Modern English in entropy-rescaled PCA distances

Where do modern English groups lie among ancient British, continental Germanic and neighbouring modern groups on a PCA map, and which statements about their neighbours survive a change of reference panel, of map and of origin? Centroids of 31 groups (three modern English groups, modern Orkney, Orcadians, Scottish and French, 19 groups of England from the Neolithic to the early modern period and five continental Germanic groups) are placed on principal-component axes built from present-day European and West Asian reference populations. Distances are read on the conventional map C and on the entropy-rescaled maps A and B (Taleb et al.¹) over nine settings (three reference panels, three maps), 16 origin shifts of the rescaled maps and a bootstrap of individuals. The three modern English groups are on the Late-Middle Iron Age side of both Saxon England and the continental Saxon group of Anderten in every setting (for Kent against Anderten the bootstrap interval of Q includes 1 on map C); Saxon England and Anderten cannot be told apart by any target (the interval of Q includes 1). The nearest group overall depends on the setting for two of the three English groups. Of 72 declared statements, 36 are stable, 8 panel-stable and 28 depend on the setting. Positions on a map are not ancestry shares.

Modern English genomes can be placed on a map of ancient groups from England and from the continental Germanic region, but one map answers little. The position of a group depends on the reference populations that build the axes, on the map (conventional or entropy-rescaled) and on the origin from which distances are read. This article asks a narrower question. For three modern English groups, with modern Orkney as a comparison, which statements about the nearest group, about the side of a pair of groups and about the position between two groups give the same answer in every setting, and which depend on the setting?

The analysis is a PCA-distance series of the Nexju platform on the custom dataset English_DG_studynachbau_1240K_v4 (base aadr-v66-ext-3). 31 groups with 1,252 individuals are projected (Table 1). Four of them are targets: Modern English, Kent; Modern English, Cornwall; Modern English, Human Origins; and Modern Orkney (GBR). The others are comparison groups. Three comparison groups carry most contrasts and are named in short in the text: England Late-Middle Iron Age (74), England Saxon (84), called Saxon England, and Germany Medieval Saxon, Anderten (6), called the continental Saxon group or Anderten. No reference population of the axes appears as a group.

Reference panels. The axes are built from present-day reference populations. Panel WE8 has eight (Basque, Spanish, North Italian, Sardinian, Finnish, Russian, Adygei and Druze; 234 individuals; 992,097 SNPs). The groups are projected onto it with shrinkmode (the base panel) and, as a second setting, with plain projection. Panel EU5 has five (Basque, Spanish, North Italian, Finnish and Russian; 141 individuals; 970,217 SNPs; shrinkmode). Each PCA has six components. On WE8 the first two carry 1.31 % and 1.07 % of the variance (eigenvalues 3.04 and 2.50); on EU5 the second and third carry 1.40 % and 1.39 %.

Maps. Map C is the conventional PCA map. Maps A and B are the entropy-rescaled maps of Taleb et al.¹; on map A each individual is transformed to sgn(ρ)·(−½ ln(1−ρ²))·√λ from the origin and a group is the mean of its transformed individuals. Three panels times three maps give nine settings.

Declared statements. Before the run the following were declared: four targets, nine classes of groups for the nearest group (Modern English, Modern Scotland and France, Neolithic Britain, Bronze Age Britain, Iron Age Britain, Roman Britain, Early medieval England, Medieval England, Continental Germanic), six contrasts of two groups, three pairs for betweenness and six cluster pairs. Each statement is counted over the nine settings and over 16 origin shifts of the rescaled maps (±0.5 and ±1 on each of the two axes of maps A and B). A statement is stable when the same answer holds in at least 8 of the 9 settings and at least 13 of the 16 shifts, and depends when no answer reaches these counts. A statement that reaches them but is not counted as stable is labelled panel-stable. In Fig. 4 every stable row has a bootstrap interval of Q that excludes 1 on all three maps and every panel-stable row has one that includes 1 on at least one map; the three panel-stable betweenness statements of Fig. 6 hold in all nine settings and all 16 shifts.

Q is the distance of a target to the first group of a contrast over its distance to the second, so the target is on the side of the first group when Q is below 1. Intervals are 95 % bootstrap intervals (2,000 resamples of individuals within groups) on the base panel.

