Post on 20-Oct-2014
description
1
Chapter 6 The 2k Factorial Design
2
6.1 Introduction
• The special cases of the general factorial design • k factors and each factor has only two levels• Levels:
– quantitative (temperature, pressure,…), or qualitative (machine, operator,…)
– High and low– Each replicate has 2 2 = 2k observations
3
• Assumptions: (1) the factor is fixed, (2) the design is completely randomized and (3) the usual normality assumptions are satisfied
• Wildly used in factor screening experiments
4
6.2 The 22 Factorial Design • Two factors, A and B, and each factor has two
levels, low and high.• Example: the concentration of reactant v.s. the
amount of the catalyst (Page 219)
5
• “-” And “+” denote the low and high levels of a factor, respectively
• Low and high are arbitrary terms
• Geometrically, the four runs form the corners of a square
• Factors can be quantitative or qualitative, although their treatment in the final model will be different
6
• Average effect of a factor = the change in response produced by a change in the level of that factor averaged over the levels if the other factors.
• (1), a, b and ab: the total of n replicates taken at the treatment combination.
• The main effects:
AAyy
n
b
n
aab
baabn
ababn
A
2
)1(
2
)]1([2
1)]}1([]{[
2
1
BByy
n
a
n
bab
ababn
baabn
B
2
)1(
2
)]1([2
1)]}1([]{[
2
1
7
• The interaction effect:
• In that example, A = 8.33, B = -5.00 and AB = 1.67
• Analysis of Variance• The total effects:
n
ab
n
ab
baabn
ababn
AB
22
)1(
])1([2
1)]}1([]{[
2
1
baabContrast
ababContrast
baabContrast
AB
B
A
)1(
)1(
)1(
8
• Sum of squares:
ABBATE
i j
n
kijkT
AB
B
A
SSSSSSSSSS
n
yySS
n
ababSS
n
ababSS
n
baabSS
4
4
])1([
4
)]1([
4
)]1([
22
1
2
1 1
2
2
2
2
9
Response:Conversion ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]
Sum of Mean FSource Squares DF Square Value Prob > FModel 291.67 3 97.22 24.82 0.0002A 208.33 1 208.33 53.19 < 0.0001B 75.00 1 75.00 19.15 0.0024AB 8.33 1 8.33 2.13 0.1828Pure Error 31.33 8 3.92Cor Total 323.00 11
Std. Dev. 1.98 R-Squared 0.9030Mean 27.50 Adj R-Squared 0.8666C.V. 7.20 Pred R-Squared 0.7817
PRESS 70.50 Adeq Precision 11.669
The F-test for the “model” source is testing the significance of the overall model; that is, is either A, B, or AB or some combination of these effects important?
10
• Table of plus and minus signs:
I A B AB
(1) + – – +
a + + – –
b + – + –
ab + + + +
11
• The regression model:
– x1 and x2 are coded variables that represent the
two factors, i.e. x1 (or x2) only take values on –
1 and 1.– Use least square method to get the estimations
of the coefficients– For that example,
– Model adequacy: residuals (Pages 224~225) and normal probability plot (Figure 6.2)
22110 xxy
21 2
00.5
2
33.85.27ˆ xxy
12
6.3 The 23 Design
• Three factors, A, B and C, and each factor has two levels. (Figure 6.4 (a))
• Design matrix (Figure 6.4 (b))• (1), a, b, ab, c, ac, bc, abc• 7 degree of freedom: main effect = 1, and
interaction = 1
13
14
• Estimate main effect:
• Estimate two-factor interaction: the difference between the average A effects at the two levels of B
])1([4n
1
4
)1(
4
abcacaba
])1([4
1
bccbabcacaba
n
bccb
n
yy
bcabccacbaban
A
AA
n
aacbbc
n
cababc
acacbabbcabcn
AB
44
)1(
)]1([4
1
15
• Three-factor interaction:
• Contrast: Table 6.3– Equal number of plus and minus– The inner product of any two columns = 0– I is an identity element– The product of any two columns yields another
