A semi-automatic method to guide the choice of ridge parameter in ridge regression
Erika Cule, Maria De Iorio
(Submitted on 3 May 2012)

We consider the application of a popular penalised regression method, Ridge Regression, to data with very high dimensions and many more covariates than observations. Our motivation is the problem of out-of-sample prediction and the setting is high-density genotype data from a genome-wide association or resequencing study. Ridge regression has previously been shown to offer improved performance for prediction when compared with other penalised regression methods. One problem with ridge regression is the choice of an appropriate parameter for controlling the amount of shrinkage of the coefficient estimates. Here we propose a method for choosing the ridge parameter based on controlling the variance of the predicted observations in the model.
Using simulated data, we demonstrate that our method outperforms subset selection based on univariate tests of association and another penalised regression method, HyperLasso regression, in terms of improved prediction error. We extend our approach to regression problems when the outcomes are binary (representing cases and controls, as is typically the setting for genome-wide association studies) and demonstrate the method on a real data example consisting of case-control and genotype data on Bipolar Disorder, taken from the Wellcome Trust Case Control Consortium and the Genetic Association Information Network.

LMM-Lasso: A Lasso Multi-Marker Mixed Model for Association Mapping with Population Structure Correction
Barbara Rakitsch, Christoph Lippert, Oliver Stegle, Karsten Borgwardt
(Submitted on 30 May 2012)

Exploring the genetic basis of heritable traits remains one of the central challenges in biomedical research. In simple cases, single polymorphic loci explain a significant fraction of the phenotype variability. However, many traits of interest appear to be subject to multifactorial control by groups of genetic loci instead. Accurate detection of such multivariate associations is nontrivial and often hindered by limited power. At the same time, confounding influences such as population structure cause spurious association signals that result in false positive findings if they are not accounted for in the model. Here, we propose LMM-Lasso, a mixed model that allows for both, multi-locus mapping and correction for confounding effects. Our approach is simple and free of tuning parameters, effectively controls for population structure and scales to genome-wide datasets. We show practical use in genome-wide association studies and linkage mapping through retrospective analyses. In data from Arabidopsis thaliana and mouse, our method is able to find a genetic cause for significantly greater fractions of phenotype variation in 91% of the phenotypes considered. At the same time, our model dissects this variability into components that result from individual SNP effects and population structure. In addition to this increase of genetic heritability, enrichment of known candidate genes suggests that the associations retrieved by LMM-Lasso are more likely to be genuine.

Kernel Approximate Bayesian Computation for Population Genetic Inferences
Shigeki Nakagome, Kenji Fukumizu, Shuhei Mano
(Submitted on 15 May 2012)

As genomic data accumulate, Bayesian inferences can be applied to estimate evolutionary parameters. However, the complexity of stochastic models used in population genetics makes it difficult to derive the likelihoods needed for Bayesian inferences. Approximate Bayesian Computation (ABC) is an alternative approach for obtaining Bayesian inferences without likelihoods. ABC is a rejection-based method that applies a tolerance of dissimilarity between sets of summary statistics from observed and simulated data. ABC gives an exact sampler from the posterior density in the limit of zero tolerance. However, the choices for summary statistics and metrics of dissimilarity are ambiguous, and acceptance rates decrease with an increasing number of summary statistics. Therefore, it is difficult to maintain estimator consistency using ABC. In this study, we apply the kernel Bayes’ rule proposed by Fukumizu et al. (2011) to ABC. We report that kernel ABC (i) avoids the need for tolerance, (ii) upholds the consistency of estimators, and (iii) is tractable for a large number of summary statistics. We demonstrate these advantages by comparing kernel ABC with conventional ABC for population genetic inferences.

Structured Input-Output Lasso, with Application to eQTL Mapping, and a Thresholding Algorithm for Fast Estimation

Seunghak Lee, Eric P. Xing
(Submitted on 9 May 2012)

We consider the problem of learning a high-dimensional multi-task regression model, under sparsity constraints induced by presence of grouping structures on the input covariates and on the output predictors. This problem is primarily motivated by expression quantitative trait locus (eQTL) mapping, of which the goal is to discover genetic variations in the genome (inputs) that influence the expression levels of multiple co-expressed genes (outputs), either epistatically, or pleiotropically, or both. A structured input-output lasso (SIOL) model based on an intricate l1/l2-norm penalty over the regression coefficient matrix is employed to enable discovery of complex sparse input/output relationships; and a highly efficient new optimization algorithm called hierarchical group thresholding (HiGT) is developed to solve the resultant non-differentiable, non-separable, and ultra high-dimensional optimization problem. We show on both simulation and on a yeast eQTL dataset that our model leads to significantly better recovery of the structured sparse relationships between the inputs and the outputs, and our algorithm significantly outperforms other optimization techniques under the same model. Additionally, we propose a novel approach for efficiently and effectively detecting input interactions by exploiting the prior knowledge available from biological experiments.