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Transcript of Chap 6- Anova
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Chapter 6
Analysis of Variance
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Chapter Overview
Analysis of Variance (ANOVA)
F-test
Tukey-Kramer
test
One-WayANOVA Two-WayANOVA
InteractionEffects
RandomizedBlock Design
MultipleComparisons
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General ANOVA Setting
Investigator controls one or more independentvariables
factors (the characteristic which differentiates the
treatment or population from one another) Each factor contains two or more levels (the different
treatments or population)
Observe effects on the dependent variable
Response to levels of independent variable
Experimental design: the plan used to collectthe data
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Completely Randomized Design
Experimental units (subjects) are assignedrandomly to treatments
Subjects are assumed homogeneous Only one factor or independent variable
With two or more treatment levels
Analyzed by one-way analysis of variance(ANOVA)
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Examples
An experiment to study the effects of five(levels) different brands of gasoline (factors) onautomobile engine operating efficiency
An experiment to asses the effects of amounts(factors) of a particular drug on manual dexterity
Level: different setting of the factor
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One-Way Analysis of Variance
Evaluate the difference among the means of threeor more groups
Examples: Accident rates for 1st, 2nd, and 3rd shift
Expected mileage for five brands of tires
Assumptions
Populations are normally distributed Populations have equal variances
Samples are randomly and independently drawn
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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)
c3210 :H
samethearemeanspopulationtheofallNot:H1
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Partitioning the Variation
Total variation can be split into two parts:
SST = total of the squared deviations ofscores from the overall mean
(Total variation)SSA = Sum of Squares Among Groups
(Among-group variation)SSW = Sum of Squares Within Groups
(Within-group variation)
SST = SSA + SSW
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Partitioning the Variation
Total Variation = the aggregate dispersion of the individualdata values across the various factor levels (SST)
Within-Group Variation = dispersion that exists amongthe data values within a particular factor level (SSW)
Among-Group Variation = dispersion between the factorsample means (SSA)
SST = SSA + SSW
(continued)
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Partition of Total Variation
Variation Due toFactor (SSA)
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-Group Variation
Commonly referred to as: Sum of Squares Between
Sum of Squares Among
Sum of Squares Explained
Among Groups Variation
= +
d.f. = n 1
d.f. = c 1 d.f. = n c
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Total Sum of Squares
c
1j
n
1i
2
ij
j
)XX(SST
Where:
SST = Total sum of squares
c = number of groups (levels or treatments)
nj = number of observations in group j
Xij = ith observation from group j
X = grand mean (mean of all data values)
SST = SSA + SSW
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Total Variation
Group 1 Group 2 Group 3
Response, X
X
2cn
212
211 )XX(...)XX()XX(SST c
(continued)
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Among-Group Variation
Where:
SSA = Sum of squares among groups
c = number of groups
nj = sample size from group j
Xj = sample mean from group j
X = grand mean (mean of all data values)
2j
c
1j
j )XX(nSSA
SST = SSA + SSW
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Among-Group Variation
Variation Due toDifferences Among Groups
i j
2j
c
1j
j )XX(nSSA
1cSSAMSA
Mean Square Among =
SSA/degrees of freedom
(continued)
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Among-Group Variation
Group 1 Group 2 Group 3
Response, X
X1
X 2X
3X
22
22
2
11)xx(n...)xx(n)xx(nSSA cc
(continued)
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Within-Group Variation
Where:
SSW = Sum of squares within groups
c = number of groupsnj = sample size from group j
Xj = sample mean from group j
Xij
= ith observation in group j
2jij
n
1i
c
1j
)XX(SSWj
SST = SSA + SSW
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Within-Group Variation
Summing the variationwithin each group and thenadding over all groups cn
SSWMSW
Mean Square Within =
SSW/degrees of freedom
2jij
n
1i
c
1j
)XX(SSWj
(continued)
j
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Within-Group Variation
Group 1 Group 2 Group 3
Response, X
1X
2X
3X
2ccn
2212
2111 )XX(...)XX()Xx(SSW c
(continued)
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Obtaining the Mean Squares
cn
SSWMSW
1c
SSAMSA
1n
SSTMST
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One-Way ANOVA Table
Source ofVariation
dfSS MS(Variance)
AmongGroups SSA MSA =
WithinGroups
n - cSSW MSW =
Total n - 1SST =SSA+SSW
c - 1MSA
MSW
F ratio
c = number of groups
n = sum of the sample sizes from all groups
df = degrees of freedom
SSA
c - 1
SSW
n - c
F =
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One-Way ANOVAF Test Statistic
Test statistic
MSA is mean squares among groups
MSWis mean squares within groups
Degrees of freedom
df1 = c 1 (c = number of groups)
df2 = n c (n = sum of sample sizes from all populations)
MSWMSAF
H0: 1= 2= = c
H1: At least two population means are different
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Interpreting One-Way ANOVAF Statistic
The F statistic is the ratio of the amongestimate of variance and the within estimateof variance
The ratio must always be positive df1 = c-1 will typically be small
df2 = n- c will typically be large
Decision Rule:
Reject H0 if F > FU,otherwise do notreject H0
0
= .05
Reject H0Do notreject H0
FU
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One-Way ANOVAF Test Example
You want to see if threedifferent golf clubs yielddifferent distances. You
randomly select fivemeasurements from trials onan automated drivingmachine for each club. At the
0.05 significance level, isthere a difference in meandistance?
