangered, is an excellent indicator of remaining high-quality habitats. This project set the standard for several of my subsequent studies of orchid speciation and, despite my transition into emeritus status, this approach continues to dominate my current research programme.

Back to the Future

I am a competent natural historian rather than a competent statistician. Obviously, I have not rooted my entire career in principal coordinates as programmed in Genstat – it features in 40 of my 190 papers. Aware of the old saw that “if the only tool you have is a hammer, you tend to see every problem as a nail”, I’ve employed other computational tools for contrasting challenges, such as building molecular trees depicting the relationships among organisms. But the flexibility of principal coordinates qualifies them to help solve a remarkably wide range of problems, and Genstat valuably facilitates the speed and efficiency of repeatedly running such analyses. In short, this elegant little analytical routine of 1979 vintage remains my favourite, and I would surely have achieved considerably less in its absence. Happily, I still have substantial matrices patiently awaiting their turn.

References

[1] Biometrics 27: 857–872 (1971), doi 10.2307/2528823

[2] Biometrika 52: 325–338 (1966), doi 10.1093/biomet/53.3-4.325

[3] Proc. Geol. Ass. 92: 211–229 (1981), doi 10.1016/S0016-7878(81)80049-1

[4] Trans R Soc Edinb B 81: 161–194 (1990), doi 10.1017/S0263593300005228

[5] Watsonia 14: 347–376 (1983), bsbi.org.uk archive = Wats14p347.pdf

[6] Peer J 1: e218 [86 pp.] (2013), doi 10.7717/peerj.218

[7] Amer J Bot 101: 979–1001 (2014), doi 10.3732/ajb.1300430