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Identifying Differentially Expressed Gene Combinations 181Fig. 3. An illustration of the entropy of standardized eigenvalues with or without theinformation of phenotypic classes, in both crosses and shift patterns.are characterized by low entropy. Accordingly, sets of genes will be scored bythe difference between pooled and class-specific entropies, using a permutationtest to assess significance.The full entropy-based score is now developed. First calculate the two classspecificcorrelation matrices, Σ 1and Σ 2, as well as the correlation for the pooleddata Σ. The eigenvalues of each are calculated and normalized to sum to 1, sofor class k the standardized eigenvalues are Λ k= (λ k1, . . . , λ kG)/G. The entropyof the standardized eigenvalues is calculated and so that the values do notdepend on dimension, is further standardized to the maximum possible entropyvalue log 2(G). Thus, the class-specific, standardized entropy can be written asEk=−G∑ g=1λkg2log ( λ )Glog ( G)2kg(8)If, as above, X = (X 1, . . ., X G) is the expression intensity vectors for the candidateset of G genes then the entropy score can be written asENT ( X)= E−( E + E ) /21 2(9)where E is the entropy obtained from pooling the classes. The ENT score iseasily extended to K > 2 classes as followsENT( X) = E− 1 Ek.K∑k(10)