Design of experiment methodology

Design of Experiment
methodology
ANOVA、FFD、CCD、Mixture Design
Response surface design
Kung, Chun-Hao
kch800721@gmail.com
Chemical engineering
department in NCKU

Central composite design
methods(CCD)
Mixture design
methods

Know
situation
Design of
experiment
Optimized
process
Flow chart

• Help you better understand and optimize your
response.
• Used to refine models after you have determined
important factors using factorial designs
Advantages of Response surface design

Factorial Points : Estimated main factor & interaction
Axial Points : Estimated pure quadratic form
Center Points : Estimated pure Error
→ Building a quadratic response surface
→ Resolves both main effects and interactions
Central composite design (CCD)

Level Temperature (℃) Annealing time (mins)
𝟐 120 5.0
1 115 6.5
0 100 10.0
-1 85 13.5
- 𝟐 80 15.0
8
Run Temp. Time Temp. Time
1 -1 -1 85 13.5
2 -1 1 85 6.5
3 1 -1 115 13.5
4 1 1 115 6.5
5 0 0 100 10
6 0 0 100 10
7 0 0 100 10
8 0 - 𝟐 100 15
9 0 𝟐 100 5
10 - 𝟐 0 80 10
11 𝟐 0 120 10
Design matrix
Reference: Michael Grätzel, Advanced Functional Materials, 24, 3250(2014)
Effect of Annealing Temperature on Film Morphology of Organic–Inorganic
Hybrid Pervoskite Solid-State Solar Cells
120℃
5 mins
15 mins
8𝟎℃
100℃
− 2: 100 − 115 − 100 × 1.414 =80
2: 100 + 115 − 100 × 1.414 = 120

run Temp. Time Voc (V) Jsc (mA/cm2) FF PCE (%)
1 80 10 0.77 7.07 0.72 3.89
2 85 13.5 0.72 10.30 0.59 4.44
3 85 6.5 0.25 12.98 0.37 1.23
4 100 15 0.78 13.86 0.72 7.77
5 100 10 0.78 11.99 0.73 6.78
6 100 5 0.81 6.63 0.73 3.92
7 115 13.5 0.71 12.80 0.66 5.99
8 115 6.5 0.72 13.18 0.71 6.72
9 120 10 0.71 11.42 0.68 5.50
Origin data-CCD

SAS-ANOVA
Source:11
DF: 10
變異數分析
來源 DF 和平方平均值平方 F 值 Pr > F
模型 5 31.3531
0
6.27062 6.47 0.0306
誤差 5 4.84396 0.96879
已校正的
總計
10 36.1970
5
根 MSE 0.98427 R 平方 0.8662
應變平均值 5.43636 調整 R 平方 0.7324
變異係數 18.10534
Set regression equation
(model y1=x1 x2 t1 t2 t3 /noint selection=forward;)Commands :
proc : procedure
reg : regression
anova : calculate ANOVA
Y = a + bX1+c x2 + dx1
2+ ex2
2+ fx1x2
Parameters:11

參數估計值
變數 DF 參數
估計
標準
誤差
t 值 Pr > |t|
Intercept 1 6.78014 0.56827 11.93 <.0001
x1 1 1.16474 0.34802 3.35 0.0204
x2 1 -0.99064 0.34802 -2.85 0.0360
t1 1 -1.21157 0.41428 -2.92 0.0328
t2 1 -0.63640 0.41428 -1.54 0.1851
t3 1 0.98500 0.49214 2.00 0.1017
z=6.78+1.16*x-0.99*y-1.21*x2-0.63*y2+0.98.*x*y
Regression of PCE (%)
[x,y] = meshgrid(-2:0.01:2);
z=6.78014+(1.16474.*x)-(0.99064.*y)-(1.21157.*(x.^2))-(0.63640.*(y.^2))+(0.98500.*x.*y);
[C,h] = contour(x,y,z, [1,2,3,4,5,5.5,6,6.5,7,7.2]);
axis([-1.5,1.5,-1.5,1.5])
clabel(C,h);

1
1
2
2
3
3
4
4
4
4
5
5
5
5
5.5
5.5
5.5
5.5 5.5
5.5
6
6
6
6
6
6
6.5
6.5
6.5
6.5
6.5
7
7
7
7
7.2
Temperatuer(degree)
Annealingtime(mins)
85 100 115
6.5
10
13.5
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
12
Parameter Coefficient
Xtemp
1.16
Ytime
-0.99
XtempXtemp
-1.21
YtimeYtime
-0.63
XtempYtime
0.99
Main factor:
Interaction: Temp.-Time
90℃ 110℃
z=6.78+1.16*x-0.99*y-1.21*x2-0.63*y2+0.98.*x*y
>
Regression of PCE (%)

14
 Model is fixed
 Algebra equation
 Time-consuming
 Model reduction
 SAS regression
 Interaction effect
 Save time
0.00 0.25 0.50 0.75 1.00
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
IBr
Cl
Binary Design (A) Ternary Design (B)
Modified mixture design methods

Advantages of mixture design
• Designs for these experiments are useful
because many product design and development
activities in industrial situations involve
formulations or mixtures.

