Generalized fiducial methods for testing quantitative trait locus effects in genetic backcross studies

Figures & data

Table 1. Probabilities of QTL genotypes.

	C	$Pr(C=i)$	$Pr\left(QQ\right|C=i)$	$Pr\left(Qq\right|C=i)$
MM/NN	1	$(1-r)/2$	1	0
Mm/NN	2	r/2	$r_1/r$	$r_2/r$
MM/Nn	3	r/2	$r_2/r$	$r_1/r$
Mm/Nn	4	$(1-r)/2$	0	1

Table 2. Type I errors (%) and standard errors (%) of the five methods.

$\alpha =0.05$	d = 5					d = 10					d = 20
n	GVJ	GVM	GVW	GVMV	LR	GVJ	GVM	GVW	GVMV	LR	GVJ	GVM	GVW	GVMV	LR
$f_1=f_2=N(0,1)$
30	5.18	4.83	5.12	5.26	7.95	4.78	4.54	4.79	5.07	7.16	5.01	4.82	4.87	5.22	7.57
	(0.221)	(0.214)	(0.220)	(0.223)	(0.271)	(0.213)	(0.208)	(0.214)	(0.219)	(0.258)	(0.218)	(0.214)	(0.215)	(0.222)	(0.265)
50	4.85	4.71	4.90	5.12	6.33	5.38	5.03	5.17	5.49	6.35	4.60	4.43	4.56	4.80	5.81
	(0.215)	(0.212)	(0.216)	(0.220)	(0.244)	(0.226)	(0.219)	(0.221)	(0.228)	(0.244)	(0.209)	(0.206)	(0.209)	(0.214)	(0.234)
100	4.92	4.93	4.93	5.00	5.13	4.98	4.80	4.87	4.95	5.30	5.03	4.97	4.87	5.02	5.43
	(0.216)	(0.217)	(0.217)	(0.218)	(0.221)	(0.218)	(0.214)	(0.215)	(0.217)	(0.224)	(0.219)	(0.217)	(0.215)	(0.218)	(0.227)
200	5.38	5.24	5.24	5.42	5.43	5.05	4.97	5.01	5.05	5.24	5.18	5.06	5.04	5.03	5.31
	(0.226)	(0.223)	(0.223)	(0.226)	(0.227)	(0.219)	(0.217)	(0.218)	(0.219)	(0.223)	(0.222)	(0.219)	(0.219)	(0.219)	(0.224)
300	5.15	5.09	5.21	5.18	5.24	5.16	4.99	4.95	5.05	5.22	5.09	5.05	5.14	5.28	5.24
	(0.221)	(0.220)	(0.222)	(0.222)	(0.223)	(0.221)	(0.218)	(0.217)	(0.219)	(0.222)	(0.220)	(0.219)	(0.221)	(0.224)	(0.223)
$f_1=f_2=N(0,1)$ under ${\sigma }_1={\sigma }_2=\sigma$
30	5.11	4.80	4.94	5.35	7.08	4.95	4.57	4.74	5.07	6.63	5.21	4.83	5.15	5.38	7.07
	(0.220)	(0.214)	(0.217)	(0.225)	(0.257)	(0.217)	(0.209)	(0.213)	(0.219)	(0.249)	(0.222)	(0.214)	(0.221)	(0.226)	(0.256)
50	4.94	4.75	4.89	5.09	5.91	5.47	5.27	5.34	5.49	6.33	4.77	4.67	4.72	4.94	5.53
	(0.217)	(0.213)	(0.216)	(0.220)	(0.236)	(0.227)	(0.223)	(0.221)	(0.228)	(0.244)	(0.213)	(0.211)	(0.212)	(0.217)	(0.229)
100	5.19	4.83	4.94	5.06	5.40	5.02	4.86	4.80	4.92	5.17	5.10	5.10	5.10	5.17	5.60
	(0.222)	(0.214)	(0.217)	(0.219)	(0.227)	(0.218)	(0.215)	(0.214)	(0.216)	(0.221)	(0.220)	(0.220)	(0.220)	(0.221)	(0.230)
200	5.30	5.24	5.20	5.20	5.48	5.00	5.02	4.94	5.08	5.07	5.07	4.98	4.97	5.09	5.37
	(0.224)	(0.223)	(0.222)	(0.222)	(0.228)	(0.218)	(0.218)	(0.217)	(0.220)	(0.219)	(0.219)	(0.218)	(0.217)	(0.220)	(0.225)
