Table 2 Quantitative characteristics and stability of identified multivariate linear mixture models tested on full and undersampled dataset.

From: Multivariate linear mixture models for the prediction of febrile seizure risk and recurrence: a prospective case–control study

	Dataset size	100%	90%	80%	70%	60%	50%
Model1	Model detection rate [%]	*66.0	*20.8	11.2	6.4	3.2	2.2
	Total number of identified models	6	61	90	121	142	160
	Height regression coefficient	0.0049 ± 0.0002	0.0051 ± 0.0007	0.0052 ± 0.0008	0.0055 ± 0.0009	0.0059 ± 0.0011	0.0067 ± 0.0012
	Fe regression coefficient	− 0.0715 ± 0.0005	− 0.0702 ± 0.0025	− 0.0687 ± 0.0035	− 0.0684 ± 0.0049	− 0.0664 ± 0.0053	− 0.0654 ± 0.0074
	Fer regression coefficient	0.0025 ± 0.0001	0.0028 ± 0.0003	0.0031 ± 0.0004	0.0033 ± 0.0005	0.0037 ± 0.0007	0.0040 ± 0.0007
	UIBC regression coefficient	0.0119 ± 0.0004	0.0129 ± 0.0010	0.0136 ± 0.0015	0.0145 ± 0.0017	0.0153 ± 0.0023	0.0165 ± 0.0028
	F-statistics	38.41 ± 0.66	35.00 ± 2.43	31.54 ± 2.92	28.96 ± 3.31	25.45 ± 3.41	23.17 ± 3.83
	Root mean square error	0.6239 ± 0.0024	0.6221 ± 0.0094	0.6206 ± 0.0128	0.6162 ± 0.0163	0.6144 ± 0.0197	0.6012 ± 0.0247
	Explained variance R² [%]	46.04 ± 0.42	46.61 ± 1.62	46.75 ± 2.20	47.95 ± 2.75	48.61 ± 3.28	50.92 ± 4.04
	Pearson correlation (y₁ vs y_p1)	0.643 ± 0.000	0.646 ± 0.012	0.648 ± 0.017	0.656 ± 0.021	0.663 ± 0.026	0.677 ± 0.031
	Non-seizure/seizure separating threshold	0.5744 ± 0.0317	0.6853 ± 0.0793	0.7355 ± 0.1091	0.8442 ± 0.1372	0.9531 ± 0.1674	1.0655 ± 0.2197
	Training: sensitivity	95.49 ± 1.61	93.90 ± 4.98	95.51 ± 4.60	92.68 ± 5.92	91.39 ± 6.18	93.40 ± 6.53
	Training: specificity	69.43 ± 1.15	70.90 ± 4.15	68.95 ± 5.25	72.24 ± 6.39	73.91 ± 7.31	71.25 ± 8.07
	Testing: sensitivity		87.32 ± 15.00	89.65 ± 12.12	84.72 ± 12.09	80.26 ± 13.38	83.27 ± 12.29
	Testing: specificity		67.36 ± 18.25	65.29 ± 14.61	66.71 ± 10.79	67.45 ± 10.67	66.43 ± 9.29
Model2	Model detection rate [%]	*100.0	*72.0	*50.9	*24.9	14.2	6.8
	Total number of identified models	1	44	64	127	160	209
	Age regression coefficient	− 0.0050 ± 0.0002	− 0.0052 ± 0.0007	− 0.0056 ± 0.0009	− 0.0060 ± 0.0010	− 0.0064 ± 0.0011	− 0.0072 ± 0.0014
	Height regression coefficient	0.0036 ± 0.0002	0.0036 ± 0.0005	0.0038 ± 0.0006	0.0040 ± 0.0007	0.0043 ± 0.0008	0.0048 ± 0.0010
	satFe regression coefficient	− 3.2236 ± 0.0455	− 3.1911 ± 0.2123	− 3.1108 ± 0.2796	− 3.0829 ± 0.3491	− 2.9630 ± 0.3964	− 2.8432 ± 0.4344
	UIBC regression coefficient	0.0093 ± 0.0003	0.0094 ± 0.0011	0.0098 ± 0.0015	0.0100 ± 0.0016	0.0108 ± 0.0019	0.0113 ± 0.0020
	F− statistics	28.82 ± 0.63	25.12 ± 2.25	22.85 ± 2.59	21.00 ± 3.00	19.20 ± 3.24	17.28 ± 3.37
	Root mean square error	0.3620 ± 0.0019	0.3638 ± 0.0076	0.3642 ± 0.0097	0.3601 ± 0.0123	0.3558 ± 0.0150	0.3509 ± 0.0175
	Explained variance R² [%]	47.38 ± 0.54	47.47 ± 2.10	48.01 ± 2.69	49.13 ± 3.41	51.02 ± 4.08	53.10 ± 4.60
	Pearson correlation (y₂ vs y_p2)	0.660 ± 0.000	0.662 ± 0.015	0.667 ± 0.020	0.674 ± 0.026	0.691 ± 0.031	0.704 ± 0.034
	Non− seizure/seizure separating threshold	0.3495 ± 0.0261	0.3657 ± 0.0899	0.3779 ± 0.1170	0.4135 ± 0.1332	0.4652 ± 0.1607	0.4906 ± 0.1730