Statics and regression
examples
16

17
Mixture design methodology
RegressionExperimental data Contour plot1. 2. 3.
SAS 9.3 MATLAB R2013a
0.00 0.25 0.50 0.75 1.00
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
IBr
Cl

Origin data
Ratio MACl MABr MAI Voc(V) Jsc (mA/cm2
) FF (%) PCE (%)
1 1 0 0 0.76 10.31 69% 5.42
2 0 1 0 0.97 5.54 71% 3.81
3 0 0 1 0.78 9.76 70% 5.38
4 0.33 0.333 0.333 0.86 11.54 66% 6.54
5 0.5 0.5 0 0.97 4.76 70% 3.23
6 0 0.5 0.5 0.92 5.97 66% 3.58
7 0.5 0 0.5 0.71 12.10 72% 6.12
8 0.67 0.17 0.17 0.73 12.37 71% 6.38
9 0.17 0.67 0.17 0.97 10.66 64% 6.54
10 0.17 0.17 0.67 0.85 11.46 77% 7.51
11 0.75 0.25 0 0.90 7.43 64% 4.32
12 0.25 0.75 0 0.96 7.68 57% 4.15
13 0.25 0 0.75 0.78 13.06 68% 6.89
14 0.75 0 0.25 0.80 9.93 72% 5.68
15 0 0.75 0.25 0.99 10.81 72% 7.71
16 0 0.25 0.75 0.82 10.05 73% 6.01

Forward selection
SAS Regression
run;
proc reg;
model y1=t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13
/noint selection=forward;
proc anova;
前進選擇: 步驟 1 R 平方 = 0.6006 和 C(p) = 57.8286
前進選擇: 步驟 2 R 平方 = 0.8367 和 C(p) = 17.3612
前進選擇: 步驟 7 R 平方 = 0.9792 和 C(p) = 1.7387
Click run
Model reduction

X1 t1
X2 t2
X3 t3
X1X2 t4
X1X3 t5
X2X3 t6
X1X2X3 t7
X1X2(X1-X2) t8
X1X3(X1-X3) t9
X2X3(X2-X3) t10
X1X1X2X3 t11
X1X2X2X3 t12
X1X2X3X3 t13
SAS Regression-13 parameters
變異數分析
來源 DF R
2
平均值平方 F 值 Pr > F
模型 6 515.7 85.9 73.60 <.0001
誤差 10 11.8 1.2
未校正的總計 16 527.4
變
數
參數
估計
標準
誤差
第二型 SS F 值 Pr > F
t1 5.57799 0.77588 60.35377 51.69 <.0001
t2 4.53861 0.83794 34.25800 29.34 0.0003
t3 6.53751 0.73561 92.22951 78.98 <.0001
t4 -5.83005 4.15712 2.29666 1.97 0.1911
t7 64.07944 25.58609 7.32434 6.27 0.0312
t10 11.75880 7.90392 2.58452 2.21 0.1677
R 平方 = 0.9779 和 C(p) = -0.0183

X1 t1
X2 t2
X3 t3
X1X2 t4
X1X3 t5
X2X3 t6
X1X2X3 t7
X1X1X2X3 t11
X1X2X2X3 t12
X1X2X3X3 t13
SAS Regression-10 parameters
變異數分析
來源 DF 和
平方
平均值
平方
F 值 Pr > F
模型 6 513.8 85.6 63.28 <.0001
誤差 10 13.53 1.35
未校正的總計 16 527.4
變
數
參數
估計
標準
誤差
第二型 SS F 值 Pr > F
t1 5.26373 1.01318 36.52686 26.99 0.0004
t2 5.09614 0.83751 50.10738 37.03 0.0001
t3 5.81178 0.83751 65.16845 48.15 <.0001
t4 -6.08694 4.54785 2.42432 1.79 0.2104
t5 3.33659 4.54785 0.72845 0.54 0.4800
t7 59.03570 29.04160 5.59231 4.13 0.0695
R 平方 = 0.9743 和 C(p) = 2.7857

