2Way ANOVA

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2-Way ANOVA

• David Barron
• Jesus College

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Randomized Block F Test

• 1. Experimental units are assigned randomly to
  treatments
• 2. Uses blocking variable in addition to
  treatment variable
• Permits better assessment of treatment because
  error variation is reduced
• 3. Hypotheses r No. rows, c No. columns
• Treatment
• Blocking

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Randomized Block F Test Assumptions

• 1. Normality
• Populations are normally distributed
• 2. Homogeneity of Variance
• Populations have equal variances
• 3. Independence of Errors
• Independent random samples are drawn
• 4. No Interaction Between Blocks Treatments

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Total Variation Partitioning
Total Variation
SST
Variation Due to Treatment

• SSA

Variation Due to Random Sampling
Variation Due to Blocking

• SSE

SSBL
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Definitions
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Total variation
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Among-group variation
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SSBL and SSE
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Mean-squares
F-test statistic for differences in c Means
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Summary Table
Source of
Degrees
Sum of
Mean
F
Variation
of
Squares
Square
Freedom
(Variance)
Among
c - 1
SSA
MSA
MSA
Treatments
MSE
Among
r - 1
SSBL
MSBL
MSBL
Blocks
MSE
SSW
Error
(r-1)(c-1)
SSE
MSE
Same as Completely Randomized Design
Total
rc - 1
SST
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Restaurant ratings
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ANOVA Table
SS df MS F P-value Rows 283.375 5 5
6.675 3.781835 0.020456 Columns 1787.458 3 595.8
194 39.75811 2.23E-07 Error 224.7917 15 14.9861
1 Total 2295.625 23
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Scatter plot of restaurant ratings
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Two-Way ANOVA

• 1. Tests the equality of 2 or more population
  means for several independent variables
• 2. Same results as separate One-Way ANOVA on each
  variable -- when interaction not present
• 3. Hypotheses - Differences among
• (1) Treatment means, (2) Blocking means
• and (3) Interaction effect between (1) (2)

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Two-Way ANOVA Assumptions

• 1. Normality
• Populations are normally distributed
• 2. Homogeneity of Variance
• Populations have equal variances
• 3. Independence of Errors
• Independent random samples are drawn

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Total Variation Partitioning
Total Variation
SST
Variation Due to Treatment A
Variation Due to Treatment B

• SSFA

SSFB
Variation Due to Random Sampling
Variation Due to Interaction

• SSE

SSAB
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Two-Way ANOVA Summary Table
Source of
Degrees of
Sum of
Mean
F
Variation
Freedom
Squares
Square
A
r - 1
SSFA
MSFA
MSFA
(Row)
MSE
B
c - 1
SSFB
MSFB
MSFB
(Column)
MSE
AB
(r-1)(c-1)
SSAB
MSAB
MSAB
(Interaction)
MSE
Error
rc(n-1)
SSE
MSE
Same as Other Designs
Total
rcn - 1
SST