**StATS: ****Dimension reduction in a microarray experiment (May 25, 2005)**.

Given the large number of genes in a microarray experiment, you need to find some way of looking at subsets or linear combinations of these genes. Assume that you have G genes and M microarrays and that the normalized signals are in a matrix X with G rows and M columns. Assume that information about the particular tissues (phenotypic data) is in a matrix Y with G rows and P columns.

Often you will find more interpretable results for these procedures when you center the rows of X so that they have a mean of zero, or standardize the rows of X so that they will also have a standard deviation of one.

Principal components. A simple approach that looks at linear combinations of the columns of X (the genes) that have the largest amount of variation. Alternately, you can look at linear combinations of the rows of X (the arrays) that have the largest amount of variation. These linear combinations are often interesting, and you can find it useful sometimes to plot these linear combinations versus some of the phenotypic data. There are two functions in R, prcomp and princomp, that perform principal components analysis.

Singular value decomposition. This is a tool that allows you to simultaneously look at principal components both from the perspective of the rows (genes) and the columns (arrays). A plot of the first two dimensions of the singular value decomposition is known as a biplot. There is an svd function in the base package of R that performs singular value decomposition.

Canonical correlation. This represents linear combinations of the rows (genes) or columns (arrays) of X that are maximally associated with the phenotypic data. There is a cancorr function in the base package of R that performs canonical correlation.

Partial least squares and latent root regressionare methods that try to find linear combinations of the rows of X that are the strongest predictors of the phenotypic data. The R package has libraries (pls and plsgenomics) that perform partial least squares regression.

Further reading:

Partial Least Squares (PLS). StatSoft Inc. Accessed on 2005-05-25. www.statsoft.com/textbook/stpls.htmlPartial Least Squares (PLS) Regression [PDF]. Abdi H, published in Lewis-Beck M., Bryman, A., Futing T. (Eds.) (2003). Encyclopedia of Social Sciences Research Methods. Thousand Oaks (CA): Sage. Accessed on 2005-05-25. www.utdallas.edu/~herve/Abdi-PLS-pretty.pdf

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