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ANOVA
Chap 11-1
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-2
Goals
After completing this chapter, you should be able
to:
Recognize situations in which to use analysis of variance
Understand different analysis of variance designs
Perform a single-factor hypothesis test and interpret results
Conduct and interpret post-analysis of variance pairwise
comparisons procedures
Set up and perform randomized blocks analysis
Analyze two-factor analysis of variance test with replications
results
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-3
Chapter Overview
Analysis of Variance (ANOVA)
F-test
F-testTukey-
Kramer
testFishers Least
Significant
Difference test
One-WayANOVA RandomizedComplete
Block ANOVA
Two-factorANOVA
with replication
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Chap 11-4
General ANOVA Setting
Investigator controls one or more independent
variables
Called factors (or treatment variables)
Each factor contains two or more levels (orcategories/classifications)
Observe effects on dependent variable
Response to levels of independent variable
Experimental design: the plan used to test
hypothesis
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Chap 11-5
One-Way Analysis of Variance
Evaluate the difference among the means of three
or more populations
Examples: Accident rates for 1
st
, 2
nd
, and 3
rd
shift
Assumptions
Populations are normally distributed
Populations have equal variances
Samples are randomly and independently drawn
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Chap 11-6
Completely Randomized Design
Experimental units (subjects) are assigned
randomly to treatments
Only one factor or independent variable With two or more treatment levels
Analyzed by
One-factor analysis of variance (one-way ANOVA)
Called a Balanced Design if all factor levels
have equal sample size
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Chap 11-7
Hypotheses of One-Way ANOVA
All population means are equal
i.e., no treatment effect (no variation in means among
groups)
At least one population mean is different
i.e., there is a treatment effect
Does not mean that all population means are different
(some pairs may be the same)
k3210 :H
samethearemeanspopulationtheofallNot:HA
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Chap 11-8
One-Factor ANOVA
All Means are the same:
The Null Hypothesis is True
(No Treatment Effect)
k3210 :H
sametheareallNot:H iA
321
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Chap 11-9
One-Factor ANOVA
At least one mean is different:
The Null Hypothesis is NOT true
(Treatment Effect is present)
k3210 :H
sametheareallNot:H iA
321 321
or
(continued)
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Chap 11-10
Partitioning the Variation
Total variation can be split into two parts:
SST = Total Sum of Squares
SSB = Sum of Squares Between
SSW = Sum of Squares Within
SST = SSB + SSW
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Chap 11-11
Partitioning the Variation
Total Variation = the aggregate dispersion of the individualdata values across the various factor levels (SST)
Within-Sample Variation = dispersion that exists among
the data values within a particular factor level (SSW)
Between-Sample Variation = dispersion among the factor
sample means (SSB)
SST = SSB + SSW
(continued)
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Chap 11-12
Partition of Total Variation
Variation Due toFactor (SSB)
Variation Due to RandomSampling (SSW)
Total Variation (SST)
Commonly referred to as:
Sum of Squares Within
Sum of Squares Error
Sum of Squares Unexplained
Within Groups Variation
Commonly referred to as:
Sum of Squares Between
Sum of Squares Among
Sum of Squares Explained
Among Groups Variation
= +
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Chap 11-13
Total Sum of Squares
k
i
n
j ij
i
)xx(SST1 1
2
Where:
SST = Total sum of squares
k = number of populations (levels or treatments)
ni = sample size from population i
xij = jth measurement from population i
x = grand mean (mean of all data values)
SST = SSB + SSW
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Chap 11-14
Total Variation(continued)
Group 1 Group 2 Group 3
Response, X
X
22
12
2
11)xx(...)xx()xx(SST
kkn
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Chap 11-15
Sum of Squares Between
Where:
SSB = Sum of squares between
k = number of populations
ni = sample size from population i
xi = sample mean from population i
x = grand mean (mean of all data values)
2
1
)xx(nSSB i
k
i
i
SST = SSB + SSW
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Chap 11-16
Between-Group Variation
Variation Due to
Differences Among Groups
i j
2
1
)xx(nSSB i
k
i
i
1kSSBMSB
Mean Square Between =
SSB/degrees of freedom
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Chap 11-17
Between-Group Variation(continued)
Group 1 Group 2 Group 3
Response, X
X1
X 2X
3X
22
22
2
11)xx(n...)xx(n)xx(nSSB kk
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Chap 11-18
