Selecting baseline designs using a minimum aberration criterion when some two-factor interactions are important

Figures & data

Figure 1. Model containing one interaction (k = 1).

Figure 2. Models containing two interactions (k = 2).

Figure 3. Models containing three interactions (k = 3).

Table 1. Sixteen-run MA baseline designs for the model with k = 1.

m	i	Interaction	Switched	D-efficiency	A-efficiency	($Q_2$, $Q_3$, $Q_4$)
4	3	$(1,2)$	None	0.25	2.88	(2.81, 1.16, 0.11)
5	3	$(1,5)$	1	0.25	3.19	(5.06, 2.80, 0.55)
6	4	$(2,4)$	1	0.25	3.50	(8.44, 6.08, 1.81)
7	5	$(1,6)$	1, 6	0.25	3.81	(12.38, 11.33, 4.97)
8	4	$(1,8)$	1, 8	0.25	4.13	(16.88, 19.12, 11.5)
9	3	$(1,9)$	1, 6	0.25	4.44	(24.50, 33.16, 24.09)
10	3	$(2,10)$	2, 5, 8	0.25	4.75	(32.69, 49.70, 39.75)
11	3	$(1,11)$	3, 4, 7	0.25	5.06	(41.44, 72.08, 67.34)
12	1	$(4,9)$	1, 2, 3, 12	0.25	5.38	(51.88, 100.44, 102.38)
13	1	$(2,12)$	1, 2, 3, 12, 13	0.25	5.69	(62.88, 135.88, 156.14)
14	1	$(1,14)$	1, 2, 3, 12, 13, 14	0.25	6.00	(75.00, 179.00, 228.50)

Table 2. Sixteen-run MA baseline designs for the models with k = 2.

m	Model	i	Interactions	Switched	D-efficiency	A-efficiency	($Q_2$, $Q_3$, $Q_4$)
4	2(a)	3	$(1,2),(3,4)$	None	0.30	4.44	(2.25, 1.31, 0.13)
	2(b)	3	$(1,2),(1,3)$	None	0.30	4.44	(2.25, 1.47, 0.16)
5	2(a)	4	$(1,2),(3,4)$	None	0.30	4.75	(4.50, 3.69, 0.78)
	2(b)	3	$(1,5),(2,5)$	1	0.30	4.75	(4.50, 3.19, 0.72)
6	2(a)	4	$(2,4),(3,5)$	1	0.29	5.06	(7.88, 7.27, 2.45)
	2(b)	4	$(2,4),(3,4)$	1	0.29	5.06	(7.88, 7.05, 2.36)
7	2(a)	4	$(4,6),(5,7)$	1, 6	0.29	5.38	(12.38, 13.25, 6.19)
	2(b)	5	$(1,6),(1,7)$	1, 6	0.29	5.38	(11.81, 13.50, 6.66)
8	2(a)	2	$(2,7),(6,8)$	1, 6	0.28	5.69	(18.88, 23.56, 13.31)
	2(b)	2	$(6,7),(6,8)$	1, 6	0.28	5.69	(17.44, 22.23, 13.95)
9	2(a)	2	$(4,8),(5,9)$	1, 2, 3	0.28	6.00	(25.94, 38.00, 26.64)
	2(b)	2	$(4,8),(4,9)$	1, 2, 3	0.28	6.00	(25.94, 37.75, 26.36)
10	2(a)	3	$(2,10),(4,9)$	2, 5, 8	0.28	6.31	(35.00, 59.61, 50.29)
	2(b)	3	$(2,10),(3,10)$	2, 5, 8	0.28	6.31	(35.00, 59.23, 49.39)
11	2(a)	1	$(4,8),(5,10)$	1, 2, 3	0.28	6.62	(45.75, 86.00, 79.62)
	2(b)	1	$(4,8),(4,9)$	1, 2, 3	0.28	6.62	(45.75, 86.00, 80.12)
12	2(a)	1	$(4,9),(5,10)$	1, 2, 3, 12	0.27	6.94	(57.06, 121.55, 129.09)
	2(b)	1	$(4,9),(4,10)$	1, 2, 3, 12	0.27	6.94	(57.06, 121.42, 128.84)
13	2(a)	1	$(2,12),(4,11)$	1, 2, 3, 12, 13	0.27	7.25	(69.50, 164.50, 196.81)
	2(b)	1	$(2,12),(2,13)$	1, 2, 3, 12, 13	0.27	7.25	(69.50, 164.50, 196.94)

