# Introduction to Statistical Quality Control_ 4th Edition_2_

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```					        Chapter 12
Factorial and Fractional Factorial
Experiments for Process Design and
Improvement

Introduction to Statistical Quality Control,
4th Edition
General model of a process
Controllable input
factors
x1 x2 . . . xp

Input                                         Output, y
Process

z1 z2   ...    zq

Uncontrollable input
factors

Introduction to Statistical Quality Control,
4th Edition
12-1. What is Experimental Design?
Objectives of Experimental Design
– Determine which variables (x’s) are most
influential on the response, y
– Determine where to set the influential x’s so
that y is near the nominal requirement
– Determine where to set the influential x’s so
that variability is small
– Determine where to set the influential x’s so
that the effects of the uncontrollable variables
z are minimized
Introduction to Statistical Quality Control,
4th Edition
12-1. What is Experimental Design?

Results of Experimental Design (used early in
process development):
1. Improved yield
2. Reduced variability and closer conformance
to nominal
3. Reduced development time
4. Reduced overall costs

Introduction to Statistical Quality Control,
4th Edition
12-2. Examples

Example 12-1 Characterizing a Process
• SPC has been applied to a soldering process.
Through u-charts and Pareto analysis, statistical
control has been established and the number of
defective solder joints has been reduced to 1%.
The average board contains over 2000 solder
joints, 1% may still be too large.
• Desired to reduce the defects level more.

Introduction to Statistical Quality Control,
4th Edition
12-1. What is Experimental Design?
Example 12-1 Characterizing a Process
• Note: since the process is in statistical control,
not obvious what machine adjustments will be
necessary. There are several variables that may
affect the occurrence of defects:
–   Solder temp, preheat temp, conveyor speed, flux type,
flux specific gravity, conveyor angle.
•   A designed experiment involving these factors
could help determine which factors could help
significantly reduce defects. (Screening
experiment)
.                  Introduction to Statistical Quality Control,
4th Edition
12-2. Guidelines for Designing
Experiments
Procedure for designing an experiment
1.   Recognition of and statement of the problem.
2.   Choice of factors and levels.
3.   Selection of the response variable.
4.   Performing the experiment
5.   Data analysis
6.   Conclusions and recommendations

•    #1-#3 make up pre-experimental planning
•    #2 and #3 often done simultaneously, or in reverse order.
Introduction to Statistical Quality Control,
4th Edition
12-3. Factorial Experiments

•   When there are several factors of interest
in an experiment, a factorial design
should be used.
•   A complete trial or replicate of the
experiment for all possible combinations
of the levels of the factors are
investigated.

Introduction to Statistical Quality Control,
4th Edition
12-3. Factorial Experiments
•   Main effect is the change in response produced
by a change in the level of a primary factor.
•   An interaction is present among factors if a
change in the levels of one factor influences the
effect of another factor.
•   Consider an experiment with two factors A & B
–   Interested in
•   Main effect of A
•   Main effect of B
•   Interaction effect of AB
Introduction to Statistical Quality Control,
4th Edition
12-3. Factorial Experiments
12-3.2 Statistical Analysis
•   Completely randomized design with two factors (A and
B) and n replicates.
•   The model is
y ij k     i   j  ( ) ij   ij k
where     = overall mean
i = effect of ith level of factor A
j = effect of jth level of factor B
()ij = effect of the interaction between A
and B
 = random error component
Introduction to Statistical Quality Control,
4th Edition
12-3. Factorial Experiments

12-3.2 Statistical Analysis
Factor B
1                     2                               b
1    y111, y112, …, y11n   y121, y122, …, y12n            y1b1, y1b2, …, y1bn
2    y211, y212, …, y21n   y221, y222, …, y22n …           y2b1, y2b2, …, y2bn
Factor A
                                                       
a    ya11, ya12, …, ya1n   ya21, ya22, …, ya2n …           yab1, yab2, …, yabn

Introduction to Statistical Quality Control,
4th Edition
12-3. Factorial Experiments

