^{*}

Conceived and designed the experiments: SV CCC. Analyzed the data: SV CCC. Contributed reagents/materials/analysis tools: SV JG CCC. Wrote the paper: SV CCC.

The authors have declared that no competing interests exist.

We used a bivariate (multivariate) linear mixed-effects model to estimate the narrow-sense heritability (^{2}_{g}^{2}^{2}^{2}

The narrow-sense heritability of a trait such as body-mass index is a measure of the variability of the trait between people that is accounted for by their additive genetic differences. Knowledge of these genetic differences provides insight into biological mechanisms and hence treatments for diseases. Genome-wide association studies (GWAS) survey a large set of genetic markers common to the population. They have identified several single markers that are associated with traits and diseases. However, these markers do not seem to account for all of the known narrow-sense heritability. Here we used a recently developed model to quantify the genetic information contained in GWAS for single traits and shared between traits. We specifically investigated metabolic syndrome traits that are associated with type 2 diabetes and heart disease, and we found that for the majority of these traits much of the previously unaccounted for heritability is contained within common markers surveyed in GWAS. We also computed the genetic correlation between traits, which is a measure of the genetic components shared by traits. We found that the genetic correlation between these traits could be predicted from their phenotypic correlation.

Obesity associated traits such as central adiposity, dyslipidemia, hypertension, and insulin resistance are major risk factors for type 2 diabetes and cardiovascular complications ^{2}

The main approach of GWAS has been to identify significant single-nucleotide polymorphisms (SNPs) by examining each SNP individually for significance. The ^{2}^{2}^{−8}, based on an adjusted

Alternatively, the narrow sense heritability explained by the common SNPs, _{g}^{2}^{2}_{g}^{2}_{g}^{2}^{2}^{2}

We estimated ^{2}_{g}^{2}

BMI | WHR | GLU | INS | TG | HDL | SBP | |

h^{2} |
0.34 (0.12) | 0.28 (0.12) | 0.33 (0.12) | 0.23 (0.12) | 0.47 (0.11) | 0.48 (0.11) | 0.30 (0.12) |

h_{g}^{2} |
0.14 (0.05) | 0.13 (0.05) | 0.10 (0.05) | 0.09 (0.05) | 0.16 (0.05) | 0.12 (0.05) | 0.24 (0.05) |

We first estimated ^{2}^{2}^{2}

We then compared these values to estimates for _{g}^{2}_{g}^{2}_{g}^{2}^{2}^{2}_{g}^{2}_{g}^{2}

We then estimated ^{2}_{g}^{2}^{2}^{2}_{g}^{2}_{g}^{2}^{2}^{2}^{2}

We next estimated the genetic correlations between MetS traits using a bivariate (multivariate) model (see ^{−3}). The phenotypic correlations between traits were similar for related and unrelated individuals and are shown in

