Genome-wide association studies (GWAS) with metabolic traits and metabolome-wide association studies (MWAS) with traits of biomedical relevance are powerful tools to identify the contribution of genetic, environmental and lifestyle factors to the etiology of complex diseases. Hypothesis-free testing of ratios between all possible metabolite pairs in GWAS and MWAS has proven to be an innovative approach in the discovery of new biologically meaningful associations. The p-gain statistic was introduced as an ad-hoc measure to determine whether a ratio between two metabolite concentrations carries more information than the two corresponding metabolite concentrations alone. So far, only a rule of thumb was applied to determine the significance of the p-gain.
Here we explore the statistical properties of the p-gain through simulation of its density and by sampling of experimental data. We derive critical values of the p-gain for different levels of correlation between metabolite pairs and show that B/(2*α) is a conservative critical value for the p-gain, where α is the level of significance and B the number of tested metabolite pairs.
We show that the p-gain is a well defined measure that can be used to identify statistically significant metabolite ratios in association studies and provide a conservative significance cut-off for the p-gain for use in future association studies with metabolic traits.
p-gainMetabolomicsMWASGWASGenome-wide association studiesMetabolome-wide association studies
With the advent of modern metabolomics techniques, hundreds of endogenous organic compounds (metabolites) from tissue samples, cell cultures and body fluids can now be measured in a highly standardized and often non-targeted manner. Current technologies are based on liquid chromatography–mass spectrometry (LC-MS), gas chromatography–mass spectrometry (GC-MS), flow injection analysis mass spectrometry (FIA-MS/MS) or nuclear magnetic resonance spectroscopy (NMR)
[1–3]. Genome-wide association studies (GWAS) with large numbers of metabolic traits and metabolome-wide association studies (MWAS) with a wide range of biomedical relevant traits are enabled by the newly achieved high-throughput metabolomics capabilities.
Specific ratios between selected pairs of metabolite concentrations (metabolite ratios) have been introduced in the past as biomarkers in many biomedical applications. For instance, medium-chain acyl-CoA dehydrogenase deficiency (MCADD) is detected in systematic “newborn screens” on the basis of elevated blood concentrations of octanoylcarnitine (C8) and other acylcarnitines, in combination with ratios between acylcarnitine concentrations, including hexanoylcarnitine (C6), decanoylcarnitine (C10), decenoylcarnitine (C10:1), C8/C6, C8/C10, and C8/C12 (dodecanoylcarnitine)
. The ratio between blood phenylalanine to tyrosine concentrations is used to identify heterozygous carriers of phenylketonuria (PKU) risk alleles
. Metabolite ratios are also used as biomarkers to detect specific exposures. For instance, the urinary hydroxyproline to creatinine ratio was proposed as an indicator for personal nitrogen dioxide (NO2) exposure
With modern high-throughput technologies, the concept of metabolite ratio analysis has been scaled up to systematically analyzing all possible combinations of ratios between metabolite pairs in a hypothesis-free approach. A number of recently published papers highlight the power of this approach: Altmaier et al. applied hypothesis-free metabolite ratio analysis to pre-clinical drug testing in diabetic mice. They linked ratios between sphingolipids that differ by two carbon moieties to a modified beta-oxidation and ratios between different classes of phospholipids to modified activity of enzymes in the phospholipid pathways. In a metabolite association study with smoking, Wang-Sattler et al. identified an association with ratios between ester- and ether-bond phospholipids. The biochemical properties of these phospholipids allowed pinpointing the association to the enzymatic action of alkylglycerone phosphate synthase. Using similar approaches, Altmaier et al. identified biochemically relevant associations between metabolite ratios and self-reported nutrition habits, and Suhre et al. used metabolite ratios to identify functional biomarkers for pre-clinical drug testing of FABP4 inhibitors. Gieger et al., Illig et al. and Suhre et al.
 introduced hypothesis-free testing of metabolite ratios to GWAS. They showed that using ratios can increase the power of GWAS by tens of orders of magnitude. The leading metabolic traits in 14 out of 15 genetic associations reported by Illig et al. and 20 out 37 associations by Suhre et al.
 are ratios between metabolite concentrations (Table
1). Most interestingly, they found that the biochemical nature of the associating metabolite pairs was in nearly all cases directly related to the biochemical function of an enzyme or transporter gene that was encoded at the respective loci.
