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Neighbor GWAS
Basic model
We analyzed neighbor effects in GWAS as an inverse problem of the two-dimensional Ising model, named “neighbor GWAS” hereafter (Fig. 1). We considered a situation where a plant accession has one of two alleles at each locus, and a number of accessions occupied a finite set of field sites, in a two-dimensional lattice. The allelic status at each locus was represented by x, and so the allelic status at each locus of the ith focal plant and the jth neighboring plants was designated as xi(j)∈{−1, +1}. Based on a two-dimensional Ising model, we defined a phenotype value for the ith focal individual plant yias:

$$y_i = beta _1x_i + beta _2mathop {sum }limits_{ < i,j > } x_ix_j$$
(1)

where β1 and β2 denoted self-genotype and neighbor effects, respectively. If two neighboring plants shared the same allele at a given locus, the product xixj turned into (−1) × (−1) = +1 or (+1) × (+1) = +1. If two neighbors had different alleles, the product xixj became (−1) × (+1) = −1 or (+1) × (−1) = −1. Accordingly, the effects of neighbor genotypic identity on a particular phenotype depended on the coefficient β2 and the number of the two alleles in a neighborhood. If the numbers of identical and different alleles were the same near a focal plant, these neighbors offset the sum of the products between the focal plant i and all j neighbors (mathop {sum }nolimits_{ < i,j > } x_ix_j) and exerted no effects on a phenotype. When we summed up the phenotype values for the total number of plants n and replaced it as E = −β2, H = −β1 and εI = Σyi, Eq. 1 could be transformed into ({it{epsilon }}_I = – {E}mathop {sum }nolimits_{ < i,j > } x_ix_j – {H}mathop {sum }x_i), which defined the interaction energy of a two-dimensional ferromagnetic Ising model (McCoy and Maillard 2012). The neighbor effect β2 and self-genotype effect β1 were interpreted as the energy coefficient E and external magnetic effects H, respectively. An individual plant represented a spin and the two allelic states of each locus corresponded to a north or south dipole. The positive or negative value of Σxixj indicated a ferromagnetism or paramagnetism, respectively. In this study, we did not consider the effects of allele dominance because this model was applied to inbred plants. However, heterozygotes could be processed if the neighbor covariate xixj was weighted by an estimated degree of dominance in the self-genotypic effects on a phenotype.
Association tests
For association mapping, we needed to determine β1 and β2 from the observed phenotypes and considered a confounding sample structure as advocated by previous GWAS (e.g., Kang et al. 2008; Korte and Farlow 2013). Extending the basic model (Eq. (1)), we described a linear mixed model at an individual level as:

$$y_i = beta _0 + beta _1x_i + frac{{beta _2}}{L}mathop {sum }limits_{ < i,j > }^L x_ix_j^{(s)} + u_i + e_i$$
(2)

where β0 indicated the intercept; the term β1xi represented fixed self-genotype effects as tested in standard GWAS; and β2 was the coefficient of fixed neighbor effects. The neighbor covariate (mathop {sum }nolimits_{ < i,j > }^L x_ix_j^{(s)}) indicated a sum of products for all combinations between the ith focal plant and the jth neighbor at the sth spatial scale from the focal plant i, and was scaled by the number of neighboring plants, L. The number of neighboring plants L was dependent on the spatial scale s to be referred. Variance components due to the sample structure of self and neighbor effects were modeled by a random effect (u_i in {boldsymbol{u}}) and ({boldsymbol{u}}sim {mathrm{Norm}}(textbf{0},sigma _1^2{boldsymbol{K}}_1 + sigma _2^2{boldsymbol{K}}_2)). The residual was expressed as (e_i in {boldsymbol{e}}) and ({boldsymbol{e}}sim {mathrm{Norm}}(textbf{0},sigma _e^2{boldsymbol{I}})), where I represented an identity matrix.
Variation partitioning
To estimate the proportion of phenotypic variation explained (PVE) by the self and neighbor effects, we utilized variance component parameters in linear mixed models. The n × n variance-covariance matrices represented the similarity in self-genotypes (i.e., kinship) and neighbor covariates among n individual plants as ({boldsymbol{K}}_1 = frac{1}{{q – 1}}{boldsymbol{X}}_1^{mathrm{T}}{boldsymbol{X}}_1) and ({boldsymbol{K}}_2 = frac{1}{{q – 1}}{boldsymbol{X}}_2^{mathrm{T}}{boldsymbol{X}}_2), where the elements of n plants × q markers matrix X1 and X2 consisted of explanatory variables for the self and neighbor effects as X1 = (xi) and ({boldsymbol{X}}_2 = (frac{{mathop {sum }nolimits_{ < i,j > }^L x_ix_j^{(s)}}}{L})). As we defined (x_{i(j)} in){+1, −1}, the elements of the kinship matrix K1 were scaled to represent the proportion of marker loci shared among n × n plants such that ({boldsymbol{K}}_1 = left( {frac{{k_{ij}, + ,1}}{2}} right));(sigma _1^2)and (sigma _2^2) indicated variance component parameters for the self and neighbor effects.
The individual-level formula Eq. (2) could also be converted into a conventional matrix form as:

