Download Single Factor Analysis of Variance (ANOVA)

The Situation
Test Statistic
Computing the Quantities
Single Factor Analysis of Variance
(ANOVA)
Bernd Schröder
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Analyzes Responses from Several
Experiments or Treatments
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Analyzes Responses from Several
Experiments or Treatments
1. Data is sampled from multiple populations or from
experiments with multiple treatments.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Analyzes Responses from Several
Experiments or Treatments
1. Data is sampled from multiple populations or from
experiments with multiple treatments. Multiple means
“more than two.”
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Analyzes Responses from Several
Experiments or Treatments
1. Data is sampled from multiple populations or from
experiments with multiple treatments. Multiple means
“more than two.” For two, we can use hypothesis tests (the
exact tests are not covered in this course).
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Analyzes Responses from Several
Experiments or Treatments
1. Data is sampled from multiple populations or from
experiments with multiple treatments. Multiple means
“more than two.” For two, we can use hypothesis tests (the
exact tests are not covered in this course).
2. The characteristic that differentiates
populations/treatments is called the factor.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Analyzes Responses from Several
Experiments or Treatments
1. Data is sampled from multiple populations or from
experiments with multiple treatments. Multiple means
“more than two.” For two, we can use hypothesis tests (the
exact tests are not covered in this course).
2. The characteristic that differentiates
populations/treatments is called the factor. The different
treatments or populations are the levels of the factor.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Analyzes Responses from Several
Experiments or Treatments
1. Data is sampled from multiple populations or from
experiments with multiple treatments. Multiple means
“more than two.” For two, we can use hypothesis tests (the
exact tests are not covered in this course).
2. The characteristic that differentiates
populations/treatments is called the factor. The different
treatments or populations are the levels of the factor.
3. Examples.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Analyzes Responses from Several
Experiments or Treatments
1. Data is sampled from multiple populations or from
experiments with multiple treatments. Multiple means
“more than two.” For two, we can use hypothesis tests (the
exact tests are not covered in this course).
2. The characteristic that differentiates
populations/treatments is called the factor. The different
treatments or populations are the levels of the factor.
3. Examples.
I
Testing different levels of medication/toxins etc. for effect.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Analyzes Responses from Several
Experiments or Treatments
1. Data is sampled from multiple populations or from
experiments with multiple treatments. Multiple means
“more than two.” For two, we can use hypothesis tests (the
exact tests are not covered in this course).
2. The characteristic that differentiates
populations/treatments is called the factor. The different
treatments or populations are the levels of the factor.
3. Examples.
I
I
Testing different levels of medication/toxins etc. for effect.
Testing different soil samples for mineral content.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Analyzes Responses from Several
Experiments or Treatments
1. Data is sampled from multiple populations or from
experiments with multiple treatments. Multiple means
“more than two.” For two, we can use hypothesis tests (the
exact tests are not covered in this course).
2. The characteristic that differentiates
populations/treatments is called the factor. The different
treatments or populations are the levels of the factor.
3. Examples.
I
I
I
Testing different levels of medication/toxins etc. for effect.
Testing different soil samples for mineral content.
Testing the frequency of a given allele in different
races/ethnic groups.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology
1. I populations or treatments of equal size J are to be
compared.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology
1. I populations or treatments of equal size J are to be
compared.
2. µi denotes the actual mean of the ith population.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology
1. I populations or treatments of equal size J are to be
compared.
2. µi denotes the actual mean of the ith population.
3. Null hypothesis.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology
1. I populations or treatments of equal size J are to be
compared.
2. µi denotes the actual mean of the ith population.
3. Null hypothesis. H0 : µ1 = µ2 = · · · = µI
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology
1. I populations or treatments of equal size J are to be
compared.
2. µi denotes the actual mean of the ith population.
3. Null hypothesis. H0 : µ1 = µ2 = · · · = µI (no difference,
or, no effect)
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology
1. I populations or treatments of equal size J are to be
compared.
2. µi denotes the actual mean of the ith population.
3. Null hypothesis. H0 : µ1 = µ2 = · · · = µI (no difference,
or, no effect)
4. Alternative hypothesis.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology
1. I populations or treatments of equal size J are to be
compared.
2. µi denotes the actual mean of the ith population.
3. Null hypothesis. H0 : µ1 = µ2 = · · · = µI (no difference,
or, no effect)
4. Alternative hypothesis. Ha : At least two means differ.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology
1. I populations or treatments of equal size J are to be
compared.
2. µi denotes the actual mean of the ith population.
