`bnpsd`

\[ \newcommand{\Fst}{F_{\text{ST}}} \newcommand{\ft}[1][j]{f_{#1}^T} \]

The `bnpsd`

package simulates the genotypes of an admixed population. In the Pritchard-Stephens-Donnelly (PSD) model (Pritchard, Stephens, and Donnelly 2000), admixed individuals draw their alleles with individual-specific probabilities (admixture proportions) from \(K\) intermediate subpopulations. We impose the Balding-Nichols (BN) model (Balding and Nichols 1995) to the intermediate subpopulation allele frequency, which thus evolve independently with subpopulation-specific inbreeding coefficients (\(\Fst\) values) from a common ancestral population \(T\). The kinship coefficients and generalized \(\Fst\) of the admixed individuals were derived in recent work (Ochoa and Storey 2016). A simulated admixed population generated using this package was used to benchmark kinship and \(\Fst\) estimators in the accompanying paper (Ochoa and Storey 2021). Here we briefly summarize the notation and intuition behind the key parameters (see (Ochoa and Storey 2016) for precise definitions).

There are conceptually two parts of the BN-PSD model, one that defines relatedness between individuals (under the admixture framework), while the other simulates allele frequencies and genotypes conditional on the specified parameters of relatedness/admixture.

The population structure determines how individuals are related to each other. The key parameters are the coancestry coefficients of the intermediate subpopulations and the admixture proportions of each individual for each subpopulation (\(q_{ju}\)), which are treated as fixed variables.

The `bnpsd`

functions that model population structure (not drawing allele frequencies or genotypes) admit arbitrary coancestry matrices for the intermediate subpopulations. However, the code that draws allele frequencies and genotypes requires these subpopulations to be *independent*, which means that the coancestry matrix must have zero values off-diagonal. In this setting—the only case we’ll considered in this vignette—the diagonal values of the coancestry matrix of the intermediate subpopulations are inbreeding coefficients (\(\ft[S_u]\) below), which correspond to subpopulation-specific \(\Fst\) values.

Each intermediate subpopulation \(S_u\) (\(u \in \{1, ..., K\}\)) evolved independently from a shared ancestral population \(T\) with an inbreeding coefficient denoted by \(\ft[S_u]\). Note that \(T\) is a superscript, and does not stand for exponentiation or matrix transposition. Each admixed individual \(j \in \{1, ..., n\}\) draws each allele from \(S_u\) with probability given by the admixture proportion \(q_{ju}\) (\(\sum_{u=1}^K q_{ju} = 1 \forall j\)). In this case the coancestry coefficients \(\theta^T_{jk}\) between individuals \(j,k\) (including \(j=k\) case) and the \(\Fst\) of the admixed individuals are given by: \[ \theta^T_{jk} = \sum_{u=1}^K q_{ju} q_{ku} \ft[S_u], \quad \quad \Fst = \sum_{j=1}^n \sum_{u=1}^K w_j q_{ju}^2 \ft[S_u], \] where \(0 < w_j < 1, \sum_{j=1}^n w_j = 1\) are user-defined weights for individuals (default \(w_j=\frac{1}{n} \forall j\)). Note \(\theta^T_{jk}\) equals the kinship coefficient for \(j \ne k\) and the inbreeding coefficient for \(j=k\).

The bias coefficient \(s\) is defined by \[
s = \frac{\bar{\theta}^T}{\Fst}
\] where \(\bar{\theta}^T = \sum_{j=1}^n \sum_{k=1}^n w_j w_k \theta_{jk}^T\). This \(0 < s \le 1\) approximates the proportional bias of \(\Fst\) estimators that assume independent subpopulations, and one `bnpsd`

function below fits its parameters to yield a desired \(s\).

This section details the distributions of the allele frequencies and genotypes of the various populations or individuals of the BN-PSD model.

Every biallelic locus \(i\) in the ancestral population \(T\) has an ancestral reference allele frequency denoted by \(p_i^T\). By default the `bnpsd`

code draws \[
p_i^T \sim \text{Uniform}(a, b)
\] with \(a=0.01, b=0.5\), but the code accepts \(p_i^T\) from arbitrary distributions (see below).

