Lecture Overview on ANOVA
Review
hypothesis testing; inferential statistics
z-test, t-test, independent & dependent t-test
New Stuff
Power – Ability to reject Ho
ANOVA
•
•
•
•
Analysis of Variance
Done with 3 or more groups
Playground Exercise
Complete SPSS Example
Dr. Sinn, PSYC301, The joy of 1-way ANOVA
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Power
Review: Hypothesis Testing Errors
Wrongly rejecting Ho: Chance of Type I error: α
Wrongly retaining Ho: Chance of Type II error: β
Power
Opposite of β
Power = 1- β
Ability to reject Ho (when Ho should be rejected).
Researchers want Power!
• Want ability to reject Ho; Show you were right to suspect a
difference.
• Want to show IV affects your DV.
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Error Areas
α area (where we reject the Ho,
β area (where we retain the Ho,
and we shouldn’t)
• beyond tcritical
• under Ho
and we shouldn’t)
• inside tcritical
• under Ha
Ho: μ=55
Ha: μ>55
α
α
tc
tc
Dr. Sinn, PSYC301, The joy of 1-way ANOVA
β
tc
4
Increasing Power
#1: Increase Treatment: Increase difference between groups (μ’s)
H0:
μ=55
Reality: μ=57
H0:
μ=55
Reality: μ=72
β
β
tc
tc
#2: Decrease Sampling Error: Decrease differences within groups.
β
β
tc
Dr. Sinn, PSYC301, The joy of 1-way ANOVA
tc
5
Examples of increasing power
Rat Study: IV:Caffeine Level DV:Amt. Food Found
Therapy Study: IV:Therapy (drug, talk, drug+talk, or control) DV: Improvement]
#1 Increase Treatment Effect
(Increase BG differences)
Rat study
0,3,or 6 mg
0,10,or 20 mg
Therapy study
10 therapy sessions
1 therapy session
#2 Decrease Sampling Error
(Decrease WG differences)
Rat study
Different strains of rats
Same strain of rat
Rats allowed to eat freely
Rats all unfed for 24 hours
Therapy study
Diff. types of Therapy
Same type of Therapy
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1-Way ANOVA
ANOVA
Analysis of Variance
1-way means 1 Independent Variable (IV)
Purpose:
ANOVA allows hypothesis testing with 3+ sample means
• Imagine study on interventions to help frosh make friends
• Three IV levels: Standard courses, interactive courses,
clustered courses.
ANOVA uses F-test
Strategy: Compare variability within
group to variability between groups.
F
F is ratio between two values:
Dr. Sinn, PSYC301, The joy of 1-way ANOVA
MS BG
MSW G
7
ANOVA Playground
(Download from Website)
1
Dr. Sinn, PSYC301, The joy of 1-way ANOVA
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Matching
Exercise
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Draw Conclusions from Playground
What does a large F mean?
What two things will make F large?
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Partitioning Variance
Partition
fancy word for “divide up”
ANOVA partitions variance (MS means variance)
Types of variance
Total variance = MSWG + MSBG
MSWG= sampling error (background noise)
MSBG = sampling error + treatment (includes effect of Independent Variable)
MS BG treatment  error
F

MSW G
error
If just error  F tends toward 1.0
If treatment effect F gets larger
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Example of 1-way ANOVA
Studying effect of caffeine on productivity
Does caffeine help or hurt?
IV: Level of Caffeine: 0, 10, 20 mg
DV: Number of Food Pellets Found
0 mg
2
3
1
4
2
10 mg
1
2
3
1
2
20 mg
4
4
4
4
5
5
Dr. Sinn, PSYC301, The joy of 1-way ANOVA
Number of
Food
Pellets
Found
13
SPSS Data Entry
IV
DV
Label levels
of IV so
output is
easier to
read.
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SPSS Analysis
Go to Analyze, Compare Means, & select One-way ANOVA
Put DV
here.
Put IV
here.
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SPSS Analysis, Part #2
Select this to get
descriptive statistics
like sample means &
standard deviations.
Gives you a
line graph of
the sample
means
Conducts “after
the fact” test to
compare all pairs
of sample means.
