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Materi : DOE Minggu IV


            1.   Introduction
            2.   Simple Comparative Experiments
            3.   Experiments with a Single Factor
            4.   The Randomized Complete Block Design  
            5.   The Latin Square Design
            6.   Factorial Design
            7.   The 2 k Factorial Design
            8.   Two-Level Fractional Factorial Design
            9.   Nested or Hierarchial Design
           10.   Response Surface Methods



10/22/12
Chapter 4. Randomized Blocks …

                      Controllable Factors
                        X 1 , X 2 , …, X q



            Input         Process                       Output (Y)

                                                      Randomization 
                        Z 1 , Z 2 , …, Z q
           unknown   Uncontrollable Factors                 known

                              A              A design factor that probably has an
                                             effect on the response, but we are
                          Nuisance           not interested in that effect
10/22/12                   Factors
Chapter 4. Randomized Blocks …                            (continued)



                               A              A design factor that probably has an
                                              effect on the response, but we are
                          Nuisance            not interested in that effect
                           Factors
           unknown    Controllable Factors                 known
                        X 1 , X 2 , …, X q
                                                          blocking         

            Input         Process                      Output (Y)


                        Z 1 , Z 2 , …, Z q
           unknown   Uncontrollable Factors                known
10/22/12
The Randomized Complete Block Design (RCBD)


    When the nuisance source of variability is known
     and controllable,
            blocking can be used to systematically eliminate its
             effect on the statistical comparisons among treatments

    Situations for which the RCBD is appropriate:
            Units of test equipment or machinery (often different in
             their operating characteristics and would be a typical
             bloking factor)
            Batches of raw material, people, and time (common
             nuisane sources of variability in an experiment)


10/22/12
Desain penggunaan 4 merk mesin …                     (Is it right ?)



                                Hari Kerja atau Operator
                            1         2         3          4
                            A         B         C          D
           Mesin yang       A         B         C          D
           digunakan
                            A         B         C          D
            (A,B,C,D)
                            A         B         C          D




                        Tidak dapat dipisahkan antara rata-rata
                           produktifitas mesin dari rata-rata
                         produktifitas hari ataupun operator


10/22/12
Desain penggunaan 4 merk mesin …                   (Right
design )



                          Hari Kerja atau Operator -> BLOK
                             1        2         3        4
                            A         B         C        D
           Mesin yang       B         C         D        A
            digunakan
                            C         D         A        B
            (A,B,C,D)
                            D         A         B        C




            Randomisasi secara lengkap dilakukan dalam
           BLOK yang sama, sehingga rata-rata produktifitas
           mesin dapat dipisahkan dari rata-rata produktifitas
                      hari ataupun operator

10/22/12
The Randomized Complete Block Design (RCBD)


           Block 1       Block 2       Block 3               Block b


            Y11           Y12            Y13                   Y1b
            Y21           Y22            Y23                   Y2b
                                                    …
            Y31           Y32            Y33                   Y3b
             …             …             …                      …
            Ya1           Ya2            Ya3                   Yab


            There is one observation per treatment (1, 2, …, a) in
            each block, and the order in which the treatments are
               run within each block is determined randomly.

10/22/12
The ANOVA: Structure Data and Model RCBD


    Treatment                           Block (j)                Total     Average
     (level i)         1           2                …   b          Yi.       Yi.


           1          Y11         Y12               …   Y1n        Y1.         Y1.
           2          Y21         Y22               …   Y2n        Y2.         Y2.
           ...        …            …                …   …          …           …
           a          Ya1         Ya2               …   Yan        Ya.         Ya.
     Total Y.j        Y.1         Y.2               …   Y.b        Y..         Y..