Table 3. Sixteen-run MA baseline designs for the models with k = 3.

m	Model	i	Interactions	Switched	D-efficiency	A-efficiency	($Q_2$, $Q_3$, $Q_4$)
4	3(a)	–	–	–	–	–	–
	3(b)	–	–	–	–	–	–
	3(c)	3	$(1,2),(1,3),(1,4)$	None	0.35	6.00	(1.69, 1.94, 0.25)
	3(d)	3	$(1,2),(1,3),(2,4)$	None	0.35	6.00	(1.69, 1.78, 0.22)
	3(e)	3	$(1,2),(1,3),(2,3)$	None	0.35	6.00	(1.69, 1.94, 0.25)
5	3(a)	–	–	–	–	–	–
	3(b)	4	$(1,2),(1,3),(4,5)$	None	0.34	6.31	(3.94, 4.47, 1.04)
	3(c)	3	$(1,5),(2,5),(3,5)$	1	0.34	6.31	(3.94, 3.73, 0.89)
	3(d)	4	$(1,2),(1,3),(2,4)$	None	0.34	6.31	(3.94, 4.56, 1.19)
	3(e)	4	$(1,2),(1,3),(2,3)$	None	0.34	6.31	(3.94, 4.66, 1.35)
6	3(a)	3	$(1,4),(2,5),(3,6)$	1, 4	0.33	6.63	(7.88, 8.62, 2.95)
	3(b)	4	$(2,4),(2,5),(3,6)$	1	0.33	6.63	(7.31, 8.61, 3.20)
	3(c)	4	$(2,4),(2,5),(2,6)$	1	0.33	6.63	(7.31, 8.80, 3.33)
	3(d)	4	$(2,4),(2,5),(3,4)$	1	0.33	6.63	(7.31, 8.39, 3.11)
	3(e)	4	$(2,4),(2,5),(4,5)$	2	0.33	6.63	(8.75, 9.91, 3.83)
7	3(a)	5	$(1,6),(2,5),(3,7)$	1	0.32	6.94	(14.12, 18.00, 8.91)
	3(b)	5	$(1,6),(1,7),(2,5)$	1, 6	0.32	6.94	(12.69, 16.70, 8.48)
	3(c)	2	$(1,7),(2,7),(3,7)$	1, 6	0.32	6.94	(12.38, 13.80, 5.27)
	3(d)	4	$(4,6),(4,7),(5,6)$	1, 6	0.32	6.94	(11.81, 14.75, 7.44)
	3(e)	5	$(1,6),(1,7),(6,7)$	1	0.32	6.94	(14.12, 18.64, 9.92)
8	3(a)	16	$(1,5),(2,3),(7,8)$	1, 5, 8	0.31	7.25	(20.62, 29.28, 17.53)
	3(b)	7	$(1,7),(2,7),(3,8)$	3, 4, 7	0.31	7.25	(19.69, 27.23, 15.88)
	3(c)	1	$(1,8),(2,8),(3,8)$	1, 2, 3	0.31	7.25	(18.00, 23.12, 11.31)
	3(d)	2	$(2,7),(6,7),(6,8)$	1, 6	0.31	7.25	(18.31, 25.92, 16.09)
