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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
        Adhesion Force Data
                        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
    adequacy
• 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
Source               DF        Seq SS           Adj SS           Adj MS         F       P
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
   made. (Can’t estimate all interactions).
• 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
   design in the factors feed rate (A), depth
   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
                                      AD

                                           -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
    AD           94,402.563                        94,402.563       48.79
    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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