300	5.15	5.16	5.17	5.27	5.30	5.14	5.06	5.17	5.18	5.14	5.09	5.03	5.18	5.08	5.22
	(0.221)	(0.221)	(0.221)	(0.223)	(0.224)	(0.221)	(0.219)	(0.221)	(0.222)	(0.221)	(0.220)	(0.219)	(0.222)	(0.220)	(0.222)
$f_1=f_2=\mathrm{L}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{s}(0,1)$
30	5.34	4.64	5.22	5.31	7.87	4.86	4.53	4.81	4.89	6.68	5.36	5.03	5.34	5.53	7.37
	(0.225)	(0.216)	(0.222)	(0.224)	(0.269)	(0.215)	(0.208)	(0.214)	(0.216)	(0.250)	(0.225)	(0.219)	(0.225)	(0.229)	(0.261)
50	4.87	4.56	4.76	5.06	5.99	5.31	4.87	5.02	5.33	5.85	4.76	4.54	4.56	4.85	5.74
	(0.215)	(0.209)	(0.213)	(0.219)	(0.237)	(0.224)	(0.215)	(0.218)	(0.225)	(0.235)	(0.213)	(0.208)	(0.209)	(0.215)	(0.233)
100	5.12	4.97	5.09	5.29	5.29	5.03	4.81	4.85	5.07	5.33	5.05	5.01	4.93	5.12	5.65
	(0.220)	(0.217)	(0.220)	(0.224)	(0.224)	(0.219)	(0.214)	(0.215)	(0.219)	(0.225)	(0.219)	(0.218)	(0.217)	(0.220)	(0.231)
200	5.00	4.92	4.98	4.90	5.19	4.68	4.70	4.60	4.77	4.88	4.93	4.94	5.09	4.95	5.26
	(0.218)	(0.216)	(0.217)	(0.216)	(0.222)	(0.211)	(0.212)	(0.209)	(0.213)	(0.215)	(0.217)	(0.217)	(0.220)	(0.217)	(0.223)
300	5.23	5.30	5.34	5.20	5.44	4.69	4.65	4.69	4.72	4.87	5.27	5.24	5.25	5.33	5.30
	(0.223)	(0.224)	(0.225)	(0.222)	(0.227)	(0.211)	(0.211)	(0.211)	(0.212)	(0.215)	(0.223)	(0.223)	(0.223)	(0.225)	(0.224)
$f_1=f_2=\mathrm{L}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{s}(0,1)$ under ${\sigma }_1={\sigma }_2=\sigma$
30	5.14	4.81	4.97	5.28	7.08	5.03	4.81	4.94	5.15	6.53	5.19	4.86	4.97	5.32	6.57
	(0.221)	(0.214)	(0.217)	(0.224)	(0.257)	(0.219)	(0.214)	(0.217)	(0.221)	(0.247)	(0.222)	(0.215)	(0.217)	(0.224)	(0.248)
50	5.19	4.82	4.96	5.14	5.92	4.80	4.60	4.68	4.83	5.68	4.67	4.40	4.53	4.72	5.43
	(0.222)	(0.214)	(0.217)	(0.221)	(0.236)	(0.214)	(0.209)	(0.211)	(0.214)	(0.231)	(0.211)	(0.205)	(0.208)	(0.212)	(0.227)
100	5.00	4.85	4.90	5.00	5.34	5.00	4.80	4.90	4.92	5.39	5.35	5.39	5.31	5.46	5.84
	(0.218)	(0.215)	(0.216)	(0.218)	(0.225)	(0.218)	(0.214)	(0.216)	(0.216)	(0.226)	(0.225)	(0.226)	(0.224)	(0.227)	(0.235)
200	4.99	4.91	5.05	4.98	4.94	5.12	5.19	5.15	5.19	5.12	5.30	5.08	5.18	5.33	5.53
	(0.218)	(0.216)	(0.219)	(0.218)	(0.217)	(0.220)	(0.222)	(0.221)	(0.222)	(0.220)	(0.224)	(0.220)	(0.222)	(0.225)	(0.229)
300	5.12	5.12	5.10	5.08	5.23	4.74	4.59	4.54	4.56	4.83	4.98	5.13	5.11	5.14	5.17
	(0.220)	(0.220)	(0.220)	(0.220)	(0.223)	(0.213)	(0.209)	(0.208)	(0.209)	(0.214)	(0.218)	(0.221)	(0.220)	(0.221)	(0.221)