Sum of Squares Within
Where:
SSW = Sum of squares within
k = number of populationsni = sample size from population i
xi = sample mean from population i
xij
= jth measurement from population i
2
11
)xx(SSW iij
n
j
k
i
j
SST = SSB + SSW
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Chap 11-19
Within-Group Variation
Summing the variation
within each group and then
adding over all groups
i
kNSSWMSW
Mean Square Within =
SSW/degrees of freedom
2
11
)xx(SSW iij
n
j
k
i
j
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Chap 11-20
Within-Group Variation(continued)
Group 1 Group 2 Group 3
Response, X
1X
2X
3X
22
212
2
111)xx(...)xx()xx(SSW kknk
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Chap 11-21
One-Way ANOVA Table
Source of
VariationdfSS MS
Between
Samples SSB MSB =
Within
SamplesN - kSSW MSW =
Total N - 1SST =SSB+SSW
k - 1MSB
MSW
F ratio
k = number of populations
N = sum of the sample sizes from all populations
df = degrees of freedom
SSB
k - 1
SSW
N - k
F =
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Chap 11-22
One-Factor ANOVAF Test Statistic
Test statistic
MSB is mean squares between variances
MSWis mean squares within variances
Degrees of freedom
df1 = k 1 (k = number of populations)
df2 = N k (N = sum of sample sizes from all populations)
MSWMSBF
H0: 1= 2= = k
HA: At least two population means are different
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Chap 11-23
Interpreting One-Factor ANOVAF Statistic
The F statistic is the ratio of the betweenestimate of variance and the within estimateof variance
The ratio must always be positive df1 = k-1 will typically be small
df2 = N- k will typically be large
The ratio should be close to 1 ifH0: 1= 2= = k is true
The ratio will be larger than 1 ifH0: 1= 2= = k is false
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Chap 11-24
One-Factor ANOVAF Test Example
You want to see if three
different group of woman
yield different distances in 10
seconds. You randomlyselect five measurements
from trials on a sprint. At the
.05 significance level, is there
a difference in meandistance?
group-1group-2 group-3
254 234 200
263 218 222
241 235 197237 227 206
251 216 204
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Chap 11-25
One-Factor ANOVA Example:Scatter Diagram
270
260
250
240
230
220
210200
190
Distance
1X
2X
3X
X
227.0x
205.8x226.0x249.2x 321
group-1group-2 group-3
254 234 200
263 218 222241 235 197
237 227 206
251 216 204
group1 2 3
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Chap 11-26
One-Factor ANOVA ExampleComputations
group-1 group-2 group-3
254 234 200
263 218 222
241 235 197
237 227 206
251 216 204
x1 = 249.2
x2 = 226.0
x3 = 205.8
x = 227.0
n1 = 5
n2 = 5
n3 = 5
N = 15
k = 3
SSB = 5 [ (249.2 227)2 + (226 227)2 + (205.8 227)2 ] = 4716.4
SSW = (254 249.2)2 + (263 249.2)2++ (204 205.8)2 = 1119.6
MSB = 4716.4 / (3-1) = 2358.2
MSW = 1119.6 / (15-3) = 93.325.275
93.3
2358.2F
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Chap 11-27
F= 25.275
One-Factor ANOVA ExampleSolution
H0: 1 = 2 = 3
HA: i not all equal
= .05
df1= 2 df2 = 12
Test Statistic:
Decision:
Conclusion:
Reject H0 at = 0.05
There is evidence that
at least one i differs
from the rest
0 = .05
F.05
= 3.885
Reject H0Do notreject H0
25.275
93.3
2358.2
MSW
MSBF
Critical
Value:
F
= 3.885
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Chap 11-28
SUMMARY
Groups Count Sum Average Variance
Club 1 5 1246 249.2 108.2
Club 2 5 1130 226 77.5Club 3 5 1029 205.8 94.2
ANOVA
Source of
VariationSS df MS F P-value F crit
BetweenGroups
4716.4 2 2358.2 25.275 4.99E-05 3.885
Within
Groups1119.6 12 93.3
Total 5836.0 14
ANOVA -- Single Factor:Excel Output
EXCEL: tools | data analysis | ANOVA: single factor
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Chap 11-29
The Tukey-Kramer Procedure
Tells which population means are significantlydifferent e.g.: 1 = 23
Done after rejection of equal means in ANOVA Allows pair-wise comparisons
Compare absolute mean differences with criticalrange
x1 = 2 3
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Chap 11-30
Tukey-Kramer Critical Range
where:
q
= Value from standardized range table
with k and N - k degrees of freedom for
the desired level of
MSW = Mean Square Within
ni and nj = Sample sizes from populations (levels) i and j
ji n
1
n
1
2
MSWqRangeCritical
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Chap 11-31
The Tukey-Kramer Procedure:Example
1. Compute absolute meandifferences:group-1 group-2 group-3
254 234 200
263 218 222
241 235 197
237 227 206
251 216 204
20.2205.8226.0xx
43.4205.8249.2xx
23.2226.0249.2xx
32
31
21
2. Find the q value from the table in appendix J with
k and N - k degrees of freedom forthe desired level of
3.77q
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Chap 11-32
The Tukey-Kramer Procedure:Example
5. All of the absolute mean differencesare greater than critical range.Therefore there is a significant
difference between each pair ofmeans at 5% level of significance.
16.2855
1
5
1
2
93.33.77
n
1
n
1
2
MSWqRangeCritical
ji
3. Compute Critical Range:
20.2xx
43.4xx
23.2xx
32
31
21
4. Compare:
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Chap 11-33
Tukey-Kramer in PHStat
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Chap 11-34
Randomized Complete Block ANOVA
Like One-Way ANOVA, we test for equal populationmeans (for different factor levels, for example)...
...but we want to control for possible variation from asecond factor (with two or more levels)
Used when more than one factor may influence thevalue of the dependent variable, but only one is of keyinterest
Levels of the secondary factor are called blocks
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Chap 11-35
Partitioning the Variation
Total variation can now be split into three parts:
SST = Total sum of squares
SSB = Sum of squares between factor levels
SSBL = Sum of squares between blocks
SSW = Sum of squares within levels
SST = SSB + SSBL + SSW
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Chap 11-36
Sum of Squares for Blocking
Where:
k = number of levels for this factor
b = number of blocks
xj = sample mean from the jth block
x = grand mean (mean of all data values)
2
1)xx(kSSBL j
b
j
SST = SSB + SSBL + SSW
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Chap 11-37
Partitioning the Variation
Total variation can now be split into three parts:
SST and SSB are
computed as they werein One-Way ANOVA
SST = SSB + SSBL + SSW
SSW = SST (SSB + SSBL)
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Chap 11-38
Mean Squares
1k
SSBbetweensquareMeanMSB
1b
SSBLblockingsquareMeanMSBL
)b)(k(
SSWwithinsquareMeanMSW
11
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Chap 11-39
Randomized Block ANOVA Table
Source of
VariationdfSS MS
Between
SamplesSSB MSB
Within
Samples (k1)(b-1)SSW MSW
Total N - 1SST
k - 1
MSBL
MSW
F ratio
k = number of populations N = sum of the sample sizes from all populations
b = number of blocks df = degrees of freedom
Between
Blocks
SSBL b - 1 MSBL
MSB
MSW
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Chap 11-40
Blocking Test
Blocking test: df1 = b - 1
df2 = (k 1)(b 1)
MSBL
MSW
...:H b3b2b10
equalaremeansblockallNot:HA
F =
Reject H0 if F > F
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Chap 11-41
Main Factor test: df1 = k - 1
df2 = (k 1)(b 1)
MSB
MSW
k3210 ...:H
equalaremeanspopulationallNot:HA
F =
Reject H0 if F > F
Main Factor Test
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Chap 11-42
FishersLeast Significant Difference Test
To test which population means are significantlydifferent e.g.: 1 = 23 Done after rejection of equal means in randomized
block ANOVA design
Allows pair-wise comparisons Compare absolute mean differences with critical
range
x= 1 2 3
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Chap 11-43
Fishers Least SignificantDifference (LSD) Test
where:
t/2 = Upper-tailed value from Students t-distribution
for/2 and (k -1)(n - 1) degrees of freedom
MSW = Mean square within from ANOVA table
b = number of blocks
k = number of levels of the main factor
b
2MSWtLSD /2
h f
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Chap 11-44
...etc
xx
xx
xx
32
31
21
Fishers Least SignificantDifference (LSD) Test
(continued)
b
2MSWtLSD /2
If the absolute mean differenceis greater than LSD then thereis a significant differencebetween that pair of means atthe chosen level of significance.
Compare:?LSDxxIs ji
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RCBD Example
Chap 11-45
A physical therapist wished to compare three
methods for teaching patiens to use a certain
prosthetic device. He felt that there rate oflearning would be different for patients of
different ages and wished to design and
experiment in which the influence of age couldbe taken out
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Chap 11-46
Data:
Three patients in each of five age groups were
selected to participate in the experiment, and onepatient in each age group was randomly assigned
to each of teaching methods. The methods of
instruction constitute our three treatments, and the
five groups are the blocs. The data are shown on
the table.
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Chap 11-47
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Chap 11-48
Two-Way ANOVA
Examines the effect of
Two or more factors of interest on the
dependent variable
e.g.: Percent carbonation and line speed onsoft drink bottling process
Interaction between the different levels of these
two factors
e.g.: Does the effect of one particularpercentage of carbonation depend on which
level the line speed is set?