	Training: sensitivity	83.53 ± 1.04	81.15 ± 4.19	83.30 ± 4.35	80.72 ± 5.91	82.28 ± 6.42	85.87 ± 6.79
	Training: specificity	82.89 ± 0.92	86.06 ± 4.14	84.81 ± 4.94	88.90 ± 5.16	89.72 ± 5.66	88.30 ± 7.02
	Testing: sensitivity		75.60 ± 18.66	75.50 ± 13.62	71.20 ± 12.15	70.69 ± 10.81	72.76 ± 10.10
	Testing: specificity		81.14 ± 15.07	78.56 ± 12.53	81.53 ± 9.97	79.77 ± 10.16	77.35 ± 11.19
Model3	Model detection rate [%]	*51.5	28.4	10.4	4.6	2.0	1.1
	Total number of identified models	15	73	203	293	383	506
	Height regression coefficient	− 0.0072 ± 0.0005	− 0.0070 ± 0.0007	− 0.0079 ± 0.0011	− 0.0080 ± 0.0012	− 0.0083 ± 0.0012	− 0.0088 ± 0.0012
	HGB regression coefficient	0.0129 ± 0.0009	0.0136 ± 0.0013	0.0153 ± 0.0022	0.0158 ± 0.0024	0.0171 ± 0.0028	0.0179 ± 0.0028
	satFe regression coefficient	6.1796 ± 0.5323	6.1236 ± 0.8455	6.8889 ± 1.3212	7.0798 ± 1.4197	7.8790 ± 1.7529	9.1360 ± 2.7797
	F-statistics	8.24 ± 0.83	7.41 ± 1.30	8.79 ± 2.07	8.48 ± 2.31	8.60 ± 2.54	10.17 ± 3.67
	Root mean square error	0.4182 ± 0.0055	0.4130 ± 0.0095	0.4068 ± 0.0148	0.3917 ± 0.0178	0.3799 ± 0.0218	0.3615 ± 0.0300
	Explained variance R² [%]	26.04 ± 1.93	26.37 ± 3.35	32.33 ± 4.90	35.13 ± 5.75	40.08 ± 6.62	47.23 ± 8.58
	Pearson correlation (y₃ vs y_p3)	0.441 ± 0.005	0.457 ± 0.033	0.495 ± 0.046	0.533 ± 0.053	0.577 ± 0.057	0.630 ± 0.067
	Non-recurrent/recurrent seizure separating threshold	1.4001 ± 0.0999	1.4947 ± 0.1410	1.6849 ± 0.2406	1.7372 ± 0.2945	1.9210 ± 0.3366	2.0830 ± 0.3144
	Training: sensitivity	83.86 ± 7.67	86.15 ± 10.19	88.11 ± 11.57	91.45 ± 10.93	92.50 ± 8.92	92.03 ± 9.59
	Training: specificity	58.44 ± 6.69	60.50 ± 6.80	64.23 ± 9.26	66.54 ± 10.81	69.81 ± 10.25	76.00 ± 10.77
	Testing: sensitivity		73.33 ± 44.24	69.80 ± 30.16	74.70 ± 28.62	74.29 ± 26.08	69.81 ± 24.57
	Testing: specificity		45.18 ± 27.23	44.85 ± 19.98	42.13 ± 17.47	46.57 ± 14.52	49.61 ± 12.24

All values were averaged from utilized 5000 iterations with randomized initial conditions. Values are represented as mean ± standard deviation among the iterations. In a majority of the listed quantitative measurements, the mean values are quite stable and standard deviation increases as the dataset is more undersampled.
*The bold highlighted “Model detection rate” represents that the model with listed regression coefficients has been the most often identified as the best model characterizing the data among the iterations.
The adaptive synthetic sampling matched the number of female samples in the case groups to minimize the risk of the imbalanced learning within each modeling iteration.
The separating threshold has been identified by maximizing sum of sensitivity and specificity. Then, the classifying sensitivity and specificity have been tested on the training dataset itself and on the training dataset (i.e., the samples excluded from the training due to dataset undersampling).

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Table 2 Quantitative characteristics and stability of identified multivariate linear mixture models tested on full and undersampled dataset.

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