Twolevel Factorial Designs

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Two-level Factorial Designs

• Bacteria Example
• Response Bill length
• Factors
• B Bacteria (Myco, Control)
• T Room Temp (Warm, Cold)
• I Inoculation (Eggs, Chicks)

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Two-level Factorial Designs
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Cube Plot
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Estimated Effects

• For a k-factor design with n replicates, the cell
  means are estimated as
• We can write any effect as a contrast
  interaction contrasts are obtained by
  element-wise multiplication of main effect
  contrast coefficients.

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Estimated Effects

• The resulting contrasts are mutually orthogonal.
• The contrasts (up to a scaling constant) can be
  summarized as a table of 1s.

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Orthogonal Contrast Coefficients
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Estimated Effects

• If we code contrast coefficients as 1, the
  estimated effects are
• These effects are twice the size of our usual
  ANOVA effects.

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Estimated Effects

• The sum of squares for the estimated effect can
  be computed using the sum of squares formula we
  learned for contrasts

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Estimated Effects

• Bacteria Example
• B effect(39.1938.9540.2140.78-39.77-40.23-40.3
  7-41.71)/4
• -.7375
• SSB(-.7375)2x21.088
• The entire ANOVA table for this example can be
  constructed in this way

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Testing Effects

• With replication (ngt1)
• Without replication (k large)
• Claim higher-order interactions are negligible
  and pool them
• For k6, if 3-way (and higher) interactions are
  negligible, 42 d.f. would be available for error

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Testing Effects

• Without replication--Normal Probability Plots
• If none of the effects is significant, the
  effects are orthogonal normal random variables
  with mean 0 and variance

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Testing Effects

• Because the effects are normal, they are also
  independent
• IID normal effects can be tested using a normal
  probability plot (Minitab Example)
• Yandell uses a half-normal plot
• You can pool values on the line as error and
  construct an ANOVA table

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Testing Effects

• Lenth (1989) developed a more formal test of
  effects.
• Denote the effects by ci, i1,,m.
• We say that the cis are iid N(0,t2), where t is
  their common standard error.

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Testing Effects

• Lenth develops two estimates of the common
  standard error, t, of the cis

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Testing Effects

• Though both are consistent estimates, PSE is more
  robust
• The following terms are used to test effects

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Testing Effects

• The df term was developed from a study of the
  empirical distribution of PSE2
• ME is a 1-a confidence bound for the absolute
  value of a single effect
• SME is an exact (since the effects are
  independent) simultaneous 1-a confidence bound
  for all m effects