Selected examples of published associations with hypothesis-free testing of metabolite ratios
Diabetic (db/db) versus wild type mice
Increased beta-oxidation in diabetic mice
Altmaier et al., Endocrinology, 2008
PC aa C36:3/PC aa C36:4
Genetic variance in delta-5 fatty acid desaturation
Gieger et al., PLoS Genetics, 2008
PC aa Cx:y/PC ae Cx:y
Reduced or lack of activity of the enzyme alkyl-DHAP in smokers
Wang-Sattler et al., PLoS One, 2008
PC aa C40:3/PC aa C42:5
Genetic variance in elongation of fatty acids
Illig et al., Nature Genetics, 2010
Medium chain fatty acids / long chain fatty acids
Perturbed lipid metabolism associated with diabetes
Suhre et al., PLoS One, 2010
PC aa C40:5/PC aa C40:6
Self-reported nutritional intake of polyunsaturated fatty acids
Confirmation of questionnaire based life-style parameters
Altmaier et al., Eur. J. Endocrinology, 2011
Ratios between phospholipids with lipid side chains from the C16:0, C16:1, C18:0, C18:1 pool and C20:3, C20:4, C22:4 PUFAs
Plasma, tissue (mouse) and cell lines (human) treated with FABP4 inhibitor
Molecular inhibition of FABP4 activity
Suhre et al., J Biomol Screen, 2011
Formate/ acetate in human urine
Genetic variance in N-acetylase activity
Suhre et al., Nature Genetics, 2011
Ratio between phosphorylated and unphosphorylated fibrinogen peptides
ABO, ALPL, and FUT2 genotypes
Genetic variance in fibrinogen phosphorylation
Suhre et al., Nature, 2011
In all studies pairs of metabolites were identified by a high increase in the strength of association when ratios were used. Note that all of these metabolite pairs are found to be biochemically related to the concrete biological questions of these studies (Interpretation). However, they were singled out from the large number of all possible metabolite pair combinations on the basis of the p-gain without any prior hypotheses.
Several reasons explain why metabolite ratios provide additional information in these association studies: (1) Ratios between related metabolite pairs reduce the overall biological variability in the dataset and thereby increase statistical power. For instance, study participants may have strongly varying nutrition habits, which introduce high variance in the distribution of that nutrient, but also in those of its biochemical break-down products. However, individuals who consume a higher amount of a certain nutrient also exhibit higher levels of its biochemical break-down products. Ratios between these metabolites can thus be considered as some kind of internal normalization. (2) Systematic experimental errors, such as variance in the concentration of external standards result in errors that are comparable for certain metabolite pairs. Such errors are cancelled out in ratios and thereby reduce the overall noise in the dataset. (3) Probably most importantly, when a metabolite pair is connected by a biochemical pathway, metabolite ratios approximate the corresponding reaction rate under idealized steady state assumptions. Metabolite ratios then represent a biologically most relevant entity, namely the flux through a biochemical pathway. For example, in Suhre et al., the association of SNP rs174547 at the FADS1 locus displayed a p-value of p = 2.3 × 10-21 and an explained variance of 5.2 % with concentrations of the omega-6 fatty acid 20:4, whereas the p-value of association with ratios between the fatty acids 20:4 and 20:3 was p = 9.987 × 10-66 with an explained variance of 15.3 %
. The FADS1 locus encodes a fatty acid delta-5 desaturase. This is a key enzyme in the metabolism of long chain polyunsaturated omega-3 and omega-6 fatty acids. The fatty acids 20:4 and 20:3 are the respective product and substrate pair of the FADS1 reaction . The strengthening in the association of a genetic variant in the FADS1 locus and the ratio between its substrate-product pair thus matches the biological function of the FADS1 enzyme. Had the biochemical function of FADS1 not been known, it could have been inferred from these observations. This example shows how the biochemical properties of the associated metabolite pairs can provide information on the functional background of the associations. Examples of published studies that use association with hypothesis-free testing of metabolite ratios are provided in Table 1 to further highlight this concept.