The distribution of the allele frequency at locus \(i\) in subpopulation \(S_u\), denoted by \(p_i^{S_u}\), is the BN distribution: \[ p_i^{S_u} | T \sim \text{Beta} \left( \nu_s p_i^T, \nu_s \left( 1-p_i^T \right) \right), \] where \(\nu_s = \frac{1}{\ft[S_u]}-1\). Allele frequencies for different loci and different subpopulations (\(S_u,S_v, u \ne v\)) are drawn independently.

Each admixed individual \(j\) at each locus \(i\) draws alleles from a mixture of Bernoulli distributions from each intermediate subpopulation, which is mathematically equivalent to assigning what we call individual-specific allele frequencies \(\pi_{ij}\) constructed as: \[ \pi_{ij} = \sum_{u=1}^K p_i^{S_u} q_{ju}. \] The unphased genotype \(x_{ij}\) (encoded to count the number of reference alleles) is drawn as: \[ x_{ij}|\pi_{ij} \sim \text{Binomial}(2, \pi_{ij}). \]

Let’s generate the same population structure used in the simulation of (Ochoa and Storey 2021).

```
# this package
library(bnpsd)
# for nice colors
library(RColorBrewer)
# for visualizing coancestry matrix with plot_popkin, and other handy functions
library(popkin)
# dimensions of data/model
# number of individuals (NOTE this is 10x less than in publication!)
100
n_ind <-# number of intermediate subpops
10
k_subpops <-
# define population structure
# subpopulation FST vector, up to a scalar
1 : k_subpops
inbr_subpops <-# desired bias coefficient
0.5
bias_coeff <-# desired FST for the admixed individuals
0.1
Fst <-# admixture proportions from 1D geography
admix_prop_1d_linear(
obj <-n_ind = n_ind,
k_subpops = k_subpops,
bias_coeff = bias_coeff,
coanc_subpops = inbr_subpops,
fst = Fst
) obj$admix_proportions
admix_proportions <-# rescaled inbreeding vector for intermediate subpopulations
obj$coanc_subpops
inbr_subpops <-
# get pop structure parameters of the admixed individuals
coanc_admix(admix_proportions, inbr_subpops)
coancestry <-
# verify that we got the desired FST!
fst_admix(admix_proportions, inbr_subpops)
#> [1] 0.1
# the mean of the diagonal of the coancestry matrix also equals FST
diag(coancestry)
inbr <-fst(inbr) # `fst` is a function in the popkin package
#> [1] 0.1
# verify that we got the desired `bias_coeff` too!
mean(coancestry) / Fst
#> [1] 0.5
```

```
# visualize the per-subpopulation inbreeding coefficients (FSTs)
# shrink default margins
par(mar = c(4, 4, 0, 0) + 0.2)
# colors for independent subpopulations
brewer.pal(k_subpops, "Paired")
col_subpops <-
barplot(inbr_subpops, col = col_subpops, names.arg = 1 : k_subpops, ylim = c(0, 1),
xlab = 'Subpopulation', ylab = 'Inbreeding coeff.')
```

```
# visualize the admixture proportions
# shrink default margins
par(mar = c(1, 4, 0, 0) + 0.2)
barplot(
t(admix_proportions),
col = col_subpops,
border = NA,
space = 0,
ylab = 'Admixture prop.'
)mtext('Individuals', 1)
```

```
# Visualize the coancestry matrix using "popkin"!
plot_popkin(
coancestry,# zero inner margin (plus padding) because we have no labels
mar = 0,
leg_title = 'Coancestry'
)
```

Now let’s draw all the random allele frequencies and genotypes from the population structure. The easiest way is to use `draw_all_admix`

