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?

5

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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TSS = (y - y )ij2

j=1

r

i=1

a i


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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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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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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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


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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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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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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Factor Levels: i = 1, 2, ... , a

Sample Variances: 1r 1r

)y-(y

s iii

r

1=j

2iij

2i

i

17



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


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


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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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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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


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


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


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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TSS = (y - y )ijk2

k=1

r

i=1

a

j

b

1

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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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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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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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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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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


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

Statistical Analysis

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