Table 3. Powers (%) of the five methods for Normal mixture model.

n	GVJ	GVM	GVW	GVMV	LR	GVJ	GVM	GVW	GVMV	LR	GVJ	GVM	GVW	GVMV	LR
Case I: $f_1=N(0,1)$ and $f_2=N(0.5,1)$
$\theta =0.5$	d = 5					d = 10					d = 20
30	17.7	16.7	17.2	17.6	17.0	16.8	15.8	15.8	16.7	16.4	15.2	15.0	15.0	15.8	14.8
50	27.1	26.8	27.1	27.7	28.0	26.3	25.7	25.7	27.1	26.9	25.3	24.7	24.7	25.1	26.3
100	54.1	53.3	53.4	54.5	54.7	52.6	52.1	52.3	53.7	54.2	48.5	48.3	48.4	48.6	48.5
200	86.3	86.0	86.3	86.2	87.2	83.6	83.5	83.8	84.0	84.1	80.7	80.4	80.5	80.4	80.4
$\theta =0.7$	d = 5					d = 10					d = 20
30	16.6	16.0	16.2	16.9	16.1	16.3	15.5	16.3	16.8	16.3	15.5	15.1	15.2	16.2	14.5
50	29.8	28.6	28.9	29.8	30.6	27.5	26.4	26.4	27.3	26.1	24.7	24.2	24.4	25.5	25.1
100	55.3	54.7	55.3	55.6	57.3	51.1	51.1	51.3	51.5	52.1	49.8	50.1	50.0	50.7	51.2
200	86.0	85.5	85.7	86.2	87.1	84.0	84.1	83.8	84.3	85.6	80.4	80.4	80.2	80.6	81.4
Case II: $f_1=N(0,1)$ and $f_2=N(0,{1.5}^2)$
$\theta =0.5$	d = 5					d = 10					d = 20
30	19.5	19.2	19.5	20.1	19.6	18.7	18.4	18.9	19.5	18.7	17.7	16.9	17.1	17.8	17.2
50	34.7	34.4	34.6	34.9	36.1	32.4	31.6	31.9	32.6	32.2	28.8	27.8	28.2	29.6	29.7
100	67.5	67.0	67.2	67.7	68.4	64.7	64.2	63.9	64.7	64.7	60.2	59.6	59.9	60.3	60.3
200	94.1	94.1	94.1	94.3	95.0	93.5	92.7	92.9	93.0	93.7	90.8	90.7	90.4	91.0	91.1
$\theta =0.7$	d = 5					d = 10					d = 20
30	18.8	18.4	18.5	19.2	18.5	19.0	18.8	18.5	19.7	19.3	17.0	16.5	16.5	17.4	16.0
50	34.8	34.2	34.3	34.8	35.2	32.4	31.9	32.1	32.7	32.7	30.3	29.7	30.2	30.6	31.5
100	65.8	65.2	65.6	65.9	68.2	63.4	63.0	63.1	63.6	64.3	58.7	58.2	58.7	59.3	59.3
200	94.0	94.2	94.2	94.1	94.9	93.5	93.3	93.5	93.7	93.9	91.6	91.1	91.2	91.7	92.0
Case III: $f_1=N(0,1)$ and $f_2=N(0.5,{1.5}^2)$
$\theta =0.5$	d = 5					d = 10					d = 20
30	29.6	28.7	29.1	29.8	28.3	28.5	27.9	27.8	29.0	27.9	25.0	24.5	24.7	24.9	24.0
50	51.5	50.7	50.7	51.5	50.9	50.8	50.1	50.4	51.1	48.6	49.7	49.0	48.9	50.0	49.0
100	86.0	85.8	85.8	85.9	85.2	84.3	84.2	84.4	84.3	83.6	80.9	80.2	80.1	81.2	80.0
200	99.3	99.3	99.3	99.3	99.3	99.3	99.2	99.3	99.3	99.2	98.3	98.4	98.2	98.3	97.9
$\theta =0.7$	d = 5					d = 10					d = 20
30	28.4	27.3	28.0	28.7	27.3	28.6	27.7	27.9	28.3	27.9	28.1	27.3	27.7	28.6	27.4
50	51.5	50.9	51.2	52.1	51.5	52.1	51.9	52.3	52.2	50.3	45.6	44.8	45.5	45.7	45.5
100	85.5	85.4	85.2	85.7	84.9	84.7	84.0	84.2	84.9	84.0	81.2	81.1	81.3	81.8	80.5
200	99.6	99.5	99.6	99.6	99.5	99.2	99.3	99.2	99.4	99.3	98.8	98.9	98.8	98.9	98.8

