DOE 121b General Factorial Design

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DOE 12-1b General Factorial Design
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DOE 12-1b General Factorial Design
We might expect that the Two-Factor Factorial
Design could be extended to Three-Factor (and
higher) Factorial Designs
3
DOE 12-1b General Factorial Design
We might expect that the Two-Factor Factorial
Design could be extended to Three-Factor (and
higher) Factorial Designsand wed be right!
4
DOE 12-1b General Factorial Design
We might expect that the Two-Factor Factorial
Design could be extended to Three-Factor (and
higher) Factorial Designsand wed be
right! Lets consider the Three-Factor ANOVA case.
5
DOE 12-1b General Factorial Design
We might expect that the Two-Factor Factorial
Design could be extended to Three-Factor (and
higher) Factorial Designsand wed be
right! Lets consider the Three-Factor ANOVA case.
Observation row, column, 3rd factor, repetition
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DOE 12-1b General Factorial Design
We might expect that the Two-Factor Factorial
Design could be extended to Three-Factor (and
higher) Factorial Designsand wed be
right! Lets consider the Three-Factor ANOVA case.
Mean of full population
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DOE 12-1b General Factorial Design
We might expect that the Two-Factor Factorial
Design could be extended to Three-Factor (and
higher) Factorial Designsand wed be
right! Lets consider the Three-Factor ANOVA case.
Effect on observation due to row treatment level
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DOE 12-1b General Factorial Design
We might expect that the Two-Factor Factorial
Design could be extended to Three-Factor (and
higher) Factorial Designsand wed be
right! Lets consider the Three-Factor ANOVA case.
Effect on observation due to column treatment
level
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DOE 12-1b General Factorial Design
We might expect that the Two-Factor Factorial
Design could be extended to Three-Factor (and
higher) Factorial Designsand wed be
right! Lets consider the Three-Factor ANOVA case.
Effect on observation due to 3rd treatment level
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DOE 12-1b General Factorial Design
We might expect that the Two-Factor Factorial
Design could be extended to Three-Factor (and
higher) Factorial Designsand wed be
right! Lets consider the Three-Factor ANOVA case.
Effect due to row-column interaction
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DOE 12-1b General Factorial Design
We might expect that the Two-Factor Factorial
Design could be extended to Three-Factor (and
higher) Factorial Designsand wed be
right! Lets consider the Three-Factor ANOVA case.
Effect due to row-3rd-factor interaction
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DOE 12-1b General Factorial Design
We might expect that the Two-Factor Factorial
Design could be extended to Three-Factor (and
higher) Factorial Designsand wed be
right! Lets consider the Three-Factor ANOVA case.
Effect due to column-3rd-factor interaction
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DOE 12-1b General Factorial Design
We might expect that the Two-Factor Factorial
Design could be extended to Three-Factor (and
higher) Factorial Designsand wed be
right! Lets consider the Three-Factor ANOVA case.
and this is only part of the terms that affect
the observation
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DOE 12-1b General Factorial Design
We might expect that the Two-Factor Factorial
Design could be extended to Three-Factor (and
higher) Factorial Designsand wed be
right! Lets consider the Three-Factor ANOVA case.
a 3-Factor Interaction effect
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DOE 12-1b General Factorial Design
We might expect that the Two-Factor Factorial
Design could be extended to Three-Factor (and
higher) Factorial Designsand wed be
right! Lets consider the Three-Factor ANOVA case.
the experimental error
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DOE 12-1b General Factorial Design
We might expect that the Two-Factor Factorial
Design could be extended to Three-Factor (and
higher) Factorial Designsand wed be
right! Lets consider the Three-Factor ANOVA case.
and with
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The Sums of Squares needed for the F-statistic
are easily generalized
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The Sums of Squares needed for the F-statistic
are easily generalized
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The Sums of Squares needed for the F-statistic
are easily generalized
The Sums of Squares for the main effects of
treatments
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and
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and
The Sums of Squares for the Interactions are
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and
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and
Finally, the 3-factor interaction term is
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These allow us to find the Sum of Squares of the
Error,
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These allow us to find the Sum of Squares of the
Error,
These Sums of Squares allow us to calculate
F-statistics for all direct and interaction
treatment effectsthere will be one F0 per A, B,
C, AB, BC, AC, and ABC (7 in all!).
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These allow us to find the Sum of Squares of the
Error,
These Sums of Squares allow us to calculate
F-statistics for all direct and interaction
treatment effectsthere will be one F0 per A, B,
C, AB, BC, AC, and ABC (7 in all!). For example,
to test the significance of treatment A,
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For the significance of treatment B,
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For the significance of treatment B,
For the significance of treatment C,
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For the significance of treatment B,
For the significance of treatment C,
For the significance of treatment interaction AB,
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For the significance of treatment interaction AC,
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For the significance of treatment interaction AC,
For the significance of treatment interaction BC,
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For the significance of treatment interaction AC,
For the significance of treatment interaction BC,
For the significance of treatment interaction ABC,
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For a main effect (A, B or C X), treatment X
(given an a value) will have a statistically
significant effect on the observations if
where
and X A, B, or Cwith x a, b, or cas
appropriate. (We have been using A to represent
row treatments, B to represent column treatments,
and C as the symbol for the 3rd treatment.