:

```
# number of loci in simulation (NOTE this is 30x less than in publication!)
10000
m_loci <-# draw all random Allele Freqs (AFs) and genotypes
# reuse the previous inbr_subpops, admix_proportions
draw_all_admix(
out <-admix_proportions = admix_proportions,
inbr_subpops = inbr_subpops,
m_loci = m_loci,
# NOTE by default p_subpops and p_ind are not returned, but here we will ask for them
want_p_subpops = TRUE,
# NOTE: asking for `p_ind` increases memory usage substantially,
# so don't ask for it unless you're sure you want it!
want_p_ind = TRUE
)# genotypes
out$X
X <-# ancestral AFs
out$p_anc
p_anc <-# intermediate independent subpopulation AFs
out$p_subpops
p_subpops <-# individual-specific AFs
out$p_ind p_ind <-
```

```
# inspect distribution of ancestral AFs (~ Uniform(0.01, 0.5))
# shrink default margins for these figures
par(mar = c(4, 4, 0, 0) + 0.2)
hist(p_anc, xlab = 'Ancestral AF', main = '', xlim = c(0, 1))
```

```
# distribution of intermediate population AFs
# (all subpopulations combined)
# (will be more dispersed toward 0 and 1 than ancestral AFs)
hist(p_subpops, xlab = 'Intermediate Subpopulation AF', main = '', xlim = c(0, 1))
```

```
# distribution of individual-specific AFs (admixed individuals)
# (admixture reduces differentiation, so these resemble ancestral AFs a bit more)
hist(p_ind, xlab = 'Individual-specific AF', main = '', xlim = c(0, 1))
```

```
# genotype distribution of admixed individuals
barplot(table(X), xlab = 'Genotypes', ylab = 'Frequency', col = 'white')
```

Let’s verify that the correlation structure of the genotypes matches the theoretical coancestry matrix we constructed earlier. For this we use the `popkin`

function of the package with the same name.

```
# for best estimates, group individuals into subpopulations using the geography
# this averages more individuals in estimating the minimum kinship
ceiling( ( 1 : n_ind ) / n_ind * k_subpops )
subpops <-table(subpops) # got k_subpops = 10 with 100 individuals each
#> subpops
#> 1 2 3 4 5 6 7 8 9 10
#> 10 10 10 10 10 10 10 10 10 10
# now estimate kinship using popkin
popkin(X, subpops)
kinship_estimate <-# replace diagonal with inbreeding coeffs. to match coancestry matrix
inbr_diag(kinship_estimate)
coancestry_estimate <-
# Visualize the coancestry matrix using "popkin"!
plot_popkin(
list( coancestry, coancestry_estimate ),
titles = c('Truth', 'Estimate'),
# second value is top margin, for panel titles
mar = c(0, 2.5),
leg_title = 'Coancestry'
)
```

We can also look at the structure of the two allele frequency matrices, `p_subpops`

and `p_ind`

, using `popkin_af`

, which estimates coancestry from allele frequency matrices (in contrast, `popkin`

estimates kinship from genotype matrices, and is preferred when genotypes are available).

```
popkin_af( p_subpops )
coancestry_subpops_estimate <- popkin_af( p_ind ) coancestry_ind_estimate <-
```

The first coancestry matrix is small, and matches the expectation of independent subpopulations (zero values off-diagonal) with the subpopulation inbreeding coefficients as the diagonal coancestry values:

```
plot_popkin(
list( diag( inbr_subpops ), coancestry_subpops_estimate ),
titles = c('Truth', 'Estimate'),
mar = c(0, 2.5),
leg_title = 'Coancestry'
)
```

The second coancestry matrix, estimated from individual-specific allele frequencies, is the same in expectation to the true coancestry matrix of the model:

```
plot_popkin(
list( coancestry, coancestry_ind_estimate ),
titles = c('Truth', 'Estimate'),
mar = c(0, 2.5),
leg_title = 'Coancestry'
)
```

The last coancestry matrix also equals in expectation the transformed kinship matrix estimated earlier, but this estimate from allele frequencies is less noisy. However, in real data such allele frequencies are not observed, and their estimates can vary in quality and result in biases, so the above is more a validation of the simulation than a practical estimation approach.

The random variables generated by `draw_all_admix`

above can also be generated separately using the following functions (where \(p\) is the usual variable symbol for allele frequencies):

`draw_p_anc`

(drawn ancestral allele frequencies)`draw_p_subpops`

(draw independent subpopulation allele frequencies)`make_p_ind_admix`

(construct individual-specific allele frequencies under the admixture model)`draw_genotypes_admix`

(draw genotypes under the admixture model)

These functions are provided for greater flexibility (examples follow further below). However, the joint function `draw_all_admix`

has the advantage of removing fixed loci (loci that were randomly set to 0 or 2 for all individuals, despite non-zero allele frequencies), which are re-drawn from scratch (starting from the ancestral allele frequencies).