Table 4. Powers (%) of the five methods for Logistic mixture model.

n	GVJ	GVM	GVW	GVMV	LR	GVJ	GVM	GVW	GVMV	LR	GVJ	GVM	GVW	GVMV	LR
Case IV: $f_1=\mathrm{L}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{s}(0,1)$ and $f_2=\mathrm{L}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{s}(0.5,1)$
$\theta =0.5$	d = 5					d = 10					d = 20
30	9.80	9.30	9.40	10.2	9.10	9.45	9.05	9.05	9.35	9.70	7.70	7.20	7.05	7.90	7.10
50	12.9	12.5	12.9	13.2	12.9	11.8	11.6	11.9	12.1	12.2	11.5	11.4	11.3	11.8	12.0
100	21.1	20.5	20.9	21.2	20.5	20.6	20.5	20.5	20.9	20.6	19.0	18.5	18.5	18.9	17.5
200	40.6	40.4	40.2	40.6	40.8	40.8	40.0	40.5	40.7	41.3	36.5	36.7	37.4	37.4	37.3
$\theta =0.7$	d = 5					d = 10					d = 20
30	8.10	7.70	8.10	8.25	8.35	8.55	8.25	8.75	8.90	9.10	7.85	7.20	7.55	7.90	7.45
50	12.6	12.2	12.3	12.3	13.2	12.8	12.2	12.3	12.8	12.5	10.3	9.85	10.3	10.5	11.1
100	21.4	20.9	21.4	21.5	21.4	19.4	18.9	19.3	19.3	19.5	18.4	18.0	17.6	18.6	16.6
200	40.4	40.1	40.3	40.7	40.1	39.4	39.0	39.5	39.5	40.8	34.6	34.4	34.4	35.3	34.6
Case V: $f_1=\mathrm{L}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{s}(0,1)$ and $f_2=\mathrm{L}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{s}(0,1.5)$
$\theta =0.5$	d = 5					d = 10					d = 20
30	14.3	13.9	14.7	14.7	14.2	17.0	16.3	16.8	17.0	17.8	15.8	15.0	15.5	16.0	14.1
50	26.9	26.2	26.7	27.3	28.8	27.8	26.7	27.0	27.6	27.9	23.0	22.8	23.2	23.5	24.5
100	51.2	50.8	51.7	51.5	51.6	49.5	49.1	49.4	49.9	50.2	47.1	46.2	46.4	47.1	47.0
200	84.0	83.9	83.5	83.7	83.9	81.6	81.4	81.1	81.5	82.2	77.2	76.9	77.2	77.4	76.8
$\theta =0.7$	d = 5					d = 10					d = 20
30	15.5	15.0	15.2	15.8	15.4	15.7	15.1	15.5	16.2	16.6	15.8	14.5	15.4	15.7	14.4
50	25.9	25.7	26.1	26.7	26.6	25.1	24.4	24.9	25.1	25.3	24.8	24.1	24.1	25.3	26.0
100	52.3	51.9	51.8	52.2	52.5	49.4	49.2	49.4	50.0	51.0	47.3	46.7	46.7	47.5	46.6
200	83.3	83.0	83.4	83.1	83.7	81.9	81.6	81.8	81.8	83.8	79.1	79.9	79.5	79.3	79.3
Case VI: $f_1=\mathrm{L}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{s}(0,1)$ and $f_2=\mathrm{L}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{s}(0.5,1.5)$
$\theta =0.5$	d = 5					d = 10					d = 20
30	20.6	20.1	20.3	20.6	19.2	18.4	17.5	18.0	18.9	19.1	17.1	16.9	17.3	17.9	15.9
50	33.4	32.3	33.1	33.6	33.6	31.8	30.9	31.5	31.1	30.5	30.4	30.1	29.7	31.0	30.1
100	63.3	62.8	62.6	62.7	61.5	61.9	61.9	61.8	62.3	61.5	55.5	54.9	55.3	55.7	53.3
200	91.8	91.8	91.7	91.6	91.5	90.2	89.9	90.2	90.1	90.0	87.7	87.5	87.6	88.0	86.1
$\theta =0.7$	d = 5					d = 10					d = 20
30	19.3	18.2	18.8	19.0	18.8	17.5	16.6	17.0	17.8	17.2	17.7	16.9	17.6	18.2	16.5
50	32.3	31.9	32.5	32.5	32.2	33.4	32.4	32.6	33.4	32.7	28.9	28.6	29.0	29.7	29.8
100	64.3	64.0	64.2	64.6	63.8	61.1	60.2	60.6	61.0	60.2	56.8	56.5	56.4	57.4	54.2
200	92.8	92.5	92.5	92.7	92.5	90.0	90.0	90.2	90.0	89.9	87.2	87.2	87.2	87.5	86.4