column– Orthogonal design
• Sum of squares: SS = (Contrast)2/8n
)]1([4n
1
)]}1([][][]{[4
1
ababcacbcabc
ababcacbcabcn
ABC
16
Factorial Effect
TreatmentCombination
I A B AB C AC BC ABC
(1) + – – + – + + –
a + + – – – – + +
b + – + – – + – +
ab + + + + – – – –
c + – – + + – – +
ac + + – – + + – –
bc + – + – + – + –
abc + + + + + + + +Contrast 24 18 6 14 2 4 4
Effect 3.00 2.25 0.75 1.75 0.25 0.50 0.50
Table of – and + Signs for the 23 Factorial Design (pg. 231)
17
• Example 6.1
A = carbonation, B = pressure, C = speed, y = fill deviation
18
Term Effect SumSqr % ContributionModel InterceptError A 3 36 46.1538Error B 2.25 20.25 25.9615Error C 1.75 12.25 15.7051Error AB 0.75 2.25 2.88462Error AC 0.25 0.25 0.320513Error BC 0.5 1 1.28205Error ABC 0.5 1 1.28205Error LOF 0Error P Error 5 6.41026
Lenth's ME 1.25382 Lenth's SME 1.88156
• Estimation of Factor Effects
19
• ANOVA Summary – Full Model
Response:Fill-deviation ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]
Sum of Mean FSource Squares DF Square Value Prob > FModel 73.00 7 10.43 16.69 0.0003A 36.00 1 36.00 57.60 < 0.0001B 20.25 1 20.25 32.40 0.0005C 12.25 1 12.25 19.60 0.0022AB 2.25 1 2.25 3.60 0.0943AC 0.25 1 0.25 0.40 0.5447BC 1.00 1 1.00 1.60 0.2415ABC 1.00 1 1.00 1.60 0.2415Pure Error 5.00 8 0.63Cor Total 78.00 15
Std. Dev. 0.79 R-Squared 0.9359Mean 1.00 Adj R-Squared 0.8798C.V. 79.06 Pred R-Squared 0.7436
PRESS 20.00 Adeq Precision 13.416
20
• The regression model and response surface:– The regression model:
– Response surface and contour plot (Figure 6.7)
21321 2
75.0
2
75.1
2
25.2
2
00.300.1ˆ xxxxxy
Coefficient Standard 95% CI 95% CI
Factor Estimate DF Error Low High Intercept 1.00 1 0.20 0.55 1.45 A-Carbonation 1.50 1 0.20 1.05 1.95 B-Pressure 1.13 1 0.20 0.68 1.57 C-Speed 0.88 1 0.20 0.43 1.32 AB 0.38 1 0.20 -0.072 0.82
21
• Contour & Response Surface Plots – Speed at the High Level
DESIGN-EXPERT Plot
Fill-deviationX = A: CarbonationY = B: Pressure
Design Points
Actual FactorC: Speed = 250.00
Fill-deviation
A: Carbonation
B: P
res
su
re
10.00 10.50 11.00 11.50 12.00
25.00
26.25
27.50
28.75
30.00
0.5
1.375
2.25
3.125
2 2
2 2
DESIGN-EXPERT Plot
Fill-deviationX = A: CarbonationY = B: Pressure
Actual FactorC: Speed = 250.00
-0.375
0.9375
2.25
3.5625
4.875
F
ill-
de
via
tio
n
10.00
10.50
11.00
11.50
12.00
25.00
26.25
27.50
28.75
30.00
A: Carbonation B: Pressure
22
• Refine Model – Remove Nonsignificant Factors
Response: Fill-deviation ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]
Sum of Mean FSource Squares DF Square Value Prob > FModel 70.75 4 17.69 26.84 < 0.0001A 36.00 1 36.00 54.62 < 0.0001B 20.25 1 20.25 30.72 0.0002C 12.25 1 12.25 18.59 0.0012AB 2.25 1 2.25 3.41 0.0917Residual 7.25 11 0.66LOF 2.25 3 0.75 1.20 0.3700Pure E 5.00 8 0.63C Total 78.00 15
Std. Dev. 0.81 R-Squared 0.9071Mean 1.00 Adj R-Squared 0.8733C.V. 81.18 Pred R-Squared 0.8033
PRESS 15.34 Adeq Precision 15.424
23
6.4 The General 2k Design • k factors and each factor has two levels• Interactions• The standard order for a 24 design: (1), a, b, ab, c,
ac, bc, abc, d, ad, bd, abd, cd, acd, bcd, abcd
two-factor interactions2
three-factor interactions3
1 factor interaction
k
k
k
24
• The general approach for the statistical analysis:– Estimate factor effects– Form initial model (full model)– Perform analysis of variance (Table 6.9)– Refine the model– Analyze residual– Interpret results
•
2
...
