Statistical Analysis
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Transcript of Statistical Analysis
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Statistical Analysis
Professor Lynne StokesDepartment of Statistical Science
Lecture 8Analysis of Variance
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Flow Rate Experiment
Filter Flow RatesA 0.233 0.197 0.259 0.244B 0.259 0.258 0.343 0.305C 0.183 0.284 0.264 0.258D 0.233 0.328 0.267 0.269
MGH Fig 6.1
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Flow Rate Experiment
A B C D
0.30
0.25
0.20
AverageFlowRate
Filter Type
0.35
AssignableCause
(Factor Changes)
UncontrolledUncontrolledErrorError
What is an What is an appropriate statistical comparisonappropriate statistical comparison
of the filter means?of the filter means?
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What is an What is an appropriate statistical comparisonappropriate statistical comparison
of the diet means?of the diet means?
Does not accountDoes not accountfor multiple comparisonsfor multiple comparisons
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5 Comparisons,
Some averages used
more than once (e.g.,
N/R50)
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Analysis of Variance for Single-Factor Experiments
Total Sum of Squares
TSS = (y - y )ij2
j=1
r
i=1
a i
Modelyij = + i + eij i = 1, ..., a; j = 1, ..., ri
Total Adjusted Sum of SquaresCorrected Sum of Squares
(Numerator of the Sample Variance)
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Analysis of Variance for Single-Factor Experiments
Total Sum of Squares
TSS = (y - y )ij2
j=1
r
i=1
a i
Modelyij = + i + eij i = 1, ..., a; j = 1, ..., ri
Goal Partition TSS into Components Associated with
Assignable Causes: Controllable Factors and Measured Covariates
Experimental Error: Uncontrolled Variation, Measurement Error, Unknown Systematic Causes
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Analysis of Variance for Single-Factor Experiments
TSS = (y - y )
(y - y + y - y )
ij2
j=1
r
i=1
a
ij2
j=1
r
i=1
a
i
i
i i
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Analysis of Variance for Single-Factor Experiments
TSS = (y - y )
(y - y + y - y )
(y - y ) (y - y )
ij2
j=1
r
i=1
a
ij2
j=1
r
i=1
a
2
i=1
a
ij2
j=1
r
i=1
a
i
i
i
i i
i i irShowShow
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Analysis of Variance for Single-Factor Experiments
TSS = (y - y )
(y - y + y - y )
(y - y ) (y - y )
ij2
j=1
r
i=1
a
ij2
j=1
r
i=1
a
2
i=1
a
ij2
j=1
r
i=1
a
i
i
i
i i
i i i
A E
r
SS SS
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Estimating Factor Effects
Modelyij = + i + eij i = 1, ..., a; j = 1, ..., ri
E y E e( ) ( )
E y E ei i i i( ) ( )
E y y E e ei i i i i( ) {( ) ( )}
0)ee(E)yy(E iijiij
Estimation
Assumption
E(eij ) = 0
Parameter Constraint i 0
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Analysis of Variance for Single-Factor Experiments
Main Effect Sum of Squares: SSA
Main Effects:
SSA : ri i(y - y )2
i=1
a
y - yi
Sum of Squares attributable tovariation in the effects of Factor A
Sum of Squares attributable tovariation in the effects of Factor A
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Is a pooled estimate of the error variance correct,Is a pooled estimate of the error variance correct,or just ad-hoc?or just ad-hoc?
