Robustness of sequential third-order response surface design to missing observations

Figures & data

Table 1. Levels of input factors of BBD and the simulated response.

Runs	A	B	C	DH (%)
1	1	1	0	52.91
2	1	−1	0	42.59
3	−1	1	0	37.39
4	−1	−1	0	54.69
5	0	1	1	33.18
6	0	1	−1	42.06
7	0	−1	1	55.37
8	0	−1	−1	52.37
9	1	0	1	30.81
10	1	0	−1	42.04
11	−1	0	1	35.82
12	−1	0	−1	45.44
13	0	0	0	51.18
14	0	0	0	51.10
15	0	0	0	51.49
16	0	0	0	51.45

Table 2. ANOVA of the fitted second-order response surface model.

Source	Sum of Squares	Df	Mean Square	F-value	p-value	Fit Statistics
Model	850.80	9	94.53	4.46	0.0413*	R^2	0.87
A	3.10	1	3.10	0.1464	0.7152^NS	Adj.R^2	0.67
B	194.77	1	194.77	9.19	0.0230*	BIC	106.29
C	89.26	1	89.26	4.21	0.0859^NS	AIC	142.56
AB	190.68	1	190.68	9.00	0.0240*
AC	0.6417	1	0.6417	0.0303	0.8676^NS
BC	35.30	1	35.30	1.67	0.2442^NS
A^2	135.17	1	135.17	6.38	0.0449*
B^2	7.91	1	7.91	0.3733	0.5636^NS
C^2	193.97	1	193.97	9.16	0.0232*
Residual	127.10	6	21.18
Lack of Fit	126.98	3	42.33	1112.49	< 0.0001**
Pure Error	0.1141	3	0.0380
Total	977.90	15

*Significant at p<0.05; **significant at p<0.01 level.

^NS Non-significant.

Table 3. Levels of input factors of additional runs and the simulated response.

Runs	A	B	C	DH (%)
17	1.01	1.01	1.01	32.84
18	1.01	1.01	−1.01	53.33
19	1.01	−1.01	1.01	28.35
20	−1.01	1.01	1.01	26.56
21	−1.01	−1.01	1.01	47.94
22	1.01	−1.01	−1.01	33.46
23	−1.01	1.01	−1.01	43.35
24	−1.01	−1.01	−1.01	54.22
25	1.7494	0	0	24.80
26	−1.7494	0	0	43.37
27	0	1.7494	0	65.28
28	0	−1.7494	0	46.86
29	0	0	1.7494	24.74
30	0	0	−1.7494	34.64

Table 4. ANOVA of third-order response surface model considering runs 1–30.

Source	Sum of Squares	df	Mean Square	F-value	p-value	Fit Statistics
Model	3269.34	19	172.07	2656.17	<0.0001**	R^2	0.99
A	18.32	1	18.32	282.84	<0.0001**	Adj.R^2	0.99
B	177.95	1	177.95	2746.88	<0.0001*	BIC	38.10
C	0.0351	1	0.0351	0.5425	0.4783^NS	AIC	103.41
AB	591.32	1	591.32	9127.92	<0.0001**
AC	1.42	1	1.42	21.98	0.0009**
BC	118.95	1	118.95	1836.23	<0.0001**
A^2	505.32	1	505.32	7800.44	< 0.0001**
B^2	39.58	1	39.58	610.93	<0.0001**
C^2	785.16	1	785.16	12120.13	<0.0001**
ABC	2.97	1	2.97	45.91	<0.0001**
A^2B	135.70	1	135.70	2094.70	<0.0001**
A^2C	56.55	1	56.55	873.00	<0.0001**
AB^2	1.51	1	1.51	23.32	0.0007**
AC^2	39.00	1	39.00	602.08	<0.0001**
B^2C	1.36	1	1.36	21.04	0.0010**
BC^2	0.6722	1	0.6722	10.38	0.0092**
A^3	63.82	1	63.82	985.14	<0.0001**
B^3	258.93	1	258.93	3996.94	<0.0001**
C^3	5.66	1	5.66	87.39	<0.0001**
Residual	0.6478	10	0.0648
Lack of Fit	0.5337	7	0.0762	2.00	0.3054^NS
Pure Error	0.1141	3	0.0380
Total	3269.99	29

*Significant at p<0.05; **significant at p<0.01 level.

