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pulver: an R package for parallel ultrarapid pvalue computation for linear regression interaction terms
BMC Bioinformatics volume 18, Article number: 429 (2017)
Abstract
Background
Genomewide association studies allow us to understand the genetics of complex diseases. Human metabolism provides information about the diseasecausing mechanisms, so it is usual to investigate the associations between genetic variants and metabolite levels. However, only considering genetic variants and their effects on one trait ignores the possible interplay between different “omics” layers. Existing tools only consider singlenucleotide polymorphism (SNP)–SNP interactions, and no practical tool is available for largescale investigations of the interactions between pairs of arbitrary quantitative variables.
Results
We developed an R package called pulver to compute pvalues for the interaction term in a very large number of linear regression models. Comparisons based on simulated data showed that pulver is much faster than the existing tools. This is achieved by using the correlation coefficient to test the nullhypothesis, which avoids the costly computation of inversions. Additional tricks are a rearrangement of the order, when iterating through the different “omics” layers, and implementing this algorithm in the fast programming language C++. Furthermore, we applied our algorithm to data from the German KORA study to investigate a realworld problem involving the interplay among DNA methylation, genetic variants, and metabolite levels.
Conclusions
The pulver package is a convenient and rapid tool for screening huge numbers of linear regression models for significant interaction terms in arbitrary pairs of quantitative variables. pulver is written in R and C++, and can be downloaded freely from CRAN at https://cran.rproject.org/web/packages/pulver/.
Background
Hundreds of genetic variants associated with complex human diseases and traits have been identified by genomewide association studies (GWAS) [1,2,3,4]. However, most GWAS only considered univariate models with one outcome and one independent variable, thereby ignoring possible interactions between different quantitative “omics” data [5], such as DNA methylation, genetic variations, mRNA levels, or protein levels. For example, studies observed associations between specific epigeneticgenetic interactions and a phenotype [6,7,8]. The lack of publications analyzing genomewide interactions may result because of the high computational cost of running linear regressions for all possible pairs of “omics” data. Understanding the interplay among different “omics” layers can provide important insights into biological pathways that underlie health and disease [9].
Previous interaction analyses in genomewide studies mainly considered interactions between singlenucleotide polymorphisms (SNPs), which led to the development of several rapid analysis tools. For example, BiForce [10] is a standalone Java program that integrates bitwise computing with multithreaded parallelization; SPHINX [11] is a framework for genomewide association mapping that finds SNPs and SNP–SNP interactions using a piecewise linear model; and epiGPU [12] calculates contingency tablebased approximate tests using consumerlevel graphics cards.
Several rapid programs are also available for calculating linear regressions without interaction terms. For example, OmicABEL [13] efficiently exploits the structure of the data but does not allow the inclusion of an interaction term. The R package MatrixEQTL [14] computes linear regressions very quickly based on matrix operations. This package also allows for testing for interaction between a set of independent variables and one fixed covariate. However, interactions between arbitrary pairs of quantitative covariates would require iteration over covariates, which is quite inefficient.
Thus, our R package called pulver is the first tool to allow the user to compute pvalues for interaction terms in huge numbers of linear regressions in a practical amount of time. The acronym pulver denotes parallel ultrarapid pvalue computation for linear regression interaction terms.
We benchmarked the performance of our implemented method using simulated data. Furthermore, we applied our algorithm to “omics” data from the Cooperative Health Research in the Region of Augsburg (KORA) F4 study (DNA methylation, genetic variants, and metabolite levels).
KORA comprises a series of independent populationbased epidemiological surveys and followup studies of participants living in the region of Augsburg, Southern Germany [15].
Access to the KORA data can be requested via the KORA.Passt System (https://helmholtzmuenchen.managedotrs.com/otrs/customer.pl).
Implementation
pulver computes pvalues for the interaction term in a series of multiple linear regression models defined by covariate matrices X and Z and an outcome matrix Y, containing continuous data, e.g. metabolite levels, mRNA or proteomics data. In most cases the residuals from the phenotype adjusted for other parameters are used. All matrices must have equal number of rows, i.e., observations. For efficiency reasons, pulver does not adjust for additional covariates, instead the residuals from the phenotype adjusted for other parameters should be used.
