. 2023 Nov 10;39(12):btad680. doi: 10.1093/bioinformatics/btad680

Table 4.

Predictive performance in internal simulation.^a

									`transreg`
ρ_x	$ρ_{β}$	α	Family	${\bar{ρ}}_{x}$	$\max ({\hat{ρ}}_{β})$	`glmnet`	`glmtrans`	`xrnet`	`exp.sta`	`exp.sim`	`iso.sta`	`iso.sim`
0.95	0.60	0	Gaussian	0.08	0.32	46.1 ± 7.3	48.5 ± 6.3	45.2 ± 7.0*	45.2 ± 7.6*	45.4 ± 7.1	45.1 ± 7.5*	44.5 ± 6.8*
0.99	0.60	0	Gaussian	0.32	0.32	33.2 ± 4.8	33.2 ± 6.4	32.6 ± 4.4*	32.2 ± 4.6*	32.5 ± 4.4*	32.1 ± 4.7*	32.0 ± 4.1*
0.95	0.80	0	Gaussian	0.08	0.54	45.1 ± 6.7	47.9 ± 11.1	42.0 ± 6.4*	43.7 ± 7.1*	43.3 ± 6.9*	43.8 ± 7.1*	41.9 ± 6.5*
0.99	0.80	0	Gaussian	0.32	0.54	32.5 ± 4.8	31.8 ± 4.2	31.1 ± 4.7*	32.0 ± 4.8*	31.6 ± 4.7*	31.6 ± 4.7*	30.4 ± 3.9*
0.95	0.99	0	Gaussian	0.08	0.89	44.1 ± 5.1	42.2 ± 5.6	37.3 ± 4.4*	40.9 ± 5.3*	37.6 ± 4.5*	40.1 ± 5.7*	38.8 ± 5.7*
0.99	0.99	0	Gaussian	0.32	0.89	31.0 ± 5.1	29.4 ± 2.1	27.6 ± 3.0*	30.4 ± 5.3	28.4 ± 3.8*	29.8 ± 4.8*	29.1 ± 3.9*
0.95	0.60	1	Gaussian	0.08	0.27	37.7 ± 6.8	39.1 ± 7.2	37.9 ± 6.6	37.3 ± 7.2	38.0 ± 7.3	37.4 ± 7.0	41.5 ± 6.4 $†$
0.99	0.60	1	Gaussian	0.32	0.27	28.5 ± 2.4	29.9 ± 3.5 $†$	29.1 ± 3.4	28.8 ± 2.5	29.0 ± 2.7	28.7 ± 2.7	28.7 ± 2.5
0.95	0.80	1	Gaussian	0.08	0.45	37.5 ± 4.8	37.4 ± 5.0	36.9 ± 5.2	35.7 ± 4.2*	36.8 ± 4.7	35.9 ± 4.7*	39.5 ± 5.4 $†$
0.99	0.80	1	Gaussian	0.32	0.45	29.9 ± 2.4	29.9 ± 2.8	29.5 ± 3.4	29.1 ± 2.5	29.4 ± 3.3	29.1 ± 2.8	29.6 ± 3.8
0.95	0.99	1	Gaussian	0.08	0.87	38.0 ± 6.4	33.0 ± 5.7*	34.5 ± 7.9*	34.8 ± 8.2*	34.1 ± 6.8*	34.0 ± 7.6*	35.2 ± 8.0
0.99	0.99	1	Gaussian	0.32	0.87	30.2 ± 4.4	29.5 ± 4.5	29.2 ± 4.6	29.4 ± 4.8	29.1 ± 4.2*	28.6 ± 3.5*	29.4 ± 4.4
0.95	0.60	0	Binomial	0.08	0.32	77.1 ± 4.5	81.2 ± 6.1 $†$	76.7 ± 4.4	76.8 ± 4.8	76.1 ± 5.0	77.7 ± 4.8	77.7 ± 5.2
0.99	0.60	0	Binomial	0.32	0.32	65.5 ± 4.7	67.0 ± 5.9	65.0 ± 4.7	63.7 ± 4.8*	65.3 ± 4.4	63.8 ± 4.8*	64.0 ± 4.9
0.95	0.80	0	Binomial	0.08	0.54	74.9 ± 4.4	81.2 ± 10.7 $†$	73.5 ± 5.1*	75.0 ± 5.7	73.9 ± 5.6	74.7 ± 4.7	73.5 ± 5.4*
0.99	0.80	0	Binomial	0.32	0.54	64.3 ± 3.6	64.7 ± 4.6	61.8 ± 4.7*	63.2 ± 4.5*	63.7 ± 5.0	63.0 ± 4.7*	63.6 ± 5.1
0.95	0.99	0	Binomial	0.08	0.89	75.4 ± 5.0	74.8 ± 4.7*	69.5 ± 4.2*	72.4 ± 6.5*	71.5 ± 6.3*	71.7 ± 4.9*	71.4 ± 4.8*
0.99	0.99	0	Binomial	0.32	0.89	62.4 ± 5.0	61.8 ± 5.0	58.1 ± 4.6*	61.0 ± 6.7	58.9 ± 6.7*	60.4 ± 6.2*	59.6 ± 5.0*
0.95	0.60	1	Binomial	0.08	0.27	76.5 ± 6.0	75.8 ± 3.9	75.7 ± 4.1	75.8 ± 5.8	76.5 ± 4.9	75.9 ± 5.4	75.9 ± 2.3
0.99	0.60	1	Binomial	0.32	0.27	61.2 ± 4.4	61.5 ± 4.9	63.4 ± 6.8 $†$	63.1 ± 5.6 $†$	62.2 ± 5.7	62.7 ± 5.4 $†$	61.5 ± 4.6
0.95	0.80	1	Binomial	0.08	0.45	78.2 ± 10.5	76.6 ± 10.2	76.3 ± 9.8	75.0 ± 6.0	77.1 ± 9.0	75.2 ± 6.1	77.9 ± 6.6
0.99	0.80	1	Binomial	0.32	0.45	64.9 ± 4.9	64.8 ± 5.6	65.5 ± 2.9	66.3 ± 6.7	64.4 ± 5.2	65.2 ± 6.0	65.5 ± 4.3
0.95	0.99	1	Binomial	0.08	0.87	80.1 ± 6.0	73.3 ± 5.7*	70.7 ± 6.3*	69.2 ± 4.5*	68.8 ± 6.8*	70.1 ± 5.2*	69.3 ± 5.3*
0.99	0.99	1	Binomial	0.32	0.87	63.2 ± 5.1	62.9 ± 4.7	61.2 ± 5.6*	62.7 ± 6.1	61.1 ± 5.7*	61.8 ± 5.8	61.2 ± 6.2*