^NS Non-significant.

Table 5. Information loss and efficiency of the STORDs for v = 3.

SI. no.	Design type and runs (N)	Design points	No of sets	Radius (d^2) d = 1, $\alpha =1$	L1	E1
1	GHG (N = 54)	$\left(\begin{array}{ccc}\pm d,& \pm d,& \pm d\end{array}\right)$	1	3	0.29	0.71
		$\left(\begin{array}{ccc}\pm d\sqrt{2},& 0,& 0\end{array}\right)$	2	2	0.31	0.69
		$\left(\begin{array}{ccc}0,& 0,& 0\end{array}\right)$	4	0	0.16	0.84*
		$\left(\begin{array}{ccc}\pm 0.341564d,& \pm 1.286527d,& \pm 1.286527d\end{array}\right)$	1	3.42697	0.39	0.61
		$\left(\begin{array}{ccc}\pm 1.985406d,& 0,& 0\end{array}\right)$	1	3.94184	0.65*	0.35
2	T1 (N = 50)	$\left(\begin{array}{ccc}\pm \alpha ,& \pm \alpha ,& \pm \alpha \end{array}\right)$	2	3	0.30	0.70
		$\left(\begin{array}{ccc}\pm \alpha \sqrt{2},& 0,& 0\end{array}\right)$	2	2	0.28	0.72
		$\left(\begin{array}{ccc}\text{0},& \text{0},& \text{0}\end{array}\right)$	4	0	0.15	0.85*
		$\left(\begin{array}{ccc}\pm \alpha \sqrt{2},& \pm \alpha \sqrt{2},& 0\end{array}\right)$	1	4	0.62*	0.38
		$\left(\begin{array}{ccc}\pm 2\alpha ,& 0,& 0\end{array}\right)$	1	4	0.62*	0.38
3	T2 (N = 50)	$\left(\begin{array}{ccc}\pm \alpha ,& \pm \alpha ,& \pm \alpha \end{array}\right)$	2	3	0.30	0.70
		$\left(\begin{array}{ccc}\pm 2\alpha ,& 0,& 0\end{array}\right)$	1	4	0.62*	0.38
		$\left(\begin{array}{ccc}\pm \alpha \sqrt{2},& \pm \alpha \sqrt{2},& 0\end{array}\right)$	1	4	0.62*	0.38
		$\left(\begin{array}{ccc}\pm \alpha \sqrt{2},& 0,& 0\end{array}\right)$	2	2	0.28	0.72
		$\left(\begin{array}{ccc}\text{0},& \text{0},& \text{0}\end{array}\right)$	4	0	0.15	0.85*
4	T3 (N = 52)	$\left(\begin{array}{ccc}\pm 0.5578\alpha ,& \pm 0.5578\alpha ,& \pm 0.5578\alpha \end{array}\right)$	1	0.93342	0.45	0.55
		$\left(\begin{array}{ccc}\pm 0.4097\alpha ,& \pm 0.4097\alpha ,& \pm 0.4097\alpha \end{array}\right)$	1	0.50356	0.37	0.63
		$\left(\begin{array}{ccc}\pm \alpha ,& 0,& 0\end{array}\right)$	1	1	0.54	0.46
		$\left(\begin{array}{ccc}\pm 0.7201\alpha ,& \pm 0.7201\alpha ,& 0\end{array}\right)$	1	1.03709	0.62*	0.38
		$\left(\begin{array}{ccc}\text{0},& \text{0},& \text{0}\end{array}\right)$	4	0	0.07	0.93*
		$\left(\begin{array}{ccc}\pm 0.8564\alpha ,& 0,& 0\end{array}\right)$	1	0.73342	0.33	0.67
5	T4 (N = 56)	$\left(\begin{array}{ccc}\pm 0.5878\alpha ,& \pm 0.5878\alpha ,& \pm 0.5878\alpha \end{array}\right)$	1	1.03653	0.53*	0.47
		$\left(\begin{array}{ccc}\pm 0.2739\alpha ,& \pm 0.2739\alpha ,& \pm 0.2739\alpha \end{array}\right)$	1	0.22506	0.38	0.62
		$\left(\begin{array}{ccc}\pm \alpha ,& 0,& 0\end{array}\right)$	1	1	0.46	0.54
		$\left(\begin{array}{ccc}\pm 0,& \pm 0.6609\alpha ,& \pm 0.6609\alpha \end{array}\right)$	1	0.87358	0.32	0.68
		$\left(\begin{array}{ccc}\pm 0.9347\alpha ,& 0,& 0\end{array}\right)$	1	0.87366	0.32	0.68
		$\left(\begin{array}{ccc}\text{0},& \text{0},& \text{0}\end{array}\right)$	4	0	0.11	0.89*
6	T5 (N = 54)	$\left(\begin{array}{ccc}\pm \alpha ,& \pm \alpha ,& 0\end{array}\right)$	1	2	0.37	0.63
		$\left(\begin{array}{ccc}\pm \alpha ,& 0,& 0\end{array}\right)$	2	1	0.27	0.73
		$\left(\begin{array}{ccc}\pm 0.9357\alpha ,& \pm 0.9357\alpha ,& 0\end{array}\right)$	1	1.75107	0.28	0.72
		$\left(\begin{array}{ccc}\pm 0.8646\alpha ,& \pm 0.8646\alpha ,& \pm 0.8646\alpha \end{array}\right)$	1	2.2426	0.52	0.48
		$\left(\begin{array}{ccc}\pm 1.5653\alpha ,& 0,& 0\end{array}\right)$	1	2.45016	0.71*	0.29
		$\left(\begin{array}{ccc}\text{0},& \text{0},& \text{0}\end{array}\right)$	4	0	0.13	0.87*
7	T6 (N = 60)	$\left(\begin{array}{ccc}\pm \alpha ,& \pm \alpha ,& 0\end{array}\right)$	1	2	0.27	0.73
		$\left(\begin{array}{ccc}\pm \alpha ,& 0,& 0\end{array}\right)$	2	1	0.25	0.75
		$\left(\begin{array}{ccc}\pm 1.4078\alpha ,& \pm 0.5112\alpha ,& \pm 0.5112\alpha \end{array}\right)$	1	2.50455	0.47*	0.53
		$\left(\begin{array}{ccc}\pm 0.7832\alpha ,& \pm 0.7832\alpha ,& \pm 0.7832\alpha \end{array}\right)$	1	1.84021	0.24	0.76
		$\left(\begin{array}{ccc}\text{0},& \text{0},& \text{0}\end{array}\right)$	4	0	0.12	0.88*
8	T7 (N = 60)	$\left(\begin{array}{ccc}\pm \alpha ,& \pm \alpha ,& 0\end{array}\right)$	1	2	0.23	0.77
		$\left(\begin{array}{ccc}\pm \alpha ,& 0,& 0\end{array}\right)$	2	1	0.18	0.82*
		$\left(\begin{array}{ccc}\text{0},& \text{0},& \text{0}\end{array}\right)$	4	0	0.08	0.92
		$\left(\begin{array}{ccc}\pm 1.5205\alpha ,& \pm 0.5980\alpha ,& \pm 0.5980\alpha \end{array}\right)$	1	3.02713	0.56*	0.44
		$\left(\begin{array}{ccc}\pm 0.5980\alpha ,& \pm 0.5980\alpha ,& \pm 0.5980\alpha \end{array}\right)$	1	1.07281	0.18	0.82