3. Null hypothesis. H0 : µ1 = µ2 = · · · = µI (no difference,
or, no effect)
4. Alternative hypothesis. Ha : At least two means differ.
5. For example, if among 10 pain relievers, all have a sample
average time until pain lessens of around 20 minutes and
one has a sample average of around 10 minutes
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology
1. I populations or treatments of equal size J are to be
compared.
2. µi denotes the actual mean of the ith population.
3. Null hypothesis. H0 : µ1 = µ2 = · · · = µI (no difference,
or, no effect)
4. Alternative hypothesis. Ha : At least two means differ.
5. For example, if among 10 pain relievers, all have a sample
average time until pain lessens of around 20 minutes and
one has a sample average of around 10 minutes, then it
pretty much looks like that one is different.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology
1. I populations or treatments of equal size J are to be
compared.
2. µi denotes the actual mean of the ith population.
3. Null hypothesis. H0 : µ1 = µ2 = · · · = µI (no difference,
or, no effect)
4. Alternative hypothesis. Ha : At least two means differ.
5. For example, if among 10 pain relievers, all have a sample
average time until pain lessens of around 20 minutes and
one has a sample average of around 10 minutes, then it
pretty much looks like that one is different.
When it’s not that obvious, we need a testing procedure
(finer analysis).
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology (cont.)
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology (cont.)
6. Xi,j is the random variable that denotes the jth measurement
from the ith population/treatment group.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology (cont.)
6. Xi,j is the random variable that denotes the jth measurement
from the ith population/treatment group.
xi,j will be the observed value (“as always”)
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology (cont.)
6. Xi,j is the random variable that denotes the jth measurement
from the ith population/treatment group.
xi,j will be the observed value (“as always”)
Data is often displayed in a matrix.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology (cont.)
6. Xi,j is the random variable that denotes the jth measurement
from the ith population/treatment group.
xi,j will be the observed value (“as always”)
Data is often displayed in a matrix.
∑Jj=1 Xij
7. Individual sample means: X i· =
J
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology (cont.)
6. Xi,j is the random variable that denotes the jth measurement
from the ith population/treatment group.
xi,j will be the observed value (“as always”)
Data is often displayed in a matrix.
∑Jj=1 Xij
7. Individual sample means: X i· =
J
The dot says we summed over the second variable.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology (cont.)
6. Xi,j is the random variable that denotes the jth measurement
from the ith population/treatment group.
xi,j will be the observed value (“as always”)
Data is often displayed in a matrix.
∑Jj=1 Xij
7. Individual sample means: X i· =
J
The dot says we summed over the second variable.
2
J
X
−
X
∑
ij
i·
j=1
8. Sample variance: Si2 =
J −1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
ANOVA Terminology (cont.)
6. Xi,j is the random variable that denotes the jth measurement
from the ith population/treatment group.
xi,j will be the observed value (“as always”)
Data is often displayed in a matrix.
∑Jj=1 Xij
7. Individual sample means: X i· =
J
The dot says we summed over the second variable.
2
J
X
−
X
∑
ij
i·
j=1
8. Sample variance: Si2 =
J −1
∑Ii=1 ∑Jj=1 Xij
9. Grand mean: X ·· =
IJ
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Underlying Assumptions and Their Consequences
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Underlying Assumptions and Their Consequences
1. All populations are assumed to be normally distributed
with the same variance σ 2 .
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Underlying Assumptions and Their Consequences
1. All populations are assumed to be normally distributed
with the same variance σ 2 . Hence all Xij are normally
distributed and E(Xij ) = µi and V(Xij ) = σ 2 .
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Underlying Assumptions and Their Consequences
1. All populations are assumed to be normally distributed
with the same variance σ 2 . Hence all Xij are normally
distributed and E(Xij ) = µi and V(Xij ) = σ 2 .
2. If the largest sample standard deviation is at most twice the
smallest sample standard deviation
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Underlying Assumptions and Their Consequences
1. All populations are assumed to be normally distributed
with the same variance σ 2 . Hence all Xij are normally
distributed and E(Xij ) = µi and V(Xij ) = σ 2 .
2. If the largest sample standard deviation is at most twice the
smallest sample standard deviation, then it is (still)
reasonable to assume that the σ s are equal.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Underlying Assumptions and Their Consequences
1. All populations are assumed to be normally distributed
with the same variance σ 2 . Hence all Xij are normally
distributed and E(Xij ) = µi and V(Xij ) = σ 2 .
2. If the largest sample standard deviation is at most twice the
smallest sample standard deviation, then it is (still)
reasonable to assume that the σ s are equal.