	3(e)	5	$(1,7),(1,8),(7,8)$	1, 6	0.31	7.25	(22.06, 32.06, 19.44)
9	3(a)	2	$(4,8),(5,9),(6,7)$	1, 2, 3	0.31	7.56	(26.81, 43.77, 32.45)
	3(b)	1	$(2,8),(4,8),(6,9)$	1, 2, 3	0.31	7.56	(27.38, 42.75, 28.75)
	3(c)	1	$(2,8),(4,8),(6,8)$	1, 2, 3	0.31	7.56	(27.38, 42.25, 28.00)
	3(d)	2	$(4,8),(4,9),(5,9)$	1, 2, 3	0.31	7.56	(26.81, 43.27, 31.89)
	3(e)	5	$(1,6),(1,9),(6,9)$	1, 6, 9	0.31	7.56	(30.56, 51.09, 38.10)
10	3(a)	1	$(4,8),(5,10),(7,9)$	1, 2, 3	0.30	7.88	(37.88, 68.33, 56.28)
	3(b)	1	$(4,8),(4,9),(5,10)$	1, 2, 3	0.30	7.88	(37.88, 68.20, 56.28)
	3(c)	1	$(4,8),(4,9),(4,10)$	1, 2, 3	0.30	7.88	(37.88, 67.95, 56.28)
	3(d)	1	$(4,8),(4,9),(6,8)$	1, 2, 3	0.30	7.88	(37.88, 68.08, 56.28)
	3(e)	3	$(6,9),(6,10),(9,10)$	2, 5, 8	0.30	7.88	(41.62, 76.97, 63.80)
11	3(a)	1	$(4,8),(5,10),(6,11)$	1, 2, 3	0.30	8.19	(49.50, 100.25, 95.38)
	3(b)	1	$(4,8),(4,9),(5,10)$	1, 2, 3	0.30	8.19	(49.50, 100.25, 95.88)
	3(c)	1	$(4,8),(4,9),(4,10)$	1, 2, 3	0.30	8.19	(49.50, 100.25, 96.88)
	3(d)	1	$(4,8),(4,9),(6,8)$	1, 2, 3	0.30	8.19	(49.50, 100.25, 96.38)
	3(e)	3	$(1,6),(1,9),(6,9)$	3, 4, 7	0.30	8.19	(53.25, 112.45, 111.03)
12	3(a)	1	$(4,9),(5,10),(6,8)$	1, 2, 3, 12	0.30	8.50	(62.25, 142.62, 155.75)
	3(b)	1	$(4,9),(4,10),(7,8)$	1, 2, 3, 12	0.30	8.50	(62.25, 142.50, 155.50)
	3(c)	1	$(4,9),(4,10),(4,11)$	1, 2, 3, 12	0.30	8.50	(62.25, 142.25, 155.00)
	3(d)	1	$(4,9),(4,10),(5,10)$	1, 2, 3, 12	0.30	8.50	(62.25, 142.38, 155.25)
	3(e)	2	$(1,6),(1,10),(6,9)$	1, 6, 9, 10	0.30	8.50	(66.00, 156.00, 177.00)