Here is the step-by-step procedure for drawing AFs and genotypes in the default `draw_all_admix`

(except for the re-drawing of fixed loci):

```
# reuse the previous m_loci, inbr_subpops, admix_proportions
# draw ancestral AFs
draw_p_anc(m_loci)
p_anc <-# draw intermediate independent subpopulations AFs
draw_p_subpops(p_anc, inbr_subpops)
p_subpops <-# draw individual-specific AFs
make_p_ind_admix(p_subpops, admix_proportions)
p_ind <-# draw genotypes
draw_genotypes_admix(p_ind) X <-
```

We provide functions for these separate steps to allow for more flexibility. It is worth noting that `draw_p_anc`

can be used to draw from a Symmetric Beta distribution if the `beta`

parameter is set. In the example below, a small shape value of `beta`

\(< 1\) is used to draw a distribution with more rare minor variants.

```
# this increases the proportion of rare alleles
draw_p_anc(m_loci, beta = 0.1)
p_anc_beta <-# shrink default margins for these figures
par(mar = c(4, 4, 0, 0) + 0.2)
hist(p_anc_beta, xlab = 'Ancestral AF', main = '', xlim = c(0, 1))
```

However, if you want any other distribution, that can be implemented at this step. For a ridiculous example, the ancestral allele frequencies could be drawn from a (non-symmetric) Beta instead of using `draw_p_anc`

:

```
# this increases the proportion of rare alleles
rbeta(m_loci, shape1 = 0.1, shape2 = 1)
p_anc_beta <-# shrink default margins for these figures
par(mar = c(4, 4, 0, 0) + 0.2)
hist(p_anc_beta, xlab = 'Ancestral AF', main = '', xlim = c(0, 1))
```

You could also draw genotypes from the ancestral population or the intermediate populations:

```
# draw genotypes for one individual from the ancestral population
# use "cbind" to turn the vector p_anc into a column matrix
# ("draw_genotypes_admix" expects a matrix)
draw_genotypes_admix( cbind( p_anc ) )
X_anc <-# returns a column matrix:
dim(X_anc)
#> [1] 10000 1
# draw genotypes from intermediate populations
# draws one individual per intermediate population
draw_genotypes_admix(p_subpops) X_subpops <-
```

Here we show examples for functions that create admixture matrices for various simple population structures. The admixture scenarios implemented in `bnpsd`

are generated by these functions:

`admix_prop_1d_linear`

: Linear 1D geography`admix_prop_1d_circular`

: Circular 1D geography`admix_prop_indep_subpops`

: Independent Subpopulations

The main example (above) was for `admix_prop_1d_linear`

, the rest follow.

This is a twist on the earlier 1D geography where subpopulations and individuals are placed on a circumference, so random walks wrap around and the appropriate density is the Von Misses distribution.

Let’s generate an analogous population structure to the original “linear” example.

```
# data dimensions
# number of individuals
100
n_ind <-# number of intermediate subpops
10
k_subpops <-
# define population structure
# subpopulation FST vector, up to a scalar
1 : k_subpops
inbr_subpops <-# desired bias coefficient
0.5
bias_coeff <-# desired FST for the admixed individuals
0.1
Fst <-
# admixture proportions from *circular* 1D geography
admix_prop_1d_circular(
obj <-n_ind = n_ind,
k_subpops = k_subpops,
bias_coeff = bias_coeff,
coanc_subpops = inbr_subpops,
fst = Fst
) obj$admix_proportions
admix_proportions <- obj$coanc_subpops
inbr_subpops <-
# get pop structure parameters of the admixed individuals
coanc_admix(admix_proportions, inbr_subpops)
coancestry <-
# verify that we got the desired FST!
fst_admix(admix_proportions, inbr_subpops)
#> [1] 0.1
# verify that we got the desired bias_coeff too!
mean(coancestry) / Fst
#> [1] 0.5
```

```
# visualize the per-subpopulation inbreeding coefficients (FSTs)
# tweak margins/etc
par(mar = c(2.5, 2.5, 0.3, 0) + 0.2, lab = c(2, 1, 7), mgp = c(1.5, 0.5, 0))
# colors for independent subpopulations
brewer.pal(k_subpops, "Paired")
col_subpops <-barplot(inbr_subpops, col = col_subpops, names.arg = colnames(admix_proportions), xlab = 'Subpopulation', ylab = 'Inbr')
```

```
# visualize the admixture proportions
# tweak margins/etc
par(mar = c(1, 4, 0.4, 0) + 0.2, lab = c(2, 2, 7))
barplot(
t(admix_proportions),
col = col_subpops,
border = NA,
space = 0,
ylab = 'Admix prop'
)mtext('Individuals', 1)
```

```
# Visualize the coancestry matrix using "popkin"!
plot_popkin(
coancestry,leg_n = 3,
mar = c(0, 0.4),
leg_title = 'Coancestry'
)
```

