Rank-based regression with bootstrapping and the least square regression for analysis of small sample interrupted time-series data: simulation studies and application to illegal organ transplants

Figures & data

Table

$X_0$	$T_t$	$X_t$	$X_t\ast \left(T_t-{\displaystyle \frac{n}{2}}\right)$
1	1	0	0
1	2	0	0
$⋮$	$⋮$	$⋮$	0
1	${\displaystyle \frac{n}{2}}$	0	0
1	${\displaystyle \frac{n}{2}}+1$	1	1
1	${\displaystyle \frac{n}{2}}+2$	1	2
$⋮$	$⋮$	$⋮$	$⋮$
1	$n$	1	$n$

Table 1. Efficiency gain when no outliers exist in the data and errors follows AR (1).

		RB method				LS method
$n$	$\rho$	${\hat{\beta }}_2$	$\widehat{\mathrm{se}}({\hat{\beta }}_2)$	${\hat{\beta }}_3$	$\widehat{\mathrm{se}}({\hat{\beta }}_3)$	${\hat{\beta }}_2$	$\widehat{\mathrm{se}}({\hat{\beta }}_2)$	${\hat{\beta }}_3$	$\widehat{\mathrm{se}}({\hat{\beta }}_3)$	Efficiency gain for ${\beta }_2$ in %	Efficiency gain for ${\beta }_3$ in %
Errors ∼ N (0,1)
16	0	0.00	1.04	−0.01	0.23	0.01	1.00	−0.01	0.22	−8.62	−9.73
	0.2	−0.03	1.11	0.00	0.26	−0.03	1.08	0.00	0.25	−6.06	−5.89
	0.4	−0.04	1.21	0.00	0.29	−0.04	1.18	0.00	0.29	−4.92	−4.71
	0.6	−0.05	1.21	0.01	0.32	−0.04	1.17	0.01	0.31	−5.83	−5.52
	0.8	0.00	1.00	0.00	0.29	0.00	0.97	0.00	0.28	−5.63	−1.50
50	0	0.02	0.59	0.00	0.04	0.01	0.57	0.00	0.04	−5.91	−4.51
	0.2	0.00	0.68	0.00	0.05	0.00	0.67	0.00	0.05	−2.36	−1.86
	0.4	−0.01	0.83	0.00	0.06	−0.01	0.82	0.00	0.06	−3.51	−1.51
	0.6	0.01	0.92	0.00	0.07	0.01	0.91	0.00	0.08	−2.63	0.55
	0.8	−0.04	1.09	0.00	0.09	−0.04	1.08	0.00	0.09	−3.62	−1.97
Errors∼ Laplace (0,1)
16	0	0.03	1.35	−0.01	0.29	0.06	1.43	0.00	0.31	10.55	13.64
	0.2	−0.03	1.45	−0.01	0.35	−0.07	1.53	−0.01	0.36	10.00	6.91
	0.4	0.02	1.62	0.00	0.40	0.02	1.64	0.00	0.41	2.44	4.34
	0.6	0.10	1.71	0.02	0.45	0.10	1.68	0.02	0.45	−4.18	0.66
	0.8	0.00	1.40	−0.01	0.41	−0.01	1.35	−0.01	0.41	−7.12	−1.11
50	0	0.00	0.70	0.00	0.05	0.00	0.80	0.00	0.06	23.24	26.43
	0.2	−0.01	0.84	0.00	0.06	−0.01	0.95	0.00	0.06	21.47	19.10
	0.4	0.01	1.04	0.00	0.07	0.02	1.10	0.00	0.08	11.70	13.08
	0.6	−0.01	1.27	0.00	0.09	−0.01	1.30	0.00	0.10	4.01	7.35
	0.8	0.08	1.55	−0.01	0.12	0.09	1.54	−0.01	0.12	−1.59	0.37
Errors ∼ t (5)
16	0	0.02	1.26	−0.01	0.27	−0.02	1.32	−0.01	0.28	8.02	3.63
	0.2	−0.05	1.42	0.00	0.31	−0.06	1.44	0.00	0.31	2.61	3.17
	0.4	−0.02	1.58	0.00	0.35	−0.01	1.60	−0.01	0.36	3.16	2.61
	0.6	0.01	1.56	0.02	0.39	0.00	1.54	0.02	0.40	−2.84	0.85
	0.8	0.01	1.26	−0.01	0.38	0.01	1.22	−0.01	0.38	−7.76	0.07
50	0	0.03	0.67	0.00	0.05	0.02	0.71	0.00	0.05	11.62	17.58
	0.2	−0.02	0.80	0.00	0.06	−0.02	0.88	0.00	0.06	16.04	15.51
	0.4	−0.04	0.99	0.00	0.07	−0.03	1.04	0.00	0.07	9.75	9.85
	0.6	0.03	1.21	0.00	0.09	0.02	1.23	0.00	0.09	4.37	6.83
	0.8	0.02	1.37	0.00	0.12	0.02	1.34	0.00	0.12	−3.83	1.24

Note: SEs under both methods are estimated through bootstrapping.