$${boldsymbol{y}} = {boldsymbol{X}}{boldsymbol{beta }} + {boldsymbol{Zu}} + {boldsymbol{e}}$$
(3)

where y was an n × 1 vector of the phenotypes; X was a matrix of fixed effects, including a unit vector, self-genotype xi, neighbor covariate (({mathop {sum }nolimits_{ < i,j > }^L x_ix_j^{(s)}})/{L}), and other confounding covariates for n plants; β was a vector that represents the coefficients of the fixed effects; Z was a design matrix allocating individuals to a genotype, and became an identity matrix if all plants were different accessions; u was the random effect with Var(u) =(sigma _1^2{boldsymbol{K}}_1 + sigma _2^2{boldsymbol{K}}_2); and e was residual as Var(e) = (sigma _e^2{boldsymbol{I}}).
Because our objective was to test for neighbor effects, we needed to avoid the detection of false positive neighbor effects. The self-genotype value xi and neighbor genotypic identity (mathop {sum }nolimits_{ < i,j > }^L x_ix_j^{(s)}) would become correlated explanatory variables in a single regression model (sensu colinear) due to the minor allele frequency (MAF) and the spatial scale of s. When MAF is low, neighbors (x_j^{(s)}) are unlikely to vary in space and most plants will have similar values for neighbor identity (mathop {sum }nolimits_{ < i,j > }^L x_ix_j^{(s)}). Furthermore, if the neighbor effects range was broad enough to encompass an entire field (i.e., s→∞), the neighbor covariate and self-genotype xiwould become colinear according to the equation: (left(mathop {sum }nolimits_{ < i,j > }^L x_ix_j^{(s)}right)/L = x_ileft( {mathop {sum }nolimits_{j = 1}^L x_j^{left( s right)}} right)/L = x_ibar x_j), where (bar x_j) indicates a population-mean of neighbor genotypes and corresponds to a population-mean of self-genotype values (bar x_i), if s→∞. The standard GWAS is a subset of the neighbor GWAS and these two models become equivalent at s = 0 and (sigma _2^2) = 0. When testing the self-genotype effect β1, we recommend that the neighbor effects and its variance component (sigma _2^2) should be excluded; otherwise, the standard GWAS fails to correct a sample structure because of the additional variance component at (sigma _2^2, ne ,0). To obtain a conservative conclusion, the significance of β2 and (sigma _2^2) should be compared using the standard GWAS model based on self-effects alone.
Given the potential collinearity between the self and neighbor effects, we defined different metrics for the proportion of phenotypic variation explained (PVE) based on self or neighbor effects. Using a single-random effect model, we calculated PVE for either the self or neighbor effects as follows:
‘single’ PVEself = (sigma _1^2/(sigma _1^2 + sigma _e^2))when s and (sigma _2^2) were set at 0, or
‘single’ PVEnei = (sigma _2^2/(sigma _2^2 + sigma _e^2))when (sigma _1^2) was set at 0.
Using a two-random effect model, we could focus on one variable while considering relationships between two variables (sensu partial out) for either of the two variance components. We defined such a partial PVE as:
‘partial’ PVEself = (sigma _1^2/(sigma _1^2 + sigma _2^2 + sigma _e^2)) and
‘partial’ PVEnei = (sigma _2^2/(sigma _1^2 + sigma _2^2 + sigma _e^2)).
As the partial PVEself was equivalent to the single PVEself when s was set at 0, the net contribution of neighbor effects at s ≠ 0 was given as
