Estimating the optimal linear combination of predictors using spherically constrained optimization

Das, Priyam; De, Debsurya; Maiti, Raju; Kamal, Mona; Hutcheson, Katherine A.; Fuller, Clifton D.; Chakraborty, Bibhas; Peterson, Christine B.

doi:10.1186/s12859-022-04953-y

BMC Bioinformatics

Table 1 Performance comparison for two and three ordinal outcomes, where each class has sample size 15 and 30 for the two category case, and 15 for the three category case

From: Estimating the optimal linear combination of predictors using spherically constrained optimization

Scenarios	d	Method	Scenario 1		Scenario 2		Scenario 3
Scenarios	d	Method	ULBA	EHUM	ULBA	EHUM	ULBA	EHUM
\(M=2, N = (15,15)\)	5	SCOR	0.928 (0.06)	0.928 (0.06)	0.974 (0.04)	0.974 (0.04)	0.900 (0.08)	0.900 (0.08)
		NM	0.806 (0.16)	0.806 (0.16)	0.889 (0.14)	0.889 (0.14)	0.760 (0.11)	0.760 (0.11)
		Step-down	0.538 (0.30)	0.538 (0.30)	0.536 (0.31)	0.536 (0.31)	0.866 (0.08)	0.866 (0.08)
		Min-max	0.691 (0.10)	0.691 (0.10)	0.711 (0.13)	0.711 (0.13)	0.839 (0.08)	0.839 (0.08)
	10	SCOR	0.972 (0.04)	0.972 (0.04)	0.984 (0.03)	0.984 (0.03)	0.953 (0.06)	0.953 (0.06)
		NM	0.864 (0.15)	0.864 (0.15)	0.834 (0.18)	0.834 (0.18)	0.866 (0.09)	0.866 (0.09)
		Step-down	0.503 (0.28)	0.503 (0.28)	0.581 (0.32)	0.581 (0.32)	0.971 (0.05)	0.971 (0.05)
		Min-max	0.881 (0.07)	0.881 (0.07)	0.860 (0.08)	0.860 (0.08)	0.982 (0.02)	0.982 (0.02)
	20	SCOR	0.971 (0.04)	0.971 (0.04)	0.979 (0.04)	0.979 (0.04)	0.958 (0.06)	0.958 (0.06)
		NM	0.805 (0.25)	0.805 (0.25)	0.692 (0.26)	0.692 (0.26)	0.930 (0.07	0.930 (0.07
		Step-down	0.472 (0.33)	0.472 (0.33)	0.472 (0.32)	0.472 (0.32)	0.975 (0.04)	0.975 (0.04)
		Min-max	0.922 (0.05)	0.922 (0.05)	0.909 (0.06)	0.909 (0.06)	0.997 (0.01)	0.997 (0.01)
\(M=2, N = (30,30)\)	5	SCOR	0.950 (0.04)	0.950 (0.04)	0.982 (0.03)	0.982 (0.03)	0.923 (0.04)	0.923 (0.04)
		NM	0.931 (0.03)	0.931 (0.03)	0.970 (0.03)	0.970 (0.03)	0.773 (0.10)	0.773 (0.10)
		Step-down	0.596 (0.28)	0.596 (0.28)	0.590 (0.30)	0.590 (0.30)	0.893 (0.06)	0.893 (0.06)
		Min-max	0.804 (0.10)	0.804 (0.10)	0.873 (0.12)	0.873 (0.12)	0.865 (0.05)	0.865 (0.05)
	10	SCOR	0.990 (0.01)	0.990 (0.01)	0.995 (0.01)	0.995 (0.01)	0.989 (0.02)	0.989 (0.02)
		NM	0.978 (0.02)	0.978 (0.02)	0.983 (0.03)	0.983 (0.03)	0.872 (0.08)	0.872 (0.08)
		Step-down	0.546 (0.32)	0.546 (0.32)	0.473 (0.32)	0.473 (0.32)	0.982 (0.03)	0.982 (0.03)
		Min-max	0.926 (0.05)	0.926 (0.05)	0.945 (0.06)	0.945 (0.06)	0.982 (0.02)	0.982 (0.02)
	20	SCOR	0.991 (0.01)	0.991 (0.01)	0.984 (0.10)	0.984 (0.10)	0.985 (0.02)	0.985 (0.02)
		NM	0.991 (0.01)	0.991 (0.01)	0.992 (0.01)	0.992 (0.01)	0.929 (0.05)	0.929 (0.05)
		Step-down	0.546 (0.32)	0.546 (0.32)	0.515 (0.31)	0.515 (0.31)	0.984 (0.04)	0.984 (0.04)
		Min-max	0.961 (0.04)	0.961 (0.04)	0.972 (0.04)	0.972 (0.04)	0.997 (0.01)	0.997 (0.01)
\(M=3, N = (15,15,15)\)	5	SCOR	0.891 (0.10)	0.890 (0.10)	0.955 (0.08)	0.955 (0.08)	0.722 (0.01)	0.719 (0.10)
		NM	0.829 (0.11)	0.837 (0.11)	0.898 (0.13)	0.902 (0.13)	0.484 (0.12)	0.485 (0.12)
		Step-down	0.358 (0.32)	0.356 (0.32)	0.398 (0.38)	0.398 (0.38)	0.638 (0.12)	0.644 (0.12)
		Min-max	0.582 (0.15)	0.582 (0.15)	0.703 (0.23)	0.703 (0.23)	0.582 (0.10)	0.582 (0.10)
	10	SCOR	0.978 (0.03)	0.967 (0.10)	0.985 (0.03)	0.986 (0.03)	0.938 (0.06)	0.942 (0.05)
		NM	0.916 (0.11)	0.907 (0.14)	0.912 (0.19)	0.925 (0.19)	0.664 (0.13)	0.664 (0.13)
		Step-down	0.354 (0.34)	0.349 (0.34)	0.329 (0.36)	0.329 (0.36)	0.902 (0.08)	0.903 (0.08)
		Min-max	0.825 (0.09)	0.824 (0.09)	0.823 (0.13)	0.823 (0.13)	0.887 (0.05)	0.887 (0.05)
	15	SCOR	0.977 (0.03)	0.927 (0.22)	0.985 (0.02)	0.945 (0.20)	0.952 (0.06)	0.951 (0.06)
		NM	0.935 (0.18)	0.888 (0.27)	0.914 (0.20)	0.853 (0.29)	0.795 (0.09)	0.795 (0.09)
		Step-down	0.297 (0.35)	0.308 (0.35)	0.322 (0.35)	0.322 (0.35)	0.935 (0.08)	0.935 (0.08)
		Min-max	0.895 (0.06)	0.895 (0.06)	0.889 (0.08)	0.889 (0.08)	0.964 (0.03)	0.964 (0.03)

The empirical hypervolume under manifolds (EHUM) and upper and lower bound approach (ULBA) objective functions are maximized by the proposed Spherically Constrained Optimization Routine (SCOR) algorithm and the existing Nelder-Mead (NM), step-down, and min-max algorithms. The estimated biomarker coefficient vectors are then used to calculate the EHUM value on a new dataset of the same size generated from the corresponding model. The entire procedure is repeated 100 times, resulting in 100 simulated training and test data sets, and the mean EHUM objective function values on the test data are reported, with the standard error in the parentheses. The result for the method with the best performance is marked in bold

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