High-dimensional proportionality test of two covariance matrices and its application to gene expression data

Figures & data

Table 1. Comparison of the mean-standard deviation-ratio and the variance estimator ratio at the 5% level with $n_1=n_2=15$ and p = 100, 200, 400, 800, 1200.

	$\mathrm{E}\left(T\right)/\sqrt{\mathrm{v}\mathrm{a}\mathrm{r}\left(T\right)}$			$\widehat{\mathrm{v}\mathrm{a}\mathrm{r}\left(T\right)}/\mathrm{v}\mathrm{a}\mathrm{r}\left(T\right)$
p	HT	SS	LZ	HT	SS	LZ
Scenario I: multivariate normal distribution
100	−0.04	−0.01	−0.02	1.06	1.27	1.01
200	−0.03	−0.01	−0.05	1.13	1.17	1.03
400	−0.04	0.03	0.03	1.18	1.09	1.03
800	0.03	0.23	0.03	1.15	1.20	1.04
1200	0.03	0.50	0.06	1.12	1.14	1.15
Scenario II: multivariate t-distribution
100	−0.07	−0.22	−0.28	1.13	3.75	495
200	−0.03	−0.18	−0.72	1.14	1.63	202
400	0.00	−0.25	−0.18	1.24	4.21	144
800	0.01	−0.35	0.65	1.22	7.50	195
1200	0.05	−0.48	−0.06	1.21	7.60	100
Scenario III: multivariate mixture normal distribution
100	−0.04	−0.02	−0.03	1.12	1.27	7.91
200	−0.06	−0.03	−0.04	1.13	1.32	7.15
400	−0.01	0.16	−0.01	1.24	1.13	8.63
800	0.03	0.42	0.11	1.19	1.12	8.14
1200	0.00	0.67	−0.06	1.22	1.03	8.16

Figure 1. ROC curves of the involved tests under the three scenarios with $(n_1,n_2,p)=(15,15,800)$.

Table 2. Empirical size and power comparison at the 5% level with $n_1=n_2=15$ and p = 100, 200, 400, 800, 1200.

	$\rho =0.3$			$\rho =0.6$			$\rho =0.7$
p	HT	SS	LZ	HT	SS	LZ	HT	SS	LZ
Multivariate normal distribution
100	4.1	4.6	3.6	21	20	22	56	53	58
200	3.7	5.0	3.5	22	21	23	59	56	60
400	4.3	4.5	3.7	25	23	23	59	59	61
800	4.5	6.8	3.5	24	28	23	62	65	63
1200	5.2	13	4.6	24	37	24	62	69	62
Multivariate t-distribution
100	4.7	4.8	19	22	21	33	53	52	47
200	4.8	4.6	20	23	20	32	57	55	46
400	5.7	4.2	23	26	21	34	60	55	48
800	6.0	6.2	22	26	22	35	61	54	47
1200	6.4	8.4	22	25	25	37	63	57	51
Multivariate mixture normal distribution
100	4.2	4.6	22	25	22	35	54	55	50
200	4.8	5.9	20	21	26	32	56	58	48
400	5.8	6.3	23	26	25	36	59	60	50
800	5.6	9.1	24	26	32	37	59	69	51
1200	6.1	13	24	26	39	34	60	72	48

Table 3. Empirical size and power comparison at the 5% level with $n_1=n_2=20$ and p = 100, 200, 400, 800, 1200.

	$\rho =0.3$			$\rho =0.6$			$\rho =0.7$
p	HT	SS	LZ	HT	SS	LZ	HT	SS	LZ
Multivariate normal distribution
100	5.1	3.9	5.3	35	29	37	79	75	78
200	4.1	4.1	3.9	34	33	35	81	81	82
400	5.6	6.4	5.1	35	36	35	84	83	84
800	4.3	3.9	4.1	36	36	35	84	86	86
1200	4.2	2.6	2.8	35	37	36	86	84	82
Multivariate t-distribution
100	5.8	4.1	25	36	31	42	75	77	58
200	5.0	4.2	22	36	35	38	78	79	58
400	6.7	4.6	27	37	34	40	80	82	57
800	5.8	4.8	26	39	33	45	84	80	63
1200	4.8	4.4	26	37	34	46	83	81	63
Multivariate mixture normal distribution
100	6.5	4.2	25	36	31	45	78	76	61
200	5.1	4.3	26	36	33	42	79	78	61
400	5.9	5.0	24	38	36	43	82	85	60
800	5.2	7.1	25	37	40	43	83	85	62
1200	4.6	6.2	25	39	42	43	84	89	63

Table 4. Empirical size and power comparison at the 5% level with $n_1=n_2=30$ and p = 100, 200, 400, 800, 1200.