In order to quantify the strengthening of the association when analyzing metabolite ratios in comparison to using metabolite concentrations, the p-gain was introduced 11]. The p-gain is defined as the increase in the strength of association, expressed as the change in p-value when using ratios compared to the smaller of the two p-values when using two metabolite concentrations individually. So far, the number of analyzed metabolite concentrations was applied as an ad-hoc critical value for the p-gain. Any association that displayed a p-gain below this number was considered to have occurred by chance. This approach can merely be regarded as an intuitive rule of thumb, since a statistical determination of the distribution of the p-gain and herewith of the critical values has not yet been conducted. In this paper, we derive critical values through determination of the distribution of the p-gain and provide a density table for readout of critical values. In addition, we investigate the characteristics of the p-gain in the situation of Bonferroni correction for multiple tests.
Results and discussion
Formal definition of the p-gain
Testing ratios between two metabolite concentrations a and b should be independent of their order. It is therefore advisable to use log-scaled metabolite ratios in the tests for association. Due to the property log(a/b) = -log(b/a) this also halves the multiple testing burden. Moreover, in many of the cases we tested, the distribution of metabolite ratios was observed to be better represented by a log-normal distribution than by a normal distribution. For instance, a test of normality in the study by Illig et al. showed that in 85.1 % of the cases, the log-transformed ratio distribution was significantly better represented by a normal distribution than when untransformed ratios were used.
The p-gain was introduced in order to measure whether the association with a genetic locus is significantly stronger for a metabolite ratio than for the belonging metabolite concentrations. As notation, we use ‘p-value(M1 | X)’, short ‘P(M1)’, to reference the p-value corresponding to a test for association between a trait X (in a GWAS this would be a genetic locus represented by a SNP and in an MWAS it would be a phenotypic trait) and the metabolite M1. With this definition, the p-gain for the association of the ratio M1/M2 of metabolites M1 and M2 with a trait X is defined as
Conservative critical p-gain values for common statistics
Although the p-gain is now frequently used in MWAS and in GWAS with metabolic traits, only a rule of thumb for the determination of critical values has been applied so far. The p-gain was considered as being significant when its value exceeded the number of analyzed metabolite concentrations, that is, the number of additionally performed tests
[11–13]. Here we derive critical values of the p-gain by determination of the distribution to define a more sensible threshold. As the distribution of the p-gain depends on the correlation structure among the metabolites, conservative critical values are beneficial in case of analyzing multiple sets of metabolites, since they can be applied to all analyzed settings. For this purpose, we use a universal p-gain defined as the ratio of p-values belonging to two uncorrelated metabolites:
Critical values of the distribution of this p-gain are conservative to the critical values of the distribution of the p-gain given in equation (1), because
The variation of the distribution of the p-gain defined in equation (2) depends on the correlation between M1 and M1/M2. For example, highly correlated metabolic traits contain mainly the same information and have similar p-values in association tests. This results in p-gain values which are close to one. Hence, the variation of the distribution is small. In contrast, weakly correlated metabolic traits contain different information and may have different p-values in association tests. This results in p-gain values distributed broadly around the one. Therefore, assuming
, as it was done in equation (2), results in a distribution of the p-gain with largest possible variation and leads to the most conservative critical values.
