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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
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.
Procedure for computing 1-way ANOVA for independent samples
• Step 2: Calculate the Grand Total correction factor
GT =
=
(X)2
N
(XA+XB+XC)2
N
Procedure for computing 1-way ANOVA for independent samples
• Step 3: Compute total Sum of Squares
SStotal= X2 - GT
= (XA2+XB
2+XC2) - GT
Procedure for computing 1-way ANOVA for independent samples
• Step 4: Compute between groups Sum of Squares
SSbet= - GT
= + + - GT
(X)2
n
(XA)2
nA
(XB)2
nB
(XC)2
nC
Procedure for computing 1-way ANOVA for independent samples
• Step 5: Compute within groups Sum of Squares
SSwit= SStotal - SSbet
Procedure for computing 1-way ANOVA for independent samples
• 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)
Procedure for computing 1-way ANOVA for independent samples
• Step 7/8: Estimate the Variances & Compute F
=
=
SSbet
dfbet
SSwit
dfwit
Procedure for computing 1-way ANOVA for independent samples
• 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.
Procedure for computing 1-way ANOVA for paired samples
• 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
Procedure for computing 1-way ANOVA for paired samples
• Step 3: Compute total Sum of Squares
SStotal= X2 - GT
= (X12+X2
2+X32) - GT
Procedure for computing 1-way ANOVA for paired samples
• Step 4: Compute between trials Sum of Squares
SSbetT= - GT
= + + - GT
(XT)2
nT
(X1)2
n1
(X2)2
n2
(X3)2
n3
Procedure for computing 1-way ANOVA for paired samples
• 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
Procedure for computing 1-way ANOVA for paired samples
• Step 6: Compute interaction Sum of Squares
SSint= SStotal - (SSbetT + SSbetS)
Procedure for computing 1-way ANOVA for paired samples
• 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
Procedure for computing 1-way ANOVA for paired samples
SSbetT
dfbetT
SSint
dfint
SSbetS
dfbetS
Systematic Variance
(between subjects)
Procedure for computing 1-way ANOVA for paired samples
• 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
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
Significant
Main Effect of Gender
Not significant
Exercise*Gender Interaction
Not significant
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
Significant
Exercise*Gender Interaction
Not significant
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
Significant
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
Main Effect of Gender
Exercise*Gender Interaction
?
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
Main Effect of Gender
Exercise*Gender Interaction
?
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
= variance between means due to
= 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
(resting vs exercise)
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
(resting vs exercise)
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