	$\rho =0.3$			$\rho =0.6$			$\rho =0.7$
p	HT	SS	LZ	HT	SS	LZ	HT	SS	LZ
Multivariate normal distribution
100	3.9	4.7	3.1	60	58	60	98	98	99
200	4.2	3.6	4.1	62	61	62	98	98	99
400	5.4	5.2	4.7	64	63	64	99	98	99
800	4.7	5.0	3.8	64	60	64	99	99	99
1200	6.2	5.8	5.1	68	65	66	99	99	99
Multivariate t-distribution
100	4.4	3.9	27	58	57	53	97	98	72
200	5.7	4.8	27	62	59	49	97	98	72
400	5.2	5.6	29	62	62	51	98	99	73
800	5.9	4.5	28	61	62	51	99	99	74
1200	6.4	5.6	29	64	61	50	99	99	73
Multivariate mixture normal distribution
100	4.6	5.8	27	59	58	54	97	97	77
200	5.8	6.2	24	60	62	50	98	99	75
400	5.7	6.2	25	62	62	54	98	99	77
800	5.6	7.3	26	63	63	53	98	99	77
1200	5.8	5.4	26	63	66	54	99	99	78

Table 5. Empirical size comparison at the 5% level with the MA(2) covariance matrices with $n_1=n_2=15$, 20, 30 and p = 100, 200, 400, 800, 1200.

	$n_1=n_2=15$			$n_1=n_2=20$			$n_1=n_2=30$
p	HT	SS	LZ	HT	SS	LZ	HT	SS	LZ
Multivariate normal distribution
100	2.7	4.1	2.6	4.2	3.7	5.6	3.6	4.3	4.8
200	3.6	2.9	3.4	4.0	3.4	3.8	4.1	3.7	4.3
400	5.1	4.4	4.0	5.7	4.6	5.6	4.7	4.5	4.8
800	4.2	5.3	3.6	3.9	3.8	2.2	4.4	4.1	3.8
1200	5.8	7.8	5.8	3.1	5.6	3.0	3.9	4.3	6.6
Multivariate t-distribution
100	4.2	3.3	19	6.5	4.0	24	4.8	5.6	29
200	6.0	4.4	19	6.0	3.9	23	5.9	4.0	28
400	6.0	4.0	22	6.5	3.2	25	6.1	4.9	28
800	5.7	7.0	22	6.1	4.2	26	6.7	3.4	27
1200	6.7	7.8	23	6.1	4.9	26	6.9	5.1	30
Multivariate mixture normal distribution
100	5.0	3.5	22	5.6	3.7	27	4.8	3.5	26
200	5.7	4.0	25	6.3	4.1	26	6.3	4.4	27
400	5.8	5.8	23	6.3	4.3	28	6.2	3.2	26
800	5.4	7.0	24	5.1	7.0	26	5.9	4.7	26
1200	5.9	8.8	24	4.9	7.9	25	5.2	5.2	26

Table 6. Empirical power comparison at the 5% level with the MA(2) covariance matrices with $n_1=n_2=15$, 20, 30 and p = 100, 200, 400, 800, 1200.

	$n_1=n_2=15$			$n_1=n_2=20$			$n_1=n_2=30$
p	HT	SS	LZ	HT	SS	LZ	HT	SS	LZ
Multivariate normal distribution
100	42	40	44	65	63	66	92	92	93
200	42	41	43	67	64	67	93	93	94
400	45	44	48	67	66	68	94	94	94
800	44	47	41	66	68	67	93	93	94
1200	45	55	44	65	70	64	94	95	94
Multivariate t-distribution
100	42	40	40	65	62	56	91	92	67
200	42	40	42	64	63	50	91	92	67
400	43	40	46	64	66	51	92	92	65
800	44	45	46	64	64	51	91	94	65
1200	45	45	46	62	61	54	92	94	68
Multivariate mixture normal distribution
100	43	45	42	64	61	57	90	92	68
200	44	44	44	66	66	56	91	93	73
400	45	48	43	66	68	55	93	93	69
800	43	55	43	65	71	53	92	95	70
1200	43	61	46	62	74	53	92	94	66

Figure 2. Histograms of the p-values of the normality tests and the gene expression means, for the tumour group and the normal group, respectively.

Figure 3. Heatmaps of the sample correlation matrices of the two groups as well as the difference of the two matrices, which are constructed via the first 100 genes in the original data. (a) Normal group, (b) tumour group and (c) difference of two groups.