Variable screening in multivariate linear regression with high-dimensional covariates

Figures & data

Table 1. Five models.

Model	n	q	$p_0$	${\mathrm{\Sigma }}_x$	Entries of $B_0$
1	200	4	8	$I_p$	$N(4\log (n)/\sqrt{n},1)$
2	75	5	3	$0.5I_p+0.5{\mathbf{1}}_p{\mathbf{1}}_p^{\mathrm{\top }}$	$N(4\log (n)/\sqrt{n},1)$
3	200	3	3	${\sigma }_{kr}={0.5}^{|k-r|},1\le k,r\le p$	$\mathrm{\Gamma }(2,1)$
4	300	2	5	$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{I_{p_0},4^{-1}(I_{p-p_0}+p_0{\mathbf{1}}_{(p-p_0)}{\mathbf{1}}_{(p-p_0)}^{\mathrm{\top }}\left)\right\}$	$\exp \left(9\right)$
5	200	6	10	$I_p$	$\exp \left(9\right)$

Table 2. Measures for the finite sample performance of variable selection.

Model size	$\mathrm{M}\mathrm{S}=\left|\hat{S}\right|$
Coverage probability	$\mathrm{C}\mathrm{P}=\mathrm{P}(S\subset \hat{S})$
% of correctly fitted model	$\mathrm{C}\mathrm{F}=\mathrm{P}(S=\hat{S})$
% of correct zero	$\mathrm{C}\mathrm{Z}=(p-p_0{)}^{-1}|S^{\mathrm{c}}\cap {\hat{S}}^{\mathrm{c}}|$
% of incorrect zero	$\mathrm{I}\mathrm{Z}=p_0^{-1}|S\cap {\hat{S}}^{\mathrm{c}}|$

Table 3. Five measures of the performance of variable selection defined in Table 2 obtained by the four competing methods under various numbers of covariates (p) and signal-to-noise ratio ($R^2$) for Model 1 in Table 1 with $(n,q,p_0)=(200,4,8)$.

Method	p	$R^2$ (%)	MS	CP (%)	CF (%)	CZ (%)	IZ (%)
MCC	1000	30	38	75.9	0	98.4	24.8
		60	38	96.7	0	98.6	3.3
		90	38	99.5	0	98.6	0.5
	5000	30	38	58.2	0	99.6	41.8
		60	38	93.2	0	99.7	6.8
		90	38	98.5	0	99.7	1.5
	10,000	30	38	50.8	0	99.8	49.2
		60	38	90.6	0	99.9	9.4
		90	38	98.1	0	99.9	1.9
NCC	1000	30	38	26.2	0	97.9	73.8
		60	38	40.6	0	98.9	59.2
		90	38	50.5	0	98.2	49.5
	5000	30	38	13.9	0	99.6	86.1
		60	38	30.7	0	99.6	69.3
		90	38	41.8	0	99.6	58.2
	10,000	30	38	10.9	0	99.8	89.1
		60	38	25.6	0	99.8	74.4
		90	38	37.9	0	99.8	62.0
NFR	1000	30	1.1	1.5	0	99.9	98.5
		60	8.1	12.9	0	99.9	87.1
		90	8.3	99.7	90.9	99.9	0.3
	5000	30	2.1	0.5	0	99.9	99.4
		60	6.5	6.2	0	99.9	93.8
		90	7.8	95.3	76.9	100	4.7
	10,000	30	3.1	0.4	0	100	99.6
		60	7.4	4.6	0	100	95.4
		90	10.3	87.9	65.8	100	12.1
UFR	1000	30	0.3	4.1	0	100	95.9
		60	7.6	94.9	71.9	100	5.1
		90	8.0	100	100	100	0
	5000	30	0.1	1.1	0	100	98.9
		60	6.9	86.4	49.9	100	13.6
		90	8.0	100	99.9	100	0
	10,000	30	0.1	0.7	0	100	99.3
		60	5.9	73.8	34.2	100	26.1
		90	8.0	100	99.9	100	0

Table 4. Five measures of the performance of variable selection defined in Table 2 obtained by the four competing methods under various numbers of covariates (p) and signal to noise ratio ($R^2$) for Model 2 in Table 1 with $(n,q,p_0)=(75,5,3)$.