Table 2. Efficiency gain when outliers exist in the data and errors follow AR (1).

		RB method				LS method
$n$	$\rho$	${\hat{\beta }}_2$	$\widehat{\mathrm{se}}({\hat{\beta }}_2)$	${\hat{\beta }}_3$	$\widehat{\mathrm{se}}({\hat{\beta }}_3)$	${\hat{\beta }}_2$	$\widehat{\mathrm{se}}({\hat{\beta }}_2)$	${\hat{\beta }}_3$	$\widehat{\mathrm{se}}({\hat{\beta }}_3)$	Efficiency gain for ${\beta }_2$ in %	Efficiency gain for ${\beta }_3$ in %
Errors ∼ N (0,1)
16	0	0.06	1.59	0.00	0.37	−0.01	3.76	−0.01	0.86	82.10	81.17
	0.2	0.00	1.74	0.00	0.45	−0.13	3.93	−0.05	0.92	80.53	76.39
	0.4	−0.01	1.82	−0.01	0.45	−0.13	3.91	−0.03	0.91	78.30	75.41
	0.6	0.05	1.73	0.01	0.43	−0.05	3.88	0.00	0.89	80.16	76.02
	0.8	−0.01	1.40	−0.01	0.36	−0.06	3.83	0.01	0.92	86.69	84.82
50	0	−0.02	0.77	0.02	0.05	0.08	3.33	0.01	0.28	94.69	96.27
	0.2	−0.04	0.87	0.01	0.06	−0.09	3.27	0.00	0.28	92.93	95.25
	0.4	−0.03	0.92	0.02	0.07	−0.03	3.38	0.01	0.28	92.55	94.29
	0.6	0.00	1.05	0.02	0.08	0.08	3.43	0.01	0.30	90.71	92.70
	0.8	−0.04	1.13	0.01	0.10	−0.07	3.36	0.00	0.29	88.74	89.24
Errors ∼ Laplace (0,1)
16	0	0.22	2.00	0.02	0.49	0.27	3.81	0.06	0.91	72.45	71.24
	0.2	0.19	2.14	−0.03	0.51	0.35	4.02	−0.02	0.95	71.70	70.90
	0.4	0.05	2.21	−0.03	0.52	−0.05	3.89	−0.01	0.92	67.86	68.17
	0.6	0.02	2.24	0.00	0.55	−0.13	3.95	−0.01	0.97	67.77	68.20
	0.8	0.01	1.83	0.00	0.49	0.06	3.80	0.00	0.94	76.67	73.00
50	0	−0.03	0.92	0.02	0.06	−0.10	3.29	0.01	0.27	92.10	94.62
	0.2	−0.06	1.01	0.02	0.07	0.00	3.29	−0.01	0.28	90.53	93.12
	0.4	−0.10	1.21	0.02	0.09	−0.12	3.39	0.00	0.29	87.26	91.10
	0.6	−0.05	1.38	0.01	0.10	0.00	3.40	−0.01	0.29	83.57	87.18
	0.8	−0.05	1.66	0.01	0.13	−0.09	3.65	0.00	0.30	79.31	80.31
Errors ∼ t (5)
16	0	0.04	2.01	0.00	0.49	−0.06	3.92	0.00	0.92	73.81	72.47
	0.2	0.04	2.03	0.01	0.47	−0.04	3.94	0.01	0.94	73.53	74.95
	0.4	−0.06	2.11	0.00	0.51	0.00	4.05	0.00	0.94	72.96	70.15
	0.6	0.00	2.04	−0.01	0.50	−0.04	3.95	−0.01	0.96	73.38	72.81
	0.8	0.04	1.70	−0.01	0.45	−0.06	3.78	0.06	0.97	79.76	78.22
50	0	−0.03	0.87	0.02	0.06	0.06	3.33	−0.01	0.29	93.11	95.13
	0.2	−0.06	1.03	0.01	0.07	−0.02	3.35	−0.01	0.27	90.60	93.41
	0.4	−0.06	1.14	0.02	0.08	−0.06	3.43	0.00	0.28	88.99	92.05
	0.6	−0.02	1.35	0.01	0.10	0.13	3.48	−0.01	0.29	85.00	89.08
	0.8	−0.08	1.53	0.02	0.12	−0.07	3.57	0.01	0.30	81.63	83.55

Note: SEs under both methods are estimated through bootstrapping.

