An Investigation of the Median-Median Method of Linear Regression

Figures & data

Table 1. Manatee data divided into three regions based on x‐score. In each region, the first column x is the number of powerboat registrations, in thousands, and the second column y is the number of manatees killed.

	Region 1		Region 2		Region 3
	x	y	x	y	x	y
	447	13	513	24	614	33
	460	21	526	15	645	39
	481	24	559	34	675	43
	498	16	585	33	711	50
	512	20			719	47
median	482	20	542.5	28.5	675	43

Figure 1. Illustration of the median-median method for the manatee data.

Figure 2. Scatterplot of manatees killed by powerboats versus number of boat registrations.

Figure 3. Scatterplot of Gesell Adaptive Score versus Age at Which Child Began Speaking.

Figure 4. Scatterplot of dietary energy density versus body mass index.

Figure 5. Scatterplot of body mass index versus dietary energy density.

Table 2. Types of outliers considered in the simulation study.

	Outlier Generation	Description
a)	No Outliers	No Outliers
b)	Y13 = 0 + 1X13 + 5||	One high outlier in the middle of the middle region
c)	Y17 = 0 + 1X17 + 5|| Y18 = 0 + 1X18 - 5||	Outliers at the first two X‐values in the upper region, one high and one low
d)	Y17 = 0 + 1X17 + 5||	One high outlier at the first X‐value in the upper region
e)	Y23 = 0 + 1X23 - 5|| Y24 = 0 + 1X24 + 5||	Outliers at the two largest X‐values, one high and one low
f)	Y17 = 0 + 1X17 + 5|| Y18 = 0 + 1X18 + 5||	High outliers at the first two X‐values in the upper region
g)	Y17 = 0 + 1X17 - 5|| Y24 = 0 + 1X24 + 5||	A low outliers at the first X‐value in the upper region and a high outlier at the largest X‐value
h)	Y24 = 0 + 1X24 + 5||	A single high outlier at the largest X‐value
i)	Y23 = 0 + 1X23 + 5|| Y24 = 0 + 1X24 + 5||	High outliers at the two largest X‐values

Table 3. Monte Carlo Results: Means, standard deviations, and mean square errors (x 10³) of the least squares (LS) and median-median (MM) slope estimates for 1000 replications.

Model: Y = 0 + 1X + ; X's: Set 1; ~ N(0, 1²).

Lower region: X₁, …, X₈; Middle region: X₉, …, X₁₆; Upper region:X₁₇, …, X₂₄.

		Mean Slope Estimate		Standard Deviation		MSE (x 10^3)
Type of Outlier	Leverage	LS	MM	LS	MM	LS	MM
a) No outliers		1.0017	1.0037	0.0295	0.0509	0.873	2.604
b) One high in middle of middle region	0.0419	1.0035	1.0037	0.0295	0.0509	0.882	2.604
c) First two X‐values in upper region, one high and one low	0.0593 0.0680	0.9980	1.0291	0.0349	0.0568	1.222	4.073
d) One high at first X‐value in upper region	0.0593	1.0170	1.0295	0.0315	0.0564	1.285	4.051
e) Two largest X‐values, one high and one low	0.1375 0.1567	1.0054	0.9703	0.0472	0.0557	2.257	3.985
f) High at first two X‐values in upper region	0.0593 0.0680	1.0357	1.0571	0.0338	0.0571	2.417	6.521
g) Low at first X‐value in upper region, high at largest X‐value	0.0593 0.1567	1.0254	1.0037	0.0424	0.0511	2.442	2.625
h) High at largest X‐value	0.1567	1.0405	1.0038	0.0405	0.0510	3.280	2.615
i) High at two largest X‐values	0.1375 0.1567	1.0749	1.0047	0.0482	0.0513	7.933	2.654

Table 4. Monte Carlo Results: Means, standard deviations, and mean square errors (x 10³) of the least squares (LS) and median-median (MM) slope estimates for 1000 replications.

Model: Y = 0 + 1X + ; X's: Set 1; ~ N(0, 5²).

Lower region: X₁, …, X₈; Middle region: X₉, …, X₁₆; Upper region:X₁₇, …, X₂₄.

		Mean Slope Estimate		Standard Deviation		MSE (x 10^3)
Type of Outlier	Leverage	LS	MM	LS	MM	LS	MM
a) No outliers		1.0087	1.0141	0.1473	0.1850	21.773	34.424
b) One high in middle of middle region	0.0419	1.0173	1.0141	0.1475	0.1850	22.056	34.424
c) First two X‐values in upper region, one high and one low	0.0593 0.0680	0.9902	1.0614	0.1746	0.1948	30.581	41.717
d) One high at first X‐value in upper region	0.0593	1.0848	1.0895	0.1576	0.1898	32.029	44.034
e) Two largest X‐values, one high and one low	0.1375 0.1567	1.0272	0.9619	0.2362	0.1985	56.530	40.854
f) High at first two X‐values in upper region	0.0593 0.0680	1.1786	1.1702	0.1691	0.1952	60.493	67.071
g) Low at first X‐value in upper region, high at largest X‐value	0.0593 0.1567	1.1271	1.0155	0.2118	0.1969	61.014	39.010
h) High at largest X‐value	0.1567	1.2023	1.0423	0.2026	0.1913	81.972	38.385
i) High at two largest X‐values	0.1375 0.1567	1.3743	1.0888	0.2408	0.2009	198.085	48.246

Table 5. Monte Carlo Results: Means, standard deviations, and mean square errors (x 10³) of the least squares (LS) and median-median (MM) slope estimates for 1000 replications.

