Discussion section 227
10/26/10
ANOVA (Analysis of Variance)
Overview
•
ANOVA is a procedure we use to test the null hypothesis when we want to evaluate the mean
differences between two or more groups.
•
The main advantage of ANOVA is that we can make multiple comparisons between groups while
keeping the alpha level constant.
Hypotheses
•
H
0
: µ
1
= µ
2
= µ
3
•
H
1
: At least one population mean is different
The F statistic
•
The ratio for the F statistic is similar to the tstatistic but uses variance differences instead of
mean differences
•
Variance is computed by dividing SS (sum of squares) by df (degrees of freedom)
ܨ=
ܸܽݎ݅ܽ݊ܿ݁ ܾ݁ݐݓ݁݁݊ ݏ݈ܽ݉݁ ݉݁ܽ݊ݏ (݃ݎݑݏ)
ܸܽݎ݅ܽ݊ܿ݁ ݓ݅ݐℎ݅݊ ݃ݎݑݏ (݁ݔ݁ܿݐ݁݀ ݀ݑ݁ ݐ ܿℎܽ݊ܿ݁ ݎ ݁ݎݎݎ)
This can also be stated in another way:
ܨ=
ܶݎ݁ܽݐ݉݁݊ݐ ݂݂݁݁ܿݐ+݂݂݀݅݁ݎ݁݊ܿ݁ݏ ݀ݑ݁ ݐ ܿℎܽ݊ܿ݁
ܦ݂݂݅݁ݎ݁݊ܿ݁ݏ ݀ݑ݁ ݐ ܿℎܽ݊ܿ݁
If the null hypothesis is true, the there will be no treatment effect and the ratio would be F=1
Formulas
•
Sums of Squares
ܵܵ
௧௪
=݊σ(ܯ−ܩܯ)
ଶ
GM is the grand mean
ܵܵ
௪௧
= σܵܵ
σ(ܺ−ܯ)
ଶ
For each group
ܵܵ
௧௧
=σ(ܺ−ܩܯ)
ଶ
•
Degrees of freedom
݂݀
௧௪
=ܭ−1
K is the number of groups (means)
݂݀
௧௧
=ܰ−1=݂݀
௧௪
+݂݀
௪௧
N is the total number of people in the study
݂݀
௪௧
=σ(݊−1)=σ݂݀=ܰ−ܭ
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Notice that because the total variance is sectioned into variance
between
and
within
groups, the two
portions must then add back to the total:
SS
between
+SS
within
= ss
total
df
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 Fall '10
 Fairchild
 overweight male volunteers

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