‘net’ PVEnei = (partial PVEself + partial PVEnei) − single PVEself,
which indicated the proportion of phenotypic variation that could be explained by neighbor effects, but not by the self-genotype effects.
Simulation
To examine the model performance, we applied the neighbor GWAS to simulated phenotypes. Phenotypes were simulated using a subset of the actual A. thaliana genotypes. To evaluate the performance of the simple linear model, we assumed a complex ecological form of neighbor effects with multiple variance components controlled. The model performance was evaluated in terms of the causal variant detection and accuracy of estimates. All analyses were performed using R version 3.6.0 (R Core Team 2019).
Genotype data
To consider a realistic genetic structure in the simulation, we used part of the A. thaliana RegMap panel (Horton et al. 2012). The genotype data for 1307 accessions were downloaded from the Joy Bergelson laboratory website (http://bergelson.uchicago.edu/?page_id=790 accessed on February 9, 2017). We extracted data for chromosomes 1 and 2 with MAF at >0.1, yielding a matrix of 1307 plants with 65,226 single nucleotide polymorphisms (SNPs). Pairwise linkage disequilibrium (LD) among the loci was r2 = 0.003 [0.00–0.06: median with upper and lower 95 percentiles]. Before generating a phenotype, genotype values at each locus were standardized to a mean of zero and a variance of 1. Subsequently, we randomly selected 1,296 accessions (= 36 × 36 accessions) without any replacements for each iteration and placed them in a 36 × 72 checkered space, following the Arabidopsis experimental settings (see Fig. S1).
Phenotype simulation
To address ecological issues specific to plant neighborhood effects, we considered two extensions, namely asymmetric neighbor effects and spatial scales. Previous studies have shown that plant–plant interactions between accessions are sometimes asymmetric under herbivory (e.g., Bergvall et al. 2006; Verschut et al. 2016; Sato and Kudoh 2017) and height competition (Weiner 1990); where one focal genotype is influenced by neighboring genotypes, while another receives no neighbor effects. Such asymmetric neighbor effects can be tested by statistical interaction terms in a linear model (Bergvall et al. 2006; Sato and Kudoh 2017). Several studies have also shown that the strength of neighbor effects depends on spatial scales (Hambäck et al. 2014), and that the scale of neighbors to be analyzed relies on the dispersal ability of the causative organisms (see Hambäck et al. 2009; Sato and Kudoh 2015; Verschut et al. 2016; Ida et al. 2018 for insect and mammal herbivores; Rieux et al. 2014 for pathogen dispersal) or the size of the competing plants (Weiner 1990). We assumed the distance decay at the sth sites from a focal individual i with the decay coefficient α as (wleft( {s,alpha } right) = {mathrm{e}}^{ – alpha (s – 1)}), since such an exponential distance decay has been widely adopted in empirical studies (Devaux et al. 2007; Carrasco et al. 2010; Rieux et al. 2014; Ida et al. 2018). Therefore, we assumed a more complex model for simulated phenotypes than the model for neighbor GWAS as follows:

$$y_i = beta _0 + beta _1x_i + frac{{beta _2}}{L}mathop {sum }limits_{ < i,j > }^L w(s,alpha )x_ix_j^{(s)} + beta _{12}frac{{x_i}}{L}mathop {sum }limits_{ < i,j > }^L w(s,alpha )x_ix_j^{(s)} + u_i + e_i$$
(4)