Club 1 Club 2 Club 3254 234 200263 218 222
241 235 197237 227 206251 216 204
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One-Way ANOVA Example:Scatter Diagram
270
260
250
240
230
220
210200
190
Distance
1X
2X
3X
X
227.0x
205.8x226.0x249.2x 321
Club 1 Club 2 Club 3254 234 200263 218 222
241 235 197237 227 206251 216 204
Club1 2 3
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F= 25.275
One-Way ANOVA ExampleSolution
H0: 1 = 2 = 3
H1: j not all equal
= 0.05
df1= 2 df2 = 12
Test Statistic:
Decision:
Conclusion:
Reject H0 at = 0.05
There is evidence thatat least one j differs
from the rest
0
= .05
FU
= 3.89Reject H0Do not
reject H0
25.275
93.3
2358.2
MSW
MSAF
CriticalValue:
FU
= 3.89
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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 ofVariation
SS df MS F P-value F crit
BetweenGroups
4716.4 2 2358.2 25.275 4.99E-05 3.89
WithinGroups
1119.6 12 93.3
Total 5836.0 14
One-Way ANOVAExcel Output
EXCEL: tools | data analysis | ANOVA: single factor
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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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Tukey-Kramer Critical Range
where:
QU = Value from Studentized Range Distributionwith c and n - c degrees of freedom forthe desired level of
MSW = Mean Square Within
nj and nj= Sample sizes from groups j and j
j'j
U
n
1
n
1
2
MSWQRangeCritical
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The Tukey-Kramer Procedure:Example
1. Compute absolute meandifferences:Club 1 Club 2 Club 3
254 234 200263 218 222241 235 197237 227 206251 216 204 20.2205.8226.0xx
43.4205.8249.2xx
23.2226.0249.2xx
32
31
21
2. Find the QU value from the table with
c = 3 and (n c) = (15 3) = 12 degrees of freedomfor the desired level of ( = 0.05 used here):
3.77QU
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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.Thus, with 95% confidence we can concludethat the mean distance for club 1 is greaterthan club 2 and 3, and club 2 is greater than
club 3.
16.2855
1
5
1
2
93.33.77
n
1
n
1
2
MSWQRangeCritical
j'j
U
3. Compute Critical Range:
20.2xx
43.4xx
23.2xx
32
31
21
4. Compare:
(continued)
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The Randomized Block Design
Like One-Way ANOVA, we test for equalpopulation means (for different factor levels, forexample)...
...but we want to control for possible variationfrom a second factor (with two or more levels)
Levels of the secondary factor are called blocks
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Partitioning the Variation
Total variation can now be split into three parts:
SST = Total variationSSA = Among-Group variationSSBL = Among-Block variationSSE = Random variation
SST = SSA + SSBL + SSE
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Sum of Squares for Blocking
Where:
c = number of groups
r = number of blocks
Xi. = mean of all values in block i
X = grand mean (mean of all data values)
r
1i
2
i. )XX(cSSBL
SST = SSA + SSBL + SSE
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Partitioning the Variation
Total variation can now be split into three parts:
SST and SSA are
computed as they werein One-Way ANOVA
SST = SSA + SSBL + SSE
SSE = SST (SSA + SSBL)
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Mean Squares
1c
SSAgroupsamongsquareMeanMSA
1r
SSBLblockingsquareMeanMSBL
)1)(1(
cr
SSEMSE errorsquareMean
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Randomized Block ANOVA Table
Source ofVariation
dfSS MS
AmongBlocks
SSBL MSBL
Error
(r1)(c-1)SSE MSE
Total rc - 1SST
r - 1 MSBL
MSE
F ratio
c = number of populations rc = sum of the sample sizes from all populations
r = number of blocks df = degrees of freedom
AmongTreatments SSA c - 1 MSA
MSA
MSE
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Blocking Test
Blocking test: df1 = r 1
df2 = (r 1)(c 1)
MSBL
MSE
...:H 3.2.1.0
equalaremeansblockallNot:H1
F =
Reject H0 if F > FU
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Main Factor test: df1 = c 1
df2 = (r 1)(c 1)
MSA
MSE
c..3.2.10...:H
equalaremeanspopulationallNot:H1
F =
Reject H0 if F > FU
Main Factor Test
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The Tukey Procedure
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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etc...
xx
xx
xx
.3.2
.3.1
.2.1
The Tukey Procedure(continued)
r
MSERangeCritical uQ
If the absolute mean differenceis greater than the critical rangethen there is a significantdifference between that pair ofmeans at the chosen level ofsignificance.
