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A fast mrMLM algorithm for multi-locus genome-wide association studies

Cox Lwaka TambaiD, Yuan‐Ming ZhangiD

Abstract

BackgroundRecent developments in technology result in the generation of big data. In genome-wide association studies (GWAS), we can get tens of million SNPs that need to be tested for association with a trait of interest. Indeed, this poses a great computational challenge. There is a need for developing fast algorithms in GWAS methodologies. These algorithms must ensure high power in QTN detection, high accuracy in QTN estimation and low false positive rate.\n\nResultsHere, we accelerated mrMLM algorithm by using GEMMA idea, matrix transformations and identities. The target functions and derivatives in vector/matrix forms for each marker scanning are transformed into some simple forms that are easy and efficient to evaluate during each optimization step. All potentially associated QTNs with P-values [≤] 0.01 are evaluated in a multi-locus model by LARS algorithm and/or EM-Empirical Bayes. We call the algorithm FASTmrMLM. Numerical simulation studies and real data analysis validated the FASTmrMLM. FASTmrMLM reduces the running time in mrMLM by more than 50%. FASTmrMLM also shows high statistical power in QTN detection, high accuracy in QTN estimation and low false positive rate as compared to GEMMA, FarmCPU and mrMLM. Real data analysis shows that FASTmrMLM was able to detect more previously reported genes than all the other methods: GEMMA/EMMA, FarmCPU and mrMLM.\n\nConclusionsFASTmrMLM is a fast and reliable algorithm in multi-locus GWAS and ensures high statistical power, high accuracy of estimates and low false positive rate.\n\nAuthor SummaryThe current developments in technology result in the generation of a vast amount of data. In genome-wide association studies, we can get tens of million markers that need to be tested for association with a trait of interest. Due to the computational challenge faced, we developed a fast algorithm for genome-wide association studies. Our approach is a two stage method. In the first step, we used matrix transformations and identities to quicken the testing of each random marker effect. The target functions and derivatives which are in vector/matrix forms for each marker scanning are transformed into some simple forms that are easy and efficient to evaluate during each optimization step. In the second step, we selected all potentially associated SNPs and evaluated them in a multi-locus model. From simulation studies, our algorithm significantly reduces the computing time. The new method also shows high statistical power in detecting significant markers, high accuracy in marker effect estimation and low false positive rate. We also used the new method to identify relevant genes in real data analysis. We recommend our approach as a fast and reliable method for carrying out a multi-locus genome-wide association study.

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Claude Opus 5.5 · xhigh effort · 17 Oct 2026 · from the full text of version 1 (7 Jun 2018)

FASTmrMLM swaps mrMLM's numerical gradients and Hessians for analytic ones, cutting runtime by more than 50% while matching or slightly beating its simulated power and accuracy.

Multi-locus GWAS methods such as mrMLM (multi-locus random-SNP-effect mixed linear model) find more quantitative trait nucleotides (QTNs) than single-marker scans, but fitting a variance component for every marker is slow as panels grow towards tens of millions of SNPs. Borrowing GEMMA's idea, the paper rewrites the restricted likelihood and its derivatives in simple closed forms so each marker is fitted quickly, and adds a LARS pre-selection step before EM-Empirical Bayes.

Key results

  1. ShownFASTmrMLM needs less than 50% of mrMLM's running time; GEMMA and FarmCPU are still faster, and EMMA is far slower.

    First simulation experiment (no polygenic background), Intel Core i5-4570 CPU 3.20GHz, 7.88G memory: 6.25 h for FASTmrMLM vs 13.77 h for mrMLM, 5.12 h for FarmCPU, 2.57 h for GEMMA and 68.77 h for EMMA; the authors report the same pattern in the remaining simulations. · Computational efficiency

  2. SuggestedIn simulation, FASTmrMLM had the highest average QTN detection power, clearly above FarmCPU, GEMMA and EMMA but statistically tied with mrMLM; ROC curves favoured it for QTNs 1, 2, 4 and 6 and mrMLM for QTNs 3 and 5.

    First simulation, average power over six simulated QTNs: 68.8% FASTmrMLM, 68.6% mrMLM, 41.9% FarmCPU, 46.0% GEMMA, 46.0% EMMA; paired t-test vs FarmCPU, GEMMA and EMMA P-value = 0.004 ~ 0.012, vs mrMLM P-value = 0.688. ROC curves from 100 significance levels between 1E-8 and 1E-2; mrMLM's curves for QTNs 3 and 5 lie only slightly above FASTmrMLM's. · Statistical power; Receiver operating characteristic curve

  3. SuggestedEffect estimates from FASTmrMLM had the lowest mean squared error (MSE), significantly below GEMMA and EMMA; its edge over mrMLM and FarmCPU was not significant.