where β12 was the coefficient for asymmetry in neighbor effects. By incorporating an asymmetry coefficient, the model (Eq. (4)) can deal with cases where neighbor effects are one-sided or occur irrespective of a focal genotype (Fig. 2). Total variance components resulting from three background effects (i.e., the self, neighbor, and self-by-neighbor effects) were defined as (u_i in {boldsymbol{u}}) and ({boldsymbol{u}}sim {mathrm{Norm}}(textbf{0},sigma _1^2{boldsymbol{K}}_1 + sigma _2^2{boldsymbol{K}}_2 + sigma _{12}^2{boldsymbol{K}}_{12})). The three variance component parameters (sigma _1^2), (sigma _2^2), and (sigma _{12}^2), determined the relative importance of the self-genotype, neighbor, and asymmetric neighbor effects in ui. Given the elements of n plants × q marker explanatory matrix with ({boldsymbol{X}}_{12} = (frac{{x_i}}{L}mathop {sum }nolimits_{ < i,j > }^L w(s,alpha )x_ix_j^{(s)})), the similarity in asymmetric neighbor effects was calculated as ({boldsymbol{K}}_{12} = frac{1}{q-1}{boldsymbol{X}}_{12}^{mathrm{T}}{boldsymbol{X}}_{12}). To control phenotypic variations, we further partitioned the proportion of phenotypic variation into those explained by the major-effect genes and variance components PVEβ + PVEu, major-effect genes alone PVEβ, and residual error PVEe, where PVEβ + PVEu + PVEe = 1. The optimize function in R was used to adjust the simulated phenotypes to the given amounts of PVE.
Fig. 2: Numerical examples of the symmetric and asymmetric neighbor effects.

The intercept, distance decay, random effects, and residual errors are neglected, to simplify this scheme. a Symmetric neighbor effects represent how neighbor genotype similarity (or dissimilarity) affects the trait value of a focal individual yi regardless of its own genotype. b Asymmetric neighbor effects can represent a case in which one genotype experiences neighbor effects while the other does not (b) and a case in which the direction of the neighbor effects depends on the genotypes of a focal individual (c). The case b was considered in our simulation as it has been empirically reported (e.g., Bergvall et al. 2006; Verschut et al. 2016; Sato & Kudoh 2017).

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Parameter setting
Ten phenotypes were simulated with varying combination of the following parameters, including the distance decay coefficient α, the proportion of phenotypic variation explained by the major-effect genes PVEβ, the proportion of phenotypic variation explained by major-effect genes and variance components PVEβ + PVEu, and the relative contributions of self, symmetric neighbor, and asymmetric neighbor effects, i.e., PVEself:PVEnei:PVEs×n. We ran the simulation with different combinations, including α = 0.01, 1.0, or 3.0; PVEself:PVEnei:PVEs×n = 8:1:1, 5:4:1, or 1:8:1; and PVEβ and PVEβ + PVEu = 0.1 and 0.4, 0.3 and 0.4, 0.3 and 0.8, or 0.6 and 0.8. The maximum reference scale was fixed at s = 3. The line of simulations was repeated for 10, 50, or 300 causal SNPs to examine cases of oligogenic and polygenic control of a trait. The non-zero coefficients (i.e., signals) for the causal SNPs were randomly sampled from −1 or 1 digit and then assigned, as some causal SNPs were responsible for both the self and neighbor effects. Of the total number of causal SNPs, 15% had self, neighbor, and asymmetric neighbor effects (i.e.,β1 ≠ 0 and β2 ≠ 0 and β12 ≠ 0); another 15% had both the self and neighbor effects, but no asymmetry in the neighbor effects (β1 ≠ 0 and β2 ≠ 0 and β12 ≠ 0); another 35% had self-genotypic effects only (β1 ≠ 0); and the remaining 35% had neighbor effects alone (β2 ≠ 0). Given its biological significance, we assumed that some loci having neighbor signals possessed asymmetric interactions between the neighbors (β2 ≠ 0 and β12 ≠ 0), while the others had symmetric interactions (β2 ≠ 0 and β12 ≠ 0). Therefore, the number of causal SNPs in β12 was smaller than that in the main neighbor effects β2. According to this assumption, the variance component (sigma _{12}^2) was also assumed to be smaller than (sigma _2^2). To examine extreme conditions and strong asymmetry in neighbor effects, we additionally analyzed the cases with PVEself:PVEnei:PVEs×n = 1:0:0, 0:1:0, or 1:1:8.
Summary statistics
The simulated phenotypes were fitted by Eq. (2) to test the significance of coefficients β1 and β2, and to estimate single or partial PVEself and PVEnei. To deal with potential collinearity between xi and neighbor genotypic identity (mathop {sum }nolimits_{ < i,j > }^L x_ix_j^{(s)}), we performed likelihood ratio tests between the self-genotype effect model and the model with both self and neighbor effects, which resulted in conservative tests of significance for β2 and (sigma _2^2). The simulated phenotype values were standardized to have a mean of zero and a variance of 1, where true β was expected to match the estimated coefficients (hat beta) when multiplied by the standard deviation of non-standardized phenotype values. The likelihood ratio was calculated as the difference in deviance, i.e., −2 × log-likelihood, which is asymptotically χ2 distributed with one degree of freedom. The variance components, (sigma _1^2) and (sigma _2^2), were estimated using a linear mixed model without any fixed effects. To solve the mixed model with the two random effects, we used the average information restricted maximum likelihood (AI-REML) algorithm implemented in the lmm.aireml function in the gaston package of R (Perdry and Dandine-Roulland 2018). Subsequently, we replaced the two variance parameters (sigma _1^2) and (sigma _2^2) in Eq. (2) with their estimates (hat sigma _1^2) and (hat sigma _2^2) from the AI-REML, and performed association tests by solving a linear mixed model with a fast approximation, using eigenvalue decomposition (implemented in the lmm.diago function: Perdry and Dandine-Roulland 2018). The model likelihood was computed using the lmm.diago.profile.likelihood function. We evaluated the self and neighbor effects for association mapping based on the forward selection of the two fixed effects, β1 and β2, as described below:
1.
Computed the null likelihood with (sigma _1^2, ne ,0) and (sigma _2^2 = 0) in Eq. (2).