In the situation of the universalized p-gain (equation (2)) we can use the convolution formula for density ratios which yields a split density (see Methods):
as displayed in Figure
1 (black line). To determine critical values, we derive the cumulative distribution function of the density, i.e.
Herewith, the critical value becomes
with α denoting the level of significance. In the case of typically used α levels of 0.05, this yields a corresponding critical value for the p-gain of ten. General quantiles are provided in Table S1 (Additional file
Critical values for multiple testing
In MWAS and in GWAS with metabolomics a large number of ratios are tested in parallel. Therefore, a correction for multiple testing has to be applied. We select Bonferroni correction as the most conservative method. When admitting a type I error rate of α and applying a correction for B tests, i.e. aiming at a level of significance of
, the critical value for the p-gain then becomes
(see Methods). For example, assumption of a type I error rate of α = 0.05 leads to a critical value of
which implies that for Bonferroni correction of B tests the uncorrected critical value of ten can be multiplied by the number of tests B. Hence, the critical value of the p-gain in the situation of multiple testing is not equal to the number of analyzed metabolite concentrations, which was used so far as an ad-hoc criterion, but rather ten times the number of tested ratios.
P-gain for correlated metabolites
The case of uncorrelated metabolites (equation (2)) is conservative with respect to the p-gain as defined in equation (1). Here we analyze the density of the p-gain as defined in equation (1) for selected correlation settings. In the situation of correlated metabolic traits the convolution formula cannot be applied anymore. Thus, we simulate the density using a copula to generate the correlation among the metabolic traits. A copula is a joint probability distribution whose one-dimensional marginal distributions are uniformly distributed over the interval [0,1]. It takes the dependency among the marginal distributions into account (see Methods). Quantiles for the p-gain densities of correlated metabolic traits are provided in Table S1 (Additional file
1) for various correlation settings. It can be observed that when any of the correlations cor(M1, ratio) or cor(M2, ratio) increase, the values of the quantiles of the p-gain decrease. This observation can be explained by the fact that the variation of the p-gain can be reduced by increasing the correlation between a metabolite concentration and the ratio (i.e. cor(M1, ratio) or cor(M2, ratio)). A reduction of the variation of the p-gain leads to smaller critical values. On the other hand, for fixed cor(M1, ratio) or cor(M2, ratio), an increase in the correlation between M1 and M2 leads to an increase in the values for the p-gain quantiles when the correlation between M1 and M2 is not close to 0. Extending these observations to the most extreme case of having fully correlated metabolite concentrations which are uncorrelated with their ratio (i.e. cor(M1,M2) = 1, cor(M1, ratio) = 0, cor(M2, ratio) = 0) we get the largest critical values and thus these critical values are conservative to all correlation settings. This idealized case reduces the p-gain as defined in equation (1) to the p-gain as defined in equation (2). For this case, we derived the distribution using the convolution formula as well as through a simulation. In both cases, the simulated and calculated density as well as the belonging critical values coincided (Table S1, Figure S1 (Additional file
1)). To determine the density of the p-gain for a given correlation setting among the metabolite concentrations and their ratio, the exact distribution of the p-gain for a given metabolite ratio can be simulated using the R-script which is provided as Supplemental Material (Additional file
Dependence on sample size in real data
In order to examine the behavior of the p-gain in the situation of real data, we compute the observed correlation structure among metabolite ratios which were published in Suhre et al. (Table S2). This dataset includes nearly uncorrelated metabolites, such as the ratio between 1-methylxanthine and 4-acetamidobutanoate (association with the NAT2 locus) as well as highly correlated metabolites, such as the androsterone sulfate to epiandrosterone sulfate ratio (association with the AKR1C locus). The distributions of exemplary metabolite ratios are presented in Figure
1. As expected, the densities for correlated metabolic traits display smaller variations than the density for uncorrelated metabolic traits. The observed p-gain values in 1,768 samples of the KORA study vary between 2.79 x 103 and 1.68 x 1066 for the 20 loci published in Suhre et al. (see Table S3 (Additional file
1)). Using this dataset we conducted simulation tests to address the influence of the sample size on the observed p-gain values. We chose randomly sets of samples sizes between 100 and 2000 samples from the KORA study and calculated the p-gain for these sets. The results of this analysis illustrate the dependence of the p-gain values on the sample size (Table S3 (Additional file
1)). For example, we observe for the association between the ACADS locus and the butyrylcarnitine to propionylcarnitine ratio a median p-gain value of 1.4 x 102 for a sample size of N = 100, of 1.1 x 105 for N = 500, of 2.8 x 1010 for N = 1000, of 3.1 x 1015 for N = 1500 and of 1.4 x 1021 for N = 2000.
