Linear mixed model genome scans for many traits
Author senresearch
3 Stars
Updated Last
11 Months Ago
Started In
July 2022


BulkLMM is a Julia package for performing genome scans for multiple traits (in "Bulk" sizes) using linear mixed models (LMMs). It is suitable for eQTL mapping with thousands of traits and markers. BulkLMM also performs permutation testing for LMMs taking into account the relatedness of individuals. We use multi-threading and matrix operations to speed up computations.

The current implementation is for genome scans with one-degree of freedom tests with choices of adding additional covariates. Future releases will cover the scenario of more-than-one degrees of freedom tests.

Linear Mixed Model (LMM)

We consider the case when a univariate trait of interest is measured in a population of related individuals with kinship matrix $K$. Let the trait vector, $y$ follow the following linear model.

$$ y = X\beta + \epsilon,$$


$$V(\epsilon) = \sigma^2_g K+\sigma^2_e I.$$

where $X$ is a matrix of covariates which would include the intercept, candidate genetic markers of interest, and (optionally) any background covariates. The variance components $\sigma^2_g$ and $\sigma^2_e$ denote the genetic and random error variance components respectively.

Single trait scan

For a single trait and candidate marker, we use a likelihood ratio test to compare a model with and without the candidate genetic marker (and including the intercept and all background covariates). This process is repeated for each marker to generate the genome scan. The result is reported in LOD (log 10 of likelihood ratio) units.

Users can specify if the variance components should be estimated using ML (maximum likelihood) or REML (restricted maximum likelihood). The scans can be performed with the variance components estimated once under the null, or separately for each marker. The latter approach is slower, but more accurate.

Permutation tests for single trait

Under the null hypothesis that no individual genetic marker is associated with the trait, traits are correlated according if the kinship matrix is not identity, and the genetic variance component is non-zero. Thus, a standard permutation test where we shuffle the trait data randomly, is not appropriate. Instead, we rotate the data using the eigen decomposition of the kinship matrix, which de-correlates the data, and then shuffle the data after rescaling them by their standard deviations.

Scans for multiple traits

Scans for multiple traits are performed by running univariate LMMs for each combination of trait and marker. We are exploring algorithms for optimizing this process by judicious use of approximations.


This package uses multi-threading to speed up some operations. You will have to start Julia with mutliple threads to take advantage of this. You should use as many threads as your computer is capable of. Further speedups may be obtained by spreading (distributing) the computation across mutliple computers.

Example: application on BXD spleen expression data

We demonstrate basic usage of BulkLMM.jl through an example applying the package on the BXD mouse strains data.

First, after successfully installed the package, load it to the current Julia session by

using BulkLMM
using CSV, DelimitedFiles, DataFrames, Statistics

The BXD data are accessible through our published github repo of the BulkLMM.jl package as .csv files under the data/bxdData directory.

The raw BXD traits BXDtraits_with_missing.csvcontains missing values. After removing the missings, load the BXD traits data

bulklmmdir = dirname(pathof(BulkLMM));
pheno_file = joinpath(bulklmmdir,"..","data/bxdData/spleen-pheno-nomissing.csv");
pheno = readdlm(pheno_file, ',', header = false);
pheno_processed = pheno[2:end, 2:(end-1)].*1.0; # exclude the header, the first (transcript ID)and the last columns (sex)

Required data format for traits should be .csv or .txt files with values separated by ',', with each column being the observations of $n$ BXD strains on a particular trait and each row being the observations on all $m$ traits of a particular mouse strain.

Also load the BXD genotypes data. The raw BXD genotypes file BXDgeno_prob.csv contains even columns that each contains the complement genotype probabilities of the column immediately preceded (odd columns). Calling the function readBXDgeno will read the BXD genotype file excluding the even columns.

geno_file = joinpath(bulklmmdir,"..","data/bxdData/spleen-bxd-genoprob.csv");
geno = readdlm(geno_file, ',', header = false);
geno_processed = geno[2:end, 1:2:end] .* 1.0;

Required data format for genotypes should be .csv or .txt files with values separated by ',', with each column being the observations of genotype probabilities of $n$ BXD strains on a particular marker place and each row being the observations on all $p$ marker places of a particular mouse strain.

For the BXD data,

size(pheno_processed) # (number of strains, number of traits)
(79, 35554)
size(geno_processed) # (number of strains, number of markers)
(79, 7321)

Compute the kinship matrix $K$ from the genotype probabilities using the function calcKinship

kinship = calcKinship(geno_processed); # calculate K

Single trait scanning:

For example, to conduct genome-wide association mappings on the 1112-th trait, ran the function scan() with inputs of the trait (as a 2D-array of one column), geno matrix, and the kinship matrix.

traitID = 1112;
pheno_y = reshape(pheno_processed[:, traitID], :, 1);
@time single_results = scan(pheno_y, geno_processed, kinship);
  0.059480 seconds (80.86 k allocations: 47.266 MiB)

The output structure single_results stores the model estimates about the variance components (VC, environmental variance, heritability estimated under the null intercept model) and the lod scores. They are obtainable by

# VCs: environmental variance, heritability, genetic_variance/total_variance
(single_results.sigma2_e, single_results.h2_null)
(0.0942525841453798, 0.850587848871709)
# LOD scores calculated for a single trait under VCs estimated under the null (intercept model)

BulkLMM.jl supports permutation testing for a single trait GWAS. Simply run the function scan() and input the optional keyword argument permutation_test = true with the number of permutations passed to the keyword argument nperms = # of permutations. For example, to ask the package to do a permutation testing of 1000 permutations, do

@time single_results_perms = scan(pheno_y, geno_processed, kinship; permutation_test = true, nperms = 1000, original = false);
  0.079464 seconds (94.02 k allocations: 207.022 MiB)

(use the input original = false to suppress the default of performing genome scans on the original trait)

The output single_results_perms is a matrix of LOD scores of dimension p * nperms, with each column being the LOD scores of the $p$ markers on a permuted copy and each row being the marker-specific LOD scores on all permuted copies.