2.
Tested the self-effect, β1, by comparing with the null likelihood.

3.
Computed the self-likelihood with (hat sigma _1^2), (hat sigma _2^2), and β1 using Eq. (2).

4.
Tested the neighbor effects, β2, by comparing with the self-likelihood.

We also calculated PVE using the mixed model (Eq. (3)) without β1 and β2 as follows:
1.
Calculated single PVEself or single PVEnei by setting either (sigma _1^2) or (sigma _2^2) at 0.

2.
Tested the single PVEself or single PVEnei using the likelihood ratio between the null and one-random effect model.

3.
Calculated the partial PVEself and partial PVEnei by estimating (sigma _1^2) and (sigma _2^2) simultaneously.

4.
Tested the partial PVEself and partial PVEnei using the likelihood ratio between the two- and one-random effect model.

We inspected the model performance based on causal variant detection, PVE estimates, and effect size estimates. The true or false positive rates between the causal and non-causal SNPs were evaluated using ROC curves and area under the ROC curves (AUC) (Gage et al. 2018). An AUC of 0.5 would indicate that the GWAS has no power to detect true signals, while an AUC of 1.0 would indicate that all the top signals predicted by the GWAS agree with the true signals. In addition, the sensitivity to distinguish self or neighbor signals (i.e., either β1 ≠ 0 or β2 ≠ 0) was evaluated using the true positive rate of the ROC curves (i.e., y-axis of the ROC curve) at a stringent specificity level, where the false positive rate (x-axis of the ROC curve) = 0.05. The roc function in the pROC package (Robin et al. 2011) was used to calculate the ROC and AUC from −log10(p value). Factors affecting the AUC or sensitivity were tested by analysis-of-variance (ANOVA) for the self or neighbor effects (AUCself or AUCnei; self or neighbor sensitivity). The AUC and PVE were calculated from s = 1 (the first nearest neighbors) to s = 3 (up to the third nearest neighbors) cases. The AUC was also calculated using standard linear models without any random effects, to examine whether the linear mixed models were superior to the linear models. We also tested the neighbor GWAS model incorporating the neighbor phenotype (y_j^{(s)}) instead of (x_j^{(s)}). The accuracy of the total PVE estimates was defined as PVE accuracy = (estimated total PVE − true total PVE) / true total PVE. The accuracy of the effect size estimates was evaluated using mean absolute errors (MAE) between the true and estimated β1 or β2 for the self and neighbor effects (MAEself and MAEnei). Factors affecting the accuracy of PVE and effect size estimates were also tested using ANOVA. Misclassifications between self and neighbor signals were further evaluated by comparing p value scores between zero and non-zero coefficients. If −log10(p value) scores of zero β are the same or larger than non-zero β, it infers a risk of misspecification of the true signals.
Arabidopsis herbivory data
We applied the neighbor GWAS to field data of Arabidopsis herbivory. The procedure for this field experiment followed that of our previous experiment (Sato et al. 2019). We selected 199 worldwide accessions (Table S1) from 2029 accessions sequenced by the RegMap (Horton et al. 2012) and 1001 Genomes project (Alonso-Blanco et al. 2016). Of the 199 accessions, most were overlapped with a previous GWAS of biotic interactions (Horton et al. 2014) and half were included by a GWAS of glucosinolates (Chan et al. 2010). Eight replicates of each of the 199 accessions were first prepared in a laboratory and then transferred to the outdoor garden at the Center for Ecological Research, Kyoto University, Japan (Otsu, Japan: 35°06′N, 134°56′E, alt. ca. 200 m: Fig. S1a). Seeds were sown on Jiffy-seven pots (33-mm diameter) and stratified at a temperature of 4 °C for a week. Seedlings were cultivated for 1.5 months under a short-day condition (8 h light: 16 h dark, 20 °C). Plants were then separately potted in plastic pots (6 cm in diameter) filled with mixed soil of agricultural compost (Metro-mix 350, SunGro Co., USA) and perlite at a 3:1 ratio. Potted plants were set in plastic trays (10 × 40 cells) in a checkered pattern (Fig. S1b). In the field setting, a set of 199 accessions and an additional Col-0 accession were randomly assigned to each block without replacement (Fig. S1c). Eight replicates of these blocks were set >2 m apart from each other (Fig. S1c). Potted plants were exposed to the field environment for 3 weeks in June 2017. At the end of the experiment, the percentage of foliage eaten was scored as: 0 for no visible damage, 1 for ≤10%, 2 for >10% and ≤25%, 3 for >25% and ≤50%, 4 for >50% and ≤75%, and 5 for >75%. All plants were scored by a single person to avoid observer bias. The most predominant herbivore in this field trial was the diamond back moth (Plutella xylostella), followed by the small white butterfly (Pieris rapae). We also recorded the initial plant size and the presence of inflorescence to incorporate them as covariates. Initial plant size was evaluated by the length of the largest rosette leaf (mm) at the beginning of the field experiment and the presence of inflorescence was recorded 2 weeks after transplanting.
We estimated the variance components and performed the association tests for the leaf damage score with the neighbor covariate at s = 1 and 2. These two scales corresponded to L = 4 (the nearest four neighbors) and L = 12 (up to the second nearest neighbors), respectively, in the Arabidopsis dataset. The variation partitioning and association tests were performed using the gaston package, as mentioned above. To determine the significance of the variance component parameters, we compared the likelihood between mixed models with one or two random effects. For the genotype data, we used an imputed SNP matrix of the 2029 accessions studied by the RegMap (Horton et al. 2012) and 1001 Genomes project (Alonso-Blanco et al. 2016). Missing genotypes were imputed using BEAGLE (Browning and Browning 2009), as described by Togninalli et al. (2018) and updated on the AraGWAS Catalog (https://aragwas.1001genomes.org). Of the 10,709,466 SNPs from the full imputed matrix, we used 1,242,128 SNPs with MAF at >0.05 and LD of adjacent SNPs at r2  More