Table 3. Bias of the estimate and MSE when no outliers exist, and errors follow AR (1).

		RB method				LS method
		${\hat{\beta }}_2$		${\hat{\beta }}_3$		${\hat{\beta }}_2$		${\hat{\beta }}_3$
$n$	$\rho$	Bias	MSE	Bias	MSE	Bias	MSE	Bias	MSE
Errors ∼ N (0,1)
16	0	0.00	1.08	−0.01	0.05	0.01	0.99	−0.01	0.05
	0.2	−0.03	1.23	0.00	0.07	−0.03	1.16	0.00	0.06
	0.4	−0.04	1.45	0.00	0.09	−0.04	1.39	0.00	0.08
	0.6	−0.05	1.45	0.01	0.10	−0.04	1.37	0.01	0.10
	0.8	0.00	0.99	0.00	0.08	0.00	0.94	0.00	0.08
50	0	0.02	0.35	0.00	0.00	0.01	0.33	0.00	0.00
	0.2	0.00	0.46	0.00	0.00	0.00	0.45	0.00	0.00
	0.4	−0.01	0.69	0.00	0.00	−0.01	0.67	0.00	0.00
	0.6	0.01	0.85	0.00	0.01	0.01	0.83	0.00	0.01
	0.8	−0.04	1.20	0.00	0.01	−0.04	1.16	0.00	0.01
Errors ∼ Laplace (0,1)
16	0	0.03	1.83	−0.01	0.08	0.06	2.04	0.00	0.10
	0.2	−0.03	2.11	−0.01	0.12	−0.07	2.35	−0.01	0.13
	0.4	0.02	2.61	0.00	0.16	0.02	2.67	0.00	0.17
	0.6	0.10	2.94	0.02	0.20	0.10	2.82	0.02	0.20
	0.8	0.00	1.95	−0.01	0.17	−0.01	1.82	−0.01	0.17
50	0	0.00	0.49	0.00	0.00	0.00	0.63	0.00	0.00
	0.2	−0.01	0.71	0.00	0.00	−0.01	0.90	0.00	0.00
	0.4	0.01	1.08	0.00	0.01	0.02	1.22	0.00	0.01
	0.6	−0.01	1.62	0.00	0.01	−0.01	1.68	0.00	0.01
	0.8	0.08	2.41	−0.01	0.02	0.09	2.37	−0.01	0.02
Errors ∼ t (5)
16	0	0.02	1.60	−0.01	0.07	−0.02	1.74	−0.01	0.08
	0.2	−0.05	2.03	0.00	0.09	−0.06	2.08	0.00	0.10
	0.4	−0.02	2.49	0.00	0.13	−0.01	2.57	−0.01	0.13
	0.6	0.01	2.44	0.02	0.15	0.00	2.37	0.02	0.16
	0.8	0.01	1.60	−0.01	0.15	0.01	1.48	−0.01	0.15
50	0	0.03	0.45	0.00	0.00	0.02	0.51	0.00	0.00
	0.2	−0.02	0.65	0.00	0.00	−0.02	0.77	0.00	0.00
	0.4	−0.04	0.97	0.00	0.00	−0.03	1.08	0.00	0.01
	0.6	0.03	1.46	0.00	0.01	0.02	1.52	0.00	0.01
	0.8	0.02	1.86	0.00	0.01	0.02	1.79	0.00	0.01

Table 4. Bias of the estimate and MSE when outliers exist, and errors follow AR (1).

