N-way ANOVA

42 Pages·2010·2.43 MB·English

ANOVA
One-factor ANOVA by example
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One-factor ANOVA by visual inspection
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One-factor ANOVA H
0
H : µ = µ = µ = …
0 1 2 3
H : not all means are equal
A
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One-factor ANOVA but why not t-tests
• t-tests?
• 3+2+1 tests -> multiple comparisons
• The variance is correctly estimated
• We need a method that uses the full dataset

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One-factor ANOVA the cook book I
• Find the Within groups SS
Fx: 𝑆𝑆1 = 𝑥 − 𝑥 2 = 8.2 − 6 2 +
𝑖 𝑖
8.2 − 7 2 + 8.2 − 8 2 + 8.2 − 8 2 +
8.2 − 9 2 + 8.2 − 11 2 = 14.4
Sum the sum of squares from each group:
SS1+SS2+SS3+SS4 = 14.4+8.8+20.8+13.3
=57.8
df = 20
• Within group variance
𝑤𝑖𝑡𝑕𝑖𝑛 𝑔𝑟𝑜𝑢𝑝 𝑆𝑆 57.8
• = = = 2.9
𝑑𝑓 20

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One-factor ANOVA the cook book II
• Find the total SS
𝑆𝑆𝑡𝑜𝑡 = 𝑥 − 𝑥 2 = 140.0
𝑖
𝑖
df = 23

• Find the between group SS
𝑆𝑆𝑏𝑒𝑡𝑤𝑒𝑒𝑛 = 𝑛 𝑥 − 𝑥 2
𝑚
= 6( 8.2 − 7.5 2 + 5.8 − 7.5 2
+ 10.2 − 7.5 2 + 5.7 − 7.5 2) = 82.1
df = 3

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The ANOVA table
ANOVA
Outcome
Sum of Squares df Mean Square F Sig.
Between Groups 82,125 3 27,375 9,467 ,000
Within Groups 57,833 20 2,892
Total 139,958 23
Variance aka mean square aka s2 is simply SS/df
F is the Between SS devided by the Within SS

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Assumptions
• The data needs to be normal distributed in the
groups
• The variance needs to be equal in all groups:
homoscedasticity
• The groups needs to be independent
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Multiple comparisons procedures aka post hoc analysis
• Rejecting H only states that one or more
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pairs of means are different, but not which.
• Tukeys multiple comparisons test as an
example.

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