3. To check normality, use a normal probability plot.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Underlying Assumptions and Their Consequences
1. All populations are assumed to be normally distributed
with the same variance σ 2 . Hence all Xij are normally
distributed and E(Xij ) = µi and V(Xij ) = σ 2 .
2. If the largest sample standard deviation is at most twice the
smallest sample standard deviation, then it is (still)
reasonable to assume that the σ s are equal.
3. To check normality, use a normal probability plot.
4. If the null hypothesis µ1 = µ2 = · · · = µI is true
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Underlying Assumptions and Their Consequences
1. All populations are assumed to be normally distributed
with the same variance σ 2 . Hence all Xij are normally
distributed and E(Xij ) = µi and V(Xij ) = σ 2 .
2. If the largest sample standard deviation is at most twice the
smallest sample standard deviation, then it is (still)
reasonable to assume that the σ s are equal.
3. To check normality, use a normal probability plot.
4. If the null hypothesis µ1 = µ2 = · · · = µI is true, then all
sample averages should be close to each other.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Underlying Assumptions and Their Consequences
1. All populations are assumed to be normally distributed
with the same variance σ 2 . Hence all Xij are normally
distributed and E(Xij ) = µi and V(Xij ) = σ 2 .
2. If the largest sample standard deviation is at most twice the
smallest sample standard deviation, then it is (still)
reasonable to assume that the σ s are equal.
3. To check normality, use a normal probability plot.
4. If the null hypothesis µ1 = µ2 = · · · = µI is true, then all
sample averages should be close to each other.
5. To determine if the variation is consistent with the null
hypothesis
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Underlying Assumptions and Their Consequences
1. All populations are assumed to be normally distributed
with the same variance σ 2 . Hence all Xij are normally
distributed and E(Xij ) = µi and V(Xij ) = σ 2 .
2. If the largest sample standard deviation is at most twice the
smallest sample standard deviation, then it is (still)
reasonable to assume that the σ s are equal.
3. To check normality, use a normal probability plot.
4. If the null hypothesis µ1 = µ2 = · · · = µI is true, then all
sample averages should be close to each other.
5. To determine if the variation is consistent with the null
hypothesis, we compare a measure of the variance between
the samples
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Underlying Assumptions and Their Consequences
1. All populations are assumed to be normally distributed
with the same variance σ 2 . Hence all Xij are normally
distributed and E(Xij ) = µi and V(Xij ) = σ 2 .
2. If the largest sample standard deviation is at most twice the
smallest sample standard deviation, then it is (still)
reasonable to assume that the σ s are equal.
3. To check normality, use a normal probability plot.
4. If the null hypothesis µ1 = µ2 = · · · = µI is true, then all
sample averages should be close to each other.
5. To determine if the variation is consistent with the null
hypothesis, we compare a measure of the variance between
the samples (“between-samples” variation)
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Underlying Assumptions and Their Consequences
1. All populations are assumed to be normally distributed
with the same variance σ 2 . Hence all Xij are normally
distributed and E(Xij ) = µi and V(Xij ) = σ 2 .
2. If the largest sample standard deviation is at most twice the
smallest sample standard deviation, then it is (still)
reasonable to assume that the σ s are equal.
3. To check normality, use a normal probability plot.
4. If the null hypothesis µ1 = µ2 = · · · = µI is true, then all
sample averages should be close to each other.
5. To determine if the variation is consistent with the null
hypothesis, we compare a measure of the variance between
the samples (“between-samples” variation) to a measure of
the variation “within” the samples.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Underlying Assumptions and Their Consequences
1. All populations are assumed to be normally distributed
with the same variance σ 2 . Hence all Xij are normally
distributed and E(Xij ) = µi and V(Xij ) = σ 2 .
2. If the largest sample standard deviation is at most twice the
smallest sample standard deviation, then it is (still)
reasonable to assume that the σ s are equal.
3. To check normality, use a normal probability plot.
4. If the null hypothesis µ1 = µ2 = · · · = µI is true, then all
sample averages should be close to each other.