		RB method				LS method
		${\hat{\beta }}_2$		${\hat{\beta }}_3$		${\hat{\beta }}_2$		${\hat{\beta }}_3$
$n$	$\rho$	Bias	MSE	Bias	MSE	Bias	MSE	Bias	MSE
Errors ∼ N (0,1)
16	0	0.06	2.53	0.00	0.14	−0.01	14.12	−0.01	0.74
	0.2	0.00	3.01	0.00	0.20	−0.13	15.47	−0.05	0.86
	0.4	−0.01	3.31	−0.01	0.20	−0.13	15.28	−0.03	0.82
	0.6	0.05	2.98	0.01	0.19	−0.05	15.01	0.00	0.79
	0.8	−0.01	1.95	−0.01	0.13	−0.06	14.67	0.01	0.85
50	0	−0.02	0.59	0.02	0.00	0.08	11.07	0.01	0.08
	0.2	−0.04	0.76	0.01	0.00	−0.09	10.68	0.00	0.08
	0.4	−0.03	0.85	0.02	0.00	−0.03	11.39	0.01	0.08
	0.6	0.00	1.10	0.02	0.01	0.08	11.79	0.01	0.09
	0.8	−0.04	1.27	0.01	0.01	−0.07	11.28	0.00	0.09
Errors ∼ Laplace (0,1)
16	0	0.22	4.04	0.02	0.24	0.27	14.58	0.06	0.82
	0.2	0.19	4.61	−0.03	0.26	0.35	16.27	−0.02	0.91
	0.4	0.05	4.87	−0.03	0.27	−0.05	15.14	−0.01	0.84
	0.6	0.02	5.01	0.00	0.30	−0.13	15.57	−0.01	0.94
	0.8	0.01	3.36	0.00	0.24	0.06	14.42	0.00	0.88
50	0	−0.03	0.85	0.02	0.00	−0.10	10.79	0.01	0.07
	0.2	−0.06	1.03	0.02	0.01	0.00	10.82	−0.01	0.08
	0.4	−0.10	1.48	0.02	0.01	−0.12	11.52	0.00	0.08
	0.6	−0.05	1.90	0.01	0.01	0.00	11.53	−0.01	0.09
	0.8	−0.05	2.76	0.01	0.02	−0.09	13.31	0.00	0.09
Errors ∼ t (5)
16	0	0.04	4.03	0.00	0.24	−0.06	15.37	0.00	0.85
	0.2	0.04	4.11	0.01	0.22	−0.04	15.51	0.01	0.88
	0.4	−0.06	4.44	0.00	0.26	0.00	16.42	0.00	0.89
	0.6	0.00	4.15	−0.01	0.25	−0.04	15.57	−0.01	0.92
	0.8	0.04	2.89	−0.01	0.21	−0.06	14.25	0.06	0.95
50	0	−0.03	0.76	0.02	0.00	0.06	11.07	−0.01	0.09
	0.2	−0.06	1.06	0.01	0.01	−0.02	11.21	−0.01	0.07
	0.4	−0.06	1.30	0.02	0.01	−0.06	11.74	0.00	0.08
	0.6	−0.02	1.82	0.01	0.01	0.13	12.12	−0.01	0.09
	0.8	−0.08	2.34	0.02	0.01	−0.07	12.72	0.01	0.09

Table 5. Type I error of asymptotic test HM and F for $H_0:{\beta }_2={\beta }_3=0$ when error follows AR (1).

		Without outliers		With outliers
$n$	$\rho$	RB method	LS method	RB method	LS method
Errors ∼ N (0,1)
16	0	0.06	0.06	0.03	0.04
	0.2	0.10	0.10	0.04	0.05
	0.4	0.20	0.20	0.08	0.06
	0.6	0.33	0.33	0.13	0.08
	0.8	0.45	0.48	0.15	0.06
50	0	0.07	0.06	0.05	0.06
	0.2	0.13	0.13	0.11	0.05
	0.4	0.26	0.26	0.15	0.05
	0.6	0.46	0.47	0.30	0.07
	0.8	0.66	0.68	0.52	0.06
Errors ∼ Laplace (0,1)
16	0	0.05	0.05	0.03	0.04
	0.2	0.10	0.11	0.07	0.06
	0.4	0.20	0.20	0.09	0.06
	0.6	0.37	0.34	0.17	0.09
	0.8	0.47	0.46	0.18	0.08
50	0	0.05	0.05	0.04	0.04
	0.2	0.13	0.12	0.08	0.06
	0.4	0.25	0.25	0.17	0.07
	0.6	0.46	0.44	0.29	0.07
	0.8	0.69	0.69	0.52	0.08
Errors ∼ t (5)
16	0	0.06	0.05	0.04	0.05
	0.2	0.10	0.09	0.05	0.05
	0.4	0.21	0.22	0.09	0.07
	0.6	0.34	0.33	0.15	0.08
	0.8	0.48	0.48	0.17	0.08
50	0	0.06	0.05	0.06	0.06
	0.2	0.13	0.13	0.10	0.04
	0.4	0.28	0.27	0.17	0.05
	0.6	0.44	0.45	0.29	0.07
	0.8	0.66	0.67	0.53	0.09

Table 6. Type I error for individual t-test for testing $H_0:{\beta }_2=0$ and $H_0:{\beta }_3=0$ under AR(1).

