When many correlated traits are measured the potential exists to discover the coordinated control of these traits via genotyped polymorphisms. A common statistical approach to this problem involves assessing the relationship between each phenotype and each single nucleotide polymorphism (SNP) individually (PHN); and taking a Bonferroni correction for the effective number of independent tests conducted. Alternatively, one can apply a dimension reduction technique, such as estimation of principal components, and test for an association with the principal components of the phenotypes (PCP) rather than the individual phenotypes. Building on the work of Lange and colleagues we develop an alternative method based on the principal component of heritability (PCH). For each SNP the PCH approach reduces the phenotypes to a single trait that has a higher heritability than any other linear combination of the phenotypes. As a result, the association between a SNP and derived trait is often easier to detect than an association with any of the individual phenotypes or the PCP. When applied to unrelated subjects, PCH has a drawback. For each SNP it is necessary to estimate the vector of loadings that maximize the heritability over all phenotypes. We develop a method of iterated sample splitting that uses one portion of the data for training and the remainder for testing. This cross-validation approach maintains the type I error control and yet utilizes the data efficiently, resulting in a powerful test for association.

Resources being amassed for genome-wide association (GWA) studies include "control databases" genotyped with a large-scale SNP array. How to use these databases effectively is an open question. We develop a method to match, by genetic ancestry, controls to affected individuals (cases). The impact of this method, especially for heterogeneous human populations, is to reduce the false-positive rate, inflate other spuriously small p values, and have little impact on the p values associated with true positive loci. Thus, it highlights true positives by downplaying false positives. We perform a GWA by matching Americans with type 1 diabetes (T1D) to controls from Germany. Despite the complex study design, these analyses identify numerous loci known to confer risk for T1D.

As one approach to uncovering the genetic underpinnings of complex disease, individuals are measured at a large number of genetic variants (usually SNPs) across the genome and these SNP genotypes are assessed for association with disease status. We propose a new statistical method called Spectral-GEM for the analysis of genome-wide association studies; the goal of Spectral-GEM is to quantify the ancestry of the sample from such genotypic data. Ignoring structure due to differential ancestry can lead to an excess of spurious findings and reduce power. Ancestry is commonly estimated using the eigenvectors derived from principal component analysis (PCA). To develop an alternative to PCA we draw on connections between multidimensional scaling and spectral graph theory. Our approach, based on a spectral embedding derived from the normalized Laplacian of a graph, can produce more meaningful delineation of ancestry than by using PCA. Often the results from Spectral-GEM are straightforward to interpret and therefore useful in association analysis. We illustrate the new algorithm with an analysis of the POPRES data [Nelson et al., 2008].

Mapping human genetic variation is fundamentally interesting in fields such as anthropology and forensic inference. At the same time, patterns of genetic diversity confound efforts to determine the genetic basis of complex disease. Due to technological advances, it is now possible to measure hundreds of thousands of genetic variants per individual across the genome. Principal component analysis (PCA) is routinely used to summarize the genetic similarity between subjects. The eigenvectors are interpreted as dimensions of ancestry. We build on this idea using a spectral graph approach. In the process we draw on connections between multidimensional scaling and spectral kernel methods. Our approach, based on a spectral embedding derived from the normalized Laplacian of a graph, can produce more meaningful delineation of ancestry than by using PCA. The method is stable to outliers and can more easily incorporate different similarity measures of genetic data than PCA. We illustrate a new algorithm for genetic clustering and association analysis on a large, genetically heterogeneous sample.