Method	p	$R^2\left(\mathrm{\%}\right)$	MS	CP (%)	CF (%)	CZ (%)	IZ (%)
MCC	1000	30	18	80.7	0	98.4	19.3
		60	18	98.1	0	98.5	1.9
		90	18	99.7	0	98.5	0.3
	5000	30	18	68.1	0	99.7	31.9
		60	18	95.5	0	99.7	4.5
		90	18	99.6	0	99.7	0.4
	10,000	30	18	61.3	0	99.8	38.7
		60	18	95.1	0	99.8	4.9
		90	18	99.7	0	99.9	0.3
NCC	1000	30	18	79.7	0	96.5	23.3
		60	18	95.8	0	96.5	4.7
		90	18	98.9	0	96.5	1.4
	5000	30	18	68.9	0	99.3	39.1
		60	18	94.3	0	99.3	7.3
		90	18	98.7	0	99.3	3.4
	10,000	30	18	68.1	0	99.6	51.9
		60	18	89.9	0	99.6	10.1
		90	18	95.3	0	99.7	2.1
NFR	1000	30	1.6	7.9	0.1	99.9	92.1
		60	3.2	51.6	23.7	99.9	48.4
		90	3.4	99.2	93.2	99.9	0.8
	5000	30	1.2	3.5	0	99.9	96.5
		60	3.9	35.6	11.3	99.9	64.4
		90	4.6	97.8	88.1	99.9	2.2
	10,000	30	1.6	2.3	0	99.9	97.4
		60	3.8	28.6	6.7	99.9	71.3
		90	4.7	96.4	84.7	99.9	3.6
UFR	1000	30	1.5	45.6	11.7	99.9	54.4
		60	3.0	99.8	99.1	100	0.2
		90	3.0	100	100	100	0
	5000	30	1.0	28.8	2.4	99.9	71.2
		60	2.9	98.9	97.5	100	1.0
		90	3.0	100	100	100	0
	10,000	30	0.9	24.4	2.0	99.9	75.6
		60	3.0	98.6	95.6	100	1.4
		90	3.0	100	100	100	0

Table 5. Five measures of the performance of variable selection defined in Table 2 obtained by the four competing methods under various numbers of covariates (p) and signal-to-noise ratio ($R^2$) for Model 3 in Table 1 with $(n,q,p_0)=(200,3,3)$.

Method	p	$R^2\left(\mathrm{\%}\right)$	MS	CP (%)	CF (%)	CZ (%)	IZ (%)
MCC	1000	30	38	100	0	96.5	0
		60	38	100	0	96.5	0
		90	38	100	0	96.5	0
	5000	30	38	100	0	99.3	0
		60	38	100	0	99.3	0
		90	38	100	0	96.3	0
	10,000	30	38	99.9	0	99.6	0.1
		60	38	100	0	99.6	0
		90	38	100	0	99.6	0
NCC	1000	30	38	29.5	0	98.3	70.5
		60	38	47.8	0	98.3	52.2
		90	38	54.8	0	98.4	45.2
	5000	30	38	18.2	0	99.7	81.8
		60	38	38.1	0	99.6	61.9
		90	38	52.0	0	99.7	48.0
	10,000	30	38	13.3	0	99.8	86.7
		60	38	36.0	0	99.8	63.9
		90	38	50.6	0	99.8	49.4
NFR	1000	30	2.2	99.9	0.2	100	60.1
		60	3.0	67.8	17.8	99.9	32.2
		90	4.7	90.6	71.6	99.9	9.4
	5000	30	2.2	37.0	0	100	63.0
		60	3.0	65.0	13.7	100	35.0
		90	3.7	89.8	69.2	100	10.2
	10,000	30	2.1	36.2	0.1	100	63.8
		60	3.9	63.1	11.7	100	36.9
		90	5.7	89.2	67.4	100	10.8
UFR	1000	30	2.1	71.5	22.6	100	28.5
		60	2.9	96.4	89.1	100	3.6
		90	3.0	99.9	99.6	100	0
	5000	30	2.0	67.2	14.4	100	32.8
		60	2.8	94.1	82.3	100	5.9
		90	3.0	99.9	99.7	100	0.1
	10,000	30	1.9	63.7	10.9	100	36.3
		60	2.8	93.6	80.6	100	6.4
		90	3.0	99.9	99.5	100	0

Table 6. Five measures of the performance of variable selection defined in Table 2 obtained by the four competing methods under various numbers of covariates (p) and signal to noise ratio ($R^2$) for Model 4 in Table 1 with $(n,q,p_0)=(300,2,5)$.