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Samples
We sampled 73 adult fox skulls (Vulpes vulpes) from three separate sample groups: wild (8 F, 12 M), unselected (15 F, 8 M), and domesticated (15 F, 15 M). Domesticated and unselected skulls from the RFF experiment were generously provided by Dr. Trut and transported to Harvard in 2004. Unfortunately, we do not know how these foxes were chosen, but have no reason not to assume that they were selected randomly from both populations. Wild fox skulls in the study were sampled from the collections of the Museum of Comparative Zoology, Harvard University. All but two wild foxes were trapped in Canada east of Quebec between 1894 and 1952, with the majority (70%) between 1894 and 1900 (Table S1). We excluded from the study sample all juvenile skulls, as determined through lower third molar eruption and fusion of the cranial suture between the basioccipital and basisphenoid16, and those skulls that had evidence of damage or disease. After applying these exclusion criteria, we arrived at our final sample of 73 skulls.
3D landmarks
To prevent movement during measurement, each skull was embedded in styrofoam and secured to the workspace desk before 3D coordinates of 29 landmarks, listed, defined and displayed in Table S2 and Fig S1, were collected on the left half of each skull by a single analyst (TMK). Of these 29 landmarks, 17% are on the cranial base, 27% are on the neurocranium, and 55% are on the face. 3D landmark coordinates were measured with a Microscribe G2 (Positional Accuracy ± 0.38 mm, Revware, Inc.). This machine consists of a mobile robotic arm tipped with a probe. After calibration, the probe tip is placed on each landmark to record its XYZ coordinates. To avoid having to move the skulls during measuring and to limit the number of variables in our final GM analysis (too high a number may be a problem given our small sample size), we restricted landmark measurements to one half of each skull. We assume that the fluctuating asymmetry between each fox population is negligible and stable as has previously been shown in comparisons across a domestic-wild hybridization zone in mice17. In most cases, landmark positions were lightly marked with pencil to ensure proper probe placement.
Linear and endocranial volume measurements
Six linear measurements were taken on each skull using digital calipers (Fowler High Precision, Positional Accuracy ± 0.03 mm): total skull length, snout length, cranial vault height, cranial vault width, bi-zygomatic width and upper jaw width (Fig. 1). Endocranial volume was measured using plastic beads. Each cranium was filled up to the level of the foramen magnum and repeatedly shaken and tamped down until no more beads could be added. The beads were then funneled into a graduated cylinder to obtain a volumetric measurement.
Figure 1

Schematic diagram of the six linear measurements taken on the fox skulls. 1: Bi-zygomatic width. 2: Cranial vault width. 3: Upper jaw width. 4: Total skull length. 5: Snout length. 6: Cranial vault height.

