10
The Analysis of
Variance
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http://www.luchsinger-mathematics.ch/Var_Reduction.jpg
ANOVA: Examples
1) Do four different types of steel have the same
structural strength?
2) Does the major of the student (math,
engineering, life sciences, economics, computer
science) have an effect on the student’s grade in
STAT 511?
3) Does the percentage of alcohol in gasoline has
an effect on the mpg?
4) Does the heat retention in a house depending
on the thickness or of insulation in the attic?
ANOVA: Graphical
ANOVA: notation
Xij: jth measurement taken from the ith population
sample sizes: n1, …, nI
𝑛𝑖
𝑗=1 𝑋𝑖𝑗
𝑋𝑖. =
𝑛𝑖
2
𝑛𝑖
𝑋𝑖𝑗 − 𝑋
𝑆𝑋𝑋
𝑗=1
2
𝑆𝑖 =
=
𝑛−1
𝑛−1
nT = n1 + … + nI
𝑛𝑖
𝐼
𝑖=1 𝑗=1 𝑋𝑖𝑗
𝑋.. =
𝑛𝑇
ANOVA: Assumptions
1. All samples are independent of each other.
2. Each population or treatment distributions
are normal with E(Xij) = I.
3. Each population has the same variance
(pooled), Var(Xij) = σ2.
ANOVA test statistic
ANOVA test
F Distribution
http://www.vosesoftware.com/ModelRiskHelp/index.htm#Distributions/
Continuous_distributions/F_distribution.htm
F curve and critical value
http://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVA
Table A.9
Critical Values
for F
Distribution
(first page)
ANOVA Table: Formulas
Source
Model
(Between)
Error
(Within)
df
SS
I
I–1
 (x
i1 j1
I
i.
x..)
2
ni
2
(x

x
.)
nT – I  ij i
i1 j1
I
Total
ni
ni
2
(x

x..)
nT – 1  ij
i1 j1
MS
(Mean Square)
F
SSM SSM

dfm
I 1
MSM
MSE
SSE SSE

dfe nT  I
ANOVA Hypothesis test: Summary
H0: μ1 = μ2 =  = μI
Ha: At least one i is different
Test statistic: 𝐹 =
𝑀𝑆𝑀
𝑀𝑆𝐸
Rejection Region: F ≥ F,dfm,dfe
ANOVA: Example
An experiment was carried out to compare five
different brands of automobile oil filters with
respect to their ability to capture foreign
material. A sample of nine filters of each
brand was used. Do the filters capture the
same amount of foreign material at a 0.05
significance level?
ANOVA: Example (cont)
2. H0: 1 = 2 = 3 = 4 = 5
The true mean amount of foreign material is
the same for all of the filters
HA: at least one of the i is different
The true mean amount of foreign material
caught is not the same for all of the filters
ANOVA: Example (cont)
Source
Model
Error
Total
df
4
40
44
SS
MS
F
13.32 3.33 37.84
3.53 0.088
16.85
Example: ANOVA (cont)
7. The data does provide strong support to the
claim that the mean amount of foreign
material caught is not the same for all of the
filters.
Problem with Multiple t tests
Overall Risk of Type I Error in Using
Repeated t Tests at  = 0.05
Table A.10:
Studentized
Range
ANOVA: Example (Tukey)
An experiment was carried out to compare five
different brands of automobile oil filters with
respect to their ability to capture foreign
material. A sample of nine filters of each brand
was used. Do the filters capture the same
amount of foreign material at a 0.05 significance
level?
Which one(s) of the filters is best?
x̅1. = 14.5 x̅2. = 13.8 x̅3. = 13.3 x̅4. = 14.3 x̅5. = 13.1
ANOVA: Example (cont)
Source
Model
Error
Total
df
4
40
44
SS
MS
F
13.32 3.33 37.84
3.53 0.088
16.85
Example:
Tukey
(cont)
i–j
1–2
1–3
1–4
1–5
2–3
2–4
2–5
3–4
3–5
4–5
x̅i - x̅j
0.7
1.2
0.2
1.4
0.5
-0.5
0.7
-1.0
0.2
1.2
CI
Same?
(0.3, 1.1)
(0.8, 1.6)
(-0.2, 0.6) yes
(1.0, 1.8)
(0.1, 0.9)
(-0.9, -0.1)
(0.3, 1.1)
(-1.4, -0.6)
(-0.2, 0.2) yes
(0.8, 1.6)
Example: Tukey (cont)
x̅5.
13.1
x̅3.
13.3
x̅2.
13.8
x̅4.
14.3
x̅1.
14.5
x̅5.
13.1
x̅3.
13.3
x̅2.
13.8
x̅4.
14.3
x̅1.
14.5