BMI | WHR | GLU | INS | TG | HDL | SBP | |

BMI | 0.75 (0.16)* | 0.23 (0.24) | 0.17 (0.27) | 0.19 (0.20) | −0.12 (0.21) | 0.55 (0.24) | |

WHR | 0.52 (0.08)* | 0.35 (0.26) | 0.67 (0.26)* | 0.10 (0.22) | −0.12 (0.22) | 0.37 (0.26) | |

GLU | 0.19 (0.12) | 0.14 (0.12) | 0.69 (0.25)* | 0.21 (0.21) | −0.07 (0.21) | 0.13 (0.27) | |

INS | 0.64 (0.08)* | 0.35 (0.09)* | 0.22 (0.11) | 0.76 (0.21)* | −0.33 (0.23) | 0.29 (0.29) | |

TG | 0.29 (0.12) | 0.34 (0.12) | 0.21 (0.13) | 0.27 (0.11) | −0.59 (0.13)* | 0.21 (0.22) | |

HDL | −0.38 (0.12)* | −0.34 (0.12) | −0.22 (0.13) | −0.39 (0.11)* | −0.45 (0.11)* | −0.06 (0.23) | |

SBP | 0.11 (0.12) | 0.18 (0.11) | 0.05 (0.12) | 0.24 (0.11) | 0.10 (0.13) | −0.02 (0.13) |

BMI | WHR | GLU | INS | TG | HDL | SBP | |

BMI | 0.91 (0.18)* | 0.01 (0.32) | 0.57 (0.24) | 0.20 (0.24) | −0.15 (0.28) | 0.16 (0.20) | |

WHR | 0.44 (0.03)* | 0.09 (0.32) | 0.33 (0.31) | 0.32 (0.23) | −0.06 (0.30) | 0.17 (0.21) | |

GLU | 0.27 (0.04)* | 0.18 (0.04)* | 0.05 (0.40) | 0.07 (0.30) | −0.16 (0.34) | 0.11 (0.24) | |

INS | 0.51 (0.03)* | 0.40 (0.04)* | 0.39 (0.04)* | 0.22 (0.29) | −0.20 (0.36) | 0.20 (0.25) | |

TG | 0.31 (0.04)* | 0.33 (0.04)* | 0.20 (0.04)* | 0.43 (0.04)* | −0.57 (0.19)* | 0.002 (0.19) | |

HDL | −0.34 (0.04)* | −0.33 (0.04)* | −0.16 (0.04)* | −0.39 (0.04)* | −0.51 (0.03)* | −0.03 (0.22) | |

SBP | 0.25 (0.05)* | 0.18 (0.05)* | 0.17 (0.05)* | 0.22 (0.04)* | 0.21 (0.05)* | −0.04 (0.05) |

BMI | WHR | GLU | INS | TG | HDL | SBP | |

BMI | 0.59 (0.04)* | 0.20 (0.04)* | 0.49 (0.04)* | 0.24 (0.04)* | −0.26 (0.04)* | 0.25 (0.04)* | |

WHR | 0.51 (0.01)* | 0.21 (0.04)* | 0.43 (0.04)* | 0.23 (0.04)* | −0.24 (0.04)* | 0.23 (0.04)* | |

GLU | 0.24 (0.01)* | 0.17 (0.01)* | 0.34 (0.04)* | 0.21 (0.04)* | −0.15 (0.04)* | 0.07 (0.04) | |

INS | 0.52 (0.01)* | 0.39 (0.01)* | 0.35 (0.01)* | 0.42 (0.04)* | −0.35 (0.04)* | 0.25 (0.04)* | |

TG | 0.30 (0.01)* | 0.33 (0.01)* | 0.19 (0.01)* | 0.40 (0.01)* | −0.52 (0.04)* | 0.14 (0.04)* | |

HDL | −0.32 (0.01)* | −0.30 (0.01)* | −0.15 (0.01)* | −0.37 (0.01)* | −0.52 (0.01)* | −0.04 (0.04) | |

SBP | 0.23 (0.01)* | 0.18 (0.01)* | 0.15 (0.01)* | 0.21 (0.01)* | 0.16 (0.01)* | −0.04 (0.01)* |

We validated our genetic correlation estimates using bivariate models for each pair of traits by analyzing all 7 MetS traits simultaneously for the unrelated individuals in a single multivariate model. This 7 trait multivariate model was much more expensive computationally so we used a less stringent convergence rule. The results were similar to the bivariate model (see

We then examined the relationship between the genetic and phenotypic correlations (see _{p}_{g}^{−7}). For unrelated individuals, we found that the phenotypic correlations were proportional to the genetic correlations with a proportionality constant of 0.85 (s.e. = 0.19 ; two-tail ^{−4}). The direct proportionality between _{p}_{g}_{g}/r_{p}

We used a recently developed approach to analyzing GWAS data and provided new estimates for the total amount of additive genetic information contained in the common SNPs for MetS traits. The approach uses a linear mixed-effects model to estimate the additive genetic variances and correlations between traits. The model relies on knowing the genetic relationships between the individuals analyzed. Previously, this had been obtained from family pedigrees. Visscher et al. ^{2}_{g}^{2}^{2}_{g}^{2}_{g}^{2}^{2}