Table 5. Powers (%) of the five methods for mixture model under ${\sigma }_1={\sigma }_2=\sigma$.

n	GVJ	GVM	GVW	GVMV	LR	GVJ	GVM	GVW	GVMV	LR	GVJ	GVM	GVW	GVMV	LR
Case I: $f_1=N(0,1)$ and $f_2=N(0.5,1)$
$\theta =0.5$	d = 5					d = 10					d = 20
30	23.0	22.4	22.7	23.4	22.2	23.5	22.4	22.8	24.1	22.8	21.4	21.0	21.0	21.5	19.6
50	37.7	36.4	37.1	38.1	37.2	36.7	36.3	36.7	37.2	33.8	34.0	33.2	33.3	34.0	34.1
100	66.9	66.5	66.8	67.0	65.9	65.4	65.3	65.1	65.8	63.6	61.3	60.9	60.9	61.6	59.6
200	93.5	93.4	93.5	93.5	93.3	91.3	91.0	91.1	91.1	90.8	88.5	88.2	88.2	88.5	87.1
$\theta =0.7$	d = 5					d = 10					d = 20
30	21.3	20.9	21.5	22.1	20.8	21.9	21.8	21.8	22.4	21.4	21.9	20.6	21.5	22.1	19.4
50	39.9	39.3	39.6	40.1	38.7	36.6	35.9	36.7	37.2	34.7	32.8	32.7	32.9	33.6	33.2
100	67.3	67.1	67.2	67.7	66.8	64.4	64.3	64.3	64.7	63.0	61.6	61.2	61.3	61.8	60.7
200	92.9	92.8	92.8	92.7	92.6	91.7	91.6	91.4	91.6	91.4	89.5	89.3	89.1	89.8	88.4
Case III: $f_1=N(0,1)$ and $f_2=N(0.5,{1.5}^2)$
$\theta =0.5$	d = 5					d = 10					d = 20
30	17.4	17.0	17.3	17.8	16.6	15.9	14.9	15.6	16.3	15.3	13.4	13.0	13.4	13.8	12.4
50	25.3	24.6	24.9	25.5	24.2	25.7	25.4	25.5	26.1	23.7	26.7	26.4	26.1	27.0	26.7
100	46.2	45.9	45.9	46.4	45.3	45.7	45.3	45.5	45.7	44.5	41.4	41.4	41.4	41.7	40.4
200	76.8	76.7	76.7	77.0	76.0	74.6	74.5	74.3	74.7	73.3	70.0	69.6	69.6	70.0	68.3
$\theta =0.7$	d = 5					d = 10					d = 20
30	16.5	15.8	16.2	16.7	15.8	15.4	15.1	15.4	16.0	15.1	16.0	15.1	15.8	16.2	14.7
50	25.6	25.1	25.5	26.0	25.4	25.1	24.6	24.6	25.3	23.2	22.8	22.3	23.0	23.1	22.8
100	49.8	49.6	49.9	49.8	49.4	46.1	45.2	45.4	46.3	45.4	42.8	42.3	42.3	43.0	41.0
200	77.0	76.9	77.2	77.0	76.5	74.6	74.3	74.4	74.4	74.2	76.8	76.7	76.7	77.0	76.0
