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. 2021 Aug 6;37(21):3889–3895. doi: 10.1093/bioinformatics/btab576

Table 1.

Mean loss of different models (re-scaled mean squared error, mean over 10 repetitions) in low-dimensional (top), sparse high-dimensional (centre) and dense high-dimensional (bottom) settings

ρx ρb ρy glmneta joinet glmnetb earth spls MRCE remMap MRFc SiER mcen GPM RMTL MTPS
0.0 0.0 0.1 20.6 20.4 21.0 25.2 21.1 20.7 32.7 52.5 21.2 23.4 21.8 21.1 22.7
0.1 0.0 0.6 21.0 21.1 20.9 25.4 21.1 21.7 33.9 41.3 21.6 22.9 20.9 21.5 22.7
0.3 0.0 0.5 21.7 21.6 21.7 24.2 21.8 21.9 27.2 38.4 21.9 22.3 21.7 22.0 24.1
0.0 0.5 0.4 21.6 21.3 21.6 24.0 21.8 22.0 41.5 44.1 21.5 22.9 21.6 21.5 23.3
0.1 0.5 0.2 21.6 21.7 21.8 28.2 21.5 21.5 23.6 47.7 22.2 23.7 21.9 22.0 23.3
0.3 0.5 0.6 21.0 21.0 21.3 25.4 22.4 20.5 27.9 33.6 21.9 21.1 21.5 21.9 21.4
0.0 0.9 0.8 20.9 20.7 20.7 21.6 21.2 21.7 23.9 41.1 21.6 23.1 20.6 20.7 21.4
0.1 0.9 0.8 20.8 20.6 20.6 23.4 21.0 21.6 23.6 37.4 21.5 22.9 20.6 20.7 22.1
0.3 0.9 0.8 20.7 20.4 20.4 22.3 21.0 21.5 23.2 32.3 20.5 21.2 20.6 20.9 22.1
0.0 0.0 0.0 24.7 22.9 29.1 49.2 26.5 100.0 41.5 98.2 31.4 27.2 100.0 30.1 29.6
0.1 0.0 0.2 26.5 25.5 29.0 35.4 21.8 100.0 37.6 84.0 38.8 26.5 100.0 67.0 30.2
0.3 0.0 0.5 26.6 26.2 28.2 32.5 37.9 100.0 49.8 57.7 36.6 25.6 100.0 46.8 29.1
0.0 0.5 0.0 28.4 23.6 30.6 48.8 23.0 100.0 47.8 97.3 27.2 30.1 100.0 32.2 34.0
0.1 0.5 0.2 26.2 24.8 29.6 39.6 34.6 100.0 42.4 84.7 45.5 26.9 100.0 65.5 29.4
0.3 0.5 0.5 26.5 26.4 30.7 42.3 39.4 100.0 33.3 59.4 47.6 27.6 100.0 42.0 30.3
0.0 0.9 0.3 27.5 24.9 28.3 26.8 23.8 100.0 41.7 97.5 32.7 28.8 100.0 28.2 32.3
0.1 0.9 0.5 26.3 25.4 27.8 27.7 23.9 100.0 35.1 83.1 32.0 27.8 100.0 28.9 30.0
0.3 0.9 0.6 25.9 26.5 26.5 31.4 33.8 100.0 36.5 57.2 34.7 26.4 100.0 30.7 28.4
0.0 0.0 0.1 89.2 89.7 89.4 143.3 89.5 100.0 100.0 99.1 94.8 97.4 100.0 86.9 89.5
0.1 0.0 0.7 27.5 25.9 28.4 80.5 27.8 100.0 42.6 61.1 29.4 35.0 100.0 27.3 27.8
0.3 0.0 0.8 22.3 22.0 22.3 50.4 21.8 100.0 42.0 37.6 23.2 25.2 100.0 23.1 22.3
0.0 0.5 0.4 89.1 91.5 89.5 165.8 88.9 100.0 100.0 99.5 92.6 96.1 100.0 90.0 99.8
0.1 0.5 0.8 28.4 26.6 29.5 73.4 27.0 100.0 64.1 61.7 28.0 33.8 100.0 28.1 28.0
0.3 0.5 0.8 21.8 21.8 21.9 51.3 21.6 100.0 58.4 37.4 23.3 24.7 100.0 22.3 23.4
0.0 0.9 0.7 90.7 89.9 91.4 146.3 91.8 100.0 100.0 99.3 90.2 99.5 100.0 92.3 96.8
0.1 0.9 0.8 28.6 26.5 29.7 73.6 26.8 100.0 58.0 62.1 27.8 33.0 100.0 27.7 30.1
0.3 0.9 0.8 22.7 22.2 22.8 47.8 22.8 100.0 45.2 38.3 23.1 25.6 100.0 22.2 22.5

Note: The first three columns indicate the correlation between inputs (ρx), the correlation between effects (ρb) and the resulting mean correlation between outputs (ρy). The other columns show the predictive performance of a univariate method (glmneta), the proposed multivariate method (joinet) and eleven other multivariate methods (glmnetb, earth, spls, MRCE, remMap, MRFc, SiER, mcen, GPM, RMTL, MTPS). For each setting (row), the colour black indicates which multivariate methods are more predictive than the univariate method (glmneta), and the underline indicates the most predictive method, based on the sharp (not rounded) numbers. aUnivariate linear regression with glmnet. bMultivariate linear regression with glmnet. cMultivariateRandomForest.