We confirmed previous findings that a large proportion of ^{2}_{g}^{2}^{2}_{g}^{2}^{2}_{g}^{2}^{2}^{2}_{g}^{2}^{2}_{g}^{2}^{2}^{2}_{g}^{2}

Our analysis of _{g}^{2}_{g}^{2}_{g}^{2}_{g}^{2}_{g}^{2}

Using the bivariate (multivariate) model

We found that the genetic correlation was directly proportional to the phenotypic correlation, which was an unexpected, empirical finding. Previously, a linear relationship between the correlations was hypothesized by Cheverud for sets of traits with common functions, and shown empirically for a number of traits

In summary, we provided evidence that the common SNPs explain large proportions of the variance for several MetS traits in agreement with previous findings for some of these traits

Our main study population was the Atherosclerosis Risk In Communities (ARIC) population. The ARIC population consists of a large sample of unrelated individuals and some families across North America. The population was recruited from four centers across the United States: Forsyth County, North Carolina; Jackson, Mississippi; Minneapolis, Minnesota; and Washington County, Maryland. For this study, we restricted our analysis to the European-American group. The population was recruited in 1987 from the general population consisting of subjects aged 45 to 64 years. The ARIC population consisted of 8,451 subjects.

Quality control and genotype calls for common SNPs were evaluated previously for ARIC using the Affymetrix Human SNP Array 6.0. We selected bilallelic autosomal markers based on the following criteria: missingness <0.05, Hardy-Weinberg equilibrium (p<10^{−6}) and minor allele frequency >0.05. Subjects with missingness >0.05 were removed. This resulted in 436,126 retained markers.

Quality control measurements from dbGAP (GENEVA ARIC Project Quality Control Report Sept 22, 2009) indicate significant population stratification between self-identified white (European-ancestory kind group) and black populations when projected onto HapMap components. Furthermore, principal-components analysis of the European-ancestory group by dbGAP showed that no component explained more than 0.1% of the population variance. For this study we only analyzed the European-ancestory group and treated it as a single population.

ARIC phenotypes were adjusted for age, sex, and study center. Only single measurements from visit 1 were used for these subjects. We only used subjects with negative diabetes status and with genotype and phenotype information for all traits. This resulted in 8,451 subjects. We standardized all the traits. We first log-transformed BMI, glucose, insulin, triglycerides, HDL, and systolic blood pressure. All laboratory measurements are under fasting conditions. Population trait statistics are in

We estimated ^{2}_{g}^{2}

We determined ^{2}

^{2}^{2}_{g}_{g}_{e}_{g}_{e}

We used the linear mixed-effects model and only unrelated individuals to estimate the additive-genetic variance attributable to the common SNPs (_{g}^{2}^{nd} cousins. For these estimates we used the same group of 5,647 unrelated individuals for all estimates in ARIC and 1,489 individuals in FHS. _{g}^{2}_{g}^{2}_{g}_{g}_{e}_{g}_{e}_{g}^{2} versus SNP number analyses were performed over allele frequency range of 0.05 to 0.5 in order of increasing and decreasing frequency.

The genetic correlation (_{g}_{g}(_{i}_{g}(_{i},t_{j}

The mean and errors for proportionality constants between the genetic and phenotypic correlations were determined by randomly sampling over the distributions of the parameter estimates (i.e. Monte Carlo method) assuming that the error around the mean parameter estimate was normally distributed and that the parameters were independent. We then fit a linear function with the y-intercept fixed at 0 (after first confirming that it was not significantly different from zero).

We assessed significance for correlation coefficients (

Significance for regression coefficient (

Preprocessing of SNPs and phenotypes was done using PLINK

We considered the following multivariate linear mixed-effects model for _{i}_{i} is an _{i}_{i}_{i}∼_{i}_{i}∼_{ij} = cov_{gij}_{n} and _{ij} = cov_{eij}_{m} and _{l}

We considered only bi-allelic SNPs in Hardy-Weinberg equilibrium. Denote the minor allele by q and the major allele by Q. Let the minor allele frequency at locus _{i}. We assign a value of 2 for genotype qq, 1 for genotype qQ and 0 for genotype QQ. The Hardy-Weinberg mean frequency for the genotype at locus _{i} and the variance is 2_{i}_{i}_{i})/(2_{i}(1−2_{i}))^{1/2} for qq, (1−2_{i})/(2_{i}(1−2_{i}))^{1/2} for qQ, and −2_{i}/(2_{i}(1−2_{i}))^{1/2} for the QQ genotype.