The independent subpopulations model, where individuals are actually unadmixed, is the most trivial form of the BN-PSD admixture model.

```
# define population structure
# we'll have k_subpops = 3, each subpopulation with these sizes:
100
n1 <- 50
n2 <- 20
n3 <-
# here's the labels (for simplicity, list all individuals of S1 first, then S2, then S3)
c(
labs <-rep.int('S1', n1),
rep.int('S2', n2),
rep.int('S3', n3)
)# data dimensions infered from labs:
# number of individuals "n_ind"
length(labs)
#> [1] 170
# number of subpopulations "k_subpops"
length(unique(labs))
k_subpops <-
k_subpops#> [1] 3
# desired admixture matrix
admix_prop_indep_subpops(labs)
admix_proportions <-
# got a numeric matrix with a single 1 value per row
# (denoting the sole subpopulation from which each individual draws its ancestry)
head(admix_proportions, 2)
#> S1 S2 S3
#> [1,] 1 0 0
#> [2,] 1 0 0
# construct the intermediate subpopulation FST vector
# the desired final FST
0.2
Fst <-# subpopulation FST vector, unnormalized so far
1 : k_subpops
inbr_subpops <-# normalized to have the desired FST
# NOTE fst is a function in the `popkin` package
inbr_subpops / fst(inbr_subpops) * Fst
inbr_subpops <-# verify FST for the intermediate subpopulations
fst(inbr_subpops)
#> [1] 0.2
# get coancestry of the admixed individuals
coanc_admix(admix_proportions, inbr_subpops)
coancestry <-# before getting FST for individuals, weigh then inversely proportional to subpop sizes
weights_subpops(labs) # function from `popkin` package
weights <-# verify FST for individuals (same as for intermediate subpops for this pop structure)
fst_admix(admix_proportions, inbr_subpops, weights)
#> [1] 0.2
```

```
# visualize the per-subpopulation inbreeding coefficients (FSTs)
# tweak margins/etc
par(mar = c(2.5, 2.5, 0, 0) + 0.2, lab = c(2, 1, 7), mgp = c(1.5, 0.5, 0))
# colors for independent subpopulations
brewer.pal(k_subpops, "Paired")
col_subpops <-barplot(inbr_subpops, col = col_subpops, names.arg = colnames(admix_proportions), xlab = 'Subpopulation', ylab = 'Inbr')
```

```
# visualize the admixture proportions
# tweak margins/etc
par(mar = c(1, 4, 0.4, 0) + 0.2, lab = c(2, 2, 7))
barplot(
t(admix_proportions),
col = col_subpops,
border = NA,
space = 0,
ylab = 'Admix prop'
)mtext('Individuals', 1)
```

```
# Visualize the coancestry matrix using "popkin"!
plot_popkin(
coancestry,leg_n = 3,
mar = c(0, 0.4),
leg_title = 'Coancestry'
)
```

Balding, D. J., and R. A. Nichols. 1995. “A Method for Quantifying Differentiation Between Populations at Multi-Allelic Loci and Its Implications for Investigating Identity and Paternity.” *Genetica* 96 (1-2): 3–12.

Ochoa, Alejandro, and John D. Storey. 2016. “FST and Kinship for Arbitrary Population Structures I: Generalized Definitions.” *bioRxiv* doi:10.1101/083915. https://doi.org/10.1101/083915.

———. 2021. “Estimating FST and Kinship for Arbitrary Population Structures.” *PLoS Genet* 17 (1): e1009241. https://doi.org/10.1371/journal.pgen.1009241.

Pritchard, J. K., M. Stephens, and P. Donnelly. 2000. “Inference of Population Structure Using Multilocus Genotype Data.” *Genetics* 155 (2): 945–59.