Principal Components of Heritability for High Dimension Quantitative Traits and General Pedigrees

Abstract

For many complex disorders, genetically relevant disease definition is still unclear. For this reason, researchers tend to collect large numbers of items related directly or indirectly to the disease diagnostic. Since the measured traits may not be all influenced by genetic factors, researchers are faced with the problem of choosing which traits or combinations of traits to consider in linkage analysis. To combine items, one can subject the data to a principal component analysis. However, when family date are collected, principal component analysis does not take family structure into account. In order to deal with these issues, Ott & Rabinowitz (1999) introduced the principal components of heritability (PCH), which capture the familial information across traits by calculating linear combinations of traits that maximize heritability. The calculation of the PCHs is based on the estimation of the genetic and the environmental components of variance. In the genetic context, the standard estimators of the variance components are Lange's maximum likelihood estimators, which require complex numerical calculations. The objectives of this paper are the following: i) to review some standard strategies available in the literature to estimate variance components for unbalanced data in mixed models; ii) to propose an ANOVA method for a genetic random effect model to estimate the variance components, which can be applied to general pedigrees and high dimensional family data within the PCH framework; iii) to elucidate the connection between PCH analysis and Linear Discriminant Analysis. We use computer simulations to show that the proposed method has similar asymptotic properties as Lange's method when the number of traits is small, and we study the efficiency of our method when the number of traits is large. A data analysis involving schizophrenia and bipolar quantitative traits is finally presented to illustrate the PCH methodology.

Global Individual Ancestry Using Principal Components for Family Data

“…For all F families we define $\mathrm{boldY}=false({{\mathrm{boldY}}^{\prime }}_1,\cdots ,{{\mathrm{boldY}}^{\prime }}_{\mathbf{F}}false)$ as a ( Np × 1) vector containing all p variables for all individuals, with $N={\sum }_{i=1}^F{n}_f$ with E ( Y ) = 1 N ⊗ μ f and Cov ( Y ) = Diag (2Φ f ) ⊗ Σ g + I N ⊗ Σ e , f = 1, …, F . This framework represents the multivariate family-based model [10, 12, 19, 20]. In this paper, the p variables represent the SNPs genotype (standardized or not) selected from the whole genome to estimate the global ancestry coefficients for individuals in family structures.…”

“…When the multivariate family-based model described above is extended for general pedigrees, Oualkacha et al proposed to use ANOVA estimators for the variance component matrices [12]. By using the sum of square and cross-product matrices S w and S b in equations (1) and (2), the estimators of the covariance matrices are written as $$true{\hat{\sum }}_g^A=\frac{{\mathbf{S}}_b/false(F-1false)-{\mathbf{S}}_w/false(N-Ffalse)}{true({\sigma }_c-\frac{{\sigma }_b}{N}true)/true(F-1true)-true({\sigma }_a-{\sigma }_{c}true)/true(N-Ftrue)},$$ $$true{\hat{\sum }}_e^A=\frac{1}{N-F}{\mathbf{S}}_w-\frac{false({\sigma }_a-{\sigma }_{c}false)}{N-F}true{\hat{\sum }}_g^A,$$ where ${\sigma }_a=true{\sum }_{f=1}^F{\sigma }_a^{(f)}$, ${\sigma }_b=true{\sum }_{f=1}^F{\sigma }_b^{(f)}$, ${\sigma }_c=true{\sum }_{f=1}^F\frac{1}{n_f}{\sigma }_b^{false(ffalse)}$, ${\sigma }_a^{false(ffalse)}=trtrue(2{\mathbf{\Phi }}^{false(ffalse)}true)$, and ${\sigma }_b^{false(ffalse)}=true{\sum }_{j=1}^{\mathrm{italicnf}}{\displaystyle {\sum }_{k=1}^{\mathit{nf}}{\mathbf{\Phi }}_{jk}^{false(ffalse)}}$…”

“…We also apply the Principal Component of Heritability (PCH) proposed by Ott and Rabinowitz [10] and extended by Oualkacha et al [12] to determine the PCs for extended pedigrees. Here, instead of looking for the linear combinations of phenotypes with maximum variance, the focus is on the linear combination of phenotypes that maximizes its heritability that accounts for the intra-family correlation.…”

“…Within the PCH framework, Wang et al [11] proposed a ridge penalized PCs based on heritability (PCH λ ) for high dimensional family data by adding a penalty to the subject-specific variation. Oualkacha et al [12] proposed an ANOVA estimator for the variance components for general pedigrees and high dimensional family data using the PCH framework. By using simulated pedigree data they showed that their proposed method yields similar results compared to the PCH and PCH λ .…”

“…The goal of our paper is to incorporate SNPs instead of quantitative traits in the PCH λ approach proposed by Oualkacha et al [12] to calculate PCs that simultaneously correct for population stratification taking into account the family structure. For admixed populations, Thornton et al [13, 14] proposed a method that takes into account the admixture in the kinship coefficients by modeling as a random effect using SNP data.…”