Zonas y figuras de protección

Large p, small n problems have become commonplace in statistical analyses of genetic data. In settings with very large p, such as studies of single nucleotide polymorphisms (SNPs), there is a multi-layered challenge, as it is necessary to develop scalable methods, which can address the computational and statistical curse of dimensionality, while leading to interpretable results that are not overly subject to false discoveries. There is a rich literature addressing problems of this type. The most common approach relies on independent screening with false discovery rate (FDR) control, but incorporating SNPs simultaneously can have substantial advantages Hoggart et al. (2008). This is typically done within an additive generalized linear model, with a penalty incorporated to allow p n. For example, L1

penalties lead to Lasso and sparse estimation (Tibshirani 1996), while L2 penalties lead to

ridge regression procedures that allow large numbers of SNPs with small coefficients (Hoerl and Kennard 1970), and the elastic net combinesL1 andL2penalties (Zou and Hastie 2005).

The assumption in such approaches is that SNPs have an additive relationship with the (transformed) mean of the response phenotype. Clearly, this assumption is very restrictive and it is natural biologically to expect interactions among the SNPs and with environmental exposures, with the density of a quantitative trait not simply shifting in mean as the important factors vary but changing in variance and shape. The focus of this article is on developing a practically implementable statistical method that allows this degree of flexibility, while additionally accommodating variable selection, covariate adjustments and dependence arising within families.

There is a considerable statistical and bioinformatics literature focused on identifying interacting predictors of a quantitative trait from among a very large number of candidates. Lou et al. (2007) developed a generalized multi factor dimensionality reduction combinatorial algorithm. Chen et al. (2007) proposed a forest-based approach to identify gene-gene interactions. Zou et al. (2010) incorporated variable selection within a Gaussian process prior for the regression function, enabling selection of a subset of interacting SNPs impacting the mean of the distribution of a quantitative response. Yi (2010) provide an overview of statistical approaches for identifying genetic interactions in high-dimensional settings, such as genome- wide association studies. Cordell (2009) reviews methods for selecting interactions between genetic loci contributing to human disease.

These methods lack the density regression flexibility, which is a key aspect we are focused on. In particular, our hypothesis is that genetic variants and environmental factors impacting the phenotype density do not simply shift the mean in a simple way, but may impact the variance and shape. Such changes in variance, skewness, and higher moments of the

phenotype density seem a natural consequence of genetic heterogeneity. By incorporating this flexibility within our statistical model, we aim to increase power to detect important SNPs and environmental factors. There is a literature on density regression, reviewed below, but current methods fail to produce a variable selection procedure that can adjust for family dependence and scale to very large p.

We propose a model based upon a tensor factorization for the weights on a set of normal kernels. This factorization depends on soft clusterings of observed categorical predictors into a lower-dimensional space. This approach implicitly accounts for interactions and gives a flexible form for the density conditional on predictors and random effects. This conditional tensor factorization for correlated data (CTFC) method accommodates the possibility that the density of the response conditional on the predictors may change in ways that are more complex than a simple shift in conditional mean.

This is an important consideration in contexts where real concern is over behavior in the tails of the conditional distribution. For example, if the quantitative trait is a measure of health like body mass index (BMI), it may be the case that certain combinations of SNP variability have minimal impact on the average BMI, but that the conditional probability of being obese vary widely with these same combinations. Figure C.1 uses data drawn from the 2001-2002 National Health and Nutrition Examination Survey (NHANES) to illustrate distributions for adults of different educational attainments; this is not variation associated with SNPs, but the concept is the same. The two groups have very similar median BMI but noticeably different 90th _{percentile BMI.}

Such considerations have motivated research into quantile regression (Koenker and Bassett 1978). Quantile regression has been used in the modeling of quantitative traits as a function of genetic variation; Ho et al. (2009) used a range of quantile regression techniques to find age-dependent gene expression patterns, and the technique has been used in the analysis of

comparative genomic hybridization data (Eilers and de Menezes 2005). In these applications, the practitioner chooses specific quantiles to focus on. In contrast, the density regression approach models all quantiles simultaneously. Our approach addresses predictor selection and density estimation in separate stages, and combines deterministic approximations with Markov Chain Monte Carlo (MCMC) techniques for inference. The proposed method makes no parametric assumptions about the form of this conditional density, instead using an adaptable nonparametric form. An important component is selection of those predictors with the most influence on the response allowing pn.