Method	p	$R^2\left(\mathrm{\%}\right)$	MS	CP (%)	CF (%)	CZ (%)	IZ (%)
MCC	1000	30	53	21.7	0	97.1	78.3
		60	53	27.8	0	97.1	72.2
		90	53	37.7	0	97.2	79.7
	5000	30	53	19.8	0	99.4	80.2
		60	53	26.7	0	99.4	73.1
		90	53	36.1	0	99.4	63.9
	10,000	30	53	17.7	0	99.7	82.3
		60	53	25.2	0	99.7	74.8
		90	53	35.1	0	99.7	64.9
NCC	1000	30	53	19.8	0	97.1	80.2
		60	53	20.7	0	97.1	79.2
		90	53	21.8	0	97.1	78.1
	5000	30	53	16.8	0	99.4	83.1
		60	53	18.6	0	99.4	81.3
		90	53	19.7	0	99.4	80.3
	10,000	30	53	16.2	0	99.7	83.8
		60	53	17.3	0	99.7	82.7
		90	53	20.0	0	99.7	80.0
NFR	1000	30	2.7	15.6	0	99.9	84.4
		60	4.9	34.3	0	99.9	65.7
		90	5.0	65.1	5.0	99.9	34.9
	5000	30	2.6	13.2	0	99.9	86.8
		60	4.8	27.5	0	99.9	72.5
		90	5.5	57.8	3.1	99.9	42.3
	10,000	30	2.0	12.1	0	99.9	87.9
		60	4.8	26.6	0	99.9	73.4
		90	6.5	54.7	1.9	99.9	45.3
UFR	1000	30	2.4	35.9	0	99.9	64.1
		60	4.4	75.8	10.3	99.9	24.2
		90	5.0	96.8	70.8	99.9	3.2
	5000	30	2.1	28.9	0	99.9	71.1
		60	4.2	70.9	7.2	99.9	29.1
		90	5.0	95.5	63.2	99.9	4.5
	10,000	30	2.0	24.7	0	99.9	75.3
		60	4.1	69.3	4.9	99.9	30.7
		90	5.0	95.6	61.9	99.9	4.4

Table 7. Five measures of the performance of variable selection defined in Table 2 obtained by the four competing methods under various numbers of covariates (p), and signal-to-noise ratio ($R^2$) for Model 5 in Table 1 with $(n,q,p_0)=(200,6,10)$.

Method	p	$R^2\left(\mathrm{\%}\right)$	MS	CP (%)	CF (%)	CZ (%)	IZ (%)
MCC	1000	30	38	85.2	0	97.0	14.8
		60	38	97.2	0	97.1	2.8
		90	38	99.8	0	97.2	0.2
	5000	30	38	70.1	0	99.4	29.9
		60	38	93.3	0	99.4	6.7
		90	38	99.5	0	99.4	0.6
	10,000	30	38	63.0	0	99.7	37.0
		60	38	90.6	0	99.7	9.4
		90	38	99.3	0	99.7	0.8
NCC	1000	30	38	61.2	0	96.8	38.8
		60	38	78.8	0	96.9	21.2
		90	38	85.8	0	97	14.2
	5000	30	38	40.8	0	99.3	59.2
		60	38	64.9	0	99.4	24.3
		90	38	75.7	0	99.4	24.3
	10,000	30	38	33.0	0	99.6	66.9
		60	38	60.0	0	99.7	39.9
		90	38	72.5	0	99.7	27.5
NFR	1000	30	3.8	7.6	0	99.9	92.4
		60	10.2	50.9	0.2	99.9	49.0
		90	10.7	96.5	63.9	99.9	3.5
	5000	30	2.5	4.3	0	99.9	95.7
		60	6.4	40.2	0.1	99.9	59.8
		90	10.6	94.6	49.9	99.9	5.4
	10,000	30	2.4	3.3	0	99.9	96.7
		60	10.6	35.3	0.1	99.9	64.7
		90	11.6	93.9	44.3	99.9	6.1
UFR	1000	30	2.7	27.3	0	100	72.7
		60	9.9	99.3	92.8	100	0.7
		90	10.0	100	99.9	100	0
	5000	30	1.3	13.3	0	100	86.7
		60	4.1	98.2	84.5	100	1.8
		90	10.0	100	99.9	100	0
	10,000	30	1.0	10.1	0	100	89.9
		60	9.8	97.8	81.3	100	2.2
		90	10.0	100	99.9	100	0