Compare:?RangeCriticalxxIs .j'.j
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Factorial Design:Two-Way ANOVA
Examines the effect of
Two factors of interest on the dependentvariable
e.g., Percent carbonation and line speed on soft drinkbottling process
Interaction between the different levels of thesetwo factors
e.g., Does the effect of one particular carbonationlevel depend on which level the line speed is set?
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Two-Way ANOVA
Assumptions
Populations are normally distributed
Populations have equal variances
Independent random samples are
drawn
(continued)
T W ANOVA
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Two-Way ANOVASources of Variation
Two Factors of interest: A and B
r = number of levels of factor A
c = number of levels of factor B
n = number of replications for each cell
n = total number of observations in all cells
(n = rcn)
Xijk = value of the kth observation of level i of
factor A and level j of factor B
T W ANOVA
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Two-Way ANOVASources of Variation
SSTTotal Variation
SSAFactor A Variation
SSBFactor B Variation
SSAB
Variation due to interactionbetween A and B
SSERandom variation (Error)
Degrees ofFreedom:
r 1
c 1
(r 1)(c 1)
rc(n 1)
n - 1
SST = SSA + SSB + SSAB + SSE
(continued)
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Two Factor ANOVA Equations
r
i
c
j
n
k
ijk XXSST1 1 1
2)(
2r
1i
..i )XX(ncSSA
2c
1j
.j. )XX(nrSSB
Total Variation:
Factor A Variation:
Factor B Variation:
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Two Factor ANOVA Equations
2
r
1i
c
1j
.j.i..ij. )XXXX(nSSAB
r
1i
c
1j
n
1k
2.ijijk )XX(SSE
Interaction Variation:
Sum of Squares Error:
(continued)
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Two Factor ANOVA Equations
where:MeanGrand
nrc
X
X
r
1i
c
1j
n
1k
ijk
r)...,2,1,(iAfactorofleveliofMeannc
X
X th
c
1j
n
1k
ijk
..i
c)...,2,1,(jBfactorofleveljofMeannr
X
X
th
r
1i
n
1k
ijk
.j.
ijcellofMeann
XX
n
1k
ijk.ij
r = number of levels of factor A
c = number of levels of factor B
n = number of replications in each cell
(continued)
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Mean Square Calculations
1r
SSAAfactorsquareMeanMSA
1c
SSBBfactorsquareMeanMSB
)1c)(1r(
SSABninteractiosquareMeanMSAB
)1'n(rc
SSEerrorsquareMeanMSE
T W ANOVA
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Two-Way ANOVA:The F Test Statistic
F Test for Factor B Effect
F Test for Interaction Effect
H0: 1.. = 2.. = 3..=
H1: Not all i.. are equal
H0: the interaction of A and B isequal to zero
H1: interaction of A and B is notzero
F Test for Factor A Effect
H0: .1. = .2. = .3.=
H1: Not all .j. are equal
Reject H0
if F > FUMSE
MSAF
MSE
MSBF
MSE
MSABF
Reject H0
if F > FU
Reject H0
if F > FU
T W ANOVA
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Two-Way ANOVASummary Table
Source ofVariation
Sum ofSquares
Degrees ofFreedom
MeanSquares
FStatistic
Factor A SSA r 1MSA
= SSA/(r 1)
MSA
MSE
Factor B SSB c 1MSB
= SSB/(c 1)
MSB
MSE
AB
(Interaction)SSAB (r 1)(c 1)
MSAB
= SSAB/ (r 1)(c 1)
MSAB
MSE
Error SSE rc(n 1)MSE =
SSE/rc(n 1)
Total SST n 1
F t f T W ANOVA
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Features of Two-Way ANOVAFTest
Degrees of freedom always add up
n-1 = rc(n-1) + (r-1) + (c-1) + (r-1)(c-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
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Example
In a study on automobile traffic and air polution , air sample taken at fourdifferent times and at five different location were analyzed to obtain the
amount of particulate matter present in the air (mg/m3).
Is there any difference in a true average amount of particulate matter
present in the air due to either different sampling times or different location
E l
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Examples:Interaction vs. No Interaction
No interaction:
Factor B Level 1
Factor B Level 3
Factor B Level 2
Factor A Levels
Factor B Level 1
Factor B Level 3
Factor B Level 2
Factor A Levels
MeanResponse
Me
anResponse
Interaction ispresent: