Two-Factor Analysis of Variance (Chapter 15.5)

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Lecture 15

• Two-Factor Analysis of Variance (Chapter 15.5)

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Oneway Analysis of Sales By StrategyComparisons
for all pairs using Tukey-Kramer
HSDAbs(Dif)-LSD Quality Price Conv. Quality
-71.768 -27.418 3.682 Price -27.418
-71.76 -40.668 Conven. 3.682
-40.668 -71.768 Positive values show pairs of
means that are significantly different.
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Randomized Blocks ANOVA - Example
Blocks
Treatments
b-1
MST / MSE
MSB / MSE
K-1
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15.5 Two-Factor Analysis of Variance -

• Example 15.3
• Suppose in Example 15.1, two factors are to be
  examined
• The effects of the marketing strategy on sales.
• Emphasis on convenience
• Emphasis on quality
• Emphasis on price
• The effects of the selected media on sales.
• Advertise on TV
• Advertise in newspapers

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Attempting one-way ANOVA

• Solution
• We may attempt to analyze combinations of levels,
  one from each factor using one-way ANOVA.
• The treatments will be
• Treatment 1 Emphasize convenience and advertise
  in TV
• Treatment 2 Emphasize convenience and advertise
  in newspapers
• .
• Treatment 6 Emphasize price and advertise in
  newspapers

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Attempting one-way ANOVA

• Solution
• The hypotheses tested are
• H0 m1 m2 m3 m4 m5 m6
• H1 At least two means differ.

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Attempting one-way ANOVA

• Solution

• In each one of six cities sales are recorded
  for ten weeks.
• In each city a different combination of
  marketing emphasis and media usage is
  employed.

• City1 City2 City3 City4 City5 City6Convn
  ce Convnce Quality Quality
  Price Price
• TV Paper TV Paper
  TV Paper

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Attempting one-way ANOVA

• Solution

Xm15-03

• The p-value .0452.
• We conclude that there is evidence that
  differences exist in the mean weekly sales
  among the six cities.

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Interesting questions no answers

• These result raises some questions
• Are the differences in sales caused by the
  different marketing strategies?
• Are the differences in sales caused by the
  different media used for advertising?
• Are there combinations of marketing strategy and
  media that interact to affect the weekly sales?

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Two-way ANOVA (two factors)

• The current experimental design cannot provide
  answers to these questions.
• A new experimental design is needed.

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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
Are there differences in the mean sales caused
by different marketing strategies?
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Two-way ANOVA (two factors)

• Test whether mean sales of Convenience,
  Quality, and Price significantly differ from
  one another.
• H0 mConv. mQuality mPrice
• H1 At least two means differ

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Two-way ANOVA (two factors)
Factor A Marketing strategy
Convenience
Quality
Price
City 1 sales
City 3 sales
City 5 sales
TV
Factor B Advertising media
City 2 sales
City 4 sales
City 6 sales
Newspapers
Are there differences in the mean sales caused
by different advertising media?
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Two-way ANOVA (two factors)
Test whether mean sales of the TV, and
Newspapers significantly differ from one
another. H0 mTV mNewspapers H1 The means
differ
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Two-way ANOVA (two factors)
Factor A Marketing strategy
Convenience
Quality
Price
City 1 sales
City 5 sales
City 3 sales
TV
Factor B Advertising media
City 2 sales
City 4 sales
City 6 sales
Newspapers
Are there differences in the mean sales caused
by interaction between marketing strategy and
advertising medium?
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Two-way ANOVA (two factors)

• Test whether mean sales of certain cells are
  different than the level expected.
• Calculation are based on the sum of square for
  interaction SS(AB)

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Difference between the levels of factor A No
difference between the levels of factor B
Difference between the levels of factor A,
and difference between the levels of factor B
no interaction
M R e e s a p n o n s e
M R e e s a p n o n s e
Level 1 of factor B
Level 1and 2 of factor B
Level 2 of factor B
Levels of factor A
Levels of factor A
1
2
3
1
2
3
M R e e s a p n o n s e
M R e e s a p n o n s e
No difference between the levels of factor
A. Difference between the levels of factor B
Interaction
Levels of factor A
Levels of factor A
1
2
3
1
2
3
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Sums of squares
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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

1. The response distributions is normal
2. The treatment variances are equal.
3. The samples are independent random samples.

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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