9	T8 (N = 60)	$\left(\begin{array}{ccc}\pm \alpha ,& \pm \alpha ,& 0\end{array}\right)$	1	2	0.25	0.75
		$\left(\begin{array}{ccc}\pm \alpha \sqrt{2},& 0,& 0\end{array}\right)$	1	2	0.24	0.76
		$\left(\begin{array}{ccc}\text{0},& \text{0},& \text{0}\end{array}\right)$	10	0	0.06	0.94*
		$\left(\begin{array}{ccc}\pm 1.5167\alpha ,& \pm 0.6037\alpha ,& \pm 0.6037\alpha \end{array}\right)$	1	3.02929	0.54*	0.46
		$\left(\begin{array}{ccc}\pm 0.5042\alpha ,& \pm 0.5042\alpha ,& \pm 0.5042\alpha \end{array}\right)$	1	0.76265	0.23	0.77
10	T9 (N = 58)	$\left(\begin{array}{ccc}\pm \alpha ,& \pm \alpha ,& 0\end{array}\right)$	1	2	0.49	0.51
		$\left(\begin{array}{ccc}\pm \alpha ,& 0,& 0\end{array}\right)$	2	1	0.28	0.72
		$\left(\begin{array}{ccc}\text{0},& \text{0},& \text{0}\end{array}\right)$	4	0	0.15	0.85*
		$\left(\begin{array}{ccc}\pm 0.9848\alpha ,& \pm 0.5748\alpha ,& \pm 0.5748\alpha \end{array}\right)$	1	1.63062	0.28	0.72
		$\left(\begin{array}{ccc}\pm 1.4453\alpha ,& 0,& 0\end{array}\right)$	1	2.08889	0.58*	0.42
11	Huda (N = 36)	$\left(\begin{array}{ccc}1.24693,& \pm 1.56366,& 0\end{array}\right)$	1	4	0.74	0.26
		$\left(\begin{array}{ccc}\text{ - 0}\text{.44504},& \pm \text{1}\text{.94986},& \text{0}\end{array}\right)$	1
		$\left(\begin{array}{ccc}\text{ - 1}\text{.80194},& \pm \text{0}\text{.86777},& \text{0}\end{array}\right)$	1
		$\left(\begin{array}{ccc}2,& \text{0},& \text{0}\end{array}\right)$	1
		$\left(\begin{array}{ccc}0.85439,& \pm \text{1}\text{.07137},& \text{0}\end{array}\right)$	1	1.8778	0.42	0.58*
		$\left(\begin{array}{ccc}\text{ - 0}\text{.30493},& \pm \text{1}\text{.33597},& \text{0}\end{array}\right)$	1
		$\left(\begin{array}{ccc}\text{ - 1}\text{.23462},& \pm \text{0}\text{.59456},& \text{0}\end{array}\right)$	1
		$\left(\begin{array}{ccc}1.37033,& \text{0},& \text{0}\end{array}\right)$	1
		$\left(\begin{array}{ccc}\pm 1.43761,& 0,& \pm 1.24500\end{array}\right)$	1	3.61676	0.55	0.45
		$\left(\begin{array}{ccc}0,& \pm 1.43761,& \pm 1.24500\end{array}\right)$	1	3.61676	0.55	0.45
		$\left(\begin{array}{ccc}\pm 1.01654,& \pm 1.01654,& \pm 1.24501\end{array}\right)$	1	3.61676	0.55	0.45
		$\left(\begin{array}{ccc}0,& 0,& \pm 2.00108\end{array}\right)$	1	4.00433	0.75*	0.25
		$\left(\begin{array}{ccc}0,& 0,& \pm 0.64054\end{array}\right)$	1	4	0.74	0.26