We derived critical values for the p-gain to determine significance in various situations. We recommend the use of metabolite ratios and the p-gain statistic when analyzing large scale metabolomics data sets and to apply the critical values with correction of multiple testing as provided in this paper. Given the success of the approach in the metabolomics field, hypothesis free testing of ratios between biologically related quantitative traits should also be considered for association studies with other ‘omics datasets.
The KORA (Cooperative Health Research in the Region of Augsburg) study is a series of independent population-based epidemiological surveys and follow-up studies of participants living in the region of Augsburg, Southern Germany
. All participants are residents of Germany with a German nationality. All participants gave signed informed consent. The study was approved by the local ethics committee (“Bayerische Landesärztekammer”). The present study includes data of the follow-up study KORA F4 (2006-2008) of the KORA S4 survey (1999-2000). For genotyping, we included 1,814 randomly selected participants of KORA F4. The KORA F4 samples were genotyped with the Affymetrix Human SNP Array 6.0 and imputed with IMPUTE v0.4.2 based on Hap Map II
We collected blood samples between 2006 and 2008 during the KORA F4 examinations. To avoid variation due to circadian rhythm, blood was drawn in the morning between 8:00 a.m. and 10:00 a.m. after a period of overnight fasting. Blood was drawn into serum gel tubes, gently inverted two times and then allowed to rest for 30 min at room temperature (18 − 25 °C) to obtain complete coagulation. The material was then centrifuged for 10 min and 2,750 g at 15 °C. Serum was divided into aliquots and kept for a maximum of 6 h at 4 °C, after which it was deep-frozen to −80 °C until analysis.
On 1,768 fasting serum samples of the KORA F4 study for which we had already genotypes available, metabolic profiling was done using ultrahigh performance liquid-phase chromatography and gas chromatography separation coupled with tandem mass spectrometry
[16–18] at Metabolon, an US-based commercial supplier of metabolic analyses. They achieved highly efficient profiling (24 minutes/sample) with low median process variability (<12 %) of more than 250 metabolites, covering over 60 biochemical pathways of human metabolism. A more detailed description of the metabolomics measurement and quality control can be found in Suhre et al..
Density of p-gain for uncorrelated metabolites (calculation)
The p-gain for two uncorrelated metabolites is defined as:
We calculated the density of the p-gain of two uncorrelated metabolites by using the convolution formula for ratios:
with P(M1) and P(M1/M2) having a uniform distribution on the interval [0,1]. Transformations lead to
The corresponding cumulative distribution is
being the significance level α, Bonferroni-corrected for B tests.
Density of the p-gain (simulation)
To determine the density of the p-gain we assumed a given correlation structure among the metabolic traits. This confers to a correlation structure among p-values corresponding to these metabolic traits. With these correlated p-values the density of the p-gain can be derived. For simulation of the variables with a given correlation structure we choose the “copula” package
[19, 20] of the R-Project Environment
. The simulated variables were marginal distributions of a multivariate distribution with a uniform distribution on the interval [0,1]. We then transformed the simulated variables with an inverse normal transformation to gain a normal distribution which is essential for linear regressions. To simulate the p-values belonging to these variables, we generated additional variables and conducted linear regressions where these additional variables were the independent and the variables simulated with the copula the dependent variables. The received p-values contain a correlation structure which belongs to the correlation structure of the metabolic traits. Out of these p-values, we calculated a density of the p-gain empirically and derived critical values for given significance levels. An R-script with the simulation commands is provided with the supplemental material.