)(2
12
2
)1()1)(1(
KABCkKABC
KABCk
KABC
Contrastn
SS
Contrastn
KABC
kbaContrast
25
6.5 A Single Replicate of the 2k Design• These are 2k factorial designs
with one observation at each corner of the “cube”
• An unreplicated 2k factorial design is also sometimes called a “single replicate” of the 2k
• If the factors are spaced too closely, it increases the chances that the noise will overwhelm the signal in the data
26
• Lack of replication causes potential problems in statistical testing– Replication admits an estimate of “pure error”
(a better phrase is an internal estimate of error)
– With no replication, fitting the full model results in zero degrees of freedom for error
• Potential solutions to this problem– Pooling high-order interactions to estimate
error (sparsity of effects principle)– Normal probability plotting of effects
(Daniels, 1959)
27
• Example 6.2 (A single replicate of the 24 design)– A 24 factorial was used to investigate the effects
of four factors on the filtration rate of a resin– The factors are A = temperature, B = pressure,
C = concentration of formaldehyde, D= stirring rate
28
29
• Estimates of the effects
Term Effect SumSqr % ContributionModel InterceptError A 21.625 1870.56 32.6397Error B 3.125 39.0625 0.681608Error C 9.875 390.062 6.80626Error D 14.625 855.563 14.9288Error AB 0.125 0.0625 0.00109057Error AC -18.125 1314.06 22.9293Error AD 16.625 1105.56 19.2911Error BC 2.375 22.5625 0.393696Error BD -0.375 0.5625 0.00981515Error CD -1.125 5.0625 0.0883363Error ABC 1.875 14.0625 0.245379Error ABD 4.125 68.0625 1.18763Error ACD -1.625 10.5625 0.184307Error BCD -2.625 27.5625 0.480942Error ABCD 1.375 7.5625 0.131959
Lenth's ME 6.74778 Lenth's SME 13.699
30
• The normal probability plot of the effectsDESIGN-EXPERT PlotFiltration Rate
A: TemperatureB: PressureC: ConcentrationD: Stirring Rate
Normal plot
No
rma
l % p
rob
ab
ility
Effect
-18.12 -8.19 1.75 11.69 21.62
1
5
10
20
30
50
70
80
90
95
99
A
CD
AC
AD
31
DESIGN-EXPERT Plot
Filtration Rate
X = A: TemperatureY = C: Concentration
C- -1.000C+ 1.000
Actual FactorsB: Pressure = 0.00D: Stirring Rate = 0.00
C: ConcentrationInteraction Graph
Filt
ratio
n R
ate
A: Temperature
-1.00 -0.50 0.00 0.50 1.00
41.7702
57.3277
72.8851
88.4426
104
DESIGN-EXPERT Plot
Filtration Rate
X = A: TemperatureY = D: Stirring Rate
D- -1.000D+ 1.000
Actual FactorsB: Pressure = 0.00C: Concentration = 0.00
D: Stirring RateInteraction Graph
Filt
ratio
n R
ate
A: Temperature
-1.00 -0.50 0.00 0.50 1.00
43
58.25
73.5
88.75
104
32
• B is not significant and all interactions involving B are negligible
• Design projection: 24 design => 23 design in A,C and D
• ANOVA table (Table 6.13)
33
Response:Filtration Rate ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]
Sum of Mean FSource Squares DF Square Value Prob >FModel 5535.81 5 1107.16 56.74 < 0.0001A 1870.56 1 1870.56 95.86 < 0.0001C 390.06 1 390.06 19.99 0.0012D 855.56 1 855.56 43.85 < 0.0001AC 1314.06 1 1314.06 67.34 < 0.0001AD 1105.56 1 1105.56 56.66 < 0.0001Residual 195.12 10 19.51Cor Total 5730.94 15
Std. Dev. 4.42 R-Squared 0.9660Mean 70.06 Adj R-Squared 0.9489C.V. 6.30 Pred R-Squared 0.9128
PRESS 499.52 Adeq Precision 20.841
34
• The regression model:
• Residual Analysis (P. 251)• Response surface (P. 252)
Final Equation in Terms of Coded Factors:
Filtration Rate =+70.06250+10.81250 * Temperature+4.93750 * Concentration+7.31250 * Stirring Rate-9.06250 * Temperature * Concentration+8.31250 * Temperature * Stirring Rate
35
DESIGN-EXPERT PlotFiltration Rate
Studentized Residuals
No
rma
l % p
rob
ab
ility
Normal plot of residuals
-1.83 -0.96 -0.09 0.78 1.65
1
5
10
20
30
50
70
80
90
95
99
36
• Half-normal plot: the absolute value of the effect estimates against the cumulative normal probabilities.