0
0.25
0.5
0.75
1 0
0.25
0.5
0.75
1
Cl
Br I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
7.4
7.2
7
6.5
6
5
6 6.5
7
5
6.5
6
0
0.25
0.5
0.75
1 0
0.25
0.5
0.75
1
Cl
Br I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
R 平方 = 0.9779 和 C(p) = -0.0183 R 平方 = 0.9743 和 C(p) = 2.7857
Different parameters compared
13 parameters 10 parameters

Contour plot- e.g. PCE
A=tril(meshgrid(0:0.001:1));
B=tril(meshgrid(1:-0.001:0)');
C=tril(1-A-B);
x=tril(0.5.*(1+C-B));
y=tril((3^0.5)*0.5.*A);
z=5.57799.*A +4.53861.*B+6.53751.*C -5.83005.*A.*B
+64.07944.*A.*B.*C+11.75880.*B.*C.*(B-C);
[C,h] = contourf(x,y,z,
[1,2,3,4,5,6,6.5,7,7.2,7.4,7.6],'LineWidth',1);
axis([0,1,0,1]);
clabel(C,h,'manual','fontsize',15);
hold on
plot([0.375,0.625],[0.6495,0.6495],'k:');
hold on
plot([0.25,0.75],[0.433,0.433],'k:');
hold on
plot([0.375,0.75],[0.6495,0],'k:');
hold on
plot([0.25,0.5],[0.433,0],'k:');
hold on
plot([0.125,0.25],[0.2165,0],'k:');
hold on
plot([0.125,0.875],[0.2165,0.2165],'k:');
hold on
plot([0.25,0.625],[0,0.6495],'k:');
hold on
plot([0.50,0.75],[0,0.433],'k:');
hold on
plot([0.75,0.875],[0,0.2165],'k:');
先建立矩
陣數列，
從0~1，
間格0.001
利用SAS迴歸得到的Eq.

A=tril(meshgrid(0:0.001:1)) B=tril(meshgrid(1:-0.001:0)');
x=tril(0.5.*(1+C-B)); y=tril((3^0.5)*0.5.*A);
C=tril(1-A-B);
z=5.58.*A +4.54.*B+6.54.*C -
5.83.*A.*B+64.08.*A.*B.*C+11.76.*B.*C.
*(B-C);
e.g. 0.5*(1+0.001-0.998) e.g. 3*0.5*0.001
Function

0
0.25
0.5
0.75
1 0
0.25
0.5
0.75
1
Cl
Br I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Contour plot- e.g. PCE
[C,h] = contourf(x,y,z, [1,2,3,4,5,6,6.5,7,7.2,7.4,7.6],'LineWidth',1);
axis([0,1,0,1]);

7
8
10
9
11
12
12.5
12.7
12
11
10
8
11
10
0
0.25
0.5
0.75
1 0
0.25
0.5
0.75
1
Cl
Br I
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
5
6
7
8
9
10
11
12
1
0.98
0.94
0.9
0.85
0.8
0.75
0.85
0.8
0
0.25
0.5
0.75
1 0
0.25
0.5
0.75
1
Cl
Br I
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.7
0.75
0.8
0.85
0.9
0.95
1
0.6
0.65
0.68
0.7
0.75
0.8
0.850.9
0.55
0.7
0.7
0
0.25
0.5
0.75
1 0
0.25
0.5
0.75
1
Cl
Br I
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.6
0.6
0.65
0.65
0.68
0.68
0.7
0.75
0.8
0.85
0.9
0.75
0.7
0.55
0
0.25
0.5
0.75
1 0
0.25
0.5
0.75
1
IBr
Cl
5
6
6.5
7
7.2
7.2
7
6.5
6
5
6
6.5
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
2
3
4
5
6
7
JscVoc FF PCE
PCE=5.6*A +4.5*B+6.5*C -5.8*A*B+64.1*A*B*C+11.8*B*C*(B-C);
SAS Regression
Jsc=9.44 X1+6.58 X2+10.75 X3-6.69 X1X2+8.07 X1X3+91.79 X1X2X3+17.10X2X3(X2-X3)
Voc=0.76 X1+0.97 X2+0.77 X3+0.38 X1X2+0.20 X2X3+101.67X1X2X3+0.23X1X2(X1-X2)+0.38
X2X3(X2-X3)-112.78 X12X2X3-99.91 X1X2
2X3-95.62 X1X2X3
2
FF=0.70 X1+0.70 X2+0.71 X3-0.26 X1X2+0.36 X1X2(X1-X2)-3.92 X1X2
2X3+4.90X1X2X3
2