(7321, 1000)
max_lods = vec(mapslices(x -> maximum(x), single_results_perms; dims = 1));
thrs = map(x -> quantile(max_lods, x), [0.05, 0.95]);

Plot the BulkLMM LOD scores of the 1112-th trait and compare with the results from running GEMMA:

Note: to get results from GEMMA, one would need to run GEMMA on a Linux machine with input files of the same trait (here the 1112-th trait, X10339113), genetic markers and the kinship matrix, and finally convert the LRT p-values into corresponding LOD scores. Alternatively, you may simply load the results we obtained by following the procedures mentioned above. The resulting LOD scores from GEMMA are a .txt file in data/bxdData/GEMMA_BXDTrait1112/gemma_lod_1112.txt.


For reproducing this figure, we need to do the following steps:

First, read in the gmap.csv and the phenocovar.csv under data/bxdData/ directory as

gmap_file = joinpath(bulklmmdir,"..","data/bxdData/gmap.csv");
gInfo = CSV.read(gmap_file, DataFrame);
phenocovar_file = joinpath(bulklmmdir,"..","data/bxdData/phenocovar.csv");
pInfo = CSV.read(phenocovar_file, DataFrame);

We would need to use some utility functions for plotting in the package named BigRiverPlots.jl, which will be released soon.

After downloading the package, run

using BigRiverPlots

# Get information for the genetic markers about which chromosome each was measured at
Chr_bxd = string.(gInfo[:, :Chr]);
Chr_bxd = reshape(Chr_bxd, :, 1);

# Get information for the genetic markers about where (in Mb length) on the chromosome each was measured at
Pos_bxd = gInfo[:, :Mb];
Pos_bxd = reshape(Pos_bxd, :, 1);

Lod_bxd = single_results.lod[1:end, :]; # load the BulkLMM LOD scores results
gemma_results_path = joinpath(bulklmmdir,"..","data/bxdData/GEMMA_BXDTrait1112/gemma_lod_1112.txt")
Lod_gemma = readdlm(gemma_results_path, '\t'); # load gemma LOD scores results available in the package

traitName = pInfo[traitID, 1] # get the trait name of the 1112-th trait

Then, to use the functions in the package BigRiverPlots.jl, run

vecSteps_bxd = BigRiverPlots.get_chromosome_steps(Pos_bxd, Chr_bxd)

# get unique chr id
v_chr_names_bxd = unique(Chr_bxd)

# generate new distances coordinates

x_bxd, y_bxd = BigRiverPlots.get_qtl_coord(Pos_bxd, Chr_bxd, Lod_bxd);
x_bxd_gemma, y_bxd_gemma = BigRiverPlots.get_qtl_coord(Pos_bxd, Chr_bxd, Lod_gemma);

qtlplot(x_bxd, y_bxd, vecSteps_bxd, v_chr_names_bxd;
        label = "BulkLMM.jl",
        xlabel = "Locus (Chromosome)", ylims = (0.0, 6.5),
        title = "Single trait $traitName LOD scores") # plot BulkLMM LODs
plot!(x_bxd, y_bxd_gemma, color = :purple, label = "GEMMA", legend = true) # plot GEMMA LODs

hline!([thrs], color = "red", linestyle=:dash, label = "") # plot thresholds...

Multiple traits scanning:

To get LODs for multiple traits, for better runtime performance, first start julia with multiple threads following Instructions for starting Julia REPL with multi-threads or switch to a multi-threaded julia kernel if using Jupyter notebooks.

Then, run the function bulkscan_null() with the matrices of traits, genome markers, kinship. The fourth required input is the number of parallelized tasks and we recommend it to be the number of julia threads.

Here, we started a 16-threaded julia and executed the program on a Linux server with the Intel(R) Xeon(R) Silver 4214 CPU @ 2.20GHz to get the LOD scores for all ~35k BXD traits:

@time multiple_results_allTraits = bulkscan_null(pheno_processed, geno_processed, kinship; nb = Threads.nthreads()).L;
 82.421037 seconds (2.86 G allocations: 710.821 GiB, 41.76% gc time)

The output multiple_results_allTraits is a matrix of LOD scores of dimension $p \times n$, with each column being the LOD scores from performing GWAS on each given trait.

(7321, 35554)

To visualize the multiple-trait scan results, we can use the plotting utility function plot_eQTLto generate the eQTL plot. The functions for plotting utilities will be available in the package BigRiverPlots.jl in the future. For now, we can easily have access to the plotting function in the script plot_utils/visuals_utils.jl, by running the following commands:

using RecipesBase, Plots, Plots.PlotMeasures, ColorSchemes
include(joinpath(bulklmmdir, "..", "plot_utils", "visuals_utils.jl"));

For the following example, we only plot the LOD scores that are above 5.0 by calling the function and specifying in the optional argument thr = 5.0:

plot_eQTL(multiple_results_allTraits, pheno, gInfo, pInfo; thr = 5.0)



The package BulkLMM.jl can be installed by running

using Pkg

To install from the Julia REPL, first press ] to enter the Pkg mode and then use:

add BulkLMM

The most recent release of the package can be obtained by running

using Pkg
Pkg.add(url = "https://github.com/senresearch/BulkLMM.jl", rev="main")

Contact, contribution and feedback

If you find any bugs, please post an issue on GitHub or contact the maintainer (Zifan Yu) directly. You may also fork the repository and send us a pull request with any contributions you wish to make.

Check out NEWS.md to see what's new in each BulkLMM.jl release.