*Indicates maximum loss of information and maximum efficiency.

Table 6. Information loss and efficiency of SATORDs.

SATORD – CLASS I				SATORD – CLASS II
Runs	Radius (d^2)	L1	E1	Runs	Radius (d^2)	L1	E1
01–16	3	0.30	0.71	01–08	3	0.81*	0.19
17–22	4	0.62*	0.38	09–14	2.82842	0.81*	0.19
23–26	0	0.15	0.85	15–18	0	0.11	0.89*
27–38	4	0.62*	0.38	19–30	2	0.65	0.35
39–50	2	0.28	0.72	31–34	0	0.11	0.89*

*Indicates maximum loss of information and maximum efficiency.

Table 7. Robustness of ABBDs.

v	Total run (N)	ABBD	Runs in each set	Parameters	${\text{L}}_{\text{1}}$	${\text{E}}_{\text{1}}$
3	30	B[3]	12	d = 1.01 e = 1.7494	0.65	0.35
		F[d]^3	8		0.81	0.19
		A[e]^3	6		0.83*	0.17
		C[0]^3	4		0.20	0.80*
4	52	B[4]	24	d = 0.61 e = 1.22	0.77*	0.23
		F[d]^4	16		0.65	0.35
		A[e]^4	8		0.65	0.35
		C[0]^4	4		0.20	0.80*
5	86	B[5]	40	d = 0.57 e = 1.17	0.73*	0.27
		F[d]^5	32		0.62	0.38
		A[e]^5	10		0.61	0.39
		C[0]^5	4		0.19	0.81*
6	176	B[6]	48	d = 0.63 e = 1.65	0.54	0.46
		F[d]^6	64		0.37	0.63
		A[e]^6	12		0.67*	0.33
		B^'[6]	48		0.54	0.46
		C[0]^6	4		0.11	0.89*
7	186	B[7]	56	e = 2.00	0.61	0.39
		A[e]^7	14		0.68*	0.32
		B^'[7]	112		0.68*	0.32
		C[0]^7	4		0.15	0.85*

*Indicates maximum loss of information and maximum efficiency.

Table 8. Robustness of AFBBDs.

V	Total run (N)	AFBBD	Runs in each set	Parameter	${\text{L}}_{\text{1}}$	${\text{E}}_{\text{1}}$
3	24	BB[3]	6	d = 1	1.00*	0.00
		F[d]^3	8	e = √3	0.95	0.05
		A[e]^3	6		0.90	0.10
		C[0]^3	4		0.25	0.75*
4	40	BB[4]	12	d = 1	1.00*	0.00
		F[d]^4	16	e = √4	0.92	0.08
		A[e]^4	8		0.92	0.08
		C[0]^4	4		0.25	0.75*
5	66	BB[5]	20	d = 1	1.00*	0.00
		F[d]^5	32	e = √5	0.81	0.19
		A[e]^5	10		0.92	0.08
		C[0]^5	4		0.13	0.88*
6	128	BB[6]	24	d = 1	0.65	0.35
		F[d]^6	64	e = √6	0.61	0.39
		A[e]^6	12		0.93*	0.07
		B^'[6]	48		0.66	0.34
		C[0]^6	4		0.25	0.75*
7	158	BB[7]	28	d = 1	1.00*	0.00
		A[e]^7	14	e = √7	0.89	0.11
		B^'[7]	112		0.71	0.29
		C[0]^7	4		0.10	0.90*

*Indicates maximum loss of information and maximum efficiency.

Table 9. Robustness of ABBDs against two missing observations.

V	Total runs (N)	ABBD	Missing two observations at different sets of design points	L2	E2
3	30	B[3] F[1.01]^3 A[1.7494]^3 C[0]^3	B[3],B[3]	0.83	0.17
			F[1.01], F[1.01]	0.96	0.04
			A[1.7494], A[1.7494]	1.00*	0.00
			C[0], C[0]	0.33	0.67*
			B[3], F[1.01]	0.94	0.06
			B[3], A[1.7494]	0.94	0.06
			F[1.01], A[1.7494]	0.97	0.03
			B[3], C[0]	0.70	0.30
			F[1.01], C[0]	0.84	0.16
			A[1.7494], C[0]	0.86	0.14
4	52	B[4] F[0.61]^4 A[1.22]^4 C[0]^4	B[4],B[4]	0.95	0.05
			F[0.61], F[0.61]	0.88	0.12
			A[1.22], A[1.22]	1.00*	0.00
			C[0], C[0]	0.40	0.60*
			B[4], F[0.61]	0.93	0.07
			B[4], A[1.22]	0.93	0.07
			B[4], C[0]	0.82	0.18
			F[0.61], A[1.22]	0.88	0.12
			F[0.61], C[0]	0.72	0.28
			A[1.22], C[0]	0.72	0.28

*Indicates maximum loss of information and maximum efficiency.