QTL analysis

This section describes a step-by-step instruction for QTL analysis.

Input data file format

The package FlxQTL does not require any particular data format. Any file readable in Julia is fine, but the input should contain traits (or phenotypes), genotypes (or genotype/allele probabilities), marker information on marker names, chromosomes, and marker positions. All inputs are types of Arrays (Float64) in Julia and should have no missing values, i.e. imputation is required if missing values exist.

Reading the data files and processing arrays

Use any Julia package able to read data files (.txt, .csv, etc.). Julia's built-in module DelimitedFiles supports "read" and "write" files. Let's try using an example dataset in FlxQTL. It is plant data: Arabidopsis thaliana in the data folder. Detailed description on the data can be referred to README in the folder.

using DelimitedFiles

pheno = readdlm("data/Arabidopsis_fitness.csv",',';skipstart=1); # skip to read the first row (column names) to obtain a matrix only

geno = readdlm("data/Arabidopsis_genotypes.csv",',';skipstart=1); 

markerinfo = readdlm("data/Arabidopsis_markerinfo_1d.csv",',';skipstart=1);

For efficient computation, the normalization of matrices is necessary. The phenotype matrix labelled as pheno here composes of a wide range of real values from 1.774 to 34.133, so that it is better to narow the range in [0,1], [-1,1], or any narrower interval for easy computation. Note that the dimension of a phenotype matrix should be the number of traits (or phenotypes) x the number of individuals, i.e. m x n.

using Statistics, StatsBase
Y=convert(Array{Float64,2},pheno'); #convert from transposed one to a Float64 matrix
Ystd=(Y.-mean(Y,dims=2))./std(Y,dims=2); # sitewise normalization

!!! Note

• If the data are skewed or have outliers, simple standadization may not resolve them. You may use a Y_huber() function to rescale the data to be less sensitve to skewness or outliers.

In the genotype data, 1, 2 indicate Italian, Swedish parents, respectively. You can rescale the genotypes for efficiency or interpretability.

geno[geno.==1.0].=0.0;geno[geno.==2.0].=1.0; # or can do geno[geno.==1.0].=-1.0 for only genome scan

For genome scan, we need restructure the standardized genotype matrix combined with marker information. Note that the genome scan in FlxQTL is implemented by CPU parallelization, so we need to add workers (or processes) before the scan starts. Depending on the computer CPU, one can add as many processes as possible. If your computer has 16 cores, then you can add 15 or little more. Note that you need to type @everywhere followed by using PackageName to dispatch the package availabe to all workers for parallel computing. The dimension of a genotype (probability) matrix should be the number of markers x the number of individuals, i.e. p x n.

using Distributed
addprocs(4) 
@everywhere using FlxQTL 
XX=Markers(markerinfo[:,1],markerinfo[:,2],markerinfo[:,3],geno') # marker names, chromosomes, marker positions, genotypes

Optionally, one can generate a trait covariate matrix (Z). The first column indicates overall mean between the two regions, and the second implies site difference: -1 for Italy, and 1 for Sweden.

Z=hcat(ones(6),vcat(-ones(3),ones(3)))
m,q = size(Z) # check the dimension

Computing a genetic relatedness matrix (GRM)

The submodule GRM contains functions for computing kinship matrices, kinshipMan, kinship4way, kinshipLin, kinshipCtr, and computing a 3D array of kinship matrices for LOCO (Leave One Chromosome Out) with a shrinkage method for nonpositive definiteness, shrinkg, shrinkgLoco, kinshipLoco. Note that the shrinkage option is only used for kinshipMan, kinship4way.

For the Arabidopsis genotype data, we will use a genetic relatedness matrix using manhattan distance measure, kinshipMan with a shrinkage with the LOCO option.

Kg = shrinkgLoco(kinshipMan,10,XX)

For no LOCO option with shrinkage,

K = shrinkg(kinshipMan,10,XX.X)

1D genome scan

The new verson of FlxQTL is operated by penalized log-likelihood function using Prior with df_prior for a error term, $\Sigma$, distributed by Inverse-Wishart distribution for numerial stability. One can also adjust them in the Keyword arguments. The default positive definite scale matrix is a large scaled matrix (Prior = cov(Y,dims=2)*3). We rather recommend controlling degrees of freedom (df_prior), i.e. $m+1 (default) \le df\_prior < 2m$, or updating the null parameter estimates (H0_up = true) for numerical stability when analyzing higher dimenional trait data if any singularity error occurrs.

!!! Note

• Setting df_prior = Int64(ceil(1.9m)) works most cases. Any tolerance (itol, tol0, tol) of base algorithms may be adjusted but is not needed in most cases.

Once all input matrices are ready, we need to proceed the eigen-decomposition to a relatedness matrix.

Tg,λg = K2eig(Kg, true) # for eigen decomposition to one kinship with LOCO

For eigen decomposition to one kinship with no LOCO option,

T,λ = K2eig(K) # no LOCO

Now start with 1D genome scan with (or without) LOCO including Z or not. For the genome scan with LOCO including Z,

LODs,B,est0 = geneScan(1,Tg,Λg,Ystd,XX,Z,true); # FlxQTL for including Z (trait covariates)

For the genome scan with LOCO excluding Z, i.e. an identity matrix, we have two options: a FlxQTL model and a conventional MLMM (slower).