Case IV: $f_1=\mathrm{L}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{s}(0,1)$ and $f_2=\mathrm{L}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{s}(0.5,1)$
$\theta =0.5$	d = 5					d = 10					d = 20
30	11.7	11.2	11.8	12.0	11.8	11.9	11.6	11.8	12.3	11.5	10.0	9.20	9.80	10.4	9.00
50	17.1	16.8	17.4	17.3	16.4	14.3	14.3	14.0	14.7	13.9	15.2	15.2	15.3	15.9	15.1
100	27.7	27.9	28.4	28.4	27.8	26.7	26.3	26.4	26.6	26.3	24.5	24.2	24.4	24.9	22.9
200	51.0	50.7	51.5	51.6	51.1	50.6	51.0	50.7	50.7	49.3	45.5	45.6	45.7	45.9	43.7
$\theta =0.7$	d = 5					d = 10					d = 20
30	11.9	11.8	12.0	12.1	11.9	11.2	10.4	10.7	11.3	10.6	11.4	10.9	11.2	11.3	11.1
50	16.0	15.9	15.8	16.1	15.5	16.2	15.4	16.1	16.6	15.6	15.2	14.7	14.6	15.2	14.9
100	27.6	27.2	27.2	27.5	26.9	27.1	26.7	27.2	26.9	26.4	23.8	23.7	23.6	23.9	22.1
200	50.5	50.8	51.1	50.9	50.5	49.8	49.3	49.7	49.8	48.0	45.6	45.2	45.0	45.6	43.0
Case VI: $f_1=\mathrm{L}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{s}(0,1)$ and $f_2=\mathrm{L}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{s}(0.5,1.5)$
$\theta =0.5$	d = 5					d = 10					d = 20
30	9.45	9.45	9.45	9.80	9.30	8.95	8.40	8.65	8.75	8.85	9.45	8.85	8.90	9.25	8.35
50	12.4	11.9	12.6	12.6	12.2	11.7	11.5	11.9	12.0	11.0	11.7	11.3	11.5	11.8	11.3
100	21.6	21.5	21.5	22.0	21.0	19.6	19.4	19.6	19.8	19.6	18.0	17.9	17.7	17.9	16.7
200	36.8	36.3	36.3	36.6	35.9	33.4	33.5	33.4	33.3	32.2	31.5	30.9	31.4	31.3	28.7
$\theta =0.7$	d = 5					d = 10					d = 20
30	9.50	9.25	9.70	9.85	9.50	8.55	8.35	8.40	8.80	8.30	9.10	8.85	9.00	9.35	8.85
50	12.3	12.2	12.5	12.6	11.6	12.5	12.1	12.4	12.7	12.4	12.2	12.0	12.1	12.3	12.8
100	20.0	19.5	19.6	20.0	19.8	18.8	18.5	18.6	19.0	18.4	18.2	17.9	18.3	18.5	16.7
200	35.8	35.6	35.9	35.9	35.5	33.9	33.9	33.7	33.9	33.7	29.5	29.4	29.5	29.8	27.2

Table 6. p-values of Shapiro–Wilk test.

