Analysis of Variance (ANOVA) Developing Study Skills and Research Methods (HL20107) Dr James Betts.

Analysis of Variance (ANOVA)

Developing Study Skills and Research Methods (HL20107)

Dr James Betts

Lecture Outline:

•Multiple Comparisons and Type I Errors

•1-way ANOVA for Unpaired data

•1-way ANOVA for Paired Data

•Factorial Research Designs.

Tim

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PlaceboGlucose

Chryssanthopoulos et al. (1994)

MalesFemales

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120 PlaceboLGIHGIGlucose

Thomas et al. (1991)

*

*P <0.05 vs. Placebo, HGI & Glucose

PlaceboLucozade

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PlaceboLucozadeGatoradePowerade


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What is Analysis of Variance?• ANOVA is an inferential test designed for use with 3 or

more data sets

• t-tests are just a form of ANOVA for 2 groups

• ANOVA only interested in establishing the existence of a statistical differences, not their direction (last slide)

• Based upon an F value (R. A. Fisher) which reflects the ratio between systematic and random/error variance…

Total Variance between means

SystematicVariance

ErrorVariance

Dependent Variable

Extraneous/Confounding

(Error) Variables

Independent Variable

Group AGroup BGroup C

Group BGroup C

Group A

Procedure for computing 1-way ANOVA for independent samples

• Step 1: Complete the tablei.e.

-square each raw score

-total the raw scores for each group

-total the squared scores for each group.


• Step 2: Calculate the Grand Total correction factor

GT =

=

(X)2

N

(XA+XB+XC)2

N


• Step 3: Compute total Sum of Squares

SStotal= X2 - GT

= (XA2+XB

2+XC2) - GT


• Step 4: Compute between groups Sum of Squares

SSbet= - GT

= + + - GT

(X)2

n

(XA)2

nA

(XB)2

nB

(XC)2

nC


• Step 5: Compute within groups Sum of Squares

SSwit= SStotal - SSbet


• Step 6: Determine the d.f. for each sum of squares

dftotal= (N - 1)

dfbet= (k - 1)

dfwit= (N - k)

SystematicVariance

(between means)

ErrorVariance

(within means)


• Step 7/8: Estimate the Variances & Compute F

=

=

SSbet

dfbet

SSwit

dfwit


• Step 9: Consult F distribution table -d1 is your df for the numerator (i.e. systematic variance)

-d2 is your df for the denominator (i.e. error variance)

ANOVA

VAR00001

.152 2 .076 .147 .865

4.635 9 .515

4.787 11

Between Groups

Within Groups

Total

Sum ofSquares df Mean Square F Sig.

Independent 1-way ANOVA: SPSS Output

Group BGroup C

Group ATrial 2Trial 3

Trial 1

Procedure for computing 1-way ANOVA for paired samples

• Step 1: Complete the tablei.e.

-square each raw score

-total the raw scores for each trial & subject

-total the squared scores for each trial & subject.


• Step 2: Calculate the Grand Total correction factor

GT =

=

= = 54.6

(X)2

N

(X1+X2+X3)2

N

(8+8.5+9.1)2

12…so GT just as

with unpaired data


• Step 3: Compute total Sum of Squares

SStotal= X2 - GT

= (X12+X2

2+X32) - GT


• Step 4: Compute between trials Sum of Squares

SSbetT= - GT

= + + - GT

(XT)2

nT

(X1)2

n1

(X2)2

n2

(X3)2

n3


• Step 5: Compute between subjects Sum of Squares

SSbetS= - GT

= + + + - GT

(XS)2

nT

(XT)2

nT

(XD)2

nD

(XH)2

nH

(XJ)2

nJ


• Step 6: Compute interaction Sum of Squares

SSint= SStotal - (SSbetT + SSbetS)


• Step 7: Determine the d.f. for each sum of squares

dftotal= (N - 1)

dfbetT= (k - 1)

dfbetS= (r - 1)

dfint= (r-1)(k-1) = dfbetT x dfbetS

• Step 8/9: Estimate the Variances & Compute F values

=

=

=

SystematicVariance

(between trials IV)

ErrorVariance


SSbetT

dfbetT

SSint

dfint

SSbetS

dfbetS

Systematic Variance

(between subjects)


• Step 10: Consult F distribution table as before

Tests of Within-Subjects Effects

Measure: MEASURE_1

.152 2 .076 .840 .477

.152 1.183 .128 .840 .439

.152 1.505 .101 .840 .457

.152 1.000 .152 .840 .427

.542 6 .090

.542 3.550 .153

.542 4.514 .120

.542 3.000 .181

Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

Sourcefactor1

Error(factor1)

Type III Sumof Squares df Mean Square F Sig.