                 Statistical model for the RCBD:                             yij
                 = µij + εij, , i = 1, 2, …, a; j = 1, 2, …, b           atau
                                                                             yij
                 = µ + τi + βj + εij
10/22/12
The ANOVA Table for a RCBD Model


           Source of     Sum of                             Mean
                                             DF                               F
           Variation     Squares                             Square
       Treatments          SST               a−1              MST      FT = MST / MSE
       Blocks              SSB               b–1              MSB      FB = MSB / MSE
       Error               SSE           (a-1)(b-1)           MSE
       Total               SSTotal           N−1

                                     a   b           Y..2
                       SS Total =    ∑∑      2
                                           Yij     −
                                                      N
                                  i =1 j =1

                             1 a 2 Y..2                           1   b       Y..2
                       SS T = ∑Yi. −    ;                   SSB =     ∑Y j2. − N
                             b i =1  N                            a   j =1
                       SSE = SS Total − SS T − SSB
10/22/12
Chapter 4. Randomized Blocks …

                                                                   People, Machine,
                                                                    and other control-
               Type of Tip        Controllable Factors             able factors are inthe
                                                                    SAME conditions
               1, 2, 3, 4            X 1 , X 2 , …, X q



             Input                      Process                      Output (Y)


           Metal coupon              Z 1 , Z 2 , …, Z q                The hardness
                                                                      testing machine

                                Uncontrollable Factors
              We wish to determine whether or not four different tips produce
              different readings on a hardness testing machine ?
10/22/12
Data from the Hardness Testing Experiment …

                                        Test Coupon (Block)
                 Type of Tip
                                  1         2         3         4

                     1           9.3       9.4       9.6      10.0
                     2           9.4       9.3       9.8       9.9
                     3           9.2       9.4       9.5       9.7
                     4           9.7       9.6      10.0      10.2




           The metal coupon differ slightly in their hardness, as might
            happen if they are taken from ingots that are product in
             different heats, the experiment units (the coupon) will
           contribute to the variabiliy observed in the hardness data.

10/22/12
Graphical Analysis: Box-Plot Data …




10/22/12
Graphical Analysis: Main Effect Plot …




10/22/12
ANOVA of RCBD: MINITAB output …




10/22/12
Wrong ANOVA : MINITAB output …




10/22/12
Data, Fits and Residual: MINITAB output …




10/22/12
Graphical comparison of means …




                                                         This plot indicates that tip
                                                            1, 2, and 3 probably
                                                         produce identical average
                                                         hardness measurements
                                                          but that tip 4 produces a
                                                             much higher mean
                                                                  hardness.


                 Tip 3   Tip 1     Tip 2         Tip 4


           9.4               9.6           9.8             10.0


10/22/12
Model Adequacy Checking: Normality test …




10/22/12
Model Adequacy Checking: Equality variance …




           Plot of residuals by tip type (treatment) and by coupon (block)



