1 Multifactor ANOVA. 2 What We Will Learn Two-factor ANOVA K ij =1 Two-factor ANOVA K ij =1...

11

Multifactor ANOVAMultifactor ANOVA

22

What We Will LearnWhat We Will Learn• Two-factor ANOVA KTwo-factor ANOVA Kijij=1=1

– InteractionInteraction– Tukey’s with multiple Tukey’s with multiple

comparisonscomparisons– Concept of randomized Concept of randomized

blocked experimentsblocked experiments– Random effects and mixed Random effects and mixed

modelsmodels

• Two-factor ANOVA KTwo-factor ANOVA Kijij>1>1– InteractionsInteractions– Tukey’sTukey’s– Mixed and random effectsMixed and random effects

• Three-factor Three-factor ANOVAANOVA– Latin SquaresLatin Squares

• 22pp Factorial Factorial ExperimentsExperiments– 2233 Experiments Experiments– 22pp>3>3– Concept of Concept of

confoundingconfounding

33

2 Factor ANOVA2 Factor ANOVA

• Factor A consists of I levelsFactor A consists of I levels

• Factor B consists of J levelsFactor B consists of J levels

• IJ different pairsIJ different pairs

• Number of observations per each Number of observations per each factor pair Kfactor pair Kijij=1=1

• Example - TiresExample - Tires

Factor B - LocationFactor A - Brand Front Right Front Left Rear Right Rear Left

GoodyearFirestone

KellyMichelon

44

TerminologyTerminology• XXijij = the rv denoting the measurement when = the rv denoting the measurement when

factor A is held at level factor A is held at level ii and factor B is held at and factor B is held at level level jj

• xxijij = actual observed value = actual observed value

• The average when factor The average when factor AA is isheld at level held at level ii

• The average when factor The average when factor BB is is held at level held at level jj

• The grand meanThe grand mean

IJ

X

X

I

XX

J

X

X

J

jij

I

i

I

iij

j

J

jij

i

11..

1.

1.

55

An Additive ModelAn Additive ModelI I parametersparameters 11, , 22, , 33……II

J J parametersparameters 11, , 22, , 33, …, …jj

ST ST ijij = = ii + + jj

ijij is the sum of an effect due to factor A at level i ( is the sum of an effect due to factor A at level i (II) and an effect due to ) and an effect due to factor B at level j (factor B at level j (jj) )

Then XThen Xijij = = ii + + jj + + ijij

Number of model parameters Number of model parameters I+J+1I+J+1

One for the errorOne for the error

ijij- - i'ji'j = ( = (ii + + jj ) - ( ) - (i'i' + + jj )= )= ii - - i’i’

The difference in mean responses for two levels of one of the factors is the The difference in mean responses for two levels of one of the factors is the same for all levels of the other factor same for all levels of the other factor

ii and and jj are not uniquely defined are not uniquely defined

66

Additive ModelAdditive Model

XXijij = = + + ii + + jj + + ij ij wherewhereand and ijij’s are independent and N(0,’s are independent and N(0,22))

Now Now = 4, = 4, 11 = -.5, = -.5, 22 = .5, = .5, 11 = -1.5, = -1.5, 22 = 1.5 = 1.5

The parameters are uniquely defined The parameters are uniquely defined Have (I-1)+(J-1)+1 = I+J-1 Have (I-1)+(J-1)+1 = I+J-1

EstimatorsEstimators

J

jji

I

ii and

11

00

1 = 2

2 = 4

1 = 1 = 2 = 5

2 = 2 = 3 = 6

..ˆ..ˆ..ˆ .. XXXXX jji

77

Two Factor HypothesisTwo Factor Hypothesis

0:

0:

0:

0:

321

321

oneleastatH

H

oneleastatH

H

aB

JioB

aA

IioA

88

Two-way ANOVATwo-way ANOVA

)1)(1(

111

111

11

1 1

2

....

2..

1 1

2.

1

2

...

2..

1 1

2.

1

2

...

1 1

2..

2

1 1

2

..

JIdf

XXXXSSBSSASSTSSE

JdfXIJ

XI

XXSSB

IdfXIJ

XJ

XXSSA

IJdfXIJ

XXXSST

I

i

J

j

jiij

I

i

J

jj

J

j

j

I

i

I

ii

J

j

i

I

i

J

jij

I

i

J

jij

99

ANOVA TableANOVA TableSource Sum of

SquaresDegrees ofFreedom

MeanSquare

F F,v1,v2 *?