Dependence of p-gain values on sample size
We determined the dependency of the p-gain of the sample size by drawing randomly between 100 and 2000 samples from the KORA data (with replacement). For each sample size, we repeated this analysis 1500 times. For all sample subsets we calculated the p-gain. We then determined the median p-gain as well as the 1st and 3rd quantile of the p-gains for each sample size.
The KORA research platform was initiated and financed by the Helmholtz Center Munich, 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. Part of this work was financed by the German National Genome Research Network (NGFN-2, NGFNPlus 01GS0823, and NGFNPlus 01GS0834) and through additional funds from the University of Ulm. Our research was supported within the Munich Center of Health Sciences (MC Health) as part of LMUinnovativ and by a grant from the BMBF to the German Center for Diabetes Research (DZD e.V.), as well as from the BMBF funded German Network for Mitochondrial Disorders (mitoNET 01GM0862) and Systems Biology of Metabotypes (SysMBo 0315494A). Furthermore, the study received funding from the European Community’s Seventh Framework Programme (FP7/2007–2013), ENGAGE project, grant agreement HEALTH-F4-2007-201413. BW is funded by ERA-NET grant 0315442A (project PathoGenoMics). JK is supported by a PhD student fellowship from the "Studienstiftung des Deutschen Volkes". KS is supported by Qatar Foundation.
Institute of Genetic Epidemiology, Helmholtz Zentrum München
Institute of Bioinformatics and Systems Biology, Helmholtz Zentrum München
Department of Genome-oriented Bioinformatics, Life and Food Science Center Weihenstephan, Technische Universität München
Institute of Epidemiology I, Helmholtz Zentrum München
Institute of Medical Informatics, Biometry and Epidemiology, Chair of Epidemiology, Ludwig-Maximilians-Universität
Faculty of Biology, Ludwig-Maximilians-Universität
Department of Physiology and Biophysics, Weill Cornell Medical College in Qatar, Education City - Qatar Foundation
Han J, Datla R, Chan S, Borchers CH: Mass spectrometry-based technologies for high-throughput metabolomics.Bioanalysis 2009,1(9):1665–1684.PubMedView Article
Koal T, Deigner HP: Challenges in mass spectrometry based targeted metabolomics.Curr Mol Med 2010,10(2):216–226.PubMedView Article
Malet-Martino M, Holzgrabe U: NMR techniques in biomedical and pharmaceutical analysis.J Pharm Biomed Anal 2011,55(1):1–15.PubMedView Article
Maier EM, Liebl B, Roschinger W, Nennstiel-Ratzel U, Fingerhut R, Olgemoller B, Busch U, Krone N, v Kries R, Roscher AA: Population spectrum of ACADM genotypes correlated to biochemical phenotypes in newborn screening for medium-chain acyl-CoA dehydrogenase deficiency.Hum Mutat 2005,25(5):443–452.PubMedView Article
Hsia DY: Phenylketonuria: the phenylalanine-tyrosine ratio in the detection of the heterozygous carrier.J Ment Defic Res 1958,2(1):8–16.PubMed
Yanagisawa Y, Nishimura H, Matsuki H, Osaka F, Kasuga H: Personal exposure and health effect relationship for NO2 with urinary hydroxyproline to creatinine ratio as indicator.Arch Environ Health 1986,41(1):41–48.PubMedView Article
Altmaier E, Ramsay SL, Graber A, Mewes HW, Weinberger KM, Suhre K: Bioinformatics analysis of targeted metabolomics–uncovering old and new tales of diabetic mice under medication.Endocrinology 2008,149(7):3478–3489.PubMedView Article
Wang-Sattler R, Yu Y, Mittelstrass K, Lattka E, Altmaier E, Gieger C, Ladwig KH, Dahmen N, Weinberger KM, Hao P, et al.: Metabolic profiling reveals distinct variations linked to nicotine consumption in humans–first results from the KORA study.PLoS One 2008,3(12):e3863.PubMedView Article
Altmaier E, Kastenmuller G, Romisch-Margl W, Thorand B, Weinberger KM, Illig T, Adamski J, Doring A, Suhre K: Questionnaire-based self-reported nutrition habits associate with serum metabolism as revealed by quantitative targeted metabolomics.Eur J Epidemiol 2011,26(2):145–156.PubMedView Article
Suhre K, Romisch-Margl W, de Angelis MH, Adamski J, Luippold G, Augustin R: Identification of a potential biomarker for FABP4 inhibition: the power of lipidomics in preclinical drug testing.J Biomol Screen 2011,16(5):467–475.PubMedView Article
Gieger C, Geistlinger L, Altmaier E, Hrabe de Angelis M, Kronenberg F, Meitinger T, Mewes HW, Wichmann HE, Weinberger KM, Adamski J, et al.: Genetics meets metabolomics: a genome-wide association study of metabolite profiles in human serum.PLoS Genet 2008,4(11):e1000282.PubMedView Article
Illig T, Gieger C, Zhai G, Romisch-Margl W, Wang-Sattler R, Prehn C, Altmaier E, Kastenmuller G, Kato BS, Mewes HW, et al.: A genome-wide perspective of genetic variation in human metabolism.Nat Genet 2010,42(2):137–141.PubMedView Article
Suhre K, Shin SY, Petersen AK, Mohney RP, Meredith D, Wagele B, Altmaier E, Deloukas P, Erdmann J, CARDIoGRAM, et al.: Human metabolic individuality in biomedical and pharmaceutical research.Nature 2011,477(7362):54–60.PubMedView Article
Lattka E, Illig T, Koletzko B, Heinrich J: Genetic variants of the FADS1 FADS2 gene cluster as related to essential fatty acid metabolism.Curr Opin Lipidol 2010,21(1):64–69.PubMedView Article
Wichmann HE, Gieger C, Illig T: KORA-gen–resource for population genetics, controls and a broad spectrum of disease phenotypes.Gesundheitswesen 2005,67(Suppl 1):S26–30.PubMedView Article
Evans AM, DeHaven CD, Barrett T, Mitchell M, Milgram E: Integrated, nontargeted ultrahigh performance liquid chromatography/electrospray ionization tandem mass spectrometry platform for the identification and relative quantification of the small-molecule complement of biological systems.Anal Chem 2009,81(16):6656–6667.PubMedView Article
Ohta T, Masutomi N, Tsutsui N, Sakairi T, Mitchell M, Milburn MV, Ryals JA, Beebe KD, Guo L: Untargeted metabolomic profiling as an evaluative tool of fenofibrate-induced toxicology in Fischer 344 male rats.Toxicol Pathol 2009,37(4):521–535.PubMedView Article
Suhre K, Meisinger C, Doring A, Altmaier E, Belcredi P, Gieger C, Chang D, Milburn MV, Gall WE, Weinberger KM, et al.: Metabolic footprint of diabetes: a multiplatform metabolomics study in an epidemiological setting.PLoS One 2010,5(11):e13953.PubMedView Article
Kojadinovic I, Yan J: Modeling Multivariate Distributions with Continuous Margins Using the copula R Package.Journal of Statistical Software 2010,34(9):1–20.
Yan J: Enjoy the Joy of Copulas: With a Package copula.Journal of Statistical Software 2007,21(4):1–21.
R Development Core Team: R: A Language and Environment for Statistical Computing. Vienna, Austria; 2010.
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