In each setting (row), we simulate 10 datasets, calculate the performance metric (mean-squared error for numerical prediction, logistic deviance for binary classification) for the test sets, express these metrics as percentages of those from prediction by the mean, and show the mean and standard deviation of these percentages. Settings: correlation parameter for features (ρ_x), correlation parameter for coefficients ( $ρ_{β}$ ), dense setting with ridge regularization ( $π = 30 %$ , α = 0) or sparse setting with lasso regularization ( $π = 5 %$ , α = 1), family of distribution (‘gaussian’ or ‘binomial’). These parameters determine (i) the mean Pearson correlation among the features in the target dataset ( ${\bar{ρ}}_{x}$ ) and (ii) the maximum Pearson correlation between the coefficients in the target dataset and the coefficients in the source datasets ( $\max ({\hat{ρ}}_{β})$ ). Methods: regularized regression (glmnet), competing transfer learning methods (glmtrans, xrnet), proposed transfer learning method (transreg) with exponential/isotonic calibration and standard/simultaneous stacking. In each setting, the colour black (grey) highlights methods that are more (less) predictive than regularized regression without transfer learning (glmnet), asterisks (daggers) indicate methods that are significantly more (less) predictive at the 5% level (one-sided Wilcoxon signed-rank test), and an underline highlights the most predictive method.