Factor A SSA df=I-1 MSA FA F,dfA,dfE

Factor B SSB df=J-1 MSB FB F,dfB,dfE

Error SSE df=(I-1)(J-1) MSE

Total SST df=IJ-1

1010

Problem 1Problem 1A\B 1 2 3 4

1 200 226 240 261

2 278 312 330 381

3 369 416 462 517

4 500 575 645 733

1111

Problem 1Problem 1

)1)(1(

111

111

11

1 1

2

....

2..

1 1

2.

1

2

...

2..

1 1

2.

1

2

...

1 1

2..

2

1 1

2

..

JIdf

XXXXSSBSSASSTSSE

JdfXIJ

XI

XXSSB

IdfXIJ

XJ

XXSSA

IJdfXIJ

XXXSST

I

i

J

j

jiij

I

i

J

jj

J

j

j

I

i

I

ii

J

j

i

I

i

J

jij

I

i

J

jij

1212

Problem 1Problem 1Source Sum of


MeanSquare

F F,v1,v2 *?




Total SST df=IJ-1

1313

SPSS Data EntrySPSS Data Entry Analyze> General Linear Model (GLM)>UnivariateAnalyze> General Linear Model (GLM)>Univariate

1414

Problem 1Problem 1

Tests of Between-Subjects Effects

Dependent Variable: HEAT_TF

2960142.938a 7 422877.563 412.248 .000

39934.188 3 13311.396 12.977 .001

324082.188 3 108027.396 105.312 .000

9232.063 9 1025.785

2969375.000 16

SourceModel

LIQUID

GAS

Error

Total

Type III Sumof Squares df Mean Square F Sig.

R Squared = .997 (Adjusted R Squared = .994)a.

1515

Your TurnYour Turn

• The effects of four types of graphite coaters The effects of four types of graphite coaters on light box readings are to be studied. As on light box readings are to be studied. As these readings might differ from day to these readings might differ from day to day, observations are to be taken on each day, observations are to be taken on each of the four types everyday for three days. of the four types everyday for three days. The order of testing of the four types on The order of testing of the four types on any given day can be randomized.any given day can be randomized.

• Analyze these data as a randomized block Analyze these data as a randomized block design and state your conclusions.design and state your conclusions.

1616

Your TurnYour Turn

Day M A K L

1 4.0 4.8 5.0 4.6

2 4.8 5.0 5.2 4.6

3 4.0 4.8 5.6 5.0

1717

Your TurnYour TurnSource Sum of


MeanSquare

F F,v1,v2 *?

Factor A

Factor B

Error

Total

1818

Multiple Comparisons in Multiple Comparisons in ANOVAANOVATukey’s Procedure (T Tukey’s Procedure (T Method)Method)

• SelectSelect • Determine QDetermine Q

– For A - QFor A - Q,I,(I-1)(J-1),I,(I-1)(J-1)

– For B - QFor B - Q,J,(I-1)(J-1),J,(I-1)(J-1)

• Determine Determine ww for factors A and B for factors A and B

• List sample means in increasing order List sample means in increasing order

• Compute the difference in each Compute the difference in each ii - -j j pair Underline pair Underline those pairs that differ by less than those pairs that differ by less than wwPairs Pairs notnot underlined are significantly different underlined are significantly different

I

MSEQw

J

MSEQw

JIJB

JIIA

)1)(1(,,

)1)(1(,,

1919

Problem 1Problem 1






• Compute the difference in each Compute the difference in each ii - -j j pair pair

• Underline those pairs that differ by less than Underline those pairs that differ by less than ww

I

MSEQw

J

MSEQw

JIJB

JIIA

)1)(1(,,

)1)(1(,,

2020

Problem 1Problem 1Multiple Comparisons


Tukey HSD

-45.5000 22.6471 .254 -140.8949 49.8949

-82.5000 22.6471 .023 -177.8949 12.8949

-136.2500* 22.6471 .001 -231.6449 -40.8551

45.5000 22.6471 .254 -49.8949 140.8949

-37.0000 22.6471 .408 -132.3949 58.3949

-90.7500 22.6471 .013 -186.1449 4.6449

82.5000 22.6471 .023 -12.8949 177.8949

37.0000 22.6471 .408 -58.3949 132.3949

-53.7500 22.6471 .152 -149.1449 41.6449

136.2500* 22.6471 .001 40.8551 231.6449

90.7500 22.6471 .013 -4.6449 186.1449

53.7500 22.6471 .152 -41.6449 149.1449

(J) LIQUID2.00

3.00

4.00

1.00

3.00

4.00

1.00

2.00

4.00

1.00

2.00

3.00

(I) LIQUID1.00

2.00

3.00

4.00

MeanDifference

(I-J) Std. Error Sig. Lower Bound Upper Bound

99% Confidence Interval

Based on observed means.