Paired 1-way ANOVA: SPSS Output

• Next week we will continue to work through some examples of 2-way ANOVA (i.e. factorial designs)

• However, you will come across 2-way ANOVA in this week’s lab class so there are a few terms & concepts that you should be aware of in advance...

Introduction to 2-way ANOVA

Factorial Designs: Technical Terms• Factor

• Levels

• Main Effect

• Interaction Effect

Factorial Designs: Multiple IV’s• Hypothesis:

– The HR response to exercise is mediated by gender

• We now have three questions to answer:

1)

2)

3)

Post Run 1 Pre Run 2 Post Run 2

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cogen

(mm

ol

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cosy

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s. k

g d

m-1)

0

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CHO CHO-PRO

Factorial Designs: Interpretation210

180

150

120

90

60

30

0

Hea

rt R

ate

(bea

tsm

in-1)

Resting Exercise

Main Effect of Exercise

Not significant

Main Effect of Gender

Not significant

Exercise*Gender Interaction

Not significant


Mu

scle

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cogen

(mm

ol

glu

cosy

l u

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s. k

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m-1)

0

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250

300

350

CHO CHO-PRO


180

150

120

90

60

30

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Hea

rt R

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(bea

tsm

in-1)

Resting Exercise


Significant


Not significant


Not significant


Mu

scle

Gly

cogen

(mm

ol

glu

cosy

l u

nit

s. k

g d

m-1)

0

50

100

150

200

250

300

350

CHO CHO-PRO


180

150

120

90

60

30

0

Hea

rt R

ate

(bea

tsm

in-1)

Resting Exercise


Not significant


Significant


Not significant


Mu

scle

Gly

cogen

(mm

ol

glu

cosy

l u

nit

s. k

g d

m-1)

0

50

100

150

200

250

300

350

CHO CHO-PRO


180

150

120

90

60

30

0

Hea

rt R

ate

(bea

tsm

in-1)

Resting Exercise


Not significant


Not Significant


Significant


Mu

scle

Gly

cogen

(mm

ol

glu

cosy

l u

nit

s. k

g d

m-1)

0

50

100

150

200

250

300

350

CHO CHO-PRO


180

150

120

90

60

30

0

Hea

rt R

ate

(bea

tsm

in-1)

Resting Exercise




?


Mu

scle

Gly

cogen

(mm

ol

glu

cosy

l u

nit

s. k

g d

m-1)

0

50

100

150

200

250

300

350

CHO CHO-PRO


180

150

120

90

60

30

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Hea

rt R

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(bea

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in-1)

Resting Exercise




?

SystematicVariance

(resting vs exercise)

ErrorVariance

(between subjects)

Systematic Variance

(male vs female)

Systematic Variance

(Interaction)

2-way mixed model ANOVA: Partitioning

= variance between means due to



= uncontrolled factors and within group differences for males vs females.

ErrorVariance

(within subjects)= uncontrolled factors plus random changes within individuals for rest vs exercise

SystematicVariance


Systematic Variance

(male vs female)Systematic

Variance(Interaction)

ErrorVariance

(within subjects)

2-way mixed model ANOVA

So for a fully unpaired design

– e.g. males vs females

&

rest group vs exercise group

…between subject variance (i.e. SD) has a negative impact upon all contrasts

ErrorVariance

(between subjects)

SystematicVariance


Systematic Variance

(am vs pm)Systematic

Variance(Interaction)

Error Variance

(within subjectsexercise)

2-way mixed model ANOVA

…but for a fully paired design

– e.g. morning vs evening

&

rest vs exercise

…between subject variance (i.e. SD) can be removed

from all contrasts.

Error Variance

(within subjectstime)Error Variance

(within subjectsinteract)

Refer back to this ‘partitioning’ in your lab class

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Analysis of Variance (ANOVA) Developing Study Skills and Research Methods (HL20107) Dr James Betts.

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