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Chap 11-49
Two-Way ANOVA
Assumptions
Populations are normally distributed
Populations have equal variances
Independent random samples are
drawn
(continued)
Two Way ANOVA
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Chap 11-50
Two-Way ANOVASources of Variation
Two Factors of interest: A and B
a = number of levels of factor Ab = number of levels of factor B
N = total number of observations in all cells
T W ANOVA
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Chap 11-51
Two-Way ANOVASources of Variation
SST
Total Variation
SSAVariation due to factor A
SSBVariation due to factor B
SSAB
Variation due to interactionbetween A and B
SSEInherent variation (Error)
Degrees of
Freedom:
a 1
b 1
(a 1)(b 1)
N ab
N - 1
SST = SSA + SSB + SSAB + SSE
(continued)
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Chap 11-52
Two Factor ANOVA Equations
a
i
b
j
n
k
ijk )xx(SST1 1 1
2
2
1
)xx(nbSSa
i
iA
2
1
)xx(naSSb
j
jB
Total Sum of Squares:
Sum of Squares Factor A:
Sum of Squares Factor B:
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Chap 11-53
Two Factor ANOVA Equations
2
1 1
)xxxx(nSSa
i
b
j
jiijAB
a
i
b
j
n
k
ijijk )xx(SSE1 1 1
2
Sum of Squares
Interaction Between
A and B:
Sum of Squares Error:
(continued)
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Chap 11-54
Two Factor ANOVA Equations
where:MeanGrand
nab
x
x
a
i
b
j
n
k
ijk
1 1 1
AfactorofleveleachofMeannb
x
x
b
j
n
k
ijk
i
1 1
BfactorofleveleachofMeanna
x
x
a
i
n
k
ijk
j
1 1
celleachofMeann
xx
n
k
ijk
ij
1
a = number of levels of factor A
b = number of levels of factor B
n = number of replications in each cell
(continued)
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Chap 11-55
Mean Square Calculations
1a
SSAfactorsquareMeanMS AA
1 b
SSBfactorsquareMeanMS BB
)b)(a(
SSninteractiosquareMeanMS ABAB
11
abN
SSEerrorsquareMeanMSE
Two-Way ANOVA:
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Chap 11-56
Two-Way ANOVA:The F Test Statistic
F Test for Factor B Main Effect
F Test for Interaction Effect
H0: A1 = A2 = A3=
HA: Not all Ai are equal
H0: factors A and B do not interactto affect the mean response
HA: factors A and B do interact
F Test for Factor A Main Effect
H0: B1 = B2 = B3=
HA: Not all Bi are equal
Reject H0
if F > FMSEMS
F A
MSE
MSF B
MSE
MSF AB
Reject H0
if F > F
Reject H0
if F > F
Two-Way ANOVA
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Chap 11-57
Two-Way ANOVASummary Table
Source of
Variation
Sum of
Squares
Degrees of
Freedom
Mean
Squares
F
Statistic
Factor A SSA a 1MSA
= SSA /(a 1)
MSA
MSE
Factor B SSB b 1MSB
= SSB /(b 1)
MSB
MSE
AB
(Interaction)
SSAB (a 1)(b 1)MSAB
= SSAB / [(a 1)(b 1)]
MSAB
MSE
Error SSE N abMSE =
SSE/(N ab)
Total SST N 1
Features of Two Way ANOVA
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Chap 11-58
Features of Two-Way ANOVAFTest
Degrees of freedom always add up
N-1 = (N-ab) + (a-1) + (b-1) + (a-1)(b-1)
Total = error + factor A + factor B + interaction
The denominator of the FTest is always the
same but the numerator is different
The sums of squares always add up
SST = SSE + SSA + SSB + SSAB
Total = error + factor A + factor B + interaction
Examples:
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Chap 11-59
Examples:Interaction vs. No Interaction
No interaction:
1 2
Factor B Level 1
Factor B Level 3
Factor B Level 2
Factor A Levels 1 2
Factor B Level 1
Factor B Level 3
Factor B Level 2
Factor A Levels
MeanResponse
Me
anResponse
Interaction is
present:
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Chapter Summary
Described one-way analysis of variance The logic of ANOVA
ANOVA assumptions
F test for difference in k means
The Tukey-Kramer procedure for multiple comparisons
Described randomized complete block designs
F test
Fishers least significant difference test for multiplecomparisons
Described two-way analysis of variance
Examined effects of multiple factors and interaction
Recommended