Full size image

Geometric morphometrics
Generalized Procrustes analysis was conducted in R v. 4.0.218 using the geomorph v. 3.3.1 package19. Landmark configurations from each specimen were translated to the origin, rescaled to centroid size, and optimally rotated (using a least-squares criterion) until the coordinates of homologous landmarks aligned as closely as possible. These steps place all specimens in the same shape space, centered on the mean shape. An orthogonal projection into a linear tangent space was applied so statistical analyses could be performed on the resulting tangent space coordinates. For Procrustes superimposition, we used the default parameters of the gpagen function in geomorph.
Repeatability analyses
Landmark measurement repeatability was evaluated through repeated measurements of three fox skulls (domesticated male ID# TM23, domesticated female ID# TF476, unselected female ID# UF1058) on ten separate occasions. In this case, repeatability encompasses both Microscribe and operator error. Generalized Procrustes Analysis (GPA) was used on these landmark coordinates to ensure that they were in the same 3D location relative to one another. The average Procrustes distance (PD) between all ten iterations of the same specimen was then compared to the average Procrustes distance within the population-sex grouping to which the skull belonged. To do this, we calculated a sensitivity ratio based on the formula: (Mean Inter-specimen PD – Mean Intra-specimen PD)/Mean Intra-specimen PD. This created a sensitivity ratio that reflects how sensitive the Microscribe measurements are with respect to the average difference among foxes of the same population-sex category. Averaging the sensitivity ratios for our three skulls, we find that the difference between replicates is roughly 3.7 times smaller than the differences within population-sex groupings. This indicates that the Microscribe G2 is robust enough to detect subtle individual differences in measured landmark coordinates. Linear and endocranial measurement repeatability was quantified through a similar method where repeat measurements were taken on 3 domesticated female fox skulls on 15 separate occasions. Sensitivity ratios were deteremined for each measurement (i.e. total skull length, snout length etc.) by calculating the standard deviation of each repeated measurement on a single specimen, averaging the three specimens’ standard deviations for that measurement, and then comparing that value to the population (domesticated female) standard deviation for that measurement. With the exception of cranial vault height (see limitations), the replicate standard deviations of each measurement were roughly a third (or less) of the population standard deviations (Table S3).
Statistical analyses
All statistical analyses were performed in R18. For all parametric inferences, we report point and interval (95% confidence) estimates of effect sizes, while for permutation-based inferences we report point estimates and p-values. All p-values involving multiple comparisons were adjusted for family-wise error using the sequential Bonferroni method.
3D shape comparisons
To test hypotheses about shape differences among the three populations of foxes, we used a permutation-based Procrustes MANOVA to regress tangent space coordinates on population identity and sex in the geomorph v. 3.3.1 package. Because we are unable to detect significant differences in allometry among populations with a permutation-based Procrustes MANOVA of tangent space coordinates on the interaction term of population identity and centroid size, the tangent space coordinates were not corrected for any scaling effects (see Supplemental information and Fig S2). Given this result, we control only for isometric size in geometric morphometric analyses (i.e. no correction for scaling in tangent space coordinates) as well as in our linear measurements. To determine how skull shape differed between fox populations, we performed pairwise comparisons of shape using Procrustes distances. We additionally performed pairwise comparisons between groups of the shape variance within a group (as assessed by the dispersion of