The mean difference is significant at the .01 level.*.

2121

Problem 1Problem 1

HEAT_TF

Tukey HSDa,b

4 336.7500

4 382.2500 382.2500

4 419.2500 419.2500

4 473.0000

.023 .013

LIQUID1.00

2.00

3.00

4.00

Sig.

N 1 2

Subset

Means for groups in homogeneous subsets are displayed.Based on Type III Sum of SquaresThe error term is Mean Square(Error) = 1025.785.

Uses Harmonic Mean Sample Size = 4.000.a.

Alpha = .01.b.

2222



Tukey HSD

-93.5000 22.6471 .011 -188.8949 1.8949

-209.2500* 22.6471 .000 -304.6449 -113.8551

-381.5000* 22.6471 .000 -476.8949 -286.1051

93.5000 22.6471 .011 -1.8949 188.8949

-115.7500* 22.6471 .003 -211.1449 -20.3551

-288.0000* 22.6471 .000 -383.3949 -192.6051

209.2500* 22.6471 .000 113.8551 304.6449

115.7500* 22.6471 .003 20.3551 211.1449

-172.2500* 22.6471 .000 -267.6449 -76.8551

381.5000* 22.6471 .000 286.1051 476.8949

288.0000* 22.6471 .000 192.6051 383.3949

172.2500* 22.6471 .000 76.8551 267.6449

(J) GAS2.00

3.00

4.00

1.00

3.00

4.00

1.00

2.00

4.00

1.00

2.00

3.00

(I) GAS1.00

2.00

3.00

4.00

MeanDifference





2323

Problem 1Problem 1HEAT_TF

Tukey HSDa,b

4 231.7500

4 325.2500

4 441.0000

4 613.2500

.011 1.000 1.000

GAS1.00

2.00

3.00

4.00

Sig.

N 1 2 3

Subset



Alpha = .01.b.

2424

Your TurnYour Turn






• Compute the difference in each Compute the difference in each ii - -j j pair pair

• Underline those pairs that differ by less than Underline those pairs that differ by less than ww

I

MSEQw

J

MSEQw

JIJB

JIIA

)1)(1(,,

)1)(1(,,

2525

Concept of Randomized Block Concept of Randomized Block DesignsDesigns• 1-way ANOVA1-way ANOVA

– II treatments treatments– IJIJ total observations or subjects total observations or subjects– JJ observations per treatment selected randomly observations per treatment selected randomly

• Observations or subjects can be heterogeneous WRT other Observations or subjects can be heterogeneous WRT other variablesvariables– Significance or non-significance due to the treatment or something Significance or non-significance due to the treatment or something

elseelse– Hence paired t-testHence paired t-test

•pre to post exams - looking at the difference of an individual’s scores; know pre to post exams - looking at the difference of an individual’s scores; know that the difference isn’t due to the subjectthat the difference isn’t due to the subject

• When When II>2 we want to perform a >2 we want to perform a randomized blockrandomized block experiment experiment

2626

Concept of Randomized Concept of Randomized Block DesignsBlock Designs

• The extra factor (block) divides the The extra factor (block) divides the IJIJ units into units into JJ groups with groups with II units within units within each groupeach group– The The II units are homogeneous WRT other factors (the block) units are homogeneous WRT other factors (the block)– Within each block, the Within each block, the II treatments are randomly selected and assigned to treatments are randomly selected and assigned to II

observations or subjectsobservations or subjects

– When social scientists use this - repeated measuresWhen social scientists use this - repeated measures– Each subject undergoes each treatment; thus, acting as their own controlEach subject undergoes each treatment; thus, acting as their own control– Could use time periods, locations, etc.Could use time periods, locations, etc.– From a large population of subjects - random effectsFrom a large population of subjects - random effects

Source Sum ofSquares

Degrees ofFreedom

MeanSquare

F F,v1,v2 *?