5. To determine if the variation is consistent with the null
hypothesis, we compare a measure of the variance between
the samples (“between-samples” variation) to a measure of
the variation “within” the samples. (Remember that we
assume all populations have the same σ ).
logo1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
Louisiana Tech University, College of Engineering and Science
The Situation
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
Test Statistic
Computing the Quantities
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
1. Mean square for treatments.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
1. Mean square for treatments.
2
2 i
J h
MSTr =
X 1· − X ·· + · · · + X I· − X ··
I −1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
1. Mean square for treatments.
2
2
2 i
J I
J h
=
X i· − X ··
MSTr =
X 1· − X ·· + · · · + X I· − X ··
∑
I − 1 i=1
I −1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
1. Mean square for treatments.
2
2
2 i
J I
J h
=
X i· − X ··
MSTr =
X 1· − X ·· + · · · + X I· − X ··
∑
I − 1 i=1
I −1
2. Mean square for error
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
1. Mean square for treatments.
2
2
2 i
J I
J h
=
X i· − X ··
MSTr =
X 1· − X ·· + · · · + X I· − X ··
∑
I − 1 i=1
I −1
2. Mean square for error MSE =
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
S12 + · · · + SI2
.
I
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
1. Mean square for treatments.
2
2
2 i
J I
J h
=
X i· − X ··
MSTr =
X 1· − X ·· + · · · + X I· − X ··
∑
I − 1 i=1
I −1
2. Mean square for error MSE =
S12 + · · · + SI2
.
I
3. The test statistic for single factor ANOVA is F =
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
MSTr
.
MSE
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
1. Mean square for treatments.
2
2
2 i
J I
J h
=
X i· − X ··
MSTr =
X 1· − X ·· + · · · + X I· − X ··
∑
I − 1 i=1
I −1
2. Mean square for error MSE =
S12 + · · · + SI2
.
I
MSTr
.
MSE
4. The J in MSTr re-scales the spread of the means back to the spread of
individual samples.
3. The test statistic for single factor ANOVA is F =
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
1. Mean square for treatments.
2
2
2 i
J I
J h
=
X i· − X ··
MSTr =
X 1· − X ·· + · · · + X I· − X ··
∑
I − 1 i=1
I −1
2. Mean square for error MSE =
S12 + · · · + SI2
.
I
MSTr
.
MSE
4. The J in MSTr re-scales the spread of the means back to the spread of
individual samples.
3. The test statistic for single factor ANOVA is F =
5. If the null hypothesis is true: E(MSTr) = E(MSE) = σ 2 .
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
1. Mean square for treatments.
2
2
2 i
J I
J h
=
X i· − X ··
MSTr =
X 1· − X ·· + · · · + X I· − X ··
∑
I − 1 i=1
I −1
2. Mean square for error MSE =
S12 + · · · + SI2
.
I
MSTr
.
MSE
4. The J in MSTr re-scales the spread of the means back to the spread of
individual samples.
3. The test statistic for single factor ANOVA is F =
5. If the null hypothesis is true: E(MSTr) = E(MSE) = σ 2 . If the null
hypothesis is false: E(MSTr) > E(MSE) = σ 2 .
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
1. Mean square for treatments.
2
2
2 i
J I
J h
=
X i· − X ··
MSTr =
X 1· − X ·· + · · · + X I· − X ··
∑
I − 1 i=1
I −1
2. Mean square for error MSE =
S12 + · · · + SI2
.
I
MSTr
.
MSE
4. The J in MSTr re-scales the spread of the means back to the spread of
individual samples.
3. The test statistic for single factor ANOVA is F =
5. If the null hypothesis is true: E(MSTr) = E(MSE) = σ 2 . If the null
hypothesis is false: E(MSTr) > E(MSE) = σ 2 .
MSTr
6. When the null hypothesis is true, the statistic F =
has an
MSE
F-distribution with ν1 = I − 1 and ν2 = I(J − 1).
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
1. Mean square for treatments.
2
2
2 i
J I
J h
=
X i· − X ··
MSTr =
X 1· − X ·· + · · · + X I· − X ··
∑
I − 1 i=1
I −1
2. Mean square for error MSE =
S12 + · · · + SI2
.
I
MSTr
.
MSE
4. The J in MSTr re-scales the spread of the means back to the spread of
individual samples.
3. The test statistic for single factor ANOVA is F =
5. If the null hypothesis is true: E(MSTr) = E(MSE) = σ 2 . If the null
hypothesis is false: E(MSTr) > E(MSE) = σ 2 .
MSTr
6. When the null hypothesis is true, the statistic F =
has an
MSE
F-distribution with ν1 = I − 1 and ν2 = I(J − 1).
7. A rejection region f > Fα,I−1,I(J−1) gives a test of significance level α.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
1. Mean square for treatments.
2
2
2 i
J I
J h
=
X i· − X ··
MSTr =
X 1· − X ·· + · · · + X I· − X ··
∑
I − 1 i=1
I −1
2. Mean square for error MSE =
S12 + · · · + SI2
.
I
MSTr
.
MSE
4. The J in MSTr re-scales the spread of the means back to the spread of
individual samples.