Model: Y = 0 + 1X + ; X's: Set 2; ~ N(0, 1²).

Lower region: X₁, …, X₈; Middle region: X₉, …, X₁₆; Upper region:X₁₇, …, X₂₄.

		Mean Slope Estimate		Standard Deviation		MSE (x 10^3)
Type of Outlier	Leverage	LS	MM	LS	MM	LS	MM
a) No outliers		1.0018	1.0040	0.0296	0.0540	0.879	2.556
b) One high in middle of middle region	0.0425	1.0053	1.0040	0.0297	0.0504	0.910	2.556
c) First two X‐values in upper region, one high and one low	0.0635 0.0635	1.0015	1.0342	0.0350	0.0569	1.227	4.407
d) One high at first X‐value in upper region	0.0635	1.0188	1.0343	0.0321	0.0567	1.384	4.391
e) Two largest X‐values, one high and one low	0.1474 0.1474	1.0022	0.9741	0.0474	0.0553	2.252	3.729
f) High at first two X‐values in upper region	0.0635 0.0635	1.0359	1.0573	0.0340	0.0572	2.445	6.555
g) Low at first X‐value in upper region, high at largest X‐value	0.0635 0.1474	1.0222	1.0039	0.0423	0.0509	2.282	2.606
h) High at largest X‐value	0.1474	1.0390	1.0042	0.0400	0.0506	3.121	2.578
i) High at two largest X‐values	0.1474 0.1474	1.0753	1.0047	0.0485	0.0507	8.022	2.593

Table 6. Monte Carlo Results: Means, standard deviations, and mean square errors (x 10³) of the least squares (LS) and median-median (MM) slope estimates for 1000 replications.

Model: Y = 0 + 1X + ; X's: Set 2; ~ N(0, 5²).

Lower region: X₁, …, X₈; Middle region: X₉, …, X₁₆; Upper region:X₁₇, …, X₂₄.

		Mean Slope Estimate		Standard Deviation		MSE (x 10^3)
Type of Outlier	Leverage	LS	MM	LS	MM	LS	MM
a) No outliers		1.0090	1.0138	0.1479	0.1849	21.955	34.378
b) One high in middle of middle region	0.0425	1.0263	1.0138	0.1485	0.1849	22.744	34.378
c) First two X‐values in upper region, one high and one low	0.0635 0.0635	1.0076	1.0628	0.1751	0.1949	30.718	41.930
d) One high at first X‐value in upper region	0.0635	1.0940	1.0876	0.1607	0.1898	34.660	44.698
e) Two largest X‐values, one high and one low	0.1474 0.1474	1.0108	0.9632	0.2371	0.1988	56.333	40.874
f) High at first two X‐values in upper region	0.0635 0.0635	1.1797	1.1700	0.1700	0.1942	61.192	66.614
g) Low at first X‐value in upper region, high at largest X‐value	0.0635 0.1474	1.1112	1.0154	0.2114	0.1980	57.055	39.441
h) High at largest X‐value	0.1474	1.1952	1.0467	0.2001	0.1917	78.143	38.930
i) High at two largest X‐values	0.1474 0.1474	1.3763	1.0886	0.2427	0.2008	200.505	48.171

Figure 6. Density estimates of the slopes for the 1000 replications for least squares and median-median estimates and for = 1 and = 5; X's: Set 1; no outliers.

Figure 7. Density estimates of the slopes for the 1000 replications for least squares and median-median estimates and for = 1 and = 5; X's: Set 1; one high and one low outlier at the end — Table 2, design (e).

Figure 8. Density estimates of the slopes for the 1000 replications for least squares and median-median estimates and for = 1 and = 5; X's: Set 1; two high outliers at the end — Table 2, design (i).

Table

Power Boat Registrations	Manatee Deaths
447	13
460	21
481	24
498	16
512	20
513	24
526	15
559	34
585	33
614	33
645	39
675	43
711	50
719	47

Table

Age (x)	Gessell Score (y)
7	113
8	104
9	91
9	96
10	83
10	83
10	100
10	100
11	100
11	84
11	102
11	86
12	105
15	95
15	102
17	121
18	93
20	87
20	94
26	71
42	57

Table

BMI (x)	DED (y)		DED (x)	BMI (y)
21.1	0.54		0.44	21.5
21.5	0.44		0.54	21.1
22.1	0.67		0.67	22.1
22.3	0.78		0.78	22.3
22.4	0.90		0.86	22.8
22.8	0.86		0.90	22.4
23.1	0.91		0.91	23.1
23.3	0.94		0.93	26.8
26.8	0.93		0.94	23.3