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10 The Analysis of Variance
http://www.luchsinger-mathematics.ch/Var_Reduction.jpg
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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?
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ANOVA: Graphical
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ANOVA: notationXij: jth measurement taken from the ith populationsample sizes: n1, …, nI
nT = n1 + … + nI
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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.
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ANOVA test statistic
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ANOVA test
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F Distribution
http://www.vosesoftware.com/ModelRiskHelp/index.htm#Distributions/Continuous_distributions/F_distribution.htm
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F curve and critical value
http://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVA
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Table A.9Critical Values
for F Distribution (first page)
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ANOVA Table: FormulasSource df SS MS
(Mean Square)F
Model(Between) I – 1
Error (Within) nT – I
Total nT – 1
inI2
i.i 1 j 1
(x x..)
inI2
iji 1 j 1
(x x..)
SSM SSMdfm I 1
T
SSE SSEdfe n I
MSMMSE
inI2
ij ii 1 j 1
(x x .)
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ANOVA Hypothesis test: Summary
H0: μ1 = μ2 = = μI
Ha: At least one i is different
Test statistic: Rejection Region: F ≥ F,dfm,dfe
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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?
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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 differentThe true mean amount of foreign material caught is not the same for all of the filters
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ANOVA: Example (cont)Source df SS MS FModel 4 13.32 3.33 37.84Error 40 3.53 0.088Total 44 16.85
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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.
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Problem with Multiple t tests
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Overall Risk of Type I Error in Using Repeated t Tests at = 0.05
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Table A.10: Studentized
Range
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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
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ANOVA: Example (cont)Source df SS MS FModel 4 13.32 3.33 37.84Error 40 3.53 0.088Total 44 16.85
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Example: Tukey (cont)
i – j x̅i - x̅j CI Same?1 – 2 0.7 (0.3, 1.1)1 – 3 1.2 (0.8, 1.6)1 – 4 0.2 (-0.2, 0.6)1 – 5 1.4 (1.0, 1.8)2 – 3 0.5 (0.1, 0.9)2 – 4 -0.5 (-0.9, -0.1)2 – 5 0.7 (0.3, 1.1)3 – 4 -1.0 (-1.4, -0.6)3 – 5 0.2 (-0.2, 0.2)4 – 5 1.2 (0.8, 1.6)
yes
yes
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Example: Tukey (cont)
x̅5. x̅3. x̅2. x̅4. x̅1.
13.1 13.3 13.8 14.3 14.5
x̅5. x̅3. x̅2. x̅4. x̅1.
13.1 13.3 13.8 14.3 14.5