- Open Access
kruX: matrix-based non-parametric eQTL discovery
© Qi et al.; licensee BioMed Central Ltd. 2014
- Received: 10 October 2013
- Accepted: 11 January 2014
- Published: 14 January 2014
The Kruskal-Wallis test is a popular non-parametric statistical test for identifying expression quantitative trait loci (eQTLs) from genome-wide data due to its robustness against variations in the underlying genetic model and expression trait distribution, but testing billions of marker-trait combinations one-by-one can become computationally prohibitive.
We developed kruX, an algorithm implemented in Matlab, Python and R that uses matrix multiplications to simultaneously calculate the Kruskal-Wallis test statistic for several millions of marker-trait combinations at once. KruX is more than ten thousand times faster than computing associations one-by-one on a typical human dataset. We used kruX and a dataset of more than 500k SNPs and 20k expression traits measured in 102 human blood samples to compare eQTLs detected by the Kruskal-Wallis test to eQTLs detected by the parametric ANOVA and linear model methods. We found that the Kruskal-Wallis test is more robust against data outliers and heterogeneous genotype group sizes and detects a higher proportion of non-linear associations, but is more conservative for calling additive linear associations.
kruX enables the use of robust non-parametric methods for massive eQTL mapping without the need for a high-performance computing infrastructure and is freely available from http://krux.googlecode.com.
- Non-parametric methods
- Matrix algebra
Genome-wide association studies have identified hundreds of DNA variants associated to complex traits including disease in human alone . To understand how these variants affect disease risk, genotype and organismal phenotype data are integrated with intermediate molecular phenotypes to reconstruct disease networks . A first step in this procedure is to identify DNA variants that underpin variations in expression levels (eQTLs) of transcripts , proteins  or metabolites . As modern technologies routinely produce genotype and expression data for a million or more single-nucleotide polymorphisms (SNPs) and ten-thousands of molecular abundance traits in a single experiment, often repeated across multiple cell or tissue types, the number of statistical tests to be performed when testing each SNP for association to each trait is huge. Furthermore, multiple testing correction requires all tests to be repeated several times on permuted data to generate an empirical null distribution. Despite being trivially parallelisable, the computational burden of testing SNP-trait associations one-by-one quickly becomes prohibitive.
Recently a new approach ("matrix-eQTL") was developed which uses the fact that the test statistics for the additive linear regression and ANOVA models can be expressed as multiplications between rescaled genotype and expression data matrices, thereby realising a dramatic speed-up compared to traditional QTL-mapping algorithms . A limitation of these models is their assumption that the expression data is always normally distributed within each genotype group. For this reason, QTL and eQTL studies have frequently used non-parametric methods which are more robust against variations in the underlying genetic model and trait distribution [7, 8]. In particular, the non-parametric Kruskal-Wallis one-way analysis of variance  does not assume normal distributions and reports small P-values if the median of at least one genotype group is significantly different from the others .
Here we report a matrix-based algorithm ("kruX"), implemented in Matlab, Python and R, to simultaneously calculate the Kruskal-Wallis test statistics for several millions of SNP-trait pairs at once that is more than ten thousand times faster than calculating them one-by-one on a human test dataset with more than 500,000 SNPs and 20,000 expression traits. Additional benefits of kruX include the explicit handling of missing values in both genotype and expression data and the support of genetic markers with any number of alleles, including variable allele numbers within a single dataset.
KruX takes as input genotype values of M genetic markers and expression levels of N transcripts, proteins or metabolites in K individuals, organised in an M × K genotype matrix G and N × K expression data matrix D. Genetic markers take values 0,1,…,ℓ, where ℓ is the maximum number of alleles (ℓ = 2 for biallelic markers), while molecular traits take continuous values. We use built-in functions of Matlab, Python and R to convert the expression data matrix D to a matrix R of data ranks, ranked independently over each row (i.e. molecular trait). KruX assumes that the input expression data has been adjusted for covariates if it is necessary to do so [10, 11] and all data quality control has been performed.
Calculation of the Kruskal-Wallis test statistic by matrix multiplication
using efficient vectorised operations. If none of the rows in D contain ties, then each entry S(n,m) equals the Kruskal-Wallis test statistic for testing trait n against marker m . For markers with less than the maximum of ℓ genotype values, 0/0 division will result in NaN columns in the intermediate matrices with entries S i (n,m)2/N i (m) for the empty genotype groups. By replacing all NaN’s by zeros before making the sum in eq. (2), the corresponding entries in S will be the correct statistics for a test with fewer than ℓ degrees of freedom. Thus we need ℓ + 1 matrix multiplications and the associated element-wise operations to calculate the test statistic values for all marker-trait combinations.