Dr. Sinn, PSYC301, The joy of 1-way ANOVA
Alpha level
still set to
.05, just like
it was with ttests.
16
SPSS Output
Descriptives
FOOD_FND
0 mg
10 mg
20 mg
Total
N
5
5
6
16
Mean
2.40
1.80
4.33
2.94
Std.
Deviation
1.140
.837
.516
1.389
Std.
Error
.510
.374
.211
.347
Min
1
1
4
1
Max
4
3
5
5
F
Sample means from 3
groups, plus mean amount of
food found overall.
MS BG
MSW G
ANOVA
Source of Variation Table
FOOD_FND
Between Groups
W ithin Groups
Total
Sum of
Squares
19.604
9.333
28.937
df
2
13
15
Mean
Square
9.802
.718
F
13.65
Dr. Sinn, PSYC301, The joy of 1-way ANOVA
Sig.
.001
17
Where does F come from?
MSWG = SSWG/dfWG = Sum of Squares / degrees of freedom
MSBG = SSBG/dfBG = Sum of Squares / degrees of freedom
Degrees of freedom
dfWG:
NT – K
dfBG:
K-1
dfTOTAL: NT – 1
(Total # of subjects - # of groups)
(# of groups – 1)
(Total # of subjects – 1)
Expectations:
If I give you df and SS, you can calculate F
You don’t have to get any SS by hand.
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SPSS Output –Post Hoc Test
FOOD_FND
No Sig. Diff. Between 0 & 10mg
Student-Newman-Keuls
GROUP
10 mg
0 mg
20 mg
Sig.
N
5
5
6
a,b
Subset for
alpha = .05
1
2
1.80
2.40
4.33
.270
1.0
4.5
4.0
Mean of FOOD_FND
Means for groups in homogeneous s ubsets3.5
are displayed.
a. Us es Harmonic Mean Sample Size = 5.294.
Rats at 20 mg found
3.0
b. The group
izes are unequal. The harmonic mean
significantly
more sfood
2.5
of the group sizes is used. Type I error levels are
than rats on 0 or 10 mg of
not guaranteed.
2.0
caffeine.
1.5
0 mg
10 mg
20 mg
GROUP
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SPSS Output– Practical Significance
ANOVA
FOOD_FND
Between Groups
W ithin Groups
Total
Sum of
Squares
19.604
9.333
28.937
df
2
13
15
Mean
Square
9.802
.718
F
13.65
Sig.
.001
η2 (“eta squared”)
Effect size statistic – indicates % of variance explained
Measures impact of IV on DV
SS BG
19.604
 

 .6775
SSTotal
28.937
2
We can explain 68% of the variance in how much food a rat finds
if we know the level of caffeine.
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Hypothesis Testing Steps
1.
Comparison: cf. three sample means.
2.
Hypothesis: Ho: μ1= μ2 = μ3
3.
Set-up: α= .05 , dfbg= K-1= 2, dfwg= NT-K = 16-3=13,
Fcrit = 3.80
4.
Fobt = 13.653
5.
Reject Ho.
Ha: Not all μ’s equal
The hypothesis was largely supported. Rats found sig.
more food on 20mg of caffeine (M=4.33) than on 0mg
(M=2.40) or 10mg (M=1.80), F(2,13) = 13.653, p <=.05.
Caffeine has a large effect on food finding behavior,
accounting for about 68% of the variance, η2 = .6775.
Dr. Sinn, PSYC301, The joy of 1-way ANOVA
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F-table
df Between Groups
df Within
Groups
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Lab #8: 1-way ANOVA
TV Problem: The hypothesis was supported.
Light TV users provided more community service
(M = 6.13) than did moderate users (M = 4.00),
who provided more than heavy users (M = 1.75),
F(2,21) = 15.963, p ≤ .05. TV accounts for about
60% of the variance in community service, η2 =
.6032.
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Follow-up Questions
Q1: Variance within group? MSwg = 2.399
Q2: Variance between groups? MSbg=38.292
Q3: Replacing heavy scores with 4,5,4,5,6,5,4,3
would decrease the difference between groups
because the heavy users would then difference less
from the other groups.
Q4: Decreasing between group differences
(decreasing treