Two-factor Analysis of Variance (Chapter 15.5)

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Lecture 16

• Two-factor Analysis of Variance (Chapter 15.5)
• Homework 4 has been posted. It is due Friday,
  March 21st.

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Two-way ANOVA (two factors)
Convenience
Quality
Price
City 1 sales
City3 sales
City 5 sales
TV
City 2 sales
City 4 sales
City 6 sales
Newspapers
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Main Effects

• Marginal mean of level of factor A The mean of
  the level of factor A across all levels of factor
  B.
• The main effects of factor A refer to how the
  marginal means of levels of factor A change as
  the level of A change
• In the absence of interactions, the main effects
  have a straightforward interpretation What
  happens to the mean as we change the level of
  factor A and keep the level of factor B fixed.

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Interactions

• There is an interaction between A and B if the
  difference in means for the different levels of
  factor A changes as the level of factor B
  changes.
• If there are interactions, the main effects no
  longer have a clear interpretation. Need to
  examine the means of all combinations of levels
  of A and B (e.g., by using an interaction plot).

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F tests for the Two-way ANOVA

• Test for the difference between the levels of the
  main factors A and
  B

SS(A)/(a-1)
SS(B)/(b-1)
SSE/(n-ab)
Rejection region F gt Fa,a-1 ,n-ab
F gt Fa, b-1, n-ab

• Test for interaction between factors A and B

SS(AB)/(a-1)(b-1)
Rejection region F gt Fa,(a-1)(b-1),n-ab
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Required conditions

• The response distributions are normal
• The treatment variances are equal.
• The samples are independent simple random
  samples.
• Note There are ab populations (and samples),
  one for each combination of levels of factor A
  and B.

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F tests for the Two-way ANOVA

• Example 15.3 continued( Xm15-03)

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F tests for the Two-way ANOVA

• Example 15.3 continued
• Test of the difference in mean sales between the
  three marketing strategies
• H0 mconv. mquality mprice
• H1 At least two mean sales are different

Factor A Marketing strategies
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F tests for the Two-way ANOVA

• Example 15.3 continued
• Test of the difference in mean sales between the
  three marketing strategies
• H0 mconv. mquality mprice
• H1 At least two mean sales are different
• F MS(Marketing strategy)/MSE 5.33
• Fcritical Fa,a-1,n-ab F.05,3-1,60-(3)(2)
  3.17 (p-value .0077)
• At 5 significance level there is evidence to
  infer that differences in weekly sales exist
  among the marketing strategies.

MS(A)/MSE
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F tests for the Two-way ANOVA

• Example 15.3 - continued
• Test of the difference in mean sales between the
  two advertising media
• H0 mTV. mNespaper
• H1 The two mean sales differ

Factor B Advertising media
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F tests for the Two-way ANOVA

• Example 15.3 - continued
• Test of the difference in mean sales between the
  two advertising media
• H0 mTV. mNespaper
• H1 The two mean sales differ
• F MS(Media)/MSE 1.42
• Fcritical Fa,a-1,n-ab F.05,2-1,60-(3)(2)
  4.02 (p-value .2387)
• At 5 significance level there is insufficient
  evidence to infer that differences in weekly
  sales exist between the two advertising media.

MS(B)/MSE
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F tests for the Two-way ANOVA

• Example 15.3 - continued
• Test for interaction between factors A and B
• H0 mTVconv. mTVquality mnewsp.price
• H1 At least two means differ

Interaction AB MarketingMedia
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F tests for the Two-way ANOVA

• Example 15.3 - continued
• Test for interaction between factor A and B
• H0 mTVconv. mTVquality mnewsp.price
• H1 At least two means differ
• F MS(MarketingMedia)/MSE .09
• Fcritical Fa,(a-1)(b-1),n-ab
  F.05,(3-1)(2-1),60-(3)(2) 3.17 (p-value .9171)
• At 5 significance level there is insufficient
  evidence to infer that the two factors interact
  to affect the mean weekly sales.

MS(AB)/MSE
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Randomized Blocks vs. Two-Way ANOVA

• The randomized block design is a special case of
  two-way ANOVA in which the blocks are the second
  factor and the number of replications is 1.
• However, in analyzing a randomized blocks design,
  we assume that there are no interactions.
• Also, in a randomized block design, blocking is
  specifically performed to reduce variation and
  there is no interest in the block effect itself.
  In the general two-way design, the effect of both
  of the factors is of interest.

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Advantages of two-way ANOVA

• When interested in studying the effects of two
  factors, two-way designs offer great advantages
  over several single-factor studies.
• Example Researchers want to determine the
  influence of dietary minerals on blood pressure.
  Rats receive diets prepared with varying amounts
  of calcium and varying amounts of magnesium, but
  with all other ingredients of the diets the same.
  There are three levels of calcium (low, medium
  and high) and three levels of magnesium.

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Two Designs

• Budget allows 90 rats to be studied.
• Two-way design Give each combination of calcium
  and magnesium to 9 rats (requires 81 total rats)
• Two one-way designs For the first experiment,
  give each of the three levels of calcium with a
  medium level of magnesium to 15 rats. For the
  second experiment, give each of the three levels
  of magnesium with a medium level of calcium to 15
  rats (requires 90 total rats)

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Advantages of two-way ANOVA

• In two-way experiment, 27 rats are assigned to
  each of the three calcium diets. In the one-way
  experiment, there are only 15 rats assigned to
  each of the calcium diets.
• For studying the marginal means of calcium, the
  two-way design can be more efficient because it
  is a block design with magnesium levels as
  blocks.
• The two-way design allows interactions between
  calcium and magnesium to be studied.

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Advantages of two-way designs compared to one-way
designs

• It is more efficient to study two factors
  simultaneously rather than separately.
• For studying the effect of one factor, the
  two-way design is like a randomized block design
  and inherits block designs advantages if second
  factor influences the response
• We can investigate interactions between factors.

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Practice Problems

• 15.48,15.72
• To format the data files, use cut and paste to
  copy labels. Then use tables, stack.