11-w11-stats250-bgunderson-chapter-16-intro-to
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Transcript 11-w11-stats250-bgunderson-chapter-16-intro-to
Author(s): Brenda Gunderson, Ph.D., 2011
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Analysis of Variance (page 169)
Method to compare means of two or more normal
populations based on independent random samples when
population variances assumed to be equal.
ANALYSIS OF VARIANCE (ANOVA)
Extension of two indep samples POOLED t-test
Recall two independent samples t-test:
H0: _____________________
Assumptions …
t statistic with _______________ df.
ANOVA Assumptions
Popul 1
Popul 2
Popul k
Assumptions:
Each sample is a ... random sample.
The k random samples are ... independent.
For each popul, the model for response is a ... normal model.
The k population variances are .... Equal.
ANOVA Hypotheses
page 170
H0: _______________________________________
Ha: _______________________________________
One possible Ha picture
ANOVA Hypotheses
Question: Why call a technique for testing the
equality of the means “analysis of VARIANCE”?
Answer: We are going to compare two estimators
of 2, the common population variance.
• MS Groups (Mean Square between Groups)
• MSE (Mean Square Within or due to Error):
F Test Statistic
These two estimates are used to form the F statistic:
F
Variation among sample means
Natural variation within groups
MS Groups
MSE
Think about this form … which MS is in numerator?
which MS is in the denominator?
If this F ratio is too __________________
we would reject the null hypothesis.
How will you fill in the blank?
Read through the Logic of ANOVA on your own!
Computing the F Test Statistic
…
page 173
Data from Population1
Data from Population 2
Data from Population k
X 11
X 12
X 21
X 22
X k1
X k2
X 1n1
X 2n2
X knk
Step 1: Calculate the mean and variance for each sample:
xi , si2
Step 2: Calculate the overall sample mean (using all N observations): x
Computing the F Test Statistic
Step 3: Calculate the sum of squares between groups:
2
SS Groups groups ni xi x
Step 4: Calculate the sum of squares within groups
(due to error):
SSE groups ni 1si2
Step 5: Optional - Calculate the total sum of squares:
2
SS Total values xij x
SSTotal = SSGroups + SSE
Computing the F Test Statistic
Step 6: Fill in the ANOVA table:
Source
Sum of Squares
Groups
SS Groups
Error
(Within)
SSE
Total
SS Total
DF
Mean Square
F
The p-value for an F Test
If H0, is true, then the F statistic has an F(k-1, N-k) distribution.
Use SPSS to provide p-value
Know how ANOVA table is
constructed
Be able to sketch picture
of p-value for an F-test
Yellow Card on ANOVA
Try It! Comparing 3 Drugs
page 175
Quantitative response (smaller more favorable)
N = 19 patients, randomly assigned to one of three drugs
Data from Drug 1
Data from Drug 2
Data from Drug 3
7.3
7.1
5.8
8.2
10.6
6.5
10.1
11.2
8.8
6.0
9.0
4.9
9.5
8.5
7.9
10.9
8.5
7.8
5.2
What should we do first?
What assumption is best checked with this graph?
That each sample is random
That the 3 samples are independent
That each population has a normal model
That the 3 populations have equal variance
A)
B)
C)
D)
12
11
10
9
8
7
6
5
4
N=
DRUG
5
7
7
1.00
2.00
3.00
State the hypotheses for ANOVA
bottom pg 175
H0: _______________________________________
Ha: _______________________________________
Computing the F Test Statistic
Data from Drug 1
Data from Drug 2
Data from Drug 3
7.3
7.1
5.8
8.2
10.6
6.5
10.1
11.2
8.8
6.0
9.0
4.9
9.5
8.5
7.9
10.9
8.5
7.8
5.2
Step 1: Calculate the mean and variance for each sample:
x1
s12
x2
s 22
x3
s32
Computing the F Test Statistic
Data from Drug 1
Data from Drug 2
Data from Drug 3
7.3
7.1
5.8
8.2
10.6
6.5
10.1
11.2
8.8
6.0
9.0
4.9
9.5
8.5
7.9
10.9
8.5
7.8
5.2
Step 2: Calculate the overall sample mean (using all N observations):
x
Computing the F Test Statistic
Step 3: Calculate the sum of squares between groups:
2
SS Groups groups ni xi x
Computing the F Test Statistic
Step 4: Calculate the sum of squares within groups
(due to error): SSE
n 1s 2
groups
i
i
Computing the F Test Statistic
Step 5: Optional - Calculate the total sum of squares:
SS Total values xij x
2
No thank you …
But we do know …
SSTotal = SSGroups + SSE
Computing the F Test Statistic
Step 6: Fill in the ANOVA table:
Source
Groups
Error
(Within)
Total
Sum of Squares
DF
Mean Square
F