Block SSBlk df=J-1 MSBlk FBlk F,dfBlk,dfE


Total SST df=IJ-1

2727

Problem 2Problem 2Subject

Seat 1 2 3 4 5 6 7 8 9

1 12 10 7 7 8 9 8 7 9

2 15 14 14 11 11 11 12 11 13

3 12 13 13 10 8 11 12 8 10

4 10 12 9 9 7 10 11 7 84

2828

Why Block?Why Block?• If the experimental units are heterogeneousIf the experimental units are heterogeneous

– Then there will be a larger variance in MSEThen there will be a larger variance in MSE– Blocking minimizes the random (MSE) variance by Blocking minimizes the random (MSE) variance by

accounting some of the error to the effects due to the accounting some of the error to the effects due to the subjects/experimental unitssubjects/experimental units

– Thus, MSE (the random error) will be smaller allowing us to Thus, MSE (the random error) will be smaller allowing us to determine if there is significance in the main effect or determine if there is significance in the main effect or treatment.treatment.

– So…. We may be able to determine if the null hypothesis So…. We may be able to determine if the null hypothesis should be rejected.should be rejected.

– There is a cost…. The MSE will have fewer degrees of There is a cost…. The MSE will have fewer degrees of freedom since some of those df’s need to go to the Block freedom since some of those df’s need to go to the Block error (the larger the df, the smaller the MSE) error (the larger the df, the smaller the MSE)

2929

Models for Random EffectsModels for Random Effects• 1-way ANOVA - Random Effects Model1-way ANOVA - Random Effects Model

• 2-way ANOVA - Mixed Effects Model2-way ANOVA - Mixed Effects Model– XXijij = = + + ii + + BBjj + + ijij

– Distributions of Distributions of BBjj & & ijij are N(0, are N(0, 22,B ,B ), ),

N(0, N(0, 22,,,,) respectively) respectively

– Hypothesis TestHypothesis Test

I

ii

1

0

0: 2 BoBH 0:

0: 321

oneleastatH

H

aA

IioA

3030

2-Factor ANOVA 2-Factor ANOVA KKijij>1>1• Here, the responses are notHere, the responses are not

additiveadditive

• There is something going onThere is something going onbetween factor A and factor Bbetween factor A and factor B

• When additivity doesn’t applyWhen additivity doesn’t applywe have an interactionwe have an interaction

• Additivity allows us to obtain an unbiased estimator for MSEAdditivity allows us to obtain an unbiased estimator for MSE

• Need to have more than one observation per cell to find a Need to have more than one observation per cell to find a unbiased estimator for MSE when interactions may be present unbiased estimator for MSE when interactions may be present

• KKijij>1 >1 – K- is a constant number per cell (each cell has same number of K- is a constant number per cell (each cell has same number of

observations)observations)

3131

Parameters for Fixed Effects Parameters for Fixed Effects Model w/InteractionModel w/Interaction

IJ

I

J

J

jij

I

i

I

iij

j

J

jij

i

11..

1.

1.

ii = = i.i.-- = = the effect of the effect of factor A at level factor A at level II

jj = = ..j j -- = = the effect of the effect of factor B at level factor B at level JJ

ij ij is the interaction is the interaction parameters parameters

ijij = = ijij – ( – ( + + ii + + jj))

so the individual means are so the individual means are represented belowrepresented below

ijij = = + + ii + + jj + + ijij

3232

HypothesesHypotheses

0:

,0:

0:

0:

0:

0:

321

321

ijaA

ijoAB

aB

IoB

aA

IoA

oneleastatH

jiallforH

oneleastatH

H

oneleastatH

H

• The model is The model is additive if all additive if all ijij=0=0

• Order of testingOrder of testing– Interaction firstInteraction first– Main effectsMain effects

• Sometimes results Sometimes results can be confusingcan be confusing

3333

The Model with Interactions The Model with Interactions • XXijkijk = the rv denoting the = the rv denoting the

measurement when measurement when factor A is held at level factor A is held at level ii and factor B is held at and factor B is held at level level j j given given kk observations for each observations for each ij ij levelslevels

• xxijkijk = actual observed = actual observed valuevalue

• XXijkijk = = + + ii + + jj + + ij ij + + ijij

ij ij are N(0,are N(0, 22))

3434

The Model with Interactions The Model with Interactions

)1)(1(

1

)1)(1(

111

111

11

1 1 1 1

2.

1

22.

1

1 1

2

........1

2...

1 1

2..

1

2

.....

1

2...

1 1

2..

1

2

.....

1

1 1

2...

