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Identifying Differentially Expressed Gene Combinations 175are of interest. However, extensions to more than two classes and more thantwo genes are discussed in the context of specific approaches. The notation(1, 2) is used to represents both classes combined, and the subscript k is usedto indicate class. In the basic case, k can be 1, 2, or (1, 2). A vector of G genespecific expression values is denoted by X 1. . . X Gand often by X, Y, and Z,when there are only three genes. Then, for example X = (x 1, . . ., x nk) are thegene-expression levels of gene 1 in samples from class k, were n kis the numberof such samples. Uppercase X’s generally refer to a vector of expression levels,whereas lowercase x’s refer to the levels in individual samples.3.1.2. Correlation-Based Scoring FunctionsThe correlation between any two genes within class k is measured by theclass-conditional Pearson correlation coefficient∑nk( x − x) y − y( )i i iρ k=nk2 nk2∑ ( x − xi ) ∑ ( y − yi )iiwhich ranges between –1 and 1. If ρ k> 0, the observed expression levelsof the two genes are positively correlated (that is they move in the same direction)in samples from class k. High conditional correlation occurs when thepoints within the same phenotype line up as a straight line going upward. Ifρ k< 0, levels are negatively correlated, and if ρ k= 0 they are uncorrelated.For each pair of genes, two class-conditional correlation coefficients, ρ 1andρ 2, and one combined correlation coefficient, ρ = ρ (1, 2)can be derived usingthe available samples.The difference in the class-conditional correlations(1)S cross=ρ −ρ1 2(2)captures cross patterns (5,6). S crossincreases with an increasing absolutedifference of the conditional correlations. The largest S crossis two and can beachieved by perfect conditional correlation in both classes, with opposite signs,for example, ρ 1= 1, and ρ 2=−1. A S crossof zero occurs when correlations arethe same, irrespective of their magnitude.An important related case is L-shaped joint distributions, formed by a horizontaland a vertical ellipsoid. These can be thought of as a rotation of Fig. 2,and occur when the phenotype affects the variance of the two genes so that eachgene varies only within one class. The S crossmeasure will not capture thesepatterns, and will have difficulties with cross pattern that approach an L-shape.However, 1D comparisons of classes one gene at a time, if appropriately designed,will identify these genes (12).