		Without outliers				With outliers
		RB method		LS method		RB method		LS method
$n$	$\rho$	${\beta }_2$	${\beta }_3$	${\beta }_2$	${\beta }_3$	${\beta }_2$	${\beta }_3$	${\beta }_2$	${\beta }_3$
Errors ∼ N (0,1)
16	0	0.05	0.05	0.04	0.05	0.04	0.03	0.04	0.03
	0.2	0.09	0.08	0.08	0.10	0.04	0.06	0.05	0.05
	0.4	0.10	0.16	0.11	0.19	0.06	0.08	0.06	0.05
	0.6	0.18	0.26	0.18	0.29	0.07	0.11	0.06	0.06
	0.8	0.21	0.39	0.24	0.41	0.09	0.15	0.06	0.08
50	0	0.06	0.05	0.06	0.05	0.03	0.04	0.02	0.06
	0.2	0.10	0.10	0.10	0.11	0.06	0.06	0.03	0.07
	0.4	0.18	0.19	0.20	0.21	0.08	0.12	0.03	0.08
	0.6	0.29	0.34	0.30	0.34	0.13	0.16	0.04	0.07
	0.8	0.45	0.53	0.47	0.55	0.23	0.30	0.04	0.09
Errors ∼ Laplace (0,1)
16	0	0.06	0.05	0.06	0.04	0.04	0.05	0.05	0.05
	0.2	0.09	0.10	0.09	0.10	0.05	0.06	0.04	0.05
	0.4	0.13	0.17	0.15	0.18	0.08	0.10	0.06	0.07
	0.6	0.19	0.28	0.17	0.28	0.10	0.13	0.06	0.08
	0.8	0.25	0.40	0.24	0.44	0.11	0.17	0.06	0.09
50	0	0.05	0.05	0.05	0.05	0.04	0.05	0.03	0.07
	0.2	0.09	0.13	0.08	0.12	0.05	0.07	0.03	0.06
	0.4	0.17	0.19	0.18	0.20	0.09	0.13	0.03	0.07
	0.6	0.29	0.33	0.28	0.33	0.16	0.21	0.03	0.07
	0.8	0.43	0.54	0.43	0.54	0.26	0.34	0.05	0.07
Errors ∼ t (5)
16	0	0.04	0.06	0.05	0.05	0.04	0.05	0.04	0.04
	0.2	0.08	0.10	0.09	0.10	0.06	0.06	0.05	0.05
	0.4	0.14	0.17	0.13	0.19	0.06	0.08	0.05	0.06
	0.6	0.17	0.28	0.18	0.30	0.08	0.13	0.05	0.08
	0.8	0.23	0.39	0.23	0.41	0.07	0.16	0.05	0.08
50	0	0.05	0.06	0.06	0.05	0.04	0.04	0.03	0.07
	0.2	0.10	0.10	0.10	0.10	0.05	0.05	0.03	0.06
	0.4	0.19	0.21	0.19	0.21	0.07	0.11	0.02	0.07
	0.6	0.30	0.30	0.30	0.33	0.14	0.22	0.03	0.06
	0.8	0.44	0.52	0.44	0.52	0.26	0.35	0.04	0.11

Table 7. Comparison of SE estimated through asymptotic and bootstrap approaches when no outliers.

		RB method				LS method
		Bootstrap SE		Asymptotic SE		Bootstrap SE		Asymptotic SE		Analytical SE
$n$	$\rho$	${\hat{\beta }}_2$	${\hat{\beta }}_3$	${\hat{\beta }}_2$	${\hat{\beta }}_3$	${\hat{\beta }}_2$	${\hat{\beta }}_3$	${\hat{\beta }}_2$	${\hat{\beta }}_3$	${\hat{\beta }}_2$	${\hat{\beta }}_3$
Errors ∼ N (0,1)
16	0	1.05	0.23	1.10	0.24	1.00	0.22	1.01	0.22	1.01	0.22
	0.2	1.13	0.27	1.03	0.22	1.09	0.26	0.94	0.20	1.11	0.25
	0.4	1.18	0.29	0.92	0.20	1.16	0.28	0.84	0.18	1.17	0.29
	0.6	1.17	0.33	0.79	0.17	1.14	0.32	0.71	0.15	1.16	0.32
	0.8	0.93	0.29	0.56	0.12	0.90	0.29	0.50	0.11	0.95	0.30
50	0	0.59	0.04	0.59	0.04	0.58	0.04	0.56	0.04	0.57	0.04
	0.2	0.66	0.05	0.57	0.04	0.64	0.05	0.55	0.04	0.67	0.05
	0.4	0.78	0.06	0.56	0.04	0.77	0.06	0.54	0.04	0.79	0.06
	0.6	0.94	0.07	0.52	0.04	0.93	0.07	0.50	0.03	0.95	0.07
	0.8	1.09	0.09	0.43	0.03	1.07	0.09	0.41	0.03	1.09	0.09
Errors ∼ Laplace (0,1)
16	0	1.38	0.29	1.34	0.29	1.43	0.30	1.36	0.29	2.02	0.44
	0.2	1.45	0.34	1.28	0.28	1.53	0.35	1.28	0.28	2.21	0.51
	0.4	1.70	0.39	1.19	0.26	1.70	0.40	1.18	0.25	2.35	0.58
	0.6	1.64	0.44	1.02	0.22	1.61	0.44	1.00	0.22	2.32	0.64
	0.8	1.40	0.41	0.74	0.16	1.35	0.41	0.71	0.15	1.90	0.60
50	0	0.69	0.05	0.68	0.05	0.81	0.05	0.78	0.05	1.13	0.08
	0.2	0.85	0.06	0.69	0.05	0.93	0.07	0.78	0.05	1.34	0.09
	0.4	1.05	0.08	0.70	0.05	1.11	0.08	0.74	0.05	1.59	0.12
	0.6	1.29	0.10	0.68	0.05	1.32	0.10	0.70	0.05	1.90	0.15
	0.8	1.51	0.12	0.57	0.04	1.51	0.12	0.58	0.04	2.18	0.19
Errors ∼ t (5)
16	0	1.26	0.26	1.28	0.28	1.30	0.27	1.27	0.27	1.69	0.36
	0.2	1.37	0.31	1.18	0.25	1.45	0.33	1.16	0.25	1.84	0.42
	0.4	1.53	0.37	1.12	0.24	1.56	0.38	1.07	0.23	1.95	0.48
	0.6	1.43	0.39	0.95	0.20	1.43	0.39	0.91	0.20	1.93	0.53
	0.8	1.30	0.38	0.69	0.15	1.27	0.38	0.66	0.14	1.58	0.50
50	0	0.68	0.05	0.68	0.05	0.73	0.05	0.72	0.05	0.94	0.07
	0.2	0.78	0.06	0.66	0.05	0.85	0.06	0.70	0.05	1.11	0.08
	0.4	0.96	0.07	0.65	0.05	1.01	0.07	0.68	0.05	1.32	0.10
	0.6	1.18	0.09	0.63	0.04	1.22	0.09	0.63	0.04	1.58	0.12
	0.8	1.43	0.12	0.53	0.04	1.42	0.12	0.53	0.04	1.82	0.16