10/22/12
MINITAB command for RCBD Analysis




10/22/12

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Modul 4. doe rcbd

  • 1. Materi : DOE Minggu IV 1. Introduction 2. Simple Comparative Experiments 3. Experiments with a Single Factor 4. The Randomized Complete Block Design  5. The Latin Square Design 6. Factorial Design 7. The 2 k Factorial Design 8. Two-Level Fractional Factorial Design 9. Nested or Hierarchial Design 10. Response Surface Methods 10/22/12
  • 2. Chapter 4. Randomized Blocks … Controllable Factors X 1 , X 2 , …, X q Input Process Output (Y) Randomization  Z 1 , Z 2 , …, Z q unknown Uncontrollable Factors known A A design factor that probably has an effect on the response, but we are Nuisance not interested in that effect 10/22/12 Factors
  • 3. Chapter 4. Randomized Blocks … (continued) A A design factor that probably has an effect on the response, but we are Nuisance not interested in that effect Factors unknown Controllable Factors known X 1 , X 2 , …, X q blocking  Input Process Output (Y) Z 1 , Z 2 , …, Z q unknown Uncontrollable Factors known 10/22/12
  • 4. The Randomized Complete Block Design (RCBD)  When the nuisance source of variability is known and controllable,  blocking can be used to systematically eliminate its effect on the statistical comparisons among treatments  Situations for which the RCBD is appropriate:  Units of test equipment or machinery (often different in their operating characteristics and would be a typical bloking factor)  Batches of raw material, people, and time (common nuisane sources of variability in an experiment) 10/22/12
  • 5. Desain penggunaan 4 merk mesin … (Is it right ?) Hari Kerja atau Operator 1 2 3 4 A B C D Mesin yang A B C D digunakan A B C D (A,B,C,D) A B C D Tidak dapat dipisahkan antara rata-rata produktifitas mesin dari rata-rata produktifitas hari ataupun operator 10/22/12
  • 6. Desain penggunaan 4 merk mesin … (Right design ) Hari Kerja atau Operator -> BLOK 1 2 3 4 A B C D Mesin yang B C D A digunakan C D A B (A,B,C,D) D A B C Randomisasi secara lengkap dilakukan dalam BLOK yang sama, sehingga rata-rata produktifitas mesin dapat dipisahkan dari rata-rata produktifitas hari ataupun operator 10/22/12
  • 7. The Randomized Complete Block Design (RCBD) Block 1 Block 2 Block 3 Block b Y11 Y12 Y13 Y1b Y21 Y22 Y23 Y2b … Y31 Y32 Y33 Y3b … … … … Ya1 Ya2 Ya3 Yab There is one observation per treatment (1, 2, …, a) in each block, and the order in which the treatments are run within each block is determined randomly. 10/22/12
  • 8. The ANOVA: Structure Data and Model RCBD Treatment Block (j) Total Average (level i) 1 2 … b Yi. Yi. 1 Y11 Y12 … Y1n Y1. Y1. 2 Y21 Y22 … Y2n Y2. Y2. ... … … … … … … a Ya1 Ya2 … Yan Ya. Ya. Total Y.j Y.1 Y.2 … Y.b Y.. Y.. Statistical model for the RCBD: yij = µij + εij, , i = 1, 2, …, a; j = 1, 2, …, b atau yij = µ + τi + βj + εij 10/22/12
  • 9. The ANOVA Table for a RCBD Model Source of Sum of Mean DF F Variation Squares Square Treatments SST a−1 MST FT = MST / MSE Blocks SSB b–1 MSB FB = MSB / MSE Error SSE (a-1)(b-1) MSE Total SSTotal N−1 a b Y..2 SS Total = ∑∑ 2 Yij − N i =1 j =1 1 a 2 Y..2 1 b Y..2 SS T = ∑Yi. − ; SSB = ∑Y j2. − N b i =1 N a j =1 SSE = SS Total − SS T − SSB 10/22/12
  • 10. Chapter 4. Randomized Blocks … People, Machine, and other control- Type of Tip Controllable Factors able factors are inthe SAME conditions 1, 2, 3, 4 X 1 , X 2 , …, X q Input Process Output (Y) Metal coupon Z 1 , Z 2 , …, Z q The hardness testing machine Uncontrollable Factors We wish to determine whether or not four different tips produce different readings on a hardness testing machine ? 10/22/12
  • 11. Data from the Hardness Testing Experiment … Test Coupon (Block) Type of Tip 1 2 3 4 1 9.3 9.4 9.6 10.0 2 9.4 9.3 9.8 9.9 3 9.2 9.4 9.5 9.7 4 9.7 9.6 10.0 10.2 The metal coupon differ slightly in their hardness, as might happen if they are taken from ingots that are product in different heats, the experiment units (the coupon) will contribute to the variabiliy observed in the hardness data. 10/22/12
  • 12. Graphical Analysis: Box-Plot Data … 10/22/12
  • 13. Graphical Analysis: Main Effect Plot … 10/22/12
  • 14. ANOVA of RCBD: MINITAB output … 10/22/12
  • 15. Wrong ANOVA : MINITAB output … 10/22/12
  • 16. Data, Fits and Residual: MINITAB output … 10/22/12
  • 17. Graphical comparison of means … This plot indicates that tip 1, 2, and 3 probably produce identical average hardness measurements but that tip 4 produces a much higher mean hardness. Tip 3 Tip 1 Tip 2 Tip 4 9.4 9.6 9.8 10.0 10/22/12
  • 18. Model Adequacy Checking: Normality test … 10/22/12
  • 19. Model Adequacy Checking: Equality variance … Plot of residuals by tip type (treatment) and by coupon (block) 10/22/12
  • 20. MINITAB command for RCBD Analysis 10/22/12