    First simulation, average MSE over six QTNs: 0.0775 FASTmrMLM, 0.0933 mrMLM, 0.2824 FarmCPU, 0.5467 GEMMA, 0.5432 EMMA; paired t-test vs GEMMA and EMMA P-value = 0.009 ~ 0.020, vs mrMLM and FarmCPU P-value = 0.110 ~ 0.806. · Mean squared errors of estimated QTN effects

  4. SuggestedFalse positives stayed rare despite the lenient LOD (log of odds) ≥ 3 cutoff: FASTmrMLM's rate was close to FarmCPU's and below those of mrMLM, GEMMA and EMMA.

    First simulation false positive rate (FPR, %): 1.80E-2 FASTmrMLM, 1.99E-2 mrMLM, 1.78E-2 FarmCPU, 3.25E-2 GEMMA and 3.25E-2 EMMA; GEMMA, EMMA and FarmCPU calls used a Bonferroni threshold of 0.05/m, and true QTNs were those within 1kb of a simulated one. · False positive rate; Monte Carlo simulation experiments

  5. SuggestedOn six Arabidopsis flowering-time traits, FASTmrMLM hits lay near more previously reported genes than any other method, and its SNP sets gave low AIC and BIC for nearly all traits (values in Table S5, not in this text).

    Known genes within 30 kb of hits for LD, LDV, SD, 0W, 2W, 4W: 14, 10, 7, 10, 10, 11 (FASTmrMLM) vs 4, 5, 1, 2, 5, 4 (mrMLM) and 5, 2, 0, 0, 3, 3 (GEMMA/EMMA); FarmCPU found none for 0W. FASTmrMLM called 17, 15, 14, 17, 14 and 15 SNPs and 7, 7, 6, 8, 7 and 5 new genes. · Real data analysis; The Arabidopsis thaliana data

Methods, data and reproducibility

Two-stage scan: each SNP is fitted as a random effect in a mixed linear model by restricted maximum likelihood (REML) and Newton-Raphson, with the polygenic-to-residual ratio fixed from the null model and Miller's identities giving fast inverses and determinants; SNPs with P ≤ 0.01 (cut to n−1 by LARS if more than n pass) enter an EM-Empirical Bayes multi-locus model where LOD ≥ 3 calls a QTN.

  • Simulated traits on 10000 SNPs (2000 per chromosome) sampled from the 216,130-SNP Atwell et al. panel; six QTNs; no, polygenic or epistatic background; 1000 runs per method · Arabidopsis thaliana · n = 199 inbred lines
  • Six flowering-time traits (LD, LDV, SD, 0W, 2W, 4W) from Atwell et al. · Arabidopsis thaliana
Methods & tools
FASTmrMLM, mrMLM (R package), GEMMA, EMMA, FarmCPU, LARS (lars R package), SCAD (ncvreg R package), adaptive Lasso (parcor R package), R
Assay conditions & compute
Timings on an Intel Core i5-4570 CPU 3.20GHz with 7.88G memory: 6.25 h (FASTmrMLM) vs 13.77 h (mrMLM) for the first simulation experiment.
Reproducibility gaps
FASTmrMLM is said to run through the R package mrMLM and simulated datasets are in Additional files, but the download links for the software, the comparators and the Atwell et al. phenotypes were lost in extraction; no software versions are given.
Limitations (5)
  • GEMMA (2.57 h) and FarmCPU (5.12 h) ran faster than FASTmrMLM (6.25 h); the authors justify the extra time by better power and estimation accuracy. · stated by the authors
  • Gains over mrMLM are small and not significant (power P-value = 0.688; MSE vs mrMLM and FarmCPU P-value = 0.110 ~ 0.806), and mrMLM had the higher ROC curves for QTNs 3 and 5. · stated by the authors
  • FASTmrMLM and mrMLM are judged at LOD ≥ 3 while GEMMA, EMMA and FarmCPU use Bonferroni 0.05/m, so the methods are compared at different thresholds; the authors note that a looser criterion in theory raises both power and FPR. · stated by the authors
  • Every test uses one Arabidopsis panel (199 lines; 10000 sampled SNPs in simulation), so speed at the tens-of-millions-of-SNPs scale that motivates the method is not demonstrated. · noted in this summary
  • Real-data support is a count of previously reported genes within 30 kb of hits, with no independent validation; the FarmCPU gene counts list more values than traits, and results for simulations 2 and 3 sit in figures and tables absent from this text. · noted in this summary

Who should read:Read this if you run multi-locus GWAS on inbred plant panels with mrMLM-style random-SNP-effect models and want a faster first-stage scan, benchmarked against GEMMA, EMMA and FarmCPU.

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