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Identifying Differentially Expressed Gene Combinations 173Fig. 2. Artificial example of joint differential expression (cross) of a pair of genes.The format is the same as that of Fig. 1. For this data set S cross= 1.64 and S shift= –0.11.A second example is shown in Fig. 2. Herein there is no obvious demarcationin space, and again, neither gene alone predicts the phenotype. However, in samplesfrom the grey phenotypes, the expression of the two genes are negatively correlated,whereas a positive correlation occurs in samples from the black ()phenotype. Therefore, this will be referred to as a “cross” pattern. Biologically,this pattern could occur when two genes are involved in a common process in onephenotype, but perform separate or complementary activities in the other.Alternatively, it could reflect an “on/off” mechanism. If both genes are off (expressionvalues 0), the black phenotypeis observed, whereas if only one of the genes is on, the grey phenotype is observed.Both cross and shift patterns may only be identified when both genes are consideredat the same time. This motivates defining joint differential expression asa departure from the null hypothesis of identical joint distributions, coupled witha weaker or no departure from the null hypothesis in the 1D marginal distributions.This definition is not entirely precise, because the term “weaker” needs to bespecified. It is designed to guide the search toward gene pairs (or more broadlysets) that would not be identified by one-gene-at-a-time searches, whereas still