Sparse canonical correlation analysis, with applications to genomic data

Transcription

1 Sparse canonical correlation analysis, with applications to genomic data Daniela M. Witten and Robert Tibshirani June 15, 2009

2 The framework Suppose that we have n observations on p 1 + p 2 variables, and the variables are naturally partitioned into two groups of p 1 and p 2 variables, respectively.

3 The framework Suppose that we have n observations on p 1 + p 2 variables, and the variables are naturally partitioned into two groups of p 1 and p 2 variables, respectively. Let X 1 R n p 1 correspond to the first set of variables, and let X 2 R n p 2 correspond to the second set of variables.

4 The framework Suppose that we have n observations on p 1 + p 2 variables, and the variables are naturally partitioned into two groups of p 1 and p 2 variables, respectively. Let X 1 R n p 1 correspond to the first set of variables, and let X 2 R n p 2 correspond to the second set of variables. Assume that the columns of X 1 and X 2 have been standardized to have mean zero and standard deviation one.

5 The data

6 Canonical correlation analysis Canonical correlation analysis (CCA) is a classical method in statistics.

7 Canonical correlation analysis Canonical correlation analysis (CCA) is a classical method in statistics. We can seek w 1 R p 1 and w 2 R p 2 that maximize correlation between X 1 w 1 and X 2 w 2 ; that is, maximize w1,w 2 w T 1 X T 1 X 2 w 2 subject to w T 1 X T 1 X 1 w 1 = w T 2 X T 2 X 2 w 2 = 1.

8 Canonical correlation analysis Canonical correlation analysis (CCA) is a classical method in statistics. We can seek w 1 R p 1 and w 2 R p 2 that maximize correlation between X 1 w 1 and X 2 w 2 ; that is, maximize w1,w 2 w T 1 X T 1 X 2 w 2 subject to w T 1 X T 1 X 1 w 1 = w T 2 X T 2 X 2 w 2 = 1. CCA is not appropriate when p 1, p 2 n or p 1, p 2 n.

9 The data when p 1, p 2 n

10 Sparse Canonical Correlation Analysis maximize w1,w 2 w T 1 X T 1 X 2 w 2 subject to w T 1 X T 1 X 1 w 1 = w T 2 X T 2 X 2 w 2 = 1. We impose L 1 constraints onto w 1 and w 2 : that is, w 1 1 c 1 and w 2 1 c 2. We assume that X T 1 X 1 = I and X T 2 X 2 = I.

11 Sparse Canonical Correlation Analysis The sparse CCA criterion is maximize w1,w 2 w T 1 X T 1 X 2 w 2 subject to w 1 2 1, w 2 2 1, w 1 1 c 1, w 2 1 c 2.

12 Sparse Canonical Correlation Analysis The sparse CCA criterion is maximize w1,w 2 w T 1 X T 1 X 2 w 2 subject to w 1 2 1, w 2 2 1, w 1 1 c 1, w 2 1 c 2. For c 1 and c 2 small, this results in w 1 and w 2 sparse: many of the elements of w 1 and w 2 will exactly equal zero.

13 Sparse CCA Algorithm 1. Begin with an intial value for w 2 R p Iterate until convergence: w 1 argmax w1 w1 T X T 1 X 2 w 2 subject to w 1 2 1, w 1 1 c 1. w 2 argmax w2 w1 T X T 1 X 2 w 2 subject to w 2 2 1, w 2 1 c 2.

14 Sparse CCA Algorithm 1. Begin with an intial value for w 2 R p Iterate until convergence: w 1 argmax w1 w1 T X T 1 X 2 w 2 subject to w 1 2 1, w 1 1 c 1. w 2 argmax w2 w1 T X T 1 X 2 w 2 subject to w 2 2 1, w 2 1 c 2. Each update takes the form w 1 S(XT 1 X 2w 2, 1 ) S(X T 1 X 2w 2, 1 ) 2 where 1 0 is chosen so that w 1 1 = c 1. Here, S is the soft-thresholding operator: S(x, a) = sgn(x)( x a) +.

15 Genomic Data Recently, it has become increasingly common for researchers to use multiple assays to obtain measurements on a single set of patient samples. For instance, a data set might consist of gene expression and DNA copy number measurements on the same set of samples.