The log of the likelihood function is given by^{−1}_{ij}_{ij}

We solved the REML equations using an EM algorithm ^{−4}. We also checked that the rate of change of the square of the covariance predictions was less than 10^{−8}. We checked our results against the software developed by Yang et al. (GCTA)

For the multivariate model, we transformed to a coordinate system where the covariance matrices were diagonal _{j} be the set of phenotypes for individual ^{−1}). The transformed genetic covariances are given by _{t}

In our computations, we used both the direct EM algorithm and the canonically transformed algorithm because even though the transformed algorithm was in principle faster, it sometimes had poor convergence properties if the initial guess was not sufficiently close to the maximum likelihood value. We ensured that both give the same results. For computational efficiency, the results shown are computed from the bivariate model for the different trait pairs. We confirmed our results with a multivariate model that included all traits.

Our error estimates were given by the inverse of the Fisher information matrix _{ij} was set equal to cov_{ji}) and with block elements (that are not all contiguous) given by

Height _{g}^{2} versus number of SNPs by sampling the allele frequency from 0.05 to 0.5 (red = low to high, blue = high to low, black = using all SNPs). A) _{g}^{2} estimates for height relative to the number of SNPs (mean and s.e.). B) Standard error versus number of SNPs.

(TIF)

Genetic correlation coefficient for unrelated individuals versus the genetic correlation coefficients for related individuals. Shown are the mean and standard errors. Dashed line is the least squares fit with the y-intercept fixed at 0 estimated using a Monte Carlo method (slope = 0.44).

(TIF)

A) Genetic correlation coefficients versus the phenotypic correlation coefficients for related individuals. Shown are the mean and standard errors. Dashed line is the least squares fit with the y-intercept fixed at 0 estimated using a Monte Carlo method (slope = 1.2). B) Genetic correlation coefficients versus the phenotype correlation coefficients for unrelated individuals. Shown are the mean and standard errors. Dashed line is the least squares fit with the y-intercept fixed at 0 estimated using a Monte Carlo method (slope = 0.85).

(TIF)

Atherosclerosis Risk in Communities Study (ARIC) population statistics by sex; mean (sd; minimum-maximum). BMI = body-mass index, WC = waist circumference, WHR = waist-to-hip ratio, GLU = fasting glucose, INS = fasting insulin, TG = fasting triglycerides, HDL = fasting high-density lipoprotein, SBP = systolic blood pressure.

(DOCX)

Framingham Heart Study (FHS) population statistics.

(DOCX)

Genetic and residual covariance estimates for the ARIC population among related individuals. Mean and standard error of genetic (upper triangle) and residual (lower triangle) covariance estimates from the univariate (diagonals) and bivariate (off-diagonals) REML model.

(DOCX)

Genetic and residual covariance estimates for the ARIC population among unrelated individuals. Mean and standard error of genetic (upper triangle) and residual (lower triangle) covariance estimates from the univariate (diagonals) and bivariate (off-diagonals) REML model.

(DOCX)

Genetic (upper triangle) and residual (lower triangle) correlations among unrelated individuals in the ARIC population based on simultaneous analysis of all MetS traits. Mean and standard error of the Pearson correlation coefficient for genetic correlations (upper triangle) and residual correlations (lower triangle). An asterisk indicates significance with p<0.05 adjusted for 21 hypotheses using the two-tailed hypothesis test and normal distribution of the Fisher transformed correlation coefficient.

(DOCX)

Genetic (upper triangle) and residual (lower triangle) covariances among unrelated individuals in the ARIC population based on simultaneous analysis of all MetS traits. Mean and standard error.

(DOCX)

We thank Peter Visscher and Marc Reitman for insightful comments.