3. The test statistic for single factor ANOVA is F =
5. If the null hypothesis is true: E(MSTr) = E(MSE) = σ 2 . If the null
hypothesis is false: E(MSTr) > E(MSE) = σ 2 .
MSTr
6. When the null hypothesis is true, the statistic F =
has an
MSE
F-distribution with ν1 = I − 1 and ν2 = I(J − 1).
7. A rejection region f > Fα,I−1,I(J−1) gives a test of significance level α.
8. For p-values, use the area to the right of the test statistic.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Example. Perform an ANOVA on
the enclosed test data to see if the
“true average performances” can be
considered equal.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Example. Perform an ANOVA on
the enclosed test data to see if the
“true average performances” can be
considered equal.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Keeping Track of the Data
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Keeping Track of the Data
The key to ANOVA (by hand) is orderly bookkeeping.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Keeping Track of the Data
The key to ANOVA (by hand) is orderly bookkeeping.
Also remember that all this was done before computers.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Keeping Track of the Data
The key to ANOVA (by hand) is orderly bookkeeping.
Also remember that all this was done before computers. So
anything that could save a few operations
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Keeping Track of the Data
The key to ANOVA (by hand) is orderly bookkeeping.
Also remember that all this was done before computers. So
anything that could save a few operations, or help minimize
rounding errors
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Keeping Track of the Data
The key to ANOVA (by hand) is orderly bookkeeping.
Also remember that all this was done before computers. So
anything that could save a few operations, or help minimize
rounding errors, was appreciated.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
Test Statistic
Computing the Quantities
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
I
Computing the Quantities
J
1. Grand total: x·· = ∑ ∑ xij
i=1 j=1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
I
Computing the Quantities
J
1. Grand total: x·· = ∑ ∑ xij
i=1 j=1
I
J
I
J
2. Total sum of squares: SST = ∑ ∑ (xij − x·· )2 = ∑ ∑ xij2 −
i=1 j=1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
i=1 j=1
1 2
x
IJ ··
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
I
Computing the Quantities
J
1. Grand total: x·· = ∑ ∑ xij
i=1 j=1
I
J
I
J
2. Total sum of squares: SST = ∑ ∑ (xij − x·· )2 = ∑ ∑ xij2 −
i=1 j=1
i=1 j=1
I
J
3. Treatment sum of squares: SSTr = ∑ ∑ (xi· − x·· )2 =
i=1 j=1
1 2
x
IJ ··
1 I 2 1 2
∑ xi· − IJ x·· ,
J i=1
J
where xi· = ∑ xij
j=1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
I
Computing the Quantities
J
1. Grand total: x·· = ∑ ∑ xij
i=1 j=1
I
J
I
J
2. Total sum of squares: SST = ∑ ∑ (xij − x·· )2 = ∑ ∑ xij2 −
i=1 j=1
i=1 j=1
I
J
3. Treatment sum of squares: SSTr = ∑ ∑ (xi· − x·· )2 =
i=1 j=1
1 2
x
IJ ··
1 I 2 1 2
∑ xi· − IJ x·· ,
J i=1
J
where xi· = ∑ xij
j=1
I
J
4. Error sum of squares: SSE = ∑ ∑ (xij − xi· )2
i=1 j=1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
I
Computing the Quantities
J
1. Grand total: x·· = ∑ ∑ xij
i=1 j=1
I
J
I
J
2. Total sum of squares: SST = ∑ ∑ (xij − x·· )2 = ∑ ∑ xij2 −
i=1 j=1
i=1 j=1
I
J
3. Treatment sum of squares: SSTr = ∑ ∑ (xi· − x·· )2 =
i=1 j=1
1 2
x
IJ ··
1 I 2 1 2
∑ xi· − IJ x·· ,
J i=1
J
where xi· = ∑ xij
j=1
I
J
4. Error sum of squares: SSE = ∑ ∑ (xij − xi· )2
i=1 j=1
5. MSTr =
SSTr
I −1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
I
Computing the Quantities
J
1. Grand total: x·· = ∑ ∑ xij
i=1 j=1
I
J
I
J
2. Total sum of squares: SST = ∑ ∑ (xij − x·· )2 = ∑ ∑ xij2 −
i=1 j=1
i=1 j=1
I
J
3. Treatment sum of squares: SSTr = ∑ ∑ (xi· − x·· )2 =
i=1 j=1
1 2
x
IJ ··
1 I 2 1 2
∑ xi· − IJ x·· ,
J i=1
J
where xi· = ∑ xij
j=1
I
J
4. Error sum of squares: SSE = ∑ ∑ (xij − xi· )2
i=1 j=1
5. MSTr =
SSTr
(What happened to J?