Fig. 1 and Fig. 2 show the 31 group centroids on the base panel (WE8, shrinkmode), on the rescaled maps A and B and on the conventional map C. The statements below are not a visual reading of these maps. Each is a count over the nine settings and the 16 origin shifts, so that a result which holds on the base panel only is reported as depending on the setting.

Nearest groups. Fig. 3 reads each target by the class of its single nearest group. Modern English, Human Origins is nearest to a modern English group in all nine settings and all 16 shifts. Kent is nearest to Roman Britain in seven settings and all shifts, and to Medieval England in the two settings of EU5 on maps A and B. Cornwall is nearest to Iron Age Britain on map C of every panel and to a modern English group on the rescaled maps A and B (six settings). Orkney is nearest to Modern Scotland and France in seven settings and to Bronze Age Britain in the EU5 maps A and B. Only Human Origins reaches the threshold of 8 of 9 settings for its most frequent class.

Table 2 gives the nearest group within each class. For all four targets it is the same group in all nine settings and all 16 shifts for the Neolithic class (England Neolithic, Megalithic) and the Bronze Age class (England Late Bronze Age). In the Roman class it is England Roman for the three English groups and England Imperial Roman, Britannia for Orkney, in all nine settings for Cornwall, Human Origins and Orkney and in eight of nine for Kent. Within the Iron Age class for Kent, Cornwall and Human Origins, within the early medieval class and within the Modern Scotland and France class for all targets, the nearest group depends on the setting.

Sides. Fig. 4 reads each target against six declared pairs of groups. All four targets are nearer to England's Late-Middle Iron Age than to Neolithic (Megalithic) England (Q from 32 to 64 for the three English groups, 16 to 23 for Orkney) and than to the Beaker group (Q 9.3 to 17 and 3.0 to 5.2). Both statements hold in all nine settings and all 16 shifts, with intervals above 1 on every map.

The three English groups are also on the Late-Middle Iron Age side of Saxon England (Q 0.12 to 0.24, intervals below 1 on all maps) and of the continental Saxon group of Anderten (Q 0.14 to 0.31); for Kent the interval on map C includes 1 (Q 0.30, 0.10 to 1.04), which is why that statement is panel-stable and not stable (Fig. 5). Orkney does not show this: against Saxon England six of nine settings take the Iron Age side, against Anderten seven of nine take the Anderten side, and the intervals include 1.

Saxon England against Anderten is not resolved for any target. All four lie on the Anderten side in all nine settings and all 16 shifts (Q 1.20 to 1.42), but the bootstrap interval of Q includes 1 on every map (lower limits 0.54 to 0.79). French against Orcadians: Kent and Orkney are on the Orcadian side in all nine settings (Q 2.2 to 3.2 and 7.7 to 12.5). For Cornwall and Human Origins the settings split two to seven, and for Human Origins the interval includes 1 on every map.

Betweenness. Fig. 6 asks whether a target lies between two groups on the map. The three modern English groups lie between Orcadians and French in all nine settings and all 16 shifts (status panel-stable). Orkney, which is close to the Orcadians, does not (2 of 9 settings). Orkney lies between England's Late-Middle Iron Age and Saxon England in all nine settings and all 16 shifts (stable). For the English groups the pairs of the Iron Age with Saxon England and with Anderten depend on the setting: Kent is between them in 4 and 5 of nine settings, Cornwall and Human Origins in 3. Betweenness on a two-dimensional map is a statement about geometry, not about mixture.

Clusters. The average-linkage trees of Fig. 7 are drawn for the base panel. Table 3 counts, over the nine settings, how often two declared groups share a cluster when the tree is cut at k = 4, 6 and 9 clusters. Cornwall and England's Late-Middle Iron Age share a cluster in all nine settings at every k, and so do Modern Orkney and Modern Orcadians. Kent shares a cluster with Cornwall and with Human Origins in nine settings at k = 4 and in eight at k = 6 and 9. Kent and England Saxon share a cluster in six settings at k = 4, in one at k = 6 and in none at k = 9. Human Origins and the Medieval Cambridgeshire group share one in nine, eight and four.

Status. Of the 72 declared statements, 36 are stable, 8 panel-stable and 28 depend on the setting (Fig. 8). Nearest groups: 20 stable and 16 depend (36 statements). Sides: 15 stable, 5 panel-stable and 4 depend (24). Betweenness: 1 stable, 3 panel-stable and 8 depend (12).

