II.4 Sixteen Run Fractional Factorial Designs

1
II.4 Sixteen Run Fractional Factorial Designs

• Introduction
• Resolution Reviewed
• Design and Analysis
• Example Five Factors Affecting Centerpost Gasket
  Clipping Time
• Example / Exercise Seven Factors Affecting a
  Polymerization Process
• Discussion

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II.4 Sixteen Run Fractional Factorial Designs
Introduction

• With 16 runs, up to 15 Factors may be analyzed at
  Resolution III.
• Resolution IV is possible with 8 or fewer
  factors.
• Resolution V is possible with 5 or fewer factors.
• These designs are very useful for screening
  situations determine which factors have strong
  main effects
• 20 rule

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II.4 Sixteen Run Designs Resolution Reviewed

• Q What is a Resolution III design?
• A a design in which main effects are not
  confounded with other main effects, but at least
  one main effect is confounded with a 2-way
  interaction
• Resolution III designs are the riskiest
  fractional factorial designsbut the most useful
  for screening
• damn the interactions.full speed ahead!

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II.4 Sixteen Run DesignsResolution Reviewed

• Q What is a Resolution IV design?
• A a design in which main effects are not
  confounded with other main effects or 2-way
  interactions, but either (a) at least one main
  effect is confounded with a 3-way interaction, or
  (b) at least one 2-way interaction is confounded
  with another 2-way interaction.
• Hence, in a Resolution IV design, if 3-way and
  higher interactions are negligible, all main
  effects are estimable with no confounding.

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II.4 Sixteen Run Designs Resolution Reviewed

• Q What is a Resolution V design?
• A a design in which main effects are not
  confounded with other main effects or 2- or 3-way
  interactions, and 2-way interactions are not
  confounded with other 2-way interactions. There
  is either (a) at least one main effect confounded
  with a 4-way interaction, or (b) at least one
  2-way interaction confounded with a 3-way
  interaction.

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II.4 Sixteen Run Designs Resolution Reviewed

• Hence, in a Resolution V design, if 3-way and
  higher interactions are negligible, all main
  effects and 2-way interactions are estimable with
  no confounding.

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16 Run Signs Table
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II.4 Sixteen Run DesignsExample Five Factors
Affecting Centerpost Gasket Clipping Time

• y clip time (secs) for 16 parts from the sprue
  (injector for liquid molding process)
• Factors and levels -
• A Table No Yes
• B Shake No Yes
• C Position Sitting Standing
• D Cutter Small Large
• E Grip Unfold Fold
• Contributed by Rodney Phillips (B.S. 1994), at
  that time working for Whirlpool. This was a STAT
  506 (Intro. To DOE) project.

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Example Five Factors Affecting Centerpost
Gasket Clipping Time

• Design the Experiment associate factors with
  carefully chosen columns in the 16-run signs
  matrix to generate a design matrix
• Always associate A, B, C, D with the first four
  columns
• With five factors, E ABCD is universally
  recommended (or E -ABCD)

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Example Five Factors Affecting Centerpost
Gasket Clipping Time
Full Alias Structure for the design EABCD

• IABCDE
• ABCDE
• BACDE
• CABDE
• DABCE
• EABCD
• ABCDE
• ACBDE

ADBCE AEBCD BCADE BDACE BEACD CDABE CEABD D
EABC
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Example Five Factors Affecting Centerpost
Gasket Clipping Time
Completed Operator Report Form
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Example Five Factors Affecting Centerpost
Gasket Clipping Time
Completed Signs Table with Estimated Effects
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Example Five Factors Affecting Centerpost
Gasket Clipping Time
Normal Plot of Estimated Effects
Ordered Effects -13.48 -10.28 -6.25 -3.68
-2.72 -0.94 -0.06 0.05 0.86 0.92
1.34 2.03 2.26 2.70 3.37
ACBDE
ABCDE
EABCD
-10
0
10
-20
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Example Five Factors Affecting Centerpost
Gasket Clipping Time
Preliminary Interpretation

• The Normal Plot indicates three effects
  distinguishable from error. These are
• E ABCD (estimating EABCD)
• A BCDE (estimating ABCDE)
• AC BDE (estimating ACBDE), marginal.