I −1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
I
Computing the Quantities
J
1. Grand total: x·· = ∑ ∑ xij
i=1 j=1
I
J
I
J
2. Total sum of squares: SST = ∑ ∑ (xij − x·· )2 = ∑ ∑ xij2 −
i=1 j=1
i=1 j=1
I
J
3. Treatment sum of squares: SSTr = ∑ ∑ (xi· − x·· )2 =
i=1 j=1
1 2
x
IJ ··
1 I 2 1 2
∑ xi· − IJ x·· ,
J i=1
J
where xi· = ∑ xij
j=1
I
J
4. Error sum of squares: SSE = ∑ ∑ (xij − xi· )2
i=1 j=1
5. MSTr =
SSTr
(What happened to J? It’s the dummy sum over j!)
I −1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
I
Computing the Quantities
J
1. Grand total: x·· = ∑ ∑ xij
i=1 j=1
I
J
I
J
2. Total sum of squares: SST = ∑ ∑ (xij − x·· )2 = ∑ ∑ xij2 −
i=1 j=1
i=1 j=1
I
J
3. Treatment sum of squares: SSTr = ∑ ∑ (xi· − x·· )2 =
i=1 j=1
1 2
x
IJ ··
1 I 2 1 2
∑ xi· − IJ x·· ,
J i=1
J
where xi· = ∑ xij
j=1
I
J
4. Error sum of squares: SSE = ∑ ∑ (xij − xi· )2
i=1 j=1
SSTr
(What happened to J? It’s the dummy sum over j!)
I −1
SSE
6. MSE =
I(J − 1)
5. MSTr =
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
I
Computing the Quantities
J
1. Grand total: x·· = ∑ ∑ xij
i=1 j=1
I
J
I
J
2. Total sum of squares: SST = ∑ ∑ (xij − x·· )2 = ∑ ∑ xij2 −
i=1 j=1
i=1 j=1
I
J
3. Treatment sum of squares: SSTr = ∑ ∑ (xi· − x·· )2 =
i=1 j=1
1 2
x
IJ ··
1 I 2 1 2
∑ xi· − IJ x·· ,
J i=1
J
where xi· = ∑ xij
j=1
I
J
4. Error sum of squares: SSE = ∑ ∑ (xij − xi· )2
i=1 j=1
SSTr
(What happened to J? It’s the dummy sum over j!)
I −1
MSTr
SSE
,
F=
6. MSE =
MSE
I(J − 1)
5. MSTr =
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Are the Claimed Formulas Right?
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Are the Claimed Formulas Right?
SST
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Are the Claimed
Formulas Right?
I J
SST = ∑ ∑ (xij − x·· )2
i=1 j=1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Are the Claimed
Formulas
Right?
I J
I J
2
SST = ∑ ∑ (xij − x·· ) = ∑ ∑ xij2 − 2xij x·· + x2··
i=1 j=1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
i=1 j=1
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Are the Claimed
Formulas
Right?
I J
I J
2
SST = ∑ ∑ (xij − x·· ) = ∑ ∑ xij2 − 2xij x·· + x2··
i=1 j=1
I
=
i=1 j=1
J
∑∑
I
xij2 − 2x··
i=1 j=1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
J
I
J
∑ ∑ xij + ∑ ∑ x2··
i=1 j=1
i=1 j=1
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Are the Claimed
Formulas
Right?
I J
I J
2
SST = ∑ ∑ (xij − x·· ) = ∑ ∑ xij2 − 2xij x·· + x2··
i=1 j=1
I
=
i=1 j=1
J
∑∑
I
xij2 − 2x··
i=1 j=1
I
J
J
I
J
∑ ∑ xij + ∑ ∑ x2··
i=1 j=1
i=1 j=1
I J
2 I J
= ∑ ∑ xij2 − ∑ ∑ xij ∑ ∑ xij + IJ
IJ i=1 j=1 i=1 j=1
i=1 j=1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
1 I J
∑ ∑ xij
IJ i=1
j=1
!2
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Are the Claimed
Formulas
Right?
I J
I J
2
SST = ∑ ∑ (xij − x·· ) = ∑ ∑ xij2 − 2xij x·· + x2··
i=1 j=1
I
=
i=1 j=1
J
∑∑
I
xij2 − 2x··
i=1 j=1
J
I
J
I
J
∑ ∑ xij + ∑ ∑ x2··
i=1 j=1
i=1 j=1
I J
2 I J
= ∑ ∑ xij2 − ∑ ∑ xij ∑ ∑ xij + IJ
IJ i=1 j=1 i=1 j=1
i=1 j=1
=
I
J
∑ ∑ xij2 −
i=1 j=1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
1 I J
∑ ∑ xij
IJ i=1
j=1
!2
I J
1 I J
x
∑ ∑ ij ∑ ∑ xij
IJ i=1
j=1 i=1 j=1
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Are the Claimed
Formulas
Right?