Linear regression analysis
For every combination of columns x, y, and z from matrices X , Y, and Z, pulver fits the following multiple linear regression model:
where y is the outcome variable, x and z are covariates, and xz is the interaction (product) of covariates x and z. All variables are quantitative. We need to test the null hypothesis β _{3} = 0 against the alternative hypothesis β _{3} ≠ 0. In particular, we are not interested in estimating the coefficients β _{1} and β _{2}, which allows us to take a computational shortcut. By centering and orthogonalizing the variables, we can reduce the multiple linear regression problem into a simple linear regression without intercept. Thus, we can compute the Student’s ttest statistic for the coefficient β _{3} as a function of the Pearson’s correlation coefficient between y and the orthogonalized xz: \( t=r\sqrt{DF/\left(1{r}^2\right)} \), where DF is the degree of freedom. See the Additional file 1 for a more detailed derivation.
By computing the tstatistic based on the correlation coefficient, which has a very simple expression in the simplified model, we avoid fitting the entire model including estimating the coefficients β _{1} and β _{2}. This is much more efficient because we are actually only interested in the interaction term.
Avoiding redundant computations
Despite the computational shortcut, even more time can be saved by employing a sophisticated arrangement of the computations. The naïve approach would iterate through three nested forloops, with one for each matrix, where all computations occur in the innermost loop. However, Fig. 1 shows that some computations can be moved out of the innermost loop to avoid redundant computations.
Programming language and general information about the program
We implemented the algorithm in an R package [16] called pulver. Due to speed considerations, the core of the algorithm was implemented in C++. We used R version 3.3.1 and compiled the C++ code with gcc compiler version 4.4.7. To integrate C++ into R, we used the R package Rcpp [17] (version 0.12.7).
To determine whether C/Fortran could improve the performance compared to that of C++, we also implemented the algorithm using a combination of C and Fortran via R’s C interface.
We used OpenMP version 3.0 [18] to parallelize the middle loop. To minimize the amount of time required to coordinate parallel tasks, we inverted the order of matrices X and Z so that the middle loop could run over more variables than the outer loop, thereby maximizing the amount of work per thread.
To improve efficiency, the program does not allow covariates other than x and z. If additional covariates are required, the outcome y must be replaced by the residuals from the regression of y on the additional covariates. Missing values in the input matrices are replaced by the respective column mean.
Our pulver package can be used as a screening tool for scenarios where the number of models (number of variables in matrix X × number of variables in matrix Y × number of variables in matrix Z) is too large for conventional tools. By specifying a pvalue threshold, the results can be limited to models with interaction term pvalues below the threshold, thereby reducing the size of the output greatly. After the initial screening process, additional model characteristics for the significant models, e.g., effect estimates and standard errors, can be obtained with traditional methods such as R’s lm function.
The user can access pulver’s functionality via two functions: pulverize and pulverize_all. The pulverize function expects three numeric matrices and returns a table with pvalues for models with interaction term pvalues below the (optionally specified) pvalue threshold. The wrapper function pulverize_all expects files with names containing X, Y, and Z matrices, calls pulverize to perform the actual computation, and returns a table in the same format as pulverize. The pulverize_all function is particularly useful if the matrices are too huge to be loaded all at the same time because of the computer memory restrictions. Thus, pulverize_all gets inputs as lists of file names containing the submatrices X, Y, and Z. pulverize_all iterates through these lists and subsequently loads matrices before calling the pulverize.
Comparisons with other R tools for running linear regressions
As illustrated in Fig. 2, the inputs for the interaction analysis can be vectors or matrices. Compared to other R tools such as lm and MatrixEQTL pulver is currently the only available option for users who want all the inputs to be matrices. It is possible to adapt other tools to allmatrix inputs, but the resulting code is not optimized for this use and will be too slow for practical purposes.
Results
To benchmark the performance of pulver against other tools, we simulated X, Y, and Z matrices with different numbers of observations and variables.
We also applied pulver to real data from the KORA study.
Performance comparison using simulated data
No other tool is specialized for the type of interaction analysis described above, so we compared the speed of our R package pulver with that of R’s builtin lm function and the R package MatrixEQTL [14] (version 2.1.1) (also see Fig. 2).
To ensure a fair comparison, we did not use the parallelization feature of pulverize because it is not available in R’s lm function or MatrixEQTL. However, parallelization is possible and it leads to significant speedups, although sublinear. For benchmarking purposes, each scenario was run 200 times using the R package microbenchmark (version 1.4–2.1, https://CRAN.Rproject.org/package=microbenchmark) and the results were filtered with a pvalue threshold of 0.05.
Figure 3 shows that pulver performed better than the alternatives in all the benchmarks. Note that the benchmark results obtained for the lm function were so slow that they could not be included in the chart.