DESIGN-EXPERT PlotFiltration Rate
A: TemperatureB: PressureC: ConcentrationD: Stirring Rate
Half Normal plot
Ha
lf N
orm
al %
pro
ba
bility
|Effect|
0.00 5.41 10.81 16.22 21.63
0
20
40
60
70
80
85
90
95
97
99
A
CD
AC
AD
37
• Example 6.3 (Data transformation in a Factorial Design)
A = drill load, B = flow, C = speed, D = type of mud, y = advance rate of the drill
38
• The normal probability plot of the effect estimates
DESIGN-EXPERT Plotadv._rate
A: loadB: flowC: speedD: mud
Half Normal plot
Ha
lf N
orm
al %
pro
ba
bili
ty
|Effect|
0.00 1.61 3.22 4.83 6.44
0
20
40
60
70
80
85
90
95
97
99
B
C
D
BCBD
39
• Residual analysisDESIGN-EXPERT Plotadv._rate
Residual
No
rma
l % p
rob
ab
ility
Normal plot of residuals
-1.96375 -0.82625 0.31125 1.44875 2.58625
1
5
10
20
30
50
70
80
90
95
99
DESIGN-EXPERT Plotadv._rate
Predicted
Re
sid
ua
ls
Residuals vs. Predicted
-1.96375
-0.82625
0.31125
1.44875
2.58625
1.69 4.70 7.70 10.71 13.71
40
• The residual plots indicate that there are problems with the equality of variance assumption
• The usual approach to this problem is to employ a transformation on the response
• In this example, yy ln*
41
DESIGN-EXPERT PlotLn(adv._rate)
A: loadB: flowC: speedD: mud
Half Normal plotH
alf
No
rma
l % p
rob
ab
ility
|Effect|
0.00 0.29 0.58 0.87 1.16
0
20
40
60
70
80
85
90
95
97
99
B
C
D
Three main effects are large
No indication of large interaction effects
What happened to the interactions?
42
Response: adv._rate Transform: Natural log Constant: 0.000
ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]Sum of Mean F
Source Squares DF Square Value Prob > FModel 7.11 3 2.37 164.82 < 0.0001B 5.35 1 5.35 371.49 < 0.0001C 1.34 1 1.34 93.05 < 0.0001D 0.43 1 0.43 29.92 0.0001Residual 0.17 12 0.014Cor Total 7.29 15
Std. Dev. 0.12 R-Squared 0.9763Mean 1.60 Adj R-Squared 0.9704C.V. 7.51 Pred R-Squared 0.9579
PRESS 0.31 Adeq Precision 34.391
43
• Following Log transformation
Final Equation in Terms of Coded Factors:
Ln(adv._rate) =+1.60+0.58 * B+0.29 * C+0.16 * D
44
DESIGN-EXPERT PlotLn(adv._rate)
Residual
No
rma
l % p
rob
ab
ility
Normal plot of residuals
-0.166184 -0.0760939 0.0139965 0.104087 0.194177
1
5
10
20
30
50
70
80
90
95
99
DESIGN-EXPERT PlotLn(adv._rate)
PredictedR
es
idu
als
Residuals vs. Predicted
-0.166184
-0.0760939
0.0139965
0.104087
0.194177
0.57 1.08 1.60 2.11 2.63
45
• Example 6.4:– Two factors (A and D) affect the mean number
of defects– A third factor (B) affects variability– Residual plots were useful in identifying the
dispersion effect– The magnitude of the dispersion effects:
– When variance of positive and negative are equal, this statistic has an approximate normal distribution
)(
)(ln
2
2*
iS
iSFi
46
6.6 The Addition of Center Points to the 2k Design • Based on the idea of replicating some of the runs
in a factorial design• Runs at the center provide an estimate of error and
allow the experimenter to distinguish between two possible models:
01 1
20
1 1 1
First-order model (interaction)
Second-order model
k k k
i i ij i ji i j i
k k k k
i i ij i j ii ii i j i i
y x x x
y x x x x
47
no "curvature"F Cy y
The hypotheses are:
01
11
: 0
: 0
k
iii
k
iii
H
H
2
Pure Quad
( )F C F C
F C
n n y ySS
n n
This sum of squares has a single degree of freedom
48
• Example 6.6
5Cn
Usually between 3 and 6 center points will work well
Design-Expert provides the analysis, including the F-test for pure quadratic curvature
49
Response: yield ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]
Sum of Mean FSource Squares DF Square Value Prob > FModel 2.83 3 0.94 21.92 0.0060A 2.40 1 2.40 55.87 0.0017B 0.42 1 0.42 9.83 0.0350AB 2.500E-003 1 2.500E-003 0.058 0.8213Curvature 2.722E-003 1 2.722E-003 0.063 0.8137Pure Error 0.17 4 0.043Cor Total 3.00 8
Std. Dev. 0.21 R-Squared 0.9427Mean 40.44 Adj R-Squared 0.8996
C.V. 0.51 Pred R-Squared N/A
PRESS N/A Adeq Precision 14.234
50
• If curvature is significant, augment the design with axial runs to create a central composite design. The CCD is a very effective design for fitting a second-order response surface model