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178 Ho et al.3.1.3. LA and GeneralizationsLA, developed by Li (7), is another correlation-based method for identifyingdynamically coexpressing genes. Assume X, Y, and Z are the standardized andnormally distributed expression intensities of gene 1, 2, and 3 and let f(z) =E(X,Y |Z = z), thenLA( XYZ , ⎥ )= Ef′( Z)= Ef( ZZ ) = EXYZ ( )When applied to a set of n samples, distributional assumptions are met by substitutingstandard normal quantiles for the order statistics of each variable andthe LA score is estimated as:( )LA( XYZ , ⎥ ) = xyz + K+xyz n. (4)1 1 1 n n nPLA was developed by Li and colleagues (8) to extend LA to larger genessets. As the name suggests, PLA applies the LA method to 2D projections ofthe gene space, selecting the projection that maximizes the LA score. Considera candidate set of G genes with expression vectors X = (X 1, . . ., X G) and anadditional mediator gene Z. A 1D projection of X is a linear combination a′Xwhere a is a projection direction and has norm one. For a projection in a 2Dspace, the method requires that the two projection directions a and b be orthogonalto each other. After projection, PLA considers the LA between a′X and b′Xmediated by Z, and seeks informative projections by maximizing a′ Ε ( ZXX′) bover all pairs of orthogonal projection directions a and b.Algorithmically, this maximization can be implemented through eigenvaluedecomposition of E ( ZXX′ ): begin with the matrix Σ formed by every possibleliquid correlation with mediator Z, that is Σ gg′= LA(X g,X g′|Ζ) = E(X gX g′Z). Ifλ 1, . . ., λ Gbe the ordered eigenvalues of Σ and v 1, . . .,v Gbe the associated eigenvectors.Then the maximum absolute LA score is.PLA( X ... X ⏐Z) = ( λ − λ ) /21 G 1 G(5)The optimal projection is to the plane defined by the orthogonal vectors± ( υ− 1+υ )/ 2 and ± ( υ−1−υ)/ 2GG . To facilitate interpretation, the signs ofeach vector are determined so that the greatest gene-to-gene variation is capturedin the positive component of the vector. If the mediating variable Z is binary,perhaps representing a phenotype rather than gene expression, then the LA scoreis formally equivalent to the correlation-based measure S crossdescribed above.The simulations described in Subheading 3.2. are confined to the two-gene,two-class case, and so the LA score is represented there by S cross. Likewise, it isto be expected that application of PLA when the mediating variable is binary and