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184 Ho et al.Fig. 5. An illustration of the approach used for simulating cross patterns, when θ=π/4.Cross patterns were simulated by first drawing observations for each of the twoclasses from a bivariate normal distribution N(µ 1= 0, µ 2= 0, σ 1, σ 2, ρ); then theclass 2 data is modified by rotating the points for that class by the angle θ, whereasdata for class 1 are held fixed. An example, with a rotation of 45°, correspondingto θ =π/4 is shown in Fig. 5. The difference between classes becomes more pronouncedas θ increases until it reaches its maximum at θ=π/2 = 90°. When θ=π= 180°, the correlations are again equal. In this approach both the combined andthe class-conditional gene variances vary with θ, but the standardized eigenvalues,|ρ||ρ|λ , and λ∗ =1−∗ = 1+(the axes of ellipse) were kept fixed.1222Figure 6 demonstrates the power of the three compared methods to detectcross patterns of the type simulated. Power is shown now as a function of theangle θ between the ellipsoids of the two classes (on the x-axis) and the eigenvaluesused to simulate the bivariate normal ellipsoid (by panel). For all methods, powerincreases with both the angle and the class-conditional correlation. The ECF-statisticdepends on which variable is chosen as the conditioning variable. Whensearching for interesting pairs, it is suggested to use the largest of the two statistics.The largest ECF-statistic is the most powerful, although by a small margin,when the data is simulated to have a very long and narrow shape. Methods areessentially equivalent for low and moderate correlations, but at high correlationsthe correlation-based approach loses power on small-angle rotations comparedwith the other two alternatives. This is because the class conditional correlation ofthe rotated data changes more slowly under these circumstances than other propertiesof the joint distribution. Because the correlation measure simply comparesthe two correlations, it lacks power to detect the difference.