Table 8. Comparison of SE estimated through asymptotic and bootstrap approaches when outliers.

		RB method				LS method
		Bootstrap SE		Asymptotic SE		Bootstrap SE		Asymptotic SE		Analytical SE
$n$	$\rho$	${\hat{\beta }}_2$	${\hat{\beta }}_3$	${\hat{\beta }}_2$	${\hat{\beta }}_3$	${\hat{\beta }}_2$	${\hat{\beta }}_3$	${\hat{\beta }}_2$	${\hat{\beta }}_3$	${\hat{\beta }}_2$	${\hat{\beta }}_3$
Errors ∼ N (0,1)
16	0	1.79	0.38	1.87	0.40	3.84	0.88	3.81	0.82	1.01	0.22
	0.2	1.91	0.43	1.83	0.39	3.81	0.95	3.73	0.80	1.11	0.25
	0.4	1.63	0.40	1.70	0.37	3.64	0.86	3.69	0.80	1.17	0.29
	0.6	1.98	0.47	1.60	0.35	3.90	0.93	3.62	0.78	1.16	0.32
	0.8	1.31	0.41	1.44	0.31	3.78	0.94	3.65	0.79	0.95	0.30
50	0	0.77	0.05	0.82	0.06	3.26	0.29	3.74	0.26	0.57	0.04
	0.2	0.86	0.06	0.80	0.06	3.44	0.27	3.76	0.26	0.67	0.05
	0.4	0.90	0.07	0.77	0.05	3.26	0.28	3.74	0.26	0.79	0.06
	0.6	1.10	0.08	0.74	0.05	3.42	0.28	3.77	0.26	0.95	0.07
	0.8	1.16	0.10	0.62	0.04	3.33	0.30	3.72	0.26	1.09	0.09
Errors ∼ Laplace (0,1)
16	0	2.15	0.52	2.25	0.48	3.81	0.91	3.86	0.83	2.02	0.44
	0.2	2.11	0.52	2.13	0.46	3.92	0.95	3.87	0.84	2.21	0.51
	0.4	2.17	0.56	2.03	0.44	3.91	0.98	3.83	0.83	2.35	0.58
	0.6	2.21	0.56	1.84	0.40	4.11	0.93	3.72	0.80	2.32	0.64
	0.8	1.74	0.50	1.63	0.35	3.79	0.95	3.72	0.80	1.90	0.60
50	0	0.93	0.06	0.97	0.07	3.41	0.28	3.80	0.26	1.13	0.08
	0.2	1.04	0.07	0.97	0.07	3.39	0.28	3.81	0.26	1.34	0.09
	0.4	1.15	0.09	0.97	0.07	3.44	0.29	3.77	0.26	1.59	0.12
	0.6	1.45	0.11	0.93	0.06	3.46	0.30	3.79	0.26	1.90	0.15
	0.8	1.61	0.13	0.81	0.06	3.40	0.29	3.75	0.26	2.18	0.19
Errors ∼ t (5)
16	0	1.84	0.44	2.13	0.46	3.65	0.91	3.80	0.82	1.69	0.36
	0.2	2.04	0.49	2.03	0.44	3.97	0.93	3.82	0.82	1.84	0.42
	0.4	2.04	0.49	1.96	0.42	3.96	0.93	3.86	0.83	1.95	0.48
	0.6	2.01	0.50	1.76	0.38	3.90	0.94	3.70	0.80	1.93	0.53
	0.8	1.78	0.46	1.52	0.33	3.82	0.92	3.58	0.77	1.58	0.50
50	0	0.88	0.06	0.94	0.06	3.32	0.28	3.79	0.26	0.94	0.07
	0.2	0.95	0.07	0.92	0.06	3.45	0.29	3.81	0.26	1.11	0.08
	0.4	1.13	0.09	0.91	0.06	3.39	0.29	3.77	0.26	1.32	0.10
	0.6	1.31	0.10	0.86	0.06	3.62	0.29	3.76	0.26	1.58	0.12
	0.8	1.51	0.13	0.74	0.05	3.56	0.30	3.73	0.26	1.82	0.16