1
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3
4
4
5
5
5
5
6
6
6
6
6
6.5
6.5
6.5
6.5
6.5
7
7
7
7.2
7.2
0
0.25
0.5
0.75
0
0.25
0.5
0.75
1
I
0 0.25 0.5 0.75 1
27
0.0 0.2 0.4 0.6 0.8 1.0
-16
-14
-12
-10
-8
-6
-4
-2
0
MAPbCl0.30
Br0.15
I0.55
MAPbCl0.30
Br0.35
I0.30
MAPbCl0.30
Br0.55
I0.15
Currentdensity(mA/cm
2
)
Voltage (V)
PCE Contour plot
run Cl Br I Voc Jsc FF PCE
0.30 0.55 0.15 0.97 9.94 0.70 6.77
0.30 0.35 0.35 0.92 14.07 0.73 9.47
0.30 0.15 0.55 0.75 12.67 0.68 6.52
Cl
IBr
Verification by J-V curve

FFD
(Full Factorial Design )
28

Factors and levels for the 23 Full Factorial Design
factors
levels
+ -
A, P3HT:PCBM concentration
(wt%)
2.5 1.5
B, rpm 600 1000
C, time (s) 60 40
Full Factorial Design

1.5wt% 600rpm 40s 600rpm 60s 1000rpm 40s 1000rpm 60s
VOC (V) 0.04 0.60 0.70 0.72
JSC (mA/cm2
) 2.94 7.22 1.21 1.03
FF 0.29 0.32 0.32 0.30
PCE (%) 0.03 2.83 0.27 0.22
R.P (Ω･cm2
)104
0.00083 278.8 71.68 99.72
R.S (Ω･cm2
) 1.88 2.23 7.36 1.15
Run A B C Voc (V) Jsc (mA/cm2) FF PCE (%)
1 - + - 0.04 2.94 0.29 0.03
2 - + + 0.60 7.22 0.65 2.83
3 - - - 0.70 1.21 0.32 0.27
4 - - + 0.72 1.03 0.30 0.22
factors + -
A, concentration (wt%) 2.5 1.5
B, revolution (rpm) 600 1000
C , time (s) 60 40

2.5wt% 600rpm 40s 600rpm 60s 1000rpm 40s 1000rpm 60s
VOC (V) 0.60 0.58 0.60 0.70
JSC (mA/cm2
) 7.54 9.11 9.40 1.90
FF 0.54 0.60 0.63 0.40
PCE (%) 2.46 3.18 3.53 0.53
R.P (Ω･cm2
)104
0.88 40.91 143.42 168.52
R.S (Ω･cm2
) 2.05 2.80 2.27 7.39
5 + + - 0.60 7.54 0.54 2.46
6 + + + 0.58 9.11 0.60 3.18
7 + - - 0.60 9.40 0.63 3.53
8 + - + 0.70 1.90 0.40 0.53
factors + -
C , time (s) 60 40

Run A B C Voc (V) Jsc (mA/cm2) FF PCE (%)
1 - + - 0.04 2.94 0.29 0.03
2 - + + 0.60 7.22 0.65 2.83
3 - - - 0.70 1.21 0.32 0.27
4 - - + 0.72 1.03 0.30 0.22
5 + + - 0.60 7.54 0.54 2.46
6 + + + 0.58 9.11 0.60 3.18
7 + - - 0.60 9.40 0.63 3.53
8 + - + 0.70 1.90 0.40 0.53
factors + -
C , time (s) 60 40
Design matrix

Run A B C AB BC AC ABC PCE
1 - + - - - + + 0.03
2 - + + - + - - 2.83
3 - - - + + + - 0.27
4 - - + + - - + 0.22
5 + + - + - - - 2.46
6 + + + + + + + 3.18
7 + - - - + - + 3.53
8 + - + - - + - 0.53
effect estimate
A 1.59
B 0.99
C 0.12
AB -0.20
BC 1.64
AC -1.26
ABC 0.22
The estimate of A =
(
𝟐.𝟒𝟔+𝟑.𝟏𝟖+𝟑.𝟓𝟑+𝟎.𝟓𝟑
𝟒
) - (
𝟎.𝟎𝟑+𝟐.𝟖𝟑+𝟎.𝟐𝟕+𝟎.𝟐𝟐
𝟒
) = 1.59
Calculate the estimate

Design of experiment methodology

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