Interval	D1Mit156-D1Mit178	D1Mit7-D1Mit46	D1Mit46-D1Mit132	D1Mit132-D1Mit334	D1Mit94-D1Mit218
$y_1$	0.3646	0.6656	0.7135	0.1150	0.1797
$y_4$	0.2191	0.1619	0.2498	0.3240	0.3778
Interval	D1Mit218-D1Mit100	D1Mit14-D1Mit105	D1Mit105-D1Mit159	D1Mit159-D1Mit267	D1Mit267-D1Mit15
$y_1$	0.5102	0.1747	0.2287	0.2990	0.3924
$y_4$	0.7499	0.3630	0.8381	0.2519	0.1183

Figure 1. The trace plot of Markov chains of the four generalized p-values methods for D1Mit156-D1Mit178.

Table 7. Sample sizes of 10 intervals.

Interval	D1Mit156-D1Mit178	D1Mit7-D1Mit46	D1Mit46-D1Mit132	D1Mit132-D1Mit334	D1Mit94-D1Mit218
Total	49	26	26	25	37
$y_1$	17	16	13	11	8
$y_2$	2	2	3	3	8
$y_3$	10	0	2	2	4
$y_4$	20	8	8	9	17
Interval	D1Mit218-D1Mit100	D1Mit14-D1Mit105	D1Mit105-D1Mit159	D1Mit159-D1Mit267	D1Mit267-D1Mit15
Total	37	250	35	35	35
$y_1$	7	137	19	20	15
$y_2$	5	4	0	0	11
$y_3$	4	5	1	6	1
$y_4$	11	104	15	9	8

Figure 2. The boxplot of $\{y_{1j},j=1,\dots ,n_1\}$ and $\{y_{4j},j=1,\dots ,n_4\}$ for D1Mit267-D1Mit15.

Table 8. Five kinds of p-values in 10 intervals.

Interval	D1Mit156-D1Mit178	D1Mit7-D1Mit46	D1Mit46-D1Mit132	D1Mit132-D1Mit334	D1Mit94-D1Mit218
LR	0.1678	0.3309	0.2524	0.7606	0.3088
GVJ	0.1039	0.2627	0.6578	0.7792	0.3849
GVM	0.1078	0.2847	0.7427	0.7688	0.3996
GVW	0.1033	0.2607	0.6013	0.7554	0.4274
GVMV	0.1009	0.2779	0.6304	0.7604	0.3676
Interval	D1Mit218-D1Mit100	D1Mit14-D1Mit105	D1Mit105-D1Mit159	D1Mit159-D1Mit267	D1Mit267-D1Mit15
LR	0.7951	0.0001	0.0060	0.0088	0.0673
GVJ	0.6339	0	0.0085	0.0081	0.9175
GVM	0.6704	0	0.0104	0.0115	0.8219
GVW	0.6408	0	0.0091	0.0130	0.9872
GVMV	0.6290	0	0.0090	0.0178	0.9407

Table 9. Five kinds of p-values in 10 intervals under ${\sigma }_1={\sigma }_2=\sigma$ assumption.

Interval	D1Mit156-D1Mit178	D1Mit7-D1Mit46	D1Mit46-D1Mit132	D1Mit132-D1Mit334	D1Mit94-D1Mit218
LR	0.1699	0.9728	0.7918	0.4872	0.5442
GVJ	0.1208	0.9596	0.6928	0.4708	0.4100
GVM	0.1272	0.9764	0.7124	0.5216	0.4032
GVW	0.1268	0.9396	0.6988	0.5168	0.4092
GVMV	0.1120	0.9460	0.6796	0.5124	0.3980
Interval	D1Mit218-D1Mit100	D1Mit14-D1Mit105	D1Mit105-D1Mit159	D1Mit159-D1Mit267	D1Mit267-D1Mit15
LR	0.5022	0.0001	0.1267	0.1706	0.1313
GVJ	0.4132	0	0.1476	0.1756	0.5536
GVM	0.4316	0	0.1380	0.1852	0.4140
GVW	0.3772	0	0.1464	0.1732	0.5820
GVMV	0.4228	0	0.1384	0.1784	0.4456