12-3.2 Statistical Analysis
•        Total corrected sum of squares                                        SSB

decomposition
a   b     n                            a             SSA                b
   ( y ijk  y... )  bn  ( y i..  y... )  an  ( y. j.  y... )
2                                     2                           2

i 1 j1 k 1                       i 1                                 j1
a       b
 n   ( y ij.  y i..  y. j.  y... ) 2
SST                           i 1 j1
a        b       n
    ( y ijk  y ij. ) 2                                SSAB
i 1 j1 k 1
SSE
Introduction to Statistical Quality Control,
4th Edition
12-3. Factorial Experiments

12-3.2 Statistical Analysis
•    Total corrected sum of squares decomposition,
notation:
SST = SSA + SSB + SSAB + SSE
•    The corresponding degree of freedom
decomposition is
abn – 1 = (a – 1) + (b – 1) + (a – 1)(b – 1) + ab(n – 1)

Introduction to Statistical Quality Control,
4th Edition
12-3. Factorial Experiments
12-3.2 Statistical Analysis
Source of      Sum of    Degrees of
Variation     Squares     Freedom               Mean Square                  F0
A             SSA       a-1                       SS                        MSA
MSA  A                    F0 
a 1                      MSE
B             SSB       b-1                       SS B                      MSB
MSB                       F0 
b 1                      MSE
Interaction   SSAB      (a – 1)(b – 1)                 SS AB                MSAB
MS AB                    F0 
(a  1)( b  1)           MSE
Error         SSE       ab(n-1)                     SS E
MSE 
ab(n  1)
Total         SST       abn - 1

Introduction to Statistical Quality Control,
4th Edition
12-3. Factorial Experiments
12-3.2 Statistical Analysis
Sum of Squares Computing Formulas
2
a b n                   y...
SS T     y ijk     2

i 1 j1 k 1            abn
2
a y2             2
y...                 b y          2
y...
Main Effects     SS A   i..                         SS B           
. j.

i 1 bn          abn                  j1 an      abn
2          2
a b y               y...
SS AB                            SS A  SS B
ij.
Interaction
i 1 j1 n         abn

Error                   SSE = SST – SSA - SSB - SSAB

Introduction to Statistical Quality Control,
4th Edition
Methods
Primer      Dipping                         Spraying                    yi.
Type
1   4.0, 4.5,4.3                 5.4, 4.9,5.6                       28.7
12.8                                   15.9

2     5.6,4.9,5.4                 5.8,6.1,6.3                        34.1
15.9                                   18.2

3     3.8,3.7,4.0           5.5,5.0,5.0 15.5                         27.0
11.5
y.j         40.2                               49.6                  89.8
Introduction to Statistical Quality Control,
4th Edition
12-3. Factorial Experiments
Example 12-5
Aircraft Primer Example
2
a   b   n     y...
SST     yijk   2

i 1 j1 k 1   abn
(89.8) 2
 (4.0) 2  (4.5) 2    (5.0) 2            10.72
18
a    y i2.. y...   2
SSprimers               
i 1 bn         abn
(28.7) 2  (34.1) 2  (27.0) 2 (89.8) 2
                                          4.58
6             18
2          2
b y            y...
SSmethods               
. j.

j1 an         abn
(40.2) 2  (49.6) 2 (89.8) 2
                                 4.91
9             18
Introduction to Statistical Quality Control,
4th Edition
12-3. Factorial Experiments
Example 12-5
Aircraft Primer Example
2      2
a   b   y ij.   y...
SSint eraction                   SSprim ers  SSm ethods
i 1 j1 n   abn
(12.8) 2  (15.9) 2  (11.5) 2  (15.9) 2  (18.2) 2  (15.5) 2

3
(89.8) 2
             4.58  4.91  0.24
18

SSE = 10.72 – 4.58 – 4.91 – 0.24 = 0.99

Introduction to Statistical Quality Control,
4th Edition
12-3. Factorial Experiments
Example 12-5
Aircraft Primer Example – Table 12-4. Analysis of Variance

Sum of     Degrees of
Source of Variation   Squares     Freedom          Mean Square              F0      P-value
Primer types           4.58           2               2.29                28.63   2.71 x 10-5
Application methods    4.91           1               4.91                61.38   4.65 x 10-6
Interaction            0.24           2               0.12                1.5     0.269
Error                  0.99          12               0.08
Total                  10.72         17

Introduction to Statistical Quality Control,
4th Edition
12-3. Factorial Experiments
Example 12-5
Aircraft Primer Example – Figure 12-12. Graph of average adhesion
force versus primer types

