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

e to

Fat

igu

e (m

in)

0

20

40

60

80

100

120

140

PlaceboGlucose

Chryssanthopoulos et al. (1994)

MalesFemales

Tim

e to

Fat

igu

e (m

in)

0

20

40

60

80

100

120 PlaceboLGIHGIGlucose

Thomas et al. (1991)

*

*P <0.05 vs. Placebo, HGI & Glucose

PlaceboLucozade

Tim

e to

Fat

igu

e (m

in)

0

20

40

60

80

100

120 PlaceboLGIHGIGlucose

Thomas et al. (1991)

*

*P <0.05 vs. Placebo, HGI & Glucose

PlaceboLucozadeGatoradePowerade

PlaceboLucozadeGatoradePowerade

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

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

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

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


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




?

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

analysis of variance (anova) developing study skills and research methods (hl20107) dr james betts

Documents

way anova slide

gt slide

n slide

significant slide

njnj slide

ncnc slide

n3n3 slide

ss total ss bet slide