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Example Five Factors Affecting Centerpost
Gasket Clipping Time
Preliminary Interpretation

• Since it is unusual for four-way interactions to
  be active, the first two are attributed to E and
  A
• Since A is active, the ACBDE effect is
  attributed to AC
• We should calculate an AC interaction table and
  plot

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Example Five Factors Affecting Centerpost
Gasket Clipping Time
AC Interaction Table and Plot
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Example Five Factors Affecting Centerpost
Gasket Clipping Time
AC Interaction Table and Plot
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Example Five Factors Affecting Centerpost
Gasket Clipping Time
Interpretation

• E -13.5. Hence, the clip time is reduced an
  average of about 13.5 seconds when the worker
  uses the low level of E (the folded grip, as
  opposed to the unfolded grip). This seems to
  hold regardless of the levels of other factors (E
  does not seem to interact with anything).

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Example Five Factors Affecting Centerpost
Gasket Clipping Time
Interpretation

• The effects of A (table) and C (position) seem to
  interact. The presence of a table reduces
  average clip time, but the reduction is larger
  (16.6 seconds) when the worker is standing than
  when he/she is sitting (4.0 seconds)

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II.4 Sixteen Run DesignsExample / Exercise
Seven Factors Affecting a Polymerization Process

• y blender motor maximum amp load
• Factors and levels -
• A Mixing Speed Lo Hi
• B Batch Size Small Large
• C Final temp. Lo Hi
• D Intermed. Temp. Lo Hi
• E Addition sequence 1 2
• F Temp. of modifer Lo Hi
• G Add. Time of modifier Lo Hi
• Contributed by Solomon Bekele (Cryovac). This
  was part of a STAT 706 (graduate DOE) project.

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Example / Exercise Seven Factors Affecting a
Polymerization Process

• Design the Experiment associate additional
  factors with columns of the 16-run signs matrix

• For 6, 7, or 8 factors, we assign the additional
  factors to the 3-way interaction columns
• For this 7-factor experiment, the following
  assignment was used
• E ABC, F BCD, G ACD

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Example / Exercise Seven Factors Affecting a
Polymerization Process
Runs table
Std Order A B C D EABC GACD FBCD
1 -1 -1 -1 -1 -1 -1 -1
2 1 -1 -1 -1 1 1 -1
3 -1 1 -1 -1 1 -1 1
4 1 1 -1 -1 -1 1 1
5 -1 -1 1 -1 1 1 1
6 1 -1 1 -1 -1 -1 1
7 -1 1 1 -1 -1 1 -1
8 1 1 1 -1 1 -1 -1
9 -1 -1 -1 1 -1 1 1
10 1 -1 -1 1 1 -1 1
11 -1 1 -1 1 1 1 -1
12 1 1 -1 1 -1 -1 -1
13 -1 -1 1 1 1 -1 -1
14 1 -1 1 1 -1 1 -1
15 -1 1 1 1 -1 -1 1
16 1 1 1 1 1 1 1
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Example / Exercise Seven Factors Affecting a
Polymerization Process

• Determine the designs alias structure
• There will again be 16 rows in the full alias
  table, but now 27 128 effects (including I)!
  Each row of the full table will have 8 confounded
  effects! Here is how to start find the full
  defining relation
• Since E ABC, we have I ABCE.
• But also F BCD, so I BCDF
• Likewise G ACD, so I ACDG
• Likewise I I x I (ABCE)(BCDF) ADEF !

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Example / Exercise Seven Factors Affecting a
Polymerization Process

• Continue in this fashion until you find
• I ABCE BCDF ACDG ADEF BDEG ABFG
  CEFG
• We have verified that this design is of
  Resolution IV (why?)

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Example / Exercise Seven Factors Affecting a
Polymerization Process

• Determine the alias table multiply the defining
  relation (rearranged alphabetically here)
• I ABCE ABFG ACDG ADEF BCDF BDEG
  CEFG
• by A for the second row
• A BCE BFG CDG DEF ABCDF ABDEG ACEFG
• by B for the third row
• B ACE AFG ABCDG ABDEF CDF DEG BCEFG
• and so on after all seven main effects are done,
  start with two way interactions
• AB CE FG BCDG BDEF ACDF ADEG ABCEFG
• and so on...(what a pain!)until you have 16 rows.