Figure 1. Power with normally distributed errors and no outliers (left panel) and non-normality for errors and outliers existing (right panel).

Figure 2. Time series plot for the rate of organ trafficking per million population.

Figure 3. Time series plot for the log-rate of organ trafficking per million population.

Table 9. Results of standard ls regression analysis.

	Country	${\hat{\beta }}_2$	$\widehat{\mathrm{se}}({\hat{\beta }}_2)$	95% CI for ${\beta }_2$	${\hat{\beta }}_3$	$\widehat{\mathrm{se}}({\hat{\beta }}_3)$	95% CI for ${\beta }_3$
Outliers	Brazil	0.35**	0.138	(0.04, 0.66)	−0.04	0.034	(−0.11, 0.04)
	Egypt	0.50**	0.162	(0.13, 0.86)	−0.05	0.040	(−0.14, 0.04)
	Iran	0.15	0.119	(−0.12,0.41)	−0.14**	0.029	(−0.20, −0.07)
	Nigeria	0.41**	0.150	(0.08,0.75)	0.03	0.037	(−0.05, 0.11)
	Turkey	0.32**	0.134	(0.02,0.62)	0.01	0.033	(−0.06, 0.08)
No outliers	Colombia	0.67**	0.215	(0.19,1.45)	0.04	0.053	(−0.08, 0.15)
	India	0.21**	0.088	(0.02,0.41)	−0.05	0.022	(−0.10, 0.01)
	Mexico	0.17	0.100	(−0.05,0.39)	−0.12**	0.025	(−0.17, −0.06)
	Pakistan	0.49**	0.136	(0.18,0.79)	−0.14**	0.033	(−0.21, −0.06)
	Philippines	0.56**	0.151	(0.22,0.89)	−0.09**	0.037	(−0.17, −0.01)
	Thailand	0.29	0.141	(−0.02,0.61)	−0.15**	0.035	(−0.23, −0.07)

Table 10. Bootstrap results of standard ls regression analysis.

	Country	${\hat{\beta }}_2$	$\widehat{\mathrm{se}}({\hat{\beta }}_2)$	95% CI for ${\beta }_2$	${\hat{\beta }}_3$	$\widehat{\mathrm{se}}({\hat{\beta }}_3)$	95% CI for ${\beta }_3$
Outliers	Brazil	0.36**	0.139	(0.10, 0.63)	−0.04	0.035	(−0.10, 0.03)
	Egypt	0.49**	0.159	(0.16, 0.80)	−0.05	0.040	(−0.12, 0.03)
	Iran	0.15	0.123	(−0.11,0.38)	−0.14**	0.029	(−0.19, −0.08)
	Nigeria	0.42**	0.157	(0.12,0.71)	0.03	0.040	(−0.05, 0.10)
	Turkey	0.32**	0.130	(0.08,0.60)	0.01	0.034	(−0.06, 0.07)
No outliers	Colombia	0.67**	0.217	(0.25,1.11)	0.04	0.053	(−0.06, 0.14)
	India	0.21**	0.090	(0.04,0.39)	−0.05**	0.023	(−0.09, −0.01)
	Mexico	0.16	0.102	(−0.03,0.37)	−0.12**	0.025	(−0.17, −0.07)
	Pakistan	0.49**	0.136	(0.21,0.74)	−0.14**	0.034	(−0.20, −0.07)
	Philippines	0.56**	0.154	(0.25,0.85)	−0.09**	0.038	(−0.17, −0.02)
	Thailand	0.29**	0.138	(0.03,0.57)	−0.15**	0.034	(−0.21, −0.08)

Table 11. Results of rank-based regression analysis.