Introduction to Statistical Quality Control,
4th Edition
12-3. Factorial Experiments
12-3.3 Residual Analysis
• Residuals are important in accessing model
• The residuals from a two-factor factorial are

eijk  yijk  yijk
ˆ
 yijk  yijk

Introduction to Statistical Quality Control,
4th Edition
Residuals
Primer Type            Dipping                               Spraying

1             -.26, .23, .03                        .10, -.40, .30

2             .30, -.40, .10                        -.26, .04, .23

3             -.03, -.13, .16                      .34, -.17, -.17

Introduction to Statistical Quality Control,
4th Edition
12-3. Factorial Experiments

12-3.3 Residual Analysis (Table 12.6 also)
Residuals Versus Primer
(response is Force)

0.4

0.3

0.2

0.1
Residual

0.0

-0.1

-0.2

-0.3

-0.4

1                             2                    3

Primer

Introduction to Statistical Quality Control,
4th Edition
12-3. Factorial Experiments

12-3.3 Residual Analysis
Normal Probability Plot

.999
.99
.95
Probability

.80
.50
.20
.05
.01
.001

-0.4   -0.3   -0.2   -0.1   0.0     0.1   0.2   0.3
Residuals

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
12-4.1 The 22 Design
• 2k is the notation used to indicate that a certain
experimental design has k factors of interest,
each at two levels.
• 22 design: Two factors A and B, each at two
levels
A             B
Low                -1            -1
High               +1            +1
There are a total of four possible combinations.
Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
12-4.1 The 22 Design
•   The simplest design involves two factors A and B and n
replicates.
•   Interested in the main effect of A, the main effect of B,
and the interaction between A and B.
•   Effects are calculated by:
Average Response at high level - Average
Response at the low level.
•   A large effect would indicate a significant factor (or
interaction). (How large is large?)
•   Contrasts can be calculated and used to estimate the
effects and the sums of squares.
Introduction to Statistical Quality Control,
4th Edition
High
(+)   b                                        ab

B

Low     (1)                                     a
(-)
Low                                 High
(-)               A                 (+)

Introduction to Statistical Quality Control,
4th Edition
Understanding the effect of A
a+ab-b-(1)
High
(+)   b                                        ab
-                                                             +

B

Low     (1)                                     a
(-)
-              Low                                 High       +
(-)               A                 (+)

Introduction to Statistical Quality Control,
4th Edition
Understanding the effect of B
b+ab-a-(1)
High
(+)   b                                        ab
+                                                             +

B

Low     (1)                                     a
(-)
-              Low                                 High       -
(-)               A                 (+)

Introduction to Statistical Quality Control,
4th Edition
Understanding the effect of AB
ab+(1)-a-b
High
(+)   b                                        ab
-                                                             +

B

Low     (1)                                     a
(-)
+              Low                                 High       -
(-)               A                 (+)

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design

12-4.1 The 22 Design
•   Let the letters (1), a, b, and ab represent the
totals of all n observations taken at these design
points.
•   Effect estimate of A:
A  yA  yA
a  ab b  (1)
        
2n       2n

1
a  ab  b  (1)
2n
Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
12-4.1 The 22 Design
• Effect estimate of B:                 B  y B  y B
b  ab a  (1)
        
2n       2n

1
b  ab  a  (1)
2n
•   Effect estimate of AB:                  ab  (1) a  b
AB          
2n        2n

1
ab  (1)  a  b
2n
Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
12-4.1 The 22 Design
• For the previous effects formulas, the quantities
in brackets are called contrasts.
• For example, ContrastA = a + ab – b – (1)
• The contrasts are used to calculate the sum of
squares for the factors and interaction.
(contrast ) 2
SS 
n (contrast coefficien ) 2
ts

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
12-4.1 The 22 Design
• The sum of squares for A, B, and AB are:

SSA   
a  ab  b  (1)              2

4n

SSB 
b  ab  a  (1)2

4n

SSAB 
ab  (1)  a  b2
4n
Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
Example 12-6
•   The effect estimates for A, B, and AB for the router
example are:
A
a  ab  b  (1)
2n

1
96.1  161.1  59.7  64.4  16.64
2( 4)