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Analysis of Variance for Single-Factor Experiments
Error Sum of Squares: SSE
Residuals:
SSE : (y - y )2
i=1
a
ij ij
ri
1
y - yij i
Sum of Squares attributable touncontrolled variation
Sum of Squares attributable touncontrolled variation
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Analysis of Variance for Single-Factor Experiments
Error Sum of Squares: SSE
Factor Levels: i = 1, 2, ... , a
Sample Variances: 1r 1r
)y-(y
s iii
r
1=j
2iij
2i
i
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Analysis of Variance for Single-Factor Experiments
Error Sum of Squares: SSE
Factor Levels: i = 1, 2, ... , a
Sample Variances:
Pooled Variance Estimate:
s
s
n a
SS
n ap
ii
a
i
ii
a
ij ij
r
E
i
2 1
2
1
1
(y - y )2
i=1
a
1r 1r
)y-(y
s iii
r
1=j
2iij
2i
i
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Degrees of Freedom
Total Sum of Squares
a
1=i
r
1=j
2ij )y-(y=TSS
Constraint
a
1=i
r
1=jij 0=)y-(y
Degrees of Freedom
n -1 = ar - 1
ShowShow
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Degrees of Freedom
Main Effect Sum of Squares
Constraint
Degrees of Freedom
a -1
ShowShow
a
1=i
2iA )y-y(rSS
a
1=ii 0=)y-y(r
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Degrees of Freedom
Error Sum of Squares
Constraints
Degrees of Freedom
n – a = a(r – 1)
ShowShow
a
1=i
r
1j
2iijE )y-y(SS
a , ... 2, 1, = i 0=)y-y(r
1jiij
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Analysis of Variance Table
Source dfSum ofSquares
MeanSquares F
Factor A a-1 SSA MSA=SSA / (a-1) MSA/MSE
Error n-a SSE MSE=SSE / (n-a)Total n-1 TSS
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Analysis of Variance for the Flow Rate Data
Source dfSum ofSquares
MeanSquares F
Filters 3 0.00820 0.00273 1.86Error 12 0.01765 0.00147Total 15 0.02585
Assumptions ?Conclusions ?
Assumptions ?Conclusions ?
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Individual Individual confidence intervalsconfidence intervals
and tests are not appropriateand tests are not appropriateunlessunless
SIMULTANEOUS SIMULTANEOUS significancesignificance
levels or confidence levelslevels or confidence levelsare usedare used
(Multiple Comparisons)(Multiple Comparisons)
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Viscosity of a Chemical Process
Reactant FeedConcentration Rate I II III IV
15% 20 lb/hr 145 148 147 14025% 20 lb/hr 158 152 155 15215% 30 lb/hr 135 138 141 13925% 30 lb/hr 150 152 146 149
ReplicateTwo Factors
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Viscosity of a Chemical Process
130
140
150
160
Viscosity
15%20 lb/hr
25%20 lb/hr
15%30 lb/hr
25%30 lb/hr
Reactant Concentration / Flow Rate
Assignable causes:Assignable causes:two factor maintwo factor maineffects and theireffects and their
interactioninteraction
UncontrolledUncontrolledexperimentalexperimental
errorerror
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Viscosity of a Chemical Process
Feed RateConcentration 20 lb/ hr 30 lb/ hr Average
15% 145.00 138.25 141.6225% 154.25 149.25 151.75
Average 149.62 143.75 146.69
Average Viscosity
Main EffectsInteraction
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Viscosity of a Chemical Process :Main Effect for Concentration
130
140
150
160
Viscosity
15%20 lb/hr
25%20 lb/hr
15%30 lb/hr
25%30 lb/hr
Reactant Concentration / Flow Rate
MainEffect
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Viscosity of a Chemical Process :Main Effect for Flow Rate
130
140
150
160
Viscosity
15%20 lb/hr
25%20 lb/hr
15%30 lb/hr
25%30 lb/hr
Reactant Concentration / Flow Rate
MainEffect
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Viscosity of a Chemical Process :Flow Rate & Concentration Interaction
130
140
150
160
Viscosity
15% 25%
Reactant Concentration
20 lb/hr30 lb/hr
Interaction ?Interaction ?