Fig. 9 shows what changes the answer. Within WE8 the plain and the shrinkmode projection give the same side statements (no flip on any map; rank correlation 0.97 to 0.99), and betweenness changes in one or two statements. Between WE8 and EU5 the rank correlation falls to 0.87 to 0.88 (shrinkmode) and 0.80 to 0.84 (plain); three side statements flip on maps A and B and none on map C, and four to seven betweenness statements flip. Maps A and B agree almost completely (rank correlation 0.9995 and 0.9992 on the two WE8 projections, 1.00 on EU5; no flip). Map C differs from them (0.95 to 0.97 on WE8, 0.89 on EU5), with one side flip on each WE8 panel and four on EU5. A shift of the origin on the rescaled maps changes at most one of the 24 side statements and at most three of the 12 betweenness statements, but up to eight nearest-group statements; the largest changes come from a shift of +1 along axis 2 (eight on both maps).

What holds in every setting. All four targets are nearer to England's Late-Middle Iron Age than to Neolithic (Megalithic) England and than to the Beaker group. The three modern English groups are nearer to the Late-Middle Iron Age than to Saxon England and than to the continental Saxon group of Anderten, and they lie between Orcadians and French on the map. Within the Neolithic and the Bronze Age class every target has the same nearest group in all settings.

What depends on the setting. The nearest group overall depends on the setting for Kent, Cornwall and Orkney. The nearest Iron Age group (for the English groups), the nearest early medieval group and the nearest group of Modern Scotland and France (for all targets) are not stable. Saxon England and Anderten cannot be told apart by any target: all four lean toward Anderten in every setting, but the bootstrap interval of Q includes 1. For Orkney the contrasts of the Iron Age with either Saxon group depend on the setting, and 8 of the 12 betweenness statements depend.

What changes the answer. Side statements are least sensitive within WE8: shrinkmode against plain projection flips none, and map C against A or B flips one. EU5 differs: three side statements flip between WE8 and EU5 on maps A and B, and four between map C and the rescaled maps within EU5. Shifts of the origin change at most one side statement but up to eight nearest-group statements.

These are statements about distance on a map. Being nearer to the Late-Middle Iron Age than to Saxon England is not evidence of more Iron Age or less Saxon ancestry; that needs a model such as qpAdm, which is not used here.

The result is a position on a map, not an ancestry share, and no qpAdm model is used. The positions depend on the reference populations that build the axes: none of the populations of WE8 or EU5 is British or Germanic, the first two components of WE8 carry only 1.31 % and 1.07 % of the variance, and panel EU5 has a flat eigenvalue spectrum (PC2 1.40 %, PC3 1.39 %), so its axes are weakly determined. Several groups are small, among them Modern Scottish (3), the Anglian group (6), Imperial Roman Britannia (6), England Roman (7) and the Saxon group of Anderten (6). Their centroids move with a few individuals; the bootstrap of Q takes this into account, the nearest-group and betweenness counts do not. The thresholds of 8 of 9 settings and 13 of 16 shifts were fixed in advance and are conventions: a statement that holds in 7 of 9 settings is reported as depending on the setting. The 72 statements are drawn from the same four targets and 31 groups and are not independent. The group Modern English, Human Origins (12) holds two SGDP re-sequences of 1000 Genomes genomes.

Data and methods. Custom Nexju dataset English_DG_studynachbau_1240K_v4 (base aadr-v66-ext-3); PCA-distance run 0d6f72b9 with the base panel WE8-shrinkmode and the settings WE8-plain and EU5-shrinkmode. Panel WE8: Basque, Spanish, North Italian, Sardinian, Finnish, Russian, Adygei, Druze; 992,097 SNPs, 234 reference individuals, shrinkmode and plain projection. Panel EU5: Basque, Spanish, North Italian, Finnish, Russian; 970,217 SNPs, 141 reference individuals, shrinkmode. Six components each. Reference populations never appear as groups; all groups are projected. Targets, classes, contrasts, betweenness pairs and cluster pairs were declared before the run. Thresholds: 8 of 9 settings and 13 of 16 origin shifts; bootstrap of individuals within groups, 2,000 resamples, 95 %. No qpAdm model is used in this article, and no position in it is an ancestry share.

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