B
b  ab  a  (1)
2n

1
59.7  161.1  96.1  64.4  7.54
2( 4)
Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
Example 12-6
• The effect estimates for A, B, and AB for the
router example are:

AB 
ab  (1)  a  b
2n

1
161 .1  64 .4  96 .1  59 .7  8.71
2( 4)

Introduction to Statistical Quality Control,
4th Edition
Sums of squares
SSA=[a+ab-b-(1)]2/abn
SSA=(96.1+161.1-59.9-64.4)2/16=1107.226

SSB=[b+ab-a-(1)]2/abn
SSB=(59.7+161.1-96.1-64.4)2/16=227.256

Introduction to Statistical Quality Control,
4th Edition
Sums of squares
SSAB=[ab+(1)-a-b]2/abn
SSAB=(161.1+64.4-96.1-59.7]2/16=303.631

SST=(18.22 +18.92+…+39.92) –
(64.4+96.1+59.7+161.1)2/16 = 1709.836

Introduction to Statistical Quality Control,
4th Edition
Sums of squares
SSE by subtraction:
SSE= 1709.836-1107.226-227.256-303.631
=71.723

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
Example 12-6
• The ANOVA table for the router example is
then
Source of              Sum of   Degrees of                                     P-value
Variation             Squares    Freedom         Mean Square             F0
Bit Size (A)         1107.226        1             1107.226          185.25 1.17 x 10-8
Speed (B)             227.256        1               227.256          38.03 4.82 x 10-5
AB                    303.631        1               303.631          50.80 1.20 x 10-5
Error                  71.723       12                 5.977
Total                1709.836       15

F.025,1,12 = 6.55          2-tailed at .05     1-tailed at .05            F.05,1,12 = 4.75

Introduction to Statistical Quality Control,
4th Edition
Part of MiniTab Output
Main Effects         2       1334.48          1334.48          667.241 111.64        0.000
2-Way Interactions   1         303.63           303.63         303.631       50.80   0.000
Residual Error       12         71.72            71.72               5.977
Pure Error         12         71.72            71.72               5.977
Total                15      1709.83

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
Regression Model
•   A regression model could be fit to data from a factorial
design
y  0  1x1   2 x 2  12 x1x 2  
•   where 0 is the grand average of all observations and
each coefficient, j is  effect estimate.
•   For Example 12-6, the fitted regression model is

 16 .64        7.54        8.71 
ˆ  23 .83  
y                      x1        x 2         x1x 2
 2             2           2 
Introduction to Statistical Quality Control,
4th Edition
Using the regression model
• Predict the value with small bit (x1=-1) and
low speed (x2=-1)
• yest = 23.83 + [16.64/2](-1) + [7.54/2](-1)
+[8.71/2](-1)(-1) = 16.1

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design

Residual Analysis
• Residual plots are used to access the adequacy of
the model once again.
• Residuals are calculated using the fitted
regression model.
• The residual plots versus the factor levels,
interactions, predicted values, and a normal
probability plot are all useful in determining the
adequacy of the model and satisfaction of
assumptions.
Introduction to Statistical Quality Control,
4th Edition
Computing the residuals
• For (1) with yest = 16.1
–   estimate = actual – yest
–   e1 = 18.2 – 16.1 = 2.1
–   e2 = 18.9 – 16.1 = 2.8
–   e3 = 12.9 – 16.1 = -3.2
–   e4 = 14.4 – 16.1 = -1.7
• See Fig. 12-19 and Fig. 12-20

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design

Analysis Procedure for Factorial Designs
1. Estimate the factor effects
2. Form preliminary model
3. Test for significance of factor effects
4. Analyze residuals
5. Refine model, if necessary
6. Interpret results

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
12-4.2 The 2k Design for k  3 Factors
• When k  2, you could have a single
replicate, but some assumptions need to be
• For k = 3, the main effects and interactions
of interest are A, B, C, AB, AC, BC,
ABC.
• The main effects are again represented by
a, b, c, ab, ac, bc, abc, and (1)
Introduction to Statistical Quality Control,
4th Edition
See Fig. 12-24 for understanding
the effects

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
12-4.2 The 2k Design for k  3 Factors
Effect Estimate for A:
A  y A  y A    
1
a  ab  ac  abc  b  c  bc  (1)
4n
Effect Estimate for B:

B  y B  y B   
1
b  ab  bc  abc  a  c  ac  (1)
4n
Effect Estimate for C:
C  y C  y C   
1
c  ac  bc  abc  a  b  ab  (1)
4n

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
12-4.2 The 2k Design for k  3 Factors
Effect Estimate for AB:
AB  y AB  y AB       
1
ab  (1)  abc  c  b  a  bc  ac
4n
Effect Estimate for AC:
AC  y AC  y AC 
1
ac  (1)  abc  b  a  c  ab  bc
4n
Effect Estimate for BC:

BC  y BC   y BC     
1
bc  (1)  abc  a  b  c  ab  ac
4n

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
12-4.2 The 2k Design for k  3 Factors
Effect Estimate for ABC:
ABC  y ABC  y ABC 
1
abc  bc  ac  c  ab  b  a  (1)
4n
In general, the effects can be estimated using
Contrast
Effect 
n 2k 1
The sum of squares for any effect is

SS 
Contrast 2
n 2k
Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
Example 12-7
• An experiment was performed to
investigate the surface finish of a metal
part. The experiment is a 23 factorial
of cut (B), and tool angle (C), with n = 2
replicates.

Introduction to Statistical Quality Control,
4th Edition
Effect estimates
•   A = (1/8)(22+27+23+30-20-21-18-16) =3.375
•   B = 1.625
•   C = 0.875
•   AB = 1.375
•   AC = 0.125
•   BC = -0.625
•   ABC = 1.125

Introduction to Statistical Quality Control,
4th Edition
Sums of squares
•   A = (27)2/[2(8)] = 45.5625
•   B= (13)2/[2(8)] = 10.5625
•   :
•   Total = 92.9375
•   Error by subtraction

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
Example 12-7
Design Factors
Surface
Run                 A               B             C      Finish   Totals
1       (1)         -1             -1            -1       9, 7     16
2        a           1             -1            -1     10, 12     22
3        b          -1              1            -1      9, 11     20
4       ab           1              1            -1     12, 15     27
5        c          -1             -1             1     11, 10     21
6        ac          1             -1             1     10, 13     23
7       bc          -1              1             1      10, 8     18
8       abc          1              1             1     16, 14     30

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
Example 12-7 – Analysis of Variance Table

Source of    Sum of   Degrees of                                        P-value
Variation   Squares    Freedom         Mean Square               F0
A           45.5625        1               45.5625            18.69   2.54 x 10-3
B           10.5625        1               10.5625             4.33   0.07
C            3.0625        1                3.0625             1.26   0.29
AB           7.5625        1                7.5625             3.10   0.12
AC           0.0625        1                0.0625             0.03   0.88
BC           1.5625        1                1.5625             0.64   0.45
ABC          5.5625        1                5.5625             2.08   0.19
Error       19.5000        8                2.4375
Total       92.9375       15

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
Regression Model
•     For Example 12-7, the fitted regression model involving
only those factors found significant (A, B) and the next
significant interaction (AB) is
 3.375        1.625        1.375 
y  11.0625  
ˆ                      x1         x 2          x1 x 2
 2            2            2 
 11.0625  1.6875x1  0.8125x 2  0.6875x1x 2
•     Developing a model from a designed experiment can be
a valuable tool in determining optimal settings for the
factors.
Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
Other Methods for Judging the Significance of
Effects
• The standard error of any effect estimate in a 2k
design is                 2

ˆ
s.e.(Effect) 
n 2k  2

•   Two standard deviation limits on any estimated
effect is
Effect estimate  2[s.e.(Effect)]
Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
Other Methods for Judging the Significance of
Effects
Effect estimate  2[s.e.(Effect)]
• This interval is an approximate 95% confidence
interval on the estimated effect.
• Interpretation: If zero is contained within the
95% confidence interval, then that effect is
essentially zero and the corresponding factor is
not significant at the  = 0.05 level.