P-value calculation and empirical FDR correction
KruX takes as input a P-value threshold P c , calculates the corresponding test statistic thresholds for d degrees of freedom (d = 1,…,ℓ - 1), and identifies the entries in S which exceed the appropriate threshold value. For these entries only a P-value is calculated using the χ2 distribution. Empirical false-discovery rate (FDR) values are computed by repeating the P-value calculation (with the same P c ) multiple times on data where the columns of the expression data ranks are randomly permuted. The FDR value for any value P ≤ P c is defined as the ratio of the average number of associations with P′ ≤ P in the randomised data to the number of associations with P′ ≤ P in the real data.
Handling missing values
where Z is the N × K matrix with Z(n,k) = 0 whenever D(n,k) = NaN and 1 otherwise, produces the corrected number of individuals in the ith group of the mth marker for the nth trait. Replacing the constant K in eq. (2) by a N × M matrix K where K(n,m) is the number of non-missing samples for trait n and performing element-wise division and substraction operations then gives the correct test statistic for all pairs.
Handling missing genotype data is less easy because the expression ranks that need to be adjusted are specific to each marker-trait combination (e.g if a marker has a missing value where a trait has rank r1, then all samples with ranks r = r1 + 1,…,K need to be lowered by 1). KruX uses the fact that missing genotype values are generally due to sample quality and therefore patterns of missing values are often repeated among markers. For each unique missing value pattern, a new genotype matrix for all markers with that pattern and a new expression data matrix with the corresponding samples removed are constructed to calculate the test statistics for all affected marker gene combinations. Missing genotype data increases the computational cost of the algorithm considerably and it is recommended to limit the number of missing values by only considering markers with a sufficiently high call rate.
Handling tied data
In the presence of tied observations, the statistic in eq. (2) needs to be divided by a factor , where the summation is over all groups of ties and T = t3 - t for each group of ties, with t the number of tied data in the group . The factor T is automatically computed for each trait during the ranking step and the matrix S is therefore easily corrected using element-wise matrix operations (Matlab version only). Whereas ties are usually rare in standard gene expression datasets, the ability to handle tied data expands the scope of kruX to count-based, discretised or qualitative data types.
Since kruX needs to create intermediate matrices of size N × M, where N is the number of traits and M the number of markers, which do not usually fit into memory for large datasets, kruX supports the use of data ‘slices’ to divide the complete data into manageable chunks. In typical applications, the number of markers is one or two orders of magnitude larger than the number of traits. Therefore the default behaviour of kruX is to keep the expression data as a single matrix and simultaneously test all traits against subsets of markers. The user can provide either a slice size and kruX will process marker blocks of this size serially, or a slice size and initial marker and kruX will process a single slice starting from that marker. The latter option allows trivial parallelisation across multiple processors.
To test kruX we provide example analysis scripts and a small anonymised dataset of 2,000 randomly selected genes and markers from 100 randomly selected yeast segregants . Here we describe an application of kruX on a human dataset of 19,610 genes and 530,222 SNP markers measured in 102 whole blood samples from the Stockholm Atherosclerosis Gene Expression (STAGE) study . All SNPs in the dataset had minor allele frequency greater than 5%, no missing values and probability to be in Hardy-Weinberg equilibrium greater than 10-6.
kruX is exact and fast
The Kruskal-Wallis test is more conservative than corresponding parametric tests
The Kruskal-Wallis test is more robust and detects more non-linear associations
We have developed kruX, a software tool that uses matrix multiplications to simultaneously calculate the Kruskal-Wallis test statistics for millions of marker-trait combinations in a single operation, thereby realising a dramatic speed-up compared to calculating the test statistics one-by-one. The availability of a fast method to identify eQTL associations using a non-parametric test allowed us to assess in more detail how differences in model assumptions compared to parametric methods lead to differences in identified eQTLs. Our results on a typical human dataset indicate that the the parametric ANOVA method is highly sensitive to the presence of outlying gene expression values and SNPs with singleton genotype groups. We caution against its use without prior filtering of such outliers. Linear models reported the highest number of eQTL associations after empirical FDR correction. These are understandably biased towards additive linear associations and were also sensitive to the presence of skewed genotype group sizes, albeit to a much lesser extent than the parametric ANOVA method. The Kruskal-Wallis test on the other hand is robust against data outliers and heterogeneous genotype group sizes and detects a higher proportion of non-linear associations, but it is more conservative for calling additive linear associations than linear models, even after FDR correction.
In summary, kruX enables the use of non-parametric methods for massive eQTL mapping without the need for a high-performance computing infrastructure.
Project name: kruX
Project home page:http://krux.googlecode.com
Operating systems: Platform independent
Programming language: Matlab, R, Python
Other requirements: None
License: GNU GPL v3
Any restrictions to use by non-academics: None
This work was supported by the Freiburg Institute for Advanced Studies and Roslin Institute Strategic Grant funding from the BBSRC.
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