Estimating relationships between phenotypes and subjects drawn from admixed families.

Background: Estimating relationships among subjects in a sample, within family structures or caused by population substructure, is complicated in admixed populations. Inaccurate allele frequencies can bias both kinship estimates and tests for association between subjects and a phenotype. We analyzed the simulated and real family data from Genetic Analysis Workshop 19, and were aware of the simulation model. Results: We found that kinship estimation is more accurate when marker data include common variants whose frequencies are less variable across populations. Estimates of heritability and association vary with age for longitudinally measured traits. Accounting for local ancestry identified different true associations than those identified by a traditional approach. Principal components aid kinship estimation and tests for association, but their utility is influenced by the frequency of the markers used to generate them. Conclusions: Admixed families can provide a powerful resource for detecting disease loci, as well as analytical challenges. Allele frequencies, although difficult to adequately estimate in admixed populations, have a strong impact on the estimation of kinship, ancestry, and association with phenotypes. Approaches that acknowledge population structure in admixed families outperform those which ignore it

The genetic architecture of the human cerebral cortex

The cerebral cortex underlies our complex cognitive capabilities, yet little is known about the specific genetic loci that influence human cortical structure. To identify genetic variants that affect cortical structure, we conducted a genome-wide association meta-analysis of brain magnetic resonance imaging data from 51,665 individuals. We analyzed the surface area and average thickness of the whole cortex and 34 regions with known functional specializations. We identified 199 significant loci and found significant enrichment for loci influencing total surface area within regulatory elements that are active during prenatal cortical development, supporting the radial unit hypothesis. Loci that affect regional surface area cluster near genes in Wnt signaling pathways, which influence progenitor expansion and areal identity. Variation in cortical structure is genetically correlated with cognitive function, Parkinson's disease, insomnia, depression, neuroticism, and attention deficit hyperactivity disorder

Probability Models in Networks and Landscape Genetics

With the advent of massively parallel high-throughput sequencing, geneticists have the technology to answer many problems. What we lack are analytical tools. As the amount of data from these sequencers continues to overwhelm much of the current analytical tools, we must come up with more efficient methods for analysis. One potentially useful tool is the MM, majorize-minimize or minorize-maximize, algorithm. The MM algorithm is an optimization method suitable for high-dimensional problems. It can avoid large matrix inversions, linearize problems, and separate parameters. Additionally it deals with constraints gracefully and can turn a non-differentiable problem into a smooth one. These benefits come at the cost of iteration. In this thesis we apply the MM algorithm in the optimization of three problems. The first problem we tackle is an extension of random graph theory by Erdos. We extend the model by relaxing two of the three underlying assumptions, namely any number of edges can form between two nodes and edges form with a Poisson probability with mean dependent on the two nodes. This is aptly named a random multigraph.The next problem extends random multigraphs to include clustering. As before, any number of edges can still form between two nodes. The difference is now the number of edges formed between two nodes is Poisson distributed with mean dependent on the two nodes along with their clusters. For our last problem we place individuals onto the map using their genetic information. Using a binomial model with a nearest neighbor penalty, we estimate allele frequency surfaces for a region. With these allele frequency surfaces, we calculate the posterior probability that an individual comes from a location by a simple application of Bayes' rule and place him at his most probable location. Furthermore, with an additional model we estimate admixture coefficients of individuals across a pixellated landscape.Each of these problems contain an underlying optimization problem which is solved using the MM algorithm. To demonstrate the utility of the models we applied them to various genetic datasets including POPRES, OMIM, gene expression, protein-protein interactions, and gene-gene interactions. Each example yielded interesting results in reasonable time