Introduction to Statistical Quality Control,
4th Edition
Example
• In example 12-7
– MSE = est2 = 2.4375
– s.e.(Effect) = SQRT{2.4375/[(2)(23-2)]} = .78

Introduction to Statistical Quality Control,
4th Edition
Example
• 2 std dev limits on the effects:
–   A: 3.375 + 1.56 (does not include 0)
–   B: 1.625 + 1.56 (does not include 0)
–   C: 0.875 + 1.56 (does include 0)
–   AB: 1.375 + 1.56 (does include 0)
–   AC: 0.125 + 1.56 (does include 0)
–   BC: -0.625 + 1.56 (does include 0)
–   ABC: 1.125 + 1.56 (does include 0)

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
12-4.3 A Single Replicate of the 2k Design
• As the number of factors in a factorial
experiment increases, the number of effects that
can be estimated also increases.
• In most situations, the sparsity of effects
principle applies.
• For a large number of factors, say k > 5, it is
common practice to run only a single replicate
of the 2k design and pool or combine the higher-
order interactions in the estimate of error.
Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
Example 12-8
• Nitride etch process on a single-wafer plasma
etcher. There are four factors of interest. The
response is etch rate for silicon nitride. A single
replicate is used.

Gap         Pressure         C2F6 Flow         Power
Level      A (cm)      B (m Torr)        C (SCCM)          D (W)
Low (-)     0.80          450               125             275
High (+)    1.20          550               200             325

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
Example 12-8
A                B              C         D         Etch Rate
Run   (Gap)      (Pressure)    (C2F6 Flow)    (Power)         (Ao/min)
1        -1              -1             -1         -1              550
2         1              -1             -1         -1              669
3        -1               1             -1         -1              604
4         1               1             -1         -1              650
5        -1              -1              1         -1              633
6         1              -1              1         -1              642
7        -1               1              1         -1              601
8         1               1              1         -1              635
9        -1              -1             -1          1             1037
10        1              -1             -1          1              749
11       -1               1             -1          1             1052
12        1               1             -1          1              868
13       -1              -1              1          1             1075
14        1              -1              1          1              860
15       -1               1              1          1             1063
16        1               1              1          1              729

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
Example 12-8
• The estimated effects are found to be
A = -101.625  AD = -153.625
B = -1.625    BD = -0.625
AB = -7.875    ABD =    4.125
C=      7.375  CD = -2.125
AC = -24.875   ACD =    5.625
BC = -43.875   BCD = -25.375
ABC = -15.625 ABCD = -40.125
D = 306.125

Introduction to Statistical Quality Control,
4th Edition
Table 12-16 has the contrast constants

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
Example 12-8
• Normal probability plot of effects
Normal Probability Plot of the Effects
(response is Etch, Alpha = .05)

A:   A
D   B:   B
C:   C
D:   D
1
Normal Score

Significant
0

-1
A
Significant

-100       0           100            200   300

Effect

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design

Example 12-8
•  Normal probability plot reveals that A, D, and AD
appear to be significant.
•  To be sure that other main factors or two factor
interactions are not significant, pool the three- and four-
factor interactions to form the error mean square.
•  (NOTE: if the normal probability plot had indicated that
any of these interactions were important, they would not
be included in the error term.)

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design
Example 12-8- Analysis of Variance
Source of        Sum of     Degrees of
Variation       Squares      Freedom          Mean Square          F0
A            41,310.563         1              41,310.563       20.28
B                10.563         1                  10.563          <1
C               217.563         1                 217.563          <1
D           374,850.063                       374,850.063      183.99
AB              248.063           1               248.063          <1
AC            2,475.063           1             2,475.063        1.21
BC            7,700.063           1             7,700.063        3.78
BD                1.563                             1.563          <1
CD               18.063          1                 18.063          <1
Error        10,186.815          5              2,037.363
Total       531,420.936          15

Introduction to Statistical Quality Control,
4th Edition
12-4. 2k Factorial Design

Example 12-8- Analysis of Variance
• Factors A, D, and the interaction AD are
significant.
• The fitted regression model for this experiment
is
 101 .625        306 .125        153 .625 
y  776 .0625  
ˆ                           x1            x 2             x1x 2
    2               2               2 

where x1 represents A, x2 represents D.

Introduction to Statistical Quality Control,
4th Edition
Assignment
• We only covered pages 569 – 615
• Understand those designs and analyses in
the included pages
• Be able to work exercises like 12-1 and
12-2

Introduction to Statistical Quality Control,
4th Edition
End

Introduction to Statistical Quality Control,
4th Edition

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