In particular, for the benchmark where the number of variables in matrix Z was varied (see Fig. 3d), pulver outperformed the other methods by several orders of magnitudes, and the results obtained by MatrixEQTL could not be included in the chart. The poor performance of MatrixEQTL is because it can only handle one Z variable, which forced us to repeatedly call MatrixEQTL for every variable in the Z matrix. This type of iteration is known to be slow in R. The good performance of pulver with benchmark d is particularly notable because this benchmark reflects the intended user case for pulver where all input matrices contain many variables.
Applying pulver to the analysis of realworld data
Metabolites are small molecules in blood whose concentrations can reflect the health status of humans [19]. Therefore, it is useful to investigate the potential effects of genetic and epigenetic factors on the concentrations of metabolites.
DNA methylation denotes the attachment of a methyl group to a DNA base. Methylation occurs mostly on the cytosine nucleotides preceding a guanine nucleotide, which are also called cytosinephosphateguanine (CpG) sites [20]. DNA methylation was measured using the Illumina InfiniumHumanMethylation450 BeadChip platform, which quantifies the relative methylation of CpG sites [21].
DNA methylation was measured in whole blood so it was based on a mixture of different cell types. We employed the method described by Houseman et al. [22] and adjusted for different proportions of cell types. Thus, CpG sites were represented by their residuals after regressing on age, sex, body mass index (BMI), Houseman variables, and the first 20 principal components of the principal component analysis control probes from 450 K Illumina arrays. The control probes were used to adjust for technical confounding, where they comprised the principal components from positive control probes, which were used as quality control for different data preparation and measurement steps.
Furthermore, to avoid false positives, all CpG sites listed by Chen et al. [23] as crossreactive probes were removed. Crossreactive probes bind to repetitive sequences or cohybridize with alternate sequences that are highly homologous to the intended targets, which could lead to false signals.
In the KORA F4 study, genotyping was performed using the Affymetrix Axiom chip [24]. Genotyped SNPs were imputed with IMPUTE v2.3.0 using the 1000 Genomes reference panel.
Metabolite concentrations were measured using two different platforms: Biocrates (151 metabolites) and Metabolon (406 metabolites). Biocrates uses a kitbased, targeted quantitative by electrospray (liquid chromatography) – tandem mass spectrometry (ESI(LC) MS/MS) method. A detailed description of the data was provided previously by Illig et al. [25]. Metabolon uses nontargeted, semiquantitative liquid chromatography coupled with tandem mass spectrometry (LCMS/MS) and GCMS methods. The data were previously described in Suhre et al. [26].
Metabolites were represented by their Box–Cox transformed residuals after regressing on age, sex, and BMI. We used the R package car [27] to compute the Box–Cox transforms.
Initially, there were 345,372 CpG sites, 9,143,401 SNPs (coded as values between 0 and 2 according to an additive genetic model), and 557 metabolites in the dataset. Analyzing the complete data would have taken a very long time even with pulver.
Thus, to estimate the time required to analyze the whole dataset, we ran scenarios using all CpG sites, all metabolites, and different numbers of SNPs (100, 1000, 2000, 4000, and 5000), and extrapolated the runtime that would be required to analyze all SNPs. Due to time limitations, we ran each of the scenarios defined above only once. The estimated runtime required to analyze the complete dataset by parallelizing the work across 40 processors was 1.5 years.
Thus, we decided to only select SNPs that had previously known significant associations with at least one metabolite [1, 25]. We determined whether these signals became even stronger after adding an interaction effect between DNA methylation and SNPs.
To avoid an excessive number of false positives, the SNPs were also required to have a minor allele frequency greater than 0.05. We applied these filters separately to the Biocrates and Metabolon data. After filtering, we had 345,372 CpG sites, 117 SNPs, and 16 metabolites for Biocrates, with 345,372 CpG sites, 6406 SNPs, and 376 metabolites for Metabolon.
We were only interested in associations that remained significant after adjusting for multiple testing, so we used a pvalue threshold of \( \frac{0.05}{345372^{\ast }{117}^{\ast }16+{345372}^{\ast }{6406}^{\ast }376}={6.01}^{\ast }\ {10}^{14} \) according to Bonferroni correction.
We found 27 significant associations for metabolites from the Biocrates platform (pvalues ranging from 1.28^{∗} 10^{−29} to 5.17^{∗} 10^{−14}) and 286 significant associations for metabolites from the Metabolon platform (pvalues ranging from 1.15^{∗} 10^{−42} to 3.73^{∗} 10^{−14}). All of the significant associations involved the metabolite butyrylcarnitine as well as SNPs and CpG sites on chromosome 12 in close proximity to the ACADS gene (see Fig. 4a and b). Figure 4c shows one of the significant results (SNP rs10840791, CpG site cg21892295, and metabolite butyrylcarnitine) to illustrate how the inclusion of an interaction term in the model increased the adjusted coefficient of determination,R^{2} (calculated using the summary.lm function in R).