	Country	${\hat{\beta }}_2$	$\widehat{\mathrm{se}}({\hat{\beta }}_2)$	95% CI for ${\beta }_2$	${\hat{\beta }}_3$	$\widehat{\mathrm{se}}({\hat{\beta }}_3)$	95% CI for ${\beta }_3$
Outliers	Brazil	0.33**	0.123	(0.06, 0.61)	−0.04	0.030	(−0.11, 0.02)
	Egypt	0.48**	0.183	(0.07, 0.89)	−0.04	0.045	(−0.14, 0.06)
	Iran	0.16	0.123	(−0.11, 0.44)	−0.15**	0.030	(−0.22, −0.08)
	Nigeria	0.43**	0.148	(0.10, 0.76)	0.03	0.036	(−0.05,0.11)
	Turkey	0.34	0.158	(−0.02,0.69)	0.00	0.038	(−0.08, 0.09)
No outliers	Colombia	0.62	0.281	(−0.01, 1.24)	0.05	0.069	(−0.11, 0.20)
	India	0.20	0.123	(−0.08, 0.45)	−0.05	0.030	(−0.12, 0.02)
	Mexico	0.16	0.130	(−0.13, 0.44)	−0.12**	0.031	(−0.19, −0.05)
	Pakistan	0.50**	0.147	(0.17,0.83)	−0.12**	0.036	(−0.20, −0.04)
	Philippines	0.56**	0.127	(0.28, 0.85)	−0.09**	0.031	(−0.16, −0.02)
	Thailand	0.25	0.157	(−0.10, 0.60)	−0.15**	0.038	(−0.24, −0.06)

Table 12. Bootstrap results of rank-based regression analysis.

	Country	${\hat{\beta }}_2$	$\widehat{\mathrm{se}}({\hat{\beta }}_2)$	95% CI for ${\beta }_2$	${\hat{\beta }}_3$	$\widehat{\mathrm{se}}({\hat{\beta }}_3)$	95% CI for ${\beta }_3$
Outliers	Brazil	0.33**	0.122	(0.09, 0.58)	−0.05	0.029	(−0.10, 0.01)
	Egypt	0.48**	0.140	(0.21, 0.77)	−0.05	0.035	(−0.11, 0.02)
	Iran	0.16	0.100	(−0.03, 0.38)	−0.15**	0.023	(−0.20, −0.10)
	Nigeria	0.43**	0.134	(0.16, 0.72)	0.03	0.031	(−0.03,0.10)
	Turkey	0.37**	0.120	(0.10,0.56)	0.00	0.029	(−0.05, 0.06)
No outliers	Colombia	0.62**	0.195	(0.21, 1.00)	0.05	0.046	(−0.05, 0.14)
	India	0.19**	0.080	(0.04, 0.35)	−0.05**	0.020	(−0.09, −0.01)
	Mexico	0.15	0.090	(−0.02, 0.33)	−0.12**	0.022	(−0.17, −0.08)
	Pakistan	0.50**	0.118	(0.27,0.74)	−0.12**	0.028	(−0.18, −0.07)
	Philippines	0.56**	0.131	(0.28, 0.84)	−0.09**	0.031	(−0.16, −0.03)
	Thailand	0.25**	0.120	(0.02, 0.49)	−0.15**	0.030	(−0.21, −0.09)

Figure 4. Thailand and Turkey fitted model.

Table 13. Comparisons of HM and F-test for testing $H_0:{\beta }_2={\beta }_3=0$.

		Rank-based test			Standard LS test
	Country	Value of HM Test	p-value	Bootstrap p-value	Value of F-statistic	p-value	Bootstrap p-value
Outliers	Brazil	4.44**	0.04	0.042	3.50	0.07	0.069
	Egypt	4.36**	0.04	0.045	5.03**	0.03	0.028
	Iran	9.94**	0.00	0.004	11.27**	0.00	0.007
	Nigeria	4.50**	0.04	0.037	4.40**	0.04	0.038
	Turkey	2.89	0.10	0.133	2.98	0.10	0.105
No outliers	Colombia	4.10	0.05	0.065	5.36**	0.03	0.018
	India	3.00	0.10	0.109	4.71**	0.04	0.039
	Mexico	7.70**	0.00	0.012	12.35**	0.00	0.000
	Pakistan	10.03**	0.00	0.004	13.19**	0.00	0.003
	Philippines	9.84**	0.00	0.006	8.90**	0.01	0.005
	Thailand	8.25**	0.01	0.005	10.26**	0.00	0.001