I J
I J
2
SST = ∑ ∑ (xij − x·· ) = ∑ ∑ xij2 − 2xij x·· + x2··
i=1 j=1
I
=
i=1 j=1
J
∑∑
I
xij2 − 2x··
i=1 j=1
J
I
J
I
J
∑ ∑ xij + ∑ ∑ x2··
i=1 j=1
i=1 j=1
I J
2 I J
= ∑ ∑ xij2 − ∑ ∑ xij ∑ ∑ xij + IJ
IJ i=1 j=1 i=1 j=1
i=1 j=1
=
I
J
∑ ∑ xij2 −
i=1 j=1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
1 I J
∑ ∑ xij
IJ i=1
j=1
!2
I J
I J
1 I J
1
x
x
=
xij2 − x··2
ij ∑ ∑ ij
∑
∑
∑
∑
IJ i=1 j=1 i=1 j=1
IJ
i=1 j=1
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Are the Claimed
Formulas
Right?
I J
I J
2
SST = ∑ ∑ (xij − x·· ) = ∑ ∑ xij2 − 2xij x·· + x2··
i=1 j=1
I
=
i=1 j=1
J
∑∑
I
xij2 − 2x··
i=1 j=1
J
I
J
I
J
∑ ∑ xij + ∑ ∑ x2··
i=1 j=1
i=1 j=1
I J
2 I J
= ∑ ∑ xij2 − ∑ ∑ xij ∑ ∑ xij + IJ
IJ i=1 j=1 i=1 j=1
i=1 j=1
=
I
J
∑ ∑ xij2 −
i=1 j=1
1 I J
∑ ∑ xij
IJ i=1
j=1
!2
I J
I J
1 I J
1
x
x
=
xij2 − x··2
ij ∑ ∑ ij
∑
∑
∑
∑
IJ i=1 j=1 i=1 j=1
IJ
i=1 j=1
Treatment sum of squares: Similar.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Fundamental Identity: SST = SSTr + SSE
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Fundamental Identity: SST = SSTr + SSE
xij − x··
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Fundamental Identity: SST = SSTr + SSE
xij − x·· = (xij − xi· ) + (xi· − x·· )
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Fundamental Identity: SST = SSTr + SSE
xij − x·· = (xij − xi· ) + (xi· − x·· )
(xij − x·· )2
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Fundamental Identity: SST = SSTr + SSE
xij − x·· = (xij − xi· ) + (xi· − x·· )
(xij − x·· )2 = (xij − xi· )2 + 2 (xij − xi· ) (xi· − x·· ) + (xi· − x·· )2
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Fundamental Identity: SST = SSTr + SSE
xij − x·· = (xij − xi· ) + (xi· − x·· )
(xij − x·· )2 = (xij − xi· )2 + 2 (xij − xi· ) (xi· − x·· ) + (xi· − x·· )2
Now sum over i,j.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Fundamental Identity: SST = SSTr + SSE
xij − x·· = (xij − xi· ) + (xi· − x·· )
(xij − x·· )2 = (xij − xi· )2 + 2 (xij − xi· ) (xi· − x·· ) + (xi· − x·· )2
Now sum over i,j. The middle term drops out after summing
over j
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Fundamental Identity: SST = SSTr + SSE
xij − x·· = (xij − xi· ) + (xi· − x·· )
(xij − x·· )2 = (xij − xi· )2 + 2 (xij − xi· ) (xi· − x·· ) + (xi· − x·· )2
Now sum over i,j. The middle term drops out after summing
J
over j, because
∑ (xij − xi·) = 0.