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Example / Exercise Seven Factors Affecting a
Polymerization Process
Full Alias Structure for the 2IV7-3 design E
ABC, F BCD, G ACD

• I ABCE ABFG ACDG ADEF BCDF BDEG
  CEFG
• A BCE BFG CDG DEF ABCDF ABDEG ACEFG
• B ACE AFG CDF DEG ABCDG ABDEF BCEFG
• C ABE ADG BDF EFG ABCFG ACDEF BCDEG
• D ACG AEF BCF BEG ABCDE ABDFG CDEFG
• E ABC ADF BDG CFG ABEFG ACDEG BCDEF
• F ABG ADE BCD CEG ABCEF ACDFG BDEFG
• G ABF ACD BDE CEF ABCEG ADEFG BCDFG
• AB CE FG ACDF ADEG BCDG BDEF ABCEFG
• AC BE DG ABDF AEFG BCFG CDEF ABCDEG
• AD CG EF ABCF ABEG BCDE BDFG ACDEFG
• AE BC DF ABDG ACFG BEFG CDEG ABCDEF
• AF BG DE ABCD ACEG BCEF CDFG ABDEFG
• AG BF CD ABDE ACEF BCEG DEFG ABCDFG
• BD CF EG ABCG ABEF ACDE ADFG BCDEFG
• ABD ACF AEG BCG BEF CDE DFG ABCDEFG

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Example / Exercise Seven Factors Affecting a
Polymerization Process
Reduced Alias Structure (up to 2-way
interactions) for the 2IV7-3 design E ABC, F
BCD, G ACD

• I ABCE ABFG ACDG ADEF BCDF BDEG CEFG

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Example / Exercise Seven Factors Affecting a
Polymerization Process
Std Order Y (amps) A B C D EABC GACD FBCD
1 130 -1 -1 -1 -1 -1 -1 -1
2 232 1 -1 -1 -1 1 1 -1
3 135 -1 1 -1 -1 1 -1 1
4 235 1 1 -1 -1 -1 1 1
5 128 -1 -1 1 -1 1 1 1
6 184 1 -1 1 -1 -1 -1 1
7 133 -1 1 1 -1 -1 1 -1
8 249 1 1 1 -1 1 -1 -1
9 130 -1 -1 -1 1 -1 1 1
10 225 1 -1 -1 1 1 -1 1
11 143 -1 1 -1 1 1 1 -1
12 270 1 1 -1 1 -1 -1 -1
13 132 -1 -1 1 1 1 -1 -1
14 198 1 -1 1 1 -1 1 -1
15 138 -1 1 1 1 -1 -1 1
16 249 1 1 1 1 1 1 1
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Example / Exercise Seven Factors Affecting a
Polymerization Process
Completed Signs Table with Estimated Effects
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Example / Exercise Seven Factors Affecting a
Polymerization Process

• Analyze the Experiment as an exercise,
• construct and interpret a Normal probability
  plot of the estimated effects
• if any 2-way interactions are distinguishable
  from error, construct interaction tables and
  plots for these
• provide interpretations

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Example / Exercise Seven Factors Affecting a
Polymerization Process
Solution Normal Plot of Estimated Effects
Ordered Effects -11.1 -9.4 -7.9 -6.1 -1.9 -1.6 2.
4 3.1 4.6 7.4 7.6 9.4 16.9 24.1 96.6
A
B
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Example / Exercise Seven Factors Affecting a
Polymerization Process
Suggested Interpretation

• The effect of mixing speed is A 96.6 amps.
  Hence, when we change the mixing speed from its
  low setting to its high setting, we expect the
  motors max amp load to increase by about 97
  amps.
• The effect of batch size is B 24.1 amps.
  Hence, when we change the batch size from small
  to large, we expect the motors max amp load to
  increase by about 24 amps.
• None of the other factors seems to affect the
  motors max amp load.

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II.4 Discussion

• As in 8-run designs, we can always fold over a
  16 run fractional factorial design. There are
  several variations on this technique in
  particular, for any 16-run Resolution III design,
  it is always possible to add 16 runs in such a
  way that the pooled design is Resolution IV.
• There are a great many other fractional factorial
  designs in particular, the Plackett-Burman
  designs have runs any multiple of four
  (4,8,12,16,20, etc.) up to 100, and in n runs can
  analyze (n-1) Factors at Resolution III.

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II.4 References

• Daniel, Cuthbert (1976). Applications of
  Statistics to Industrial Experimentation. New
  York John Wiley Sons, Inc.
• Box, G.E.P. and Draper, N.R. (1987). Empirical
  Model-Building and Response Surfaces. New York
  John Wiley Sons, Inc.
• Box, G.E.P., Hunter, W. G., and Hunter, J.S.
  (1978). Statistics for Experimenters. New York
  John Wiley Sons, Inc.
• Lochner, R.H. and Matar, J.E. (1990). Designing
  for Quality. Milwaukee ASQC Quality Press.