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Analysis of Variance for Multi-Factor Experiments
Total Sum of Squares
TSS = (y - y )ijk2
k=1
r
i=1
a
j
b
1
Model
Goal Partition TSS into components associated with
Assignable Causes: main effects for Factors A &B, interaction between Factors A & B
Experimental Error: uncontrolled variation, measurement error, unknown systematic causes
yijk = + i + j + ()ij + eijk
BalancedDesign
BalancedDesign
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Analysis of Variance for Multi-Factor Experiments
TSS = (y - y )ijk2
k=1
r
i=1
a
j
b
1
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Analysis of Variance for Multi-Factor Experiments
TSS = (y - y )
(y - y + y - y )
ijk2
k=1
r
i=1
a
ijk i i2
k=1
r
i=1
a
j
b
j
b
1
1
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Analysis of Variance for Multi-Factor Experiments
TSS = (y - y )
(y - y + y - y )
(y - y ) (y - y )
ijk2
k=1
r
i=1
a
ijk i i2
k=1
r
i=1
a
i2
i=1
a
ijk2
k=1
r
i=1
a
j
b
j
b
ij
b
br
1
1
1 ShowShow
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Analysis of Variance for Multi-Factor Experiments
TSS = (y - y )
(y - y ) (y - y )
(y - y - y + y )
ijk2
k=1
r
i=1
a
i2
i=1
a2
j=1
b
ijk2
k=1
r
i=1
a
j
b
j
i jj
b
br ar
1
1 ShowShow
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Analysis of Variance for Multi-Factor Experiments
TSS = (y - y )
(y - y ) (y - y )
(y - y - y + y ) (y - y )
ijk2
k=1
r
i=1
a
i2
i=1
a2
j=1
b
ij2
i=1
a
ijk2
k=1
r
i=1
a
j
b
j
i jj
b
ijj
b
br ar
r
1
1 1
ShowShow
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Analysis of Variance for Multi-Factor Experiments
TSS = (y - y )
(y - y ) (y - y )
(y - y - y + y ) (y - y )
ijk2
k=1
r
i=1
a
i2
i=1
a2
j=1
b
ij2
i=1
a
ijk2
k=1
r
i=1
a
j
b
j
i jj
b
ijj
b
A B AB E
br ar
r
SS SS SS SS
1
1 1
Don’t memorize the formulas,understand what they measureDon’t memorize the formulas,
understand what they measure
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Analysis of Variance Table
Source dfSum ofSquares
MeanSquares F
Factor A a-1 SSA MSA=SSA / (a-1) MSA/MSE
Factor B b-1 SSB MSB=SSB/(b-1) MSB/MSE
AB (a-1)(b-1) SSAB MSAB=SSAB/(a-1)(b-1) MSAB/MSE
Error ab(r-1) SSE MSE=SSE / ab(r-1)Total n-1 TSS
Understand thedegrees of freedom
Understand thedegrees of freedom
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Viscosity Data
Source df SS MS F pConcentration 1 410.06 410.06 49.08 0.000Feed Rate 1 138.06 138.06 16.53 0.002Conc x Rate 1 3.06 3.06 0.37 0.556Error 12 100.25 8.35Total 15 651.43
Conclusions ?Conclusions ?
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Sums of Squares:Connections to Model Parameters
kjikjiijjk
kjiijjkE
jiij
jijijAB
iiA
rSS
etc
rrSS
brbrSS
,,
2
ijk,,
2i
,,
2i
,
2
,
2i
2
i2
i
ijk
2ijk
)ˆ-y()y-y(
.
)()y+y-y-y(
ˆ)y-y(
)y-(y=TSS
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Unbalanced Experiments(including rij = 0)
Calculation formulas are not correctCalculation formulas are not correct
“Sums of Squares” in computer-generated ANOVA Tables
are NOT sums of squares (can be negative)usually are not additive;
need not equal the usual calculation formula values
“Sums of Squares” in computer-generated ANOVA Tables
are NOT sums of squares (can be negative)usually are not additive;
need not equal the usual calculation formula values