LODs,B,est0 = geneScan(1,Tg,Λg,Ystd,XX,true); # FlxQTL for Z=I 

LODs,B,est0 = geneScan(1,Tg,Λg,Ystd,XX,diagm(ones(m)),true); # or equivalently, FlxQTL for Z=I 

lods,b,Est0 = geneScan(Tg,Λg,Ystd,XX,1,true); # MLMM

Or one can return $\log_{10}P$ instead of LOD scores using logP = true in the keyword argument for all the geneScan() functions.

lnP, B, est0 = geneScan(1,Tg,Λg,Ystd,XX,true;logP=true); 

Note that the first argument in geneScan is cross::Int64, which indicates a type of genotype or genotype probability. For instance, if you use a genotype matrix whose entry is one of 0,1,or 2, type 1. If you use genotype probability matrices, depending on the number of alleles or genotypes in a marker, one can type the corresponding number. i.e. 4-way cross: 4, HS DO mouse: 8 for alleles, 32 for genotypes, etc.

For no LOCO option,

LOD,B1,est00 = geneScan(1,T,λ,Ystd,XX,Z);

LOD,B1,est00 = geneScan(1,T,λ,Ystd,XX); # Z=I 

LOD,B1,est00 = geneScan(1,T,λ,Ystd,XX,diagm(ones(m))); #Z=I

lod,b1,Est00 = geneScan(T,λ,Ystd,1,XX); # MLMM

The function geneScan() has three outputs: LOD scores (LODs), effects matrix under H1 (B), and parameter estimates under H0 (est0), which is an Array{Any,1}. If you want to see null parameter esitmate in chromosome 1 for LOCO option, for instance, type est0[1].B, est0[1].loglik, est0[1].τ2, est0[1].Σ, which is a struct of Approx, if H0_up=true for higher dimensional trait data ($m\ge 16\sim 18$) in the keyword argument. The default (false) returns a struct of Result estimated by MLMM for lower dimensional traits. In particular, you can extract values from each matrix in B (3D array of matrices) to generate an effects plot. To print an effect size matrix for the third marker, type B[:,:,3], where the last dimension is the order of a marker in the genotype (probability) data.

Generating plots

The QTLplot module is currently unavailable but plotting functions will be replaced with BigRiverQTLPlots.jl soon.

Performing a permutation test

Since the statistical inference for FlxQTL relies on LOD scores, we have now added three kinds of permutation test functions including the existing no LOCO based functions to estimate thresholds for a type I error: permTest(), mlmmTest(), and permutationTest(). The first two functions are estimated with no LOCO for the flxQTL and MLMM models while the third function is with LOCO either partially or all in implementation designed for both low- and high-dimensional traits to improve threshod estimation. By setting a keyword argument, LOCO_all = true, the function estimates null parameters from the actual data with LOCO, followed by LOCO scan with permuted data to obtain the distribution of maximum LOD scores, which is better for higher dimensional traits, i.e. $m \\ge 16 \\sim 18$ depending on the data. The default setting only estimates a GRM without LOCO inside the function, but all scan with permuted data is implemented with LOCO. This is suitable for lower dimensional traits.

The first argument is nperm::Int64 to set the number of permutations for the test. For keyword arguments, pval=[0.05 0.01] is default to get thresholds of type I error rates (α), and the identity matrix, i.e. Z=diagm(ones(m)) is default. Note that permutation test is implemented by no LOCO option since the difference between no LOCO and LOCO options for many simulations would be very minor.

maxLODs, H1par_perm, cutoff = permTest(1000,1,K,Ystd,XX;Z=Z,pval=[0.05]) # cutoff at 5 %

maxlods, H1par_perm1, cutoff1 = mlmmTest(1000,1,K,Ystd,XX;pval=[0.05])  # for MLMM

Maxlods, H1par_loco, cutoff2 = permutationTest(1000,1,Kg,Ystd,XX;Z=Z) # for low dimensional traits: LOCO_all=false

2D genome scan

A gene2Scan() function has two options just as geneScan functions. Note that one needs a coarser genotype (probability) matrix since the distance between any two markers in a chromosome is very close each other, often yielding a numerical error during the operation. This will be worse when the conversional MLMM is chosen to implement. The provided data for 2D scan can be processed by the R/qtl library with a sim.geno function by picking one of the simulated data sets, where step=5 and draws=16 are set for this Arabidopsis data.

geno_2d = readdlm("data/Arabidopsis_genotypes_2d.csv",',';skipstart=1); 

markerinfo_2d = readdlm("data/Arabidopsis_markerinfo_2d.csv",',';skipstart=1);

geno_2d[geno_2d.==1.0].=0.0;geno_2d[geno_2d.==2.0].=1.0; # or can do geno_2d[geno_2d.==1.0].=-1.0 for only genome scan

X2=Markers(markerinfo_2d[:,1],markerinfo_2d[:,2],markerinfo_2d[:,3],geno_2d) # marker names, 

LOD_2d,B2,est02 = gene2Scan(1,T,Λg,Ystd,XX,true);  # for Z=I included as default in keyword arguments

lod_2d,b2,Est02 = gene2Scan(Tg,Λg,Ystd,XX,1,true) # MLMM

Or one can do with no LOCO as explained as in aforementioned 1D genome scan.