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For an interaction effect (A, B or C X, Y),
interaction treatment XY (given an a value) will
have a statistically significant effect on the
observations if
where
and X, Y A, B, or Cwith x, y a, b, or cas
appropriate. (We have been using A to represent
row treatments, B to represent column treatments,
and C as the symbol for the 3rd treatment.
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Example A soft drink bottler is interested in
obtaining more uniform fill levels in the bottles
produced by his manufacturing process. The
filling machine theoretically fills each bottle
to the correct target filling height, but in
practice there is variation around this target.
The bottler would like to better understand the
source of this variability so as to eventually be
able to reduce it.
The process engineer can control three variables
during the filling process percent carbonation
(A), operating pressure in the filling machine
(B), and the number of bottles produced per
minuteline speed (C).
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The filler pressure and the line speed are easy
to control but the carbonation isnt.
Carbonation depends on product temperature which
is harder to control.
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The filler pressure and the line speed are easy
to control but the carbonation isnt.
Carbonation depends on product temperature which
is harder to control. For the experiment, the
decision is made to control carbonation at three
levels 10, 12 and 14. Two levels are chosen
for filling pressure 25 psi and 30 psi. Two
levels are selected for line speed 200 bpm and
250 bpm bottles per minute. The experimenter
decides to do two repetitions, per set of
treatment levels. All runs (3222 24) are
done in random order.
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The filler pressure and the line speed are easy
to control but the carbonation isnt.
Carbonation depends on product temperature which
is harder to control. For the experiment, the
decision is made to control carbonation at three
levels 10, 12 and 14. Two levels are chosen
for filling pressure 25 psi and 30 psi. Two
levels are selected for line speed 200 bpm and
250 bpm bottles per minute. The experimenter
decides to do two repetitions, per set of
treatment levels. All runs (3222 24) are
done in random order. The observation made is the
average deviation from the target fill height
observed in a production run of bottles at each
set of conditionsthis is the so-called response
variable.
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The filler pressure and the line speed are easy
to control but the carbonation isnt.
Carbonation depends on product temperature which
is harder to control. For the experiment, the
decision is made to control carbonation at three
levels 10, 12 and 14. Two levels are chosen
for filling pressure 25 psi and 30 psi. Two
levels are selected for line speed 200 bpm and
250 bpm bottles per minute. The experimenter
decides to do two repetitions, per set of
treatment levels. All runs (3222 24) are
done in random order. The observation made is the
average deviation from the target fill height
observed in a production run of bottles at each
set of conditionsthis is the so-called response
variable. Positive deviations indicate fill
heights above the target, negative indicate fill
heights below the fill height target.
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Data Fill Height Deviations (mm)
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Data Fill Height Deviations (mm)
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Finding the Sums of Squares
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This could also be called the SSCarbonation.
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This could also be called the SSPressure.
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This could also be called the SSLine Speed.
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This is the SS for the Carbonation-Operating
Pressure interaction.
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The Sum of Squares for the Carbonation-Line Speed
interaction.
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This is the Sum of Squares for the Pressure-Line
Speed interaction.
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The Sum of Squares for the three-way interaction
of the Carbonation, Operating Pressure, and Line
Speed.
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The Sum of Squares for the Error is thus,
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The Sum of Squares for the Error is thus,
The significance of Carbonation in affecting fill
height deviation depends on
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Carbonation A will be significant in affecting
fill height deviation if this F0 is
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Carbonation A will be significant in affecting
fill height deviation if this F0 is
For a 0.01, using appendix 612 this
inequality becomes 178.412 gt 6.93. It is clear
that Carbonation has a significant effect on
variability in fill height along the bottling
line.
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Carbonation A will be significant in affecting
fill height deviation if this F0 is
For a 0.01, using appendix 612 this
inequality becomes 178.412 gt 6.93. It is clear
that Carbonation has a significant effect on
variability in fill height along the bottling
line. In fact, p lt 0.0001 in this case.
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The significance of filler pressure in affecting
fill height deviation depends on
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The significance of filler pressure in affecting
fill height deviation depends on
Operating pressure in the filler B will be
significant in affecting fill height deviation if
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The significance of filler pressure in affecting
fill height deviation depends on
Operating pressure in the filler B will be
significant in affecting fill height deviation if
If a 0.01 then this becomes 64.059 gt 9.33, so
pressure does cause variations in filling height.
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Continuing to check the significance of
treatments produces the following outcomes (all
for a 0.01)
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• Continuing to check the significance of
  treatments produces the following outcomes (all
  for a 0.01)
• Line Speed is significant as a treatment on fill
  height. (So all three direct effectscarbonation
  , pressure, and line speedare significant.

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• Continuing to check the significance of
  treatments produces the following outcomes (all
  for a 0.01)
• Line Speed is significant as a treatment on fill
  height. (So all three direct effectscarbonation
  , pressure, and line speedare significant.
• None of the interaction is significant at the a
  0.01 level, though the Carbonation-Operating
  Pressure would be if a were increased to 0.06.
  (None of the other interactions becomes
  significant unless a is increased to near 0.25.