Uncovering Local Trends in Genetic Effects of Multiple Phenotypes via Functional Linear Models

Recent technological advances equipped researchers with capabilities that go beyond traditional genotyping of loci known to be polymorphic in a general population. Genetic sequences of study participants can now be assessed directly. This capability removed technology-driven bias toward scoring predominantly common polymorphisms and let researchers reveal a wealth of rare and sample-specific variants. Although the relative contributions of rare and common polymorphisms to trait variation are being debated, researchers are faced with the need for new statistical tools for simultaneous evaluation of all variants within a region. Several research groups demonstrated flexibility and good statistical power of the functional linear model approach. In this work we extend previous developments to allow inclusion of multiple traits and adjustment for additional covariates. Our functional approach is unique in that it provides a nuanced depiction of effects and interactions for the variables in the model by representing them as curves varying over a genetic region. We demonstrate flexibility and competitive power of our approach by contrasting its performance with commonly used statistical tools and illustrate its potential for discovery and characterization of genetic architecture of complex traits using sequencing data from the Dallas Heart Study

Adaptive Basis Sampling for Smoothing Splines

Smoothing splines provide flexible nonparametric regression estimators. Penalized likelihood method is adopted when responses are from exponential families and multivariate models are constructed with certain analysis of variance decomposition. However, the high computational cost of smoothing splines for large data sets has hindered their wide application. We develop a new method, named adaptive basis sampling, for efficient computation of smoothing splines in super-large samples. Generally, a smoothing spline for a regression problem with sample size n can be expressed as a linear combination of n basis functions and its computational complexity is O(n³). We achieve a more scalable computation in the multivariate case by evaluating the smoothing spline using a smaller set of basis functions, obtained by an adaptive sampling scheme that uses values of the response variable. Our asymptotic analysis shows that smoothing splines computed via adaptive basis sampling converge to the true function at the same rate as full basis smoothing splines. We show that the proposed method outperforms a sampling method that does not use the values of response variable by simulation studies, and apply it to several real data examples

Model-based Boosting in R: A Hands-on Tutorial Using the R Package mboost

We provide a detailed hands-on tutorial for the R add-on package mboost. The package implements boosting for optimizing general risk functions utilizing component-wise (penalized) least squares estimates as base-learners for fitting various kinds of generalized linear and generalized additive models to potentially high-dimensional data. We give a theoretical background and demonstrate how mboost can be used to fit interpretable models of different complexity. As an example we use mboost to predict the body fat based on anthropometric measurements throughout the tutorial

Efficient estimation algorithms for large and complex data sets

The recent world-wide surge in available data allows the investigation of many new and sophisticated questions that were inconceivable just a few years ago. However, two types of data sets often complicate the subsequent analysis: Data that is simple in structure but large in size, and data that is small in size but complex in structure. These two kinds of problems also apply to biological data. For example, data sets acquired from family studies, where the data can be visualized as pedigrees, are small in size but, because of the dependencies within families, they are complex in structure. By comparison, next-generation sequencing data, such as data from chromatin immunoprecipitation followed by deep sequencing (ChIP-Seq), is simple in structure but large in size. Even though the available computational power is increasing steadily, it often cannot keep up with the massive amounts of new data that are being acquired. In these situations, ordinary methods are no longer applicable or scale badly with increasing sample size. The challenge in today’s environment is then to adapt common algorithms for modern data sets. This dissertation considers the challenge of performing inference on modern data sets, and approaches the problem in two parts: first using a problem in the field of genetics, and then using one from molecular biology. In the first part, we focus on data of a complex nature. Specifically, we analyze data from a family study on colorectal cancer (CRC). To model familial clusters of increased cancer risk, we assume inheritable but latent variables for a risk factor that increases the hazard rate for the occurrence of CRC. During parameter estimation, the inheritability of this latent variable necessitates a marginalization of the likelihood that is costly in time for large families. We first approached this problem by implementing computational accelerations that reduced the time for an optimization by the Nelder-Mead method to about 10% of a naive implementation. In a next step, we developed an expectation-maximization (EM) algorithm that works on data obtained from pedigrees. To achieve this, we used factor graphs to factorize the likelihood into a product of “local” functions, which enabled us to apply the sum-product algorithm in the E-step, reducing the computational complexity from exponential to linear. Our algorithm thus enables parameter estimation for family studies in a feasible amount of time. In the second part, we turn to ChIP-Seq data. Previously, practitioners were required to assemble a set of tools based on different statistical assumptions and dedicated to specific applications such as calling protein occupancy peaks or testing for differential occupancies between experimental conditions. In order to remove these restrictions and create a unified framework for ChIP-Seq analysis, we developed GenoGAM (Genome-wide Generalized Additive Model), which extends generalized additive models to efficiently work on data spread over a long x axis by reducing the scaling from cubic to linear and by employing a data parallelism strategy. Our software makes the well-established and flexible GAM framework available for a number of genomic applications. Furthermore, the statistical framework allows for significance testing for differential occupancy. In conclusion, I show how developing algorithms of lower complexity can open the door for analyses that were previously intractable. On this basis, it is recommended to focus subsequent research efforts on lowering the complexity of existing algorithms and design new, lower-complexity algorithms