Table 8. Five measures of the performance of variable selection defined in Table 2 obtained by the four competing methods under various sample sizes (n) and signal-to-noise ratio ($R^2$) for Model 1 in Table 1 with $(p,q,p_0)=(5000,4,8)$.

Method	n	$R^2\left(\mathrm{\%}\right)$	MS	CP (%)	CF (%)	CZ (%)	IZ (%)
MCC	100	30	22	39.3	0	99.6	60.3
		60	22	72.6	0	99.7	27.4
		90	22	92.3	0	99.7	7.7
	200	30	38	81.2	0	99.4	18.8
		60	38	95.5	0	99.4	4.5
		90	38	98.9	0	99.4	1.1
	400	30	67	96.7	0	98.8	3.3
		60	67	99.3	0	98.8	0.7
		90	67	99.8	0	98.8	0.1
NCC	100	30	22	24.7	0	99.6	75.3
		60	22	45.9	0	99.6	54.0
		90	22	57.4	0	99.6	42.6
	200	30	38	48.3	0	99.3	51.8
		60	38	65.2	0	99.3	34.8
		90	38	70.9	0	99.4	29.1
	400	30	67	62.1	0	98.8	37.9
		60	67	72.2	0	98.8	27.8
		90	67	76.7	0	98.8	23.3
NFR	100	30	2.1	1.4	0	99.9	98.6
		60	6.2	14.8	0	99.9	85.2
		90	8.3	78.8	14.2	99.9	21.2
	200	30	3.0	10.6	0	99.9	89.4
		60	8.0	50.0	0.2	99.9	49.9
		90	8.9	84.8	24.0	100	15.2
	400	30	7.7	33.3	30.0	100	66.6
		60	8.0	65.9	2.2	100	34.1
		90	9.0	86.8	30.7	100	13.1
UFR	100	30	1.0	3.2	0	100	96.8
		60	5.6	69.2	12.0	100	30.8
		90	8.0	99.9	99.1	100	0.1
	200	30	2.8	35.5	0.1	100	64.5
		60	7.6	95.6	69.8	100	4.4
		90	8.0	99.9	99.7	100	0
	400	30	6.4	79.5	12.3	100	20.5
		60	7.9	98.4	88.1	100	1.5
		90	8.0	100	100	100	0

Table 9. Selected genes for the BMD data.

method	genes
MCC	ACSL3, NIPSNAP3B, DLEU2, C1ORF61, DKK1, SOST, ABCA8,
	AFFX-M27830-M-at, RRNF216, PLIN5, PACS2, MUM1, CRYGS,
	RABEP2, PEX14, USP2, FKBP14, FAM55C, MAPK8
NCC	ACSL3, NIPSNAP3B, DLEU2, C1ORF61, DKK1, SOST, ABCA8,
	AFFX-M27830-M-at, RRNF216, PLIN5, MEPE, COPS4, CRYGS
	RABEP2, CTSE, PEX14, FKBP14, NARG1, PAIP1
NFR	ACSL3, NIPSNAP3B, DLEU2, C1ORF61, DKK1, SOST, ABCA8,
	AFFX-M27830-M-at, RNF216, PLIN5, FKBP14, FAM55C, TNK2
UFR	ACSL3, NIPSNAP3B, DLEU2, C1ORF61, DKK1, SOST, ABCA8,
	AFFX-M27830-M-at, RNF216, PLIN5