Note: “–” means that there is no such a model for the given m.

Table 4. Twenty-run MA baseline designs for the model with k = 1.

m	i	Interaction	Switched	D-efficiency	A-efficiency	($Q_2$, $Q_3$, $Q_4$)
4	1	$(1,2)$	1, 2, 4	0.20	2.39	(2.78, 1.07, 0.00)
5	8	$(3,5)$	None	0.20	2.70	(5.24, 2.62, 0.26)
6	70	$(2,5)$	1, 2, 5	0.20	3.02	(8.56, 5.78, 1.27)
7	471	$(4,6)$	1, 3, 4, 6, 7	0.21	3.35	(12.55, 11.26, 4.22)

Table 5. Twenty-run MA baseline designs for the models with k = 2.

m	Model	i	Interactions	Switched	D-efficiency	A-efficiency	($Q_2$, $Q_3$, $Q_4$)
4	2(a)	1	$(1,2),(3,4)$	1, 2, 4	0.25	3.80	(2.37, 1.22, 0.00)
	2(b)	1	$(1,2),(1,3)$	1, 2, 4	0.25	3.75	(2.30, 1.50, 0.00)
5	2(a)	10	$(1,2),(3,5)$	2, 3, 5	0.25	4.16	(5.21, 3.57, 0.29)
	2(b)	8	$(1,5),(3,5)$	None	0.25	4.13	(4.96, 3.10, 0.34)
6	2(a)	67	$(1,5),(4,6)$	2, 3, 4, 5	0.24	4.53	(8.83, 7.74, 2.34)
	2(b)	47	$(2,6),(3,6)$	1,2, 3, 5, 6	0.24	4.52	(8.62, 7.02, 1.93)
7	2(a)	403	$(1,4),(3,7)$	2, 3, 4, 6, 7	0.24	4.94	(13.42, 14.13, 6.20)
	2(b)	451	$(1,5),(2,5)$	3, 4, 5, 7	0.24	4.92	(13.23, 15.33, 7.83)

Table 6. Twenty-run MA baseline designs for the models with k = 3.

m	Model	i	Interactions	Switched	D-efficiency	A-efficiency	($Q_2$, $Q_3$, $Q_4$)
4	3(a)	–	–	–	–	–	–
	3(b)	–	–	–	–	–	–
	3(c)	1	$(1,2),(1,3),(1,4)$	1, 2, 4	0.29	5.13	(1.80, 1.98, 0.00)
	3(d)	1	$(1,2),(1,3),(2,4)$	1, 2, 4	0.29	5.19	(1.85, 1.69, 0.00)
	3(e)	1	$(1,2),(1,3),(2,3)$	None	0.29	5.11	(2.14, 2.18, 0.31)
5	3(a)	–	–	–	–	–	–
	3(b)	10	$(1,2),(3,5),(4,5)$	1, 4, 5	0.29	5.71	(4.84, 4.71, 0.54)
	3(c)	8	$(1,5),(2,5),(3,5)$	1,2 ,5	0.28	5.58	(4.75, 4.67, 0.88)
	3(d)	10	$(1,2),(1,4),(4,5)$	1,4 ,5	0.29	5.62	(4.84, 4.21, 0.36)
	3(e)	10	$(1,2),(1,5),(2,5)$	None	0.28	5.55	(5.12, 4.33, 0.89)
6	3(a)	57	$(1,4),(2,5),(3,6)$	1, 4, 6	0.28	6.20	(8.71, 9.56, 3.35)
	3(b)	46	$(1,6),(3,4),(3,5)$	1, 3, 5	0.28	6.09	(8.92, 9.85, 3.31)
	3(c)	69	$(1,5),(4,5),(5,6)$	None	0.28	6.03	(8.67, 9.60, 2.92)
	3(d)	67	$(1,5),(4,6),(5,6)$	2, 3, 4, 5	0.28	6.05	(8.71, 9.29, 3.04)
	3(e)	58	$(4,5),(4,6),(5,6)$	4, 5	0.28	5.99	(9.08, 10.35, 6.66)
7	3(a)	429	$(2,7),(3,6),(4,5)$	2	0.27	6.55	(14.07, 16.70, 7.32)
	3(b)	334	$(1,5),(2,6),(5,7)$	1, 2, 4, 5, 7	0.27	6.53	(13.91, 17.15, 8.00)
	3(c)	430	$(1,5),(1,6),(1,7)$	1, 7	0.27	6.48	(13.89, 15.91, 6.82)
	3(d)	429	$(3,5),(3,6),(4,5)$	1, 2, 4	0.27	6.55	(13.68, 17.57, 7.90)
	3(e)	425	$(4,6),(4,7),(6,7)$	5, 6, 7	0.27	6.48	(14.27, 16.33, 6.52)

Note: “–” means that there is no such a model for the given m.