The ACADS gene encodes the enzyme AcylCoA dehydrogenase, which uses butyrylcarnitine as a substrate [25], and previous studies have shown that SNPs and CpGs in this gene region are independently associated with butyrylcarnitine [1, 4, 25].
Discussion
In the case where interaction terms need to be calculated for arbitrary pairs of variables, pulver performs far better than its competitors. The time savings are achieved by avoiding redundant calculations. Thus, computationally expensive pvalues are only computed at the very end and only for results below a significance threshold determined using the (computationally cheap) Pearson’s correlation coefficient. To maximize the speedup, we recommend always specifying a pvalue threshold and using pulver as a filter to find models with significant or nearsignificant interaction terms. If a pvalue threshold is not specified, the time savings will be suboptimal and the number of results will be very high.
Thus, we recommend using a pvalue threshold to adjust for multiple testing, such as Bonferroni correction, i.e. \( \frac{0.05}{\mathrm{number}\ \mathrm{of}\ \mathrm{tests}} \) ., number of tests = number of columns in X × number of columns in Y × number of columns in Z.
The core algorithm of pulver was implemented in two languages namely, C++ and C/Fortran, to examine different performances due to programming languages. However, comparing the two different implementation of pulver reveals no striking differences. Thus, we continued to use the C++ version as it offered some useful implemented functions such as those implemented in the C++ Standard Library algorithms [28].
The package imputes missing values based on their column means. If this is not required, then we recommend using other more sophisticated methods, such as the mice package in R [29], in order to remove missing values before applying pulver.
pulver was developed as a screening tool to efficiently identify associations between the outcome, such as metabolite levels, and the interaction among two quantitative variables, such as CpGSNP interaction. Once, significant associations are identified, other information regarding the fitted models, such as slope coefficients, standard errors, or residuals, can be computed in a second step using traditional tools.
Conclusion
Our pulver package is currently the fastest implementation available for calculating pvalues for the interaction term of two quantitative variables given a huge number of linear regression models. Pulver is part of a processing pipeline focused on interaction terms in linear regression models and its main value is allowing users to conduct comprehensive screenings that are beyond the capabilities of existing tools.
Availability and requirements
Project name: pulver.
Project home page: https://cran.rproject.org/web/packages/pulver/index.html
Operating system(s): Platform independent.
Programming language: R, C++.
Other requirements: R 3.3.0 or higher.
License: GNU GPL.
Any restrictions to use by nonacademics: None.
Abbreviations
 GWAS:

Genomewide association studies
 SNP:

Singlenucleotide polymorphism
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Acknowledgements
We thank all of the participants in the KORA F4 study, everyone involved with the generation of the data, and the two anonymous reviewers for comments.
Funding
The KORA study was initiated and financed by the Helmholtz Zentrum München – German Research Center for Environmental Health, which is funded by the German Federal Ministry of Education and Research (BMBF) and by the State of Bavaria. Furthermore, KORA research was supported within the Munich Center of Health Sciences (MCHealth), LudwigMaximiliansUniversität, as part of LMUinnovativ.
Availability of data and materials
pulver can be downloaded from CRAN at https://cran.rproject.org/web/packages/pulver/.
The data used in the simulations were generated by the create_input_files function found in testing.R.
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Contributions
SM and CG designed the study. SM and CB wrote the pulver software and conducted computational benchmarking. SM, CB, SW, MN, KS, RW, MW, TM, JA, GK, KS, AP, HG, FJT, and CG contributed to the data acquisition or data analysis and interpretation of results. SM wrote the manuscript. SM, CB, SW, MN, KS, RW,MW, TM, JA, GK, KS, AP, HG, FJT, and CG contributed to the review, editing, and final approval of the manuscript.
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Ethics approval and consent to participate
The KORA study was approved by the local ethics committee (“Bayerische Landesärztekammer”, reference number: 06068).
All KORA participants gave their signed informed consent.
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Not applicable
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The authors declare that they have no competing interests.
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Additional file
Additional file 1:
Theory underlying pulver. This file describes the derivation of the tvalue computed from the beta value divided by the standard error and the correlation value. (PDF 426 kb)
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Molnos, S., Baumbach, C., Wahl, S. et al. pulver: an R package for parallel ultrarapid pvalue computation for linear regression interaction terms. BMC Bioinformatics 18, 429 (2017). https://doi.org/10.1186/s128590171838y
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Keywords
 Algorithm
 Linear regression interaction term
 SNP–CpG interaction
 Software