residuals around the mean shape for a given population)20. All pairwise comparisons were made using the RRPP v. 0.6.1 package21,22. Permutation-based p-values for the pairwise comparisons were corrected for family-wise error using the sequential Bonferroni method. To visualize changes in skull shape between populations, a principal components analysis (PCA) was performed on the tangent space coordinates. Skull warp changes along the first principal component were graphed to visualize shape changes along this axis. Size differences between populations were assessed via a linear model using a weighted least squares (WLS) estimator, where centroid size was regressed on population identity and sex. The WLS estimator allowed for separate residual variances for each combination of population and sex, so that heteroskedasticity across these groups could be accounted for in the model. Variance in centroid size was assessed with a Levene’s test based on absolute deviations from the median and was performed using the car package v. 3.0-1023.
Linear and endocranial volume comparisons
Prior to modeling linear and volumetric data, we created size-adjusted versions of our variables to account for a difference in isometric size between wild and RFF populations. Normalizing to size allows us to parse out the effects of size selection from those of selection for docility as they likely have overlapping effects on craniofacial shape. We adjusted for size by normalizing each linear measurement and the cube root of endocranial volume by centroid size. We used centroid size rather than the geometric mean of the six linear measurements because centroid size was calculated using a larger sample of craniometric landmarks and is therefore the better proxy of overall cranial size. We performed size corrections on the raw measurements instead of including a size variable in the models because it allows the size-correction to be intrinsic to each fox rather than depending on the size of every fox in the model.
To determine if there were population-level differences in size-corrected linear and volumetric variables, we used a linear model with a generalized least squares (GLS) estimator from the nlme v. 3.1-150 package24 to regress all 7 skull variables simultaneously as correlated responses on population identity and sex (see Supplementary Methods for details of estimation strategy and model specification and Figs. S3, S4). We report estimates of pairwise percent differences between population means for each skull variable. We use this method because the 7 linear and volumetric skull variables were correlated in two ways (see Fig. S3). First, they were measured on the same specimens, and second, they represent non-independent aspects of shape variation. Modeling these response variables in 7 separate general linear models (e.g., ANOVA) would result in biologically unrealistic inferences because these correlations would be artificially fixed at zero. In addition, since skull variables exhibited varying amounts of dispersion, the GLS model allowed for different residual variances for each response variable.
Sexual dimorphism comparisons
Sexual dimorphism within a species is often represented as dimorphism in size as well as shape25. Therefore, in contrast to the previous analyses, we assess the degree of sexual dimorphism in both size and shape. We used a similar GLS model to determine the degree of sexual dimorphism of the raw (non-size corrected) variables within each population. To estimate sex-specific effects, we added an additional interaction term between sex and population identity in this model. We report the degree of dimorphism using estimates of mean differences between males and females for a given skull variable, within a population. For both models using linear and volumetric data, we performed model selection for variance components and correlation structures using the Bayesian information criterion, since this has been shown to provide a good balance between parsimony and over-fitting for explanatory models26. Linear model (GLS) assumptions were checked using diagnostic plots of standardized residuals and fitted values (see Fig. S5). More