Detecting differentially expressed genes of heterogeneous and positively skewed data using half Johnson’s modified t-test

Figures & data

Table 1. Type I error rates for the half Student’s t-test and half Johnson’s modified t-test

	n0 = n1 = 20		n0 = n1 = 60
Significance level	Half Student’s t-test	Half Johnson’s modified t-test	Half Student’s t-test	Half Johnson’s modified t-test
Normal distribution
0.05	0.0492	0.0494	0.0513	0.0510
0.01	0.0102	0.0106	0.0114	0.0113
0.005	0.0051	0.0056	0.0056	0.0057
0.001	0.0010	0.0013	0.0012	0.0012
Non-normal but symmetric distribution (γ3 = 0)
0.05	0.0470	0.0382	0.0496	0.0464
0.01	0.0094	0.0053	0.0096	0.0079
0.005	0.0047	0.0020	0.0047	0.0037
0.001	0.0010	0.0002	0.0010	0.0007
Positively skewed distribution (γ3 = 0.6)
0.05	0.0534	0.0557	0.0514	0.0528
0.01	0.0126	0.0166	0.0104	0.0115
0.005	0.0068	0.0107	0.0052	0.0062
0.001	0.0017	0.0043	0.0010	0.0016
Negatively skewed distribution (γ3 = −0.6)
0.05	0.0529	0.0570	0.0508	0.0523
0.01	0.0122	0.0166	0.0113	0.0122
0.005	0.0071	0.0106	0.0059	0.0068
0.001	0.0021	0.0048	0.0015	0.0020

Figure 1. Statistical power in normal distribution.

Notes: Solid line: half Johnson’s modified t; dash line: half Student’s t.
The difference in means between case and control groups (denoted as d) was set to 0, 15, and 25. The ratio of standard deviations for case group to control group (denoted as r) was set to 1, 1.5, 2, and 2.5.

Figure 2. Statistical power in non-normal but symmetric distribution.

Notes: solid line: half Johnson’s modified t; dash line: half Student’s t.
The difference in means between case and control groups (denoted as d) was set to 0, 15, and 25. The ratio of standard deviations for case group to control group (denoted as r) was set to 1, 1.5, 2, and 2.5.

Figure 3. Statistical power in positively skewed distribution.

Notes: solid line: half Johnson’s modified t; dash line: half Student’s t.
The difference in means between case and control groups (denoted as d) was set to 0, 15, and 25. The ratio of standard deviations for case group to control group (denoted as r) was set to 1, 1.5, 2, and 2.5.

Figure 4. Statistical power in negatively skewed distribution.

Notes: solid line: half Johnson’s modified t; dash line: half Student’s t.
The difference in means between case and control groups (denoted as d) was set to 0, 15, and 25. The ratio of standard deviations for case group to control group (denoted as r) was set to 1, 1.5, 2, and 2.5.

Table 2. Number (percentage) of differentially expressed genes of studied test methods in colon cancer data

	Test methods
	Half Student’s t-test	Half Johnson’s modified t-test	Welch’s t-test	Wilcoxon test
Significance level
0.05	577 (28.9%)	632 (31.6%)	355 (17.8%)	275 (13.8%)
0.01	360 (18.0%)	422 (21.1%)	192 (9.60%)	141 (7.05%)
0.005	299 (15.0%)	368 (18.4%)	147 (7.35%)	99 (4.95%)
0.001	214 (10.7%)	292 (14.6%)	74 (3.70%)	45 (2.25%)
False discovery rate
0.05	344 (17.2%)	429 (21.5%)	117 (5.85%)	48 (2.40%)
0.005	185 (9.25%)	278 (13.9%)	3 (0.15%)	5 (0.25%)