16 Genomic Data Recently, it has become increasingly common for researchers to use multiple assays to obtain measurements on a single set of patient samples. For instance, a data set might consist of gene expression and DNA copy number measurements on the same set of samples. The Question: Can we identify sets of genes whose expression is correlated with regions of copy number change?

17 DNA copy number change

18 DNA copy number change

19 Gene expression

20 Gene expression + copy number data

21 Gene expression + copy number data We want to find regions of DNA copy number change that are highly correlated with the expression of a set of genes.

22 Gene expression + copy number data We want to find regions of DNA copy number change that are highly correlated with the expression of a set of genes. That is, we want w 1 R p 1, w 2 R p 2 Cor(X 1 w 1, X 2 w 2 ) is high. sparse such that

23 Gene expression + copy number data We want to find regions of DNA copy number change that are highly correlated with the expression of a set of genes. That is, we want w 1 R p 1, w 2 R p 2 Cor(X 1 w 1, X 2 w 2 ) is high. sparse such that We ll make a statement like 0.3*(gene 1 expression) + 0.2*(gene 2 expression) - 4*(gene 3 expression) is highly correlated with genomic loss on part of chromosome 3.

24 Idea To find regions of copy number change on chromosome 1 that are correlated with gene expression sets anywhere on the genome, run sparse CCA using copy number data on chromosome 1 and expression measurements for all genes.

25 Data set We used a publicly-available breast cancer data set of Chin et al. (2006). 89 samples with breast cancer gene expression measurements DNA copy number measurements.

26 Copy number change on chromosome 1 1

27 Genes correlated w/copy # change on chrom. 1 i Gene Chromosome Weight 1 jumping translocation breakpoint translocated promoter region (to activated MET oncogene) glyceronephosphate O-acyltransferase NADH dehydrogenase (ubiquinone) Fe-S protein nucleoporin 133kD geranylgeranyl diphosphate synthase rab3 GTPase-activating protein, non-catalytic subunit (150kD) peroxisomal biogenesis factor 11B phosphatidylinositol glycan, class C tubulin-specific chaperone e protoporphyrinogen oxidase tuftelin papillary renal cell carcinoma (translocation-associated) splicing factor 3b, subunit 4, 49kD UDP-Gal:betaGlcNAc beta 1,4- galactosyltransferase hypothetical protein FLJ hypothetical protein HSPC mitochondrial ribosomal protein L HSPC003 protein hypothetical protein FLJ CGI-78 protein chromosome 1 open reading frame hypothetical protein My

28 Results, continued 1. Similar results for other chromosomes.

29 Results, continued 1. Similar results for other chromosomes. 2. We can use a permutation approach to estimate a p-value for the w 1 and w 2 obtained.

30 Results, continued 1. Similar results for other chromosomes. 2. We can use a permutation approach to estimate a p-value for the w 1 and w 2 obtained. 3. We described the identification of cis effects. To identify trans effects, run sparse CCA using copy number data on chromosome 1 and expression measurements for all genes NOT on chromosome 1.

31 Conclusions A method for the integrative analysis of genomic data sets. An extension of sparse CCA leads to sparse multiple CCA, for the case of K > 2 data sets on a single set of observations. If an outcome measurement (e.g. survival time, cancer subtype) is available for each observation, than sparse supervised CCA can be used. Software available in R PMA package and Excel AddIn Correlate (implemented by Sam Gross)

32 References 1. Chin et al. (2006) Genomic and transcriptional aberrations linked to breast cancer pathophysiologies, Cancer Cell Parkhomenko, Trichtler, and Beyene (2009) Sparse canonical correlation analysis with application to genomic data integration, Statistical Applications in Genetics and Molecular Biology 8 Article Waaijenborg, Versewel de Witt Hamer, and Zwinderman (2008) Quantifying the association between gene expression and DNA markers by penalized canonical correlation analysis, Statistical Applications in Genetics and Molecular Biology 7 Article Witten, Tibshirani, and Hastie (2009) A penalized matrix decomposition, with applications to sparse canonical correlation analysis and principal components, Biostatistics. 5. Witten and Tibshirani (2009) Extensions of sparse canonical correlation analysis, with applications to genomic data, Statistical Applications in Genetics and Molecular Biology 8 Article 28.