DESIGN OF EXPERIMENTS

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DESIGN OF EXPERIMENTS

• Process Improvement
• Single-factor Experiments
• ANOVA
• Two-Factor Experiments

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DOE FOR PROCESS IMPROVEMENT

• How can a process be improved?
• Intuitive use of the 7 QC tools and the PDCA
  cycle
• Works well for obvious improvements
• A rigorous experiment which measures the impact
  of different "factors" on the "response
• May be needed for more complex processes
• Factors may interact and confuse the results

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DOE FOR PROCESS IMPROVEMENT

• The factors are process inputs
• Some are controllable
• Machine settings
• Material selection
• Some may be uncontrollable
• Temperature
• Customer abuse of product
• The response will be a key quality characteristic

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SINGLE-FACTOR EXPERIMENTS

• Define
• a -- number of levels of factor
• n -- number of observations at each level
• yij - response of jth observation at level i
• Suppose we want to see if cooking time affects
  the chewiness of chocolate chip cookies
• Chewiness measured by elongation
• a 4 different time settings
• n 3 observations at each level

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SINGLE-FACTOR EXPERIMENTS

• We can describe the data by the following model

• Where
• m -- overall mean
• ti -- effect of level i of factor
• eij -- random error (assumed to be normally
  distributed with equal variance)
• We want to measure the ti and determine if any
  are significant

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SINGLE-FACTOR EXPERIMENTS

• To estimate the mean and the factor effects, we
  find

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SINGLE-FACTOR EXPERIMENTS

• We then can estimate the effects as follows

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ANALYSIS OF VARIANCE

• To determine whether these estimates are
  significant, we use the "analysis of variance" to
  test
• H0 t1 t2 ... ta 0
• Against
• H1 ti ltgt 0 for some i
• Procedure
• 1) Total variance about the mean is partitioned
  into components attributed to the factor or
  error (the "sums of squares")
• 2) The "mean sum of squares" for each component
  is calculated
• 3) An F-test is performed to test for
  significance

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PARTITIONING THE VARIANCE
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PARTITIONING THE VARIANCE

• In our example,

• SST, SSF and SSE are the Total, Factor and
  Error sums of squares
• So is this a significant factor sum of squares?
  Need to find the mean sums of squares.

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THE MEAN SUMS OF SQUARES

• The mean sums of squares are calculated based
  upon the sums of squares and the corresponding
  degrees of freedom

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THE FINAL TEST

• The ratio of MSF to MSE follows the F
  distribution with a degrees of freedom in the
  numerator and a(n-1) in the denominator

• If F0 gt Fa,a-1,a(n-1), reject otherwise accept
  H0
• Since F0 9.99 gt F0.1,3,8 2.92, reject H0 at
  least one time setting makes a difference

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THE F DISTRIBUTION
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TWO-FACTOR EXPERIMENTS

• The same basic approach is followed if we have
  more than one factor
• Define
• a -- number of levels of factor A
• b -- number of levels of factor B
• n -- number of observations for each combination
  of a and b
• yijk - response of kth observation at level i for
  factor a and level j for factor b

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TWO-FACTOR EXPERIMENTS

• Suppose we want to see if syrup content as well
  as cooking temperature affects the chewiness of
  chocolate chip cookies
• Chewiness measured by elongation
• a 2 different syrup contents
• b 2 different temperature settings
• n 2 observations at each factor level
  combination

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TWO-FACTOR EXPERIMENTS

• We can describe the data by the following model

• Where
• m -- overall mean
• ai -- effect of level i of factor A
• bi -- effect of level j of factor B
• abij interaction effect of i of factor A and
  level j of factor B
• eijk -- random error (assumed to be normally
  distributed with equal variance)

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