j=1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Fundamental Identity: SST = SSTr + SSE
xij − x·· = (xij − xi· ) + (xi· − x·· )
(xij − x·· )2 = (xij − xi· )2 + 2 (xij − xi· ) (xi· − x·· ) + (xi· − x·· )2
Now sum over i,j. The middle term drops out after summing
J
over j, because
∑ (xij − xi·) = 0. Hence
j=1
SST
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Fundamental Identity: SST = SSTr + SSE
xij − x·· = (xij − xi· ) + (xi· − x·· )
(xij − x·· )2 = (xij − xi· )2 + 2 (xij − xi· ) (xi· − x·· ) + (xi· − x·· )2
Now sum over i,j. The middle term drops out after summing
J
over j, because
∑ (xij − xi·) = 0. Hence
j=1
I
J
SST = ∑ ∑ (xij − x·· )2
i=1 j=1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Fundamental Identity: SST = SSTr + SSE
xij − x·· = (xij − xi· ) + (xi· − x·· )
(xij − x·· )2 = (xij − xi· )2 + 2 (xij − xi· ) (xi· − x·· ) + (xi· − x·· )2
Now sum over i,j. The middle term drops out after summing
J
over j, because
∑ (xij − xi·) = 0. Hence
j=1
I
J
I
J
I
J
SST = ∑ ∑ (xij − x·· )2 = ∑ ∑ (xij − xi· )2 + ∑ ∑ (xi· − x·· )2
i=1 j=1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
i=1 j=1
i=1 j=1
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Fundamental Identity: SST = SSTr + SSE
xij − x·· = (xij − xi· ) + (xi· − x·· )
(xij − x·· )2 = (xij − xi· )2 + 2 (xij − xi· ) (xi· − x·· ) + (xi· − x·· )2
Now sum over i,j. The middle term drops out after summing
J
over j, because
∑ (xij − xi·) = 0. Hence
j=1
I
J
I
J
I
J
SST = ∑ ∑ (xij − x·· )2 = ∑ ∑ (xij − xi· )2 + ∑ ∑ (xi· − x·· )2 = SSE + SSTr
i=1 j=1
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
i=1 j=1
i=1 j=1
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Fundamental Identity: SST = SSTr + SSE
xij − x·· = (xij − xi· ) + (xi· − x·· )
(xij − x·· )2 = (xij − xi· )2 + 2 (xij − xi· ) (xi· − x·· ) + (xi· − x·· )2
Now sum over i,j. The middle term drops out after summing
J
over j, because
∑ (xij − xi·) = 0. Hence
j=1
I
J
I
J
I
J
SST = ∑ ∑ (xij − x·· )2 = ∑ ∑ (xij − xi· )2 + ∑ ∑ (xi· − x·· )2 = SSE + SSTr
i=1 j=1
i=1 j=1
i=1 j=1
1. SST measures the total variation of the data.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Fundamental Identity: SST = SSTr + SSE
xij − x·· = (xij − xi· ) + (xi· − x·· )
(xij − x·· )2 = (xij − xi· )2 + 2 (xij − xi· ) (xi· − x·· ) + (xi· − x·· )2
Now sum over i,j. The middle term drops out after summing
J
over j, because
∑ (xij − xi·) = 0. Hence
j=1
I
J
I
J
I
J
SST = ∑ ∑ (xij − x·· )2 = ∑ ∑ (xij − xi· )2 + ∑ ∑ (xi· − x·· )2 = SSE + SSTr
i=1 j=1
i=1 j=1
i=1 j=1
1. SST measures the total variation of the data.
2. SSE is the contribution from the variation within the
populations/treatment groups.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Fundamental Identity: SST = SSTr + SSE
xij − x·· = (xij − xi· ) + (xi· − x·· )
(xij − x·· )2 = (xij − xi· )2 + 2 (xij − xi· ) (xi· − x·· ) + (xi· − x·· )2
Now sum over i,j. The middle term drops out after summing
J
over j, because
∑ (xij − xi·) = 0. Hence
j=1
I
J
I
J
I
J
SST = ∑ ∑ (xij − x·· )2 = ∑ ∑ (xij − xi· )2 + ∑ ∑ (xi· − x·· )2 = SSE + SSTr
i=1 j=1
i=1 j=1
i=1 j=1
1. SST measures the total variation of the data.
2. SSE is the contribution from the variation within the
populations/treatment groups.
3. SSTr is the contribution from between the
populations/groups.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Example. Perform an ANOVA on
the enclosed test data to see if the
“true average performances” can be
considered equal.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Example. Perform an ANOVA on
the enclosed test data to see if the
“true average performances” can be
considered equal. Use a significance
level of α = 0.05 and also compute
the p-value.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Test Statistic
Computing the Quantities
Example. Perform an ANOVA on
the enclosed test data to see if the
“true average performances” can be
considered equal. Use a significance
level of α = 0.05 and also compute
the p-value.
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
Test Statistic
Computing the Quantities
logo1
Louisiana Tech University, College of Engineering and Science
The Situation
Bernd Schröder
Single Factor Analysis of Variance (ANOVA)
Test Statistic
Computing the Quantities
logo1
Louisiana Tech University, College of Engineering and Science