Motivated by studying the association between nutrient intake and human gut microbiome composition, we developed a method for structure-constrained sparse canonical correlation analysis (ssCCA) in a high-dimensional setting. ssCCA takes into account the phylogenetic relationships among bacteria, which provides important prior knowledge on evolutionary relationships among bacterial taxa. Our ssCCA formulation utilizes a phylogenetic structure-constrained penalty function to impose certain smoothness on the linear coefficients according to the phylogenetic relationships among the taxa. An efficient coordinate descent algorithm is developed for optimization. A human gut microbiome data set is used to illustrate this method. Both simulations and real data applications show that ssCCA performs better than the standard sparse CCA in identifying meaningful variables when there are structures in the data.
"This article considers multiple tests for which p-values cannot be computed exactly but can be approximated through Monte-Carlo methods, for instance bootstrap tests or permutation tests. This scenario occurs widely in practical situations (Chen et al., 2013; Nusinow et al., 2012; Rahmatallah et al., 2012). Throughout the article, the ideal p-value of a test is the p-value obtained by integrating over the underlying theoretical bootstrap distribution of a bootstrap test or, in case of a permutation test, by exhaustively generating all permutations. "
[Show abstract][Hide abstract] ABSTRACT: Most procedures which correct for multiple tests assume ideal p-values, for
instance the Bonferroni correction or the procedures of Bonferroni-Holm, Sidak,
Hochberg or Benjamini-Hochberg. This article considers multiple testing under
the assumption that the ideal p-values for the hypotheses under consideration
are not available and thus have to be approximated using Monte-Carlo
simulation. This scenario widely occurs in practical situations. We are
interested in obtaining the same rejections and non-rejections as the ones
obtained if the ideal p-values for all hypotheses had been available. The
contribution of this article is threefold. Firstly, it introduces a new
framework for the scenario aforementioned, both in terms of a generic algorithm
used to draw samples and an arbitrary multiple testing procedure to evaluate
the tests. We establish conditions on both the testing procedure and on the
algorithm which guarantee that the rejections and non-rejections obtained
through Monte-Carlo simulation only are identical to the ones obtained with the
ideal p-values. Secondly, by simplifying our condition for an arbitrary step-up
or step-down procedure, we extend the applicability of our framework to a
general class of step-up and step-down procedures used in practice. Thirdly, we
show how to use our framework to improve established methods without proven
properties in such a way as to yield certain theoretical guarantees on their
results. These modifications can easily be implemented in practice and lead to
a certain way of reporting classifications as three sets together with an error
bound on their correctness, demonstrated exemplarily using a real biological
"Chen etal. incorporated the group effect into an association study of nutrient intake with human gut microbiome . Both papers show an improvement when incorporating the group effect, however, a prior knowledge of group structure is needed and only the group effect of one type of data is discussed. "
[Show abstract][Hide abstract] ABSTRACT: The emergence of high-throughput genomic datasets from different sources and platforms (e.g., gene expression, single nucleotide polymorphisms (SNP), and copy number variation (CNV)) has greatly enhanced our understandings of the interplay of these genomic factors as well as their influences on the complex diseases. It is challenging to explore the relationship between these different types of genomic data sets. In this paper, we focus on a multivariate statistical method, canonical correlation analysis (CCA) method for this problem. Conventional CCA method does not work effectively if the number of data samples is significantly less than that of biomarkers, which is a typical case for genomic data (e.g., SNPs). Sparse CCA (sCCA) methods were introduced to overcome such difficulty, mostly using penalizations with l-1 norm (CCA-l1) or the combination of l-1and l-2 norm (CCA-elastic net). However, they overlook the structural or group effect within genomic data in the analysis, which often exist and are important (e.g., SNPs spanning a gene interact and work together as a group).
We propose a new group sparse CCA method (CCA-sparse group) along with an effective numerical algorithm to study the mutual relationship between two different types of genomic data (i.e., SNP and gene expression). We then extend the model to a more general formulation that can include the existing sCCA models. We apply the model to feature/variable selection from two data sets and compare our group sparse CCA method with existing sCCA methods on both simulation and two real datasets (human gliomas data and NCI60 data). We use a graphical representation of the samples with a pair of canonical variates to demonstrate the discriminating characteristic of the selected features. Pathway analysis is further performed for biological interpretation of those features.
The CCA-sparse group method incorporates group effects of features into the correlation analysis while performs individual feature selection simultaneously. It outperforms the two sCCA methods (CCA-l1 and CCA-group) by identifying the correlated features with more true positives while controlling total discordance at a lower level on the simulated data, even if the group effect does not exist or there are irrelevant features grouped with true correlated features. Compared with our proposed CCA-group sparse models, CCA-l1 tends to select less true correlated features while CCA-group inclines to select more redundant features.
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