Cluster and propensity based approximation of a network

Abstract Background The models in this article generalize current models for both correlation networks and multigraph networks. Correlation networks are widely applied in genomics research. In contrast to general networks, it is straightforward to test the statistical significance of an edge in a correlation network. It is also easy to decompose the underlying correlation matrix and generate informative network statistics such as the module eigenvector. However, correlation networks only capture the connections between numeric variables. An open question is whether one can find suitable decompositions of the similarity measures employed in constructing general networks. Multigraph networks are attractive because they support likelihood based inference. Unfortunately, it is unclear how to adjust current statistical methods to detect the clusters inherent in many data sets. Results Here we present an intuitive and parsimonious parametrization of a general similarity measure such as a network adjacency matrix. The cluster and propensity based approximation (CPBA) of a network not only generalizes correlation network methods but also multigraph methods. In particular, it gives rise to a novel and more realistic multigraph model that accounts for clustering and provides likelihood based tests for assessing the significance of an edge after controlling for clustering. We present a novel Majorization-Minimization (MM) algorithm for estimating the parameters of the CPBA. To illustrate the practical utility of the CPBA of a network, we apply it to gene expression data and to a bi-partite network model for diseases and disease genes from the Online Mendelian Inheritance in Man (OMIM). Conclusions The CPBA of a network is theoretically appealing since a) it generalizes correlation and multigraph network methods, b) it improves likelihood based significance tests for edge counts, c) it directly models higher-order relationships between clusters, and d) it suggests novel clustering algorithms. The CPBA of a network is implemented in Fortran 95 and bundled in the freely available R package PropClust

Convex clustering: an attractive alternative to hierarchical clustering.

The primary goal in cluster analysis is to discover natural groupings of objects. The field of cluster analysis is crowded with diverse methods that make special assumptions about data and address different scientific aims. Despite its shortcomings in accuracy, hierarchical clustering is the dominant clustering method in bioinformatics. Biologists find the trees constructed by hierarchical clustering visually appealing and in tune with their evolutionary perspective. Hierarchical clustering operates on multiple scales simultaneously. This is essential, for instance, in transcriptome data, where one may be interested in making qualitative inferences about how lower-order relationships like gene modules lead to higher-order relationships like pathways or biological processes. The recently developed method of convex clustering preserves the visual appeal of hierarchical clustering while ameliorating its propensity to make false inferences in the presence of outliers and noise. The solution paths generated by convex clustering reveal relationships between clusters that are hidden by static methods such as k-means clustering. The current paper derives and tests a novel proximal distance algorithm for minimizing the objective function of convex clustering. The algorithm separates parameters, accommodates missing data, and supports prior information on relationships. Our program CONVEXCLUSTER incorporating the algorithm is implemented on ATI and nVidia graphics processing units (GPUs) for maximal speed. Several biological examples illustrate the strengths of convex clustering and the ability of the proximal distance algorithm to handle high-dimensional problems. CONVEXCLUSTER can be freely downloaded from the UCLA Human Genetics web site at http://www.genetics.ucla.edu/software/

Convex clustering concepts.

<p>For clarity, we present three random data points extracted from the three classes in the Iris dataset. Black points denote the original data points <b><i>X</i></b> and blue points denote the cluster centers <b><i>U</i></b>. At <i>μ</i> = 0, <b><i>X</i></b> and <b><i>U</i></b> coincide. At intermediate <i>μ</i> values (middle figure), <b><i>U</i></b> coalesces towards its cluster center. For sufficiently large <i>μ</i>, <b><i>U</i></b> converges to cluster centers (right figure). Note that in this demonstration, only the left two points have non-zero pairwise weights <i>w</i>_<i>ij</i>. Hence, the two resulting clusters reflect the two graphs defined by the matrix of weights.</p

Convex clustering of the breast cancer samples.

<p>Points on the plot indicate data vectors projected onto the first and third principal components (PCs) of the sample. Lines trace the cluster centers as they traverse the regularization path.</p

Magnified view of the convex clustering results for the HGDP data in Europe and Central Asia.

<p>Magnified view of the convex clustering results for the HGDP data in Europe and Central Asia.</p

Hierarchical clustering of the European populations from the POPRES data.

<p>Hierarchical clustering of the European populations from the POPRES data.</p