2

11 1

2

...1

JIdf

XK

XXX

SSABSSBSSASSTSSE

JIdfXXXXSSAB

JdfXIJK

XIK

XXSSB

IdfXIJK

XJK

XXSSA

IJKdfXIJK

XXXSST

I

i

J

j

I

i

J

j i jij

K

kijkijijk

K

k

I

i

J

j

jiij

K

k

I

i

J

jj

J

j

j

K

k

I

i

I

ii

J

j

i

K

k

I

i

J

jijk

K

k

I

i

J

jijk

K

k

3535

The ANOVA TableThe ANOVA TableSource Sum ofSquares

Degrees ofFreedom

MeanSquare

F F,v1,v2 *?



A*B SSAB df=(I-1)(J-1) MSAB FAB F,dfAB,dfE

Error SSE df=IJ (K-1) MSE

Total SST df=IJK-1

3636

Problem 2Problem 2Tests of Between-Subjects Effects

Dependent Variable: CURRENT

14627.778a 5 2925.556 42.128 .000

1237688.889 1 1237688.889 17822.720 .000

1244.444 2 622.222 8.960 .004

13338.889 1 13338.889 192.080 .000

44.444 2 22.222 .320 .732

833.333 12 69.444

1253150.000 18

15461.111 17

SourceCorrected Model

Intercept

PHOSPHOR

GLASS

PHOSPHOR * GLASS

Error

Total

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

R Squared = .946 (Adjusted R Squared = .924)a.

3737

Problem 2Problem 2

Estimated Marginal Means of CURRENT

GLASS

2.001.00

Est

ima

ted

Ma

rgin

al M

ea

ns

320

300

280

260

240

220

PHOSPHOR

1.00

2.00

3.00

3838

Multiple Comparisons in Multiple Comparisons in ANOVAANOVATukey’s Procedure (T Method)Tukey’s Procedure (T Method)• HHoABoAB is not rejected and H is not rejected and HoAoA and/or H and/or HoBoB is rejected is rejected


– For A - QFor A - Q,I,IJ(K-1),I,IJ(K-1)

– For B - QFor B - Q,J,IJ(K-1),J,IJ(K-1)



• Compute the difference in each Compute the difference in each ii - -j j pair Underline those pairs pair Underline those pairs that differ by less than that differ by less than wwPairs Pairs notnot underlined are significantly different underlined are significantly different

IK

MSEQw

JK

MSEQw

KIJJB

KIJIA

)1(,,

)1(,,

3939


Dependent Variable: CURRENT

Tukey HSD

-13.3333 4.8113 .042 -30.5003 3.8336

6.6667 4.8113 .379 -10.5003 23.8336

13.3333 4.8113 .042 -3.8336 30.5003

20.0000* 4.8113 .004 2.8331 37.1669

-6.6667 4.8113 .379 -23.8336 10.5003

-20.0000* 4.8113 .004 -37.1669 -2.8331

(J) PHOSPHOR2.00

3.00

1.00

3.00

1.00

2.00

(I) PHOSPHOR1.00

2.00

3.00

MeanDifference





4040

Problem 2Problem 2CURRENT

Tukey HSDa,b

6 253.3333

6 260.0000 260.0000

6 273.3333

.379 .042

PHOSPHOR3.00

1.00

2.00

Sig.

N 1 2

Subset



Alpha = .01.b.

4141

Mixed and Random Effects Mixed and Random Effects ModelsModels• 1-way ANOVA - Random Effects Model1-way ANOVA - Random Effects Model

• 2-way ANOVA - Mixed Effects Model2-way ANOVA - Mixed Effects Model– XXijij = = + + ii + + BBjj + + GGijij + + ijij

– Distributions of Distributions of BBjj, G, Gijij & &ijij are N(0, are N(0, 22,B ,B ), N(0, ), N(0, 22

,G ,G ), N(0, ), N(0, 22,,,,) )

respectivelyrespectively– Hypothesis TestHypothesis Test

– Order of testingOrder of testing• Interaction firstInteraction first

•Main effectsMain effects

I

ii

1

0

0:

0:2

2

GoG

BoB

H

H

0:

0: 321

oneleastatH

H

aA

IioA

4242

Source Sum ofSquares

Degrees ofFreedom

MeanSquare

F F,v1,v2 *?

Factor AFixed

SSA df=I-1 MSA FAMSA/MSAB

F,dfA,dfE

Factor BRandom

SSB df=J-1 MSB FBMSB/MSE

F,dfB,dfE

A*BRandom

SSAB df=(I-1)(J-1) MSAB FABMSAB/MSE

F,dfAB,dfE

Error SSE df=IJ (K-1) MSE

Total SST df=IJK-1

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1 Multifactor ANOVA. 2 What We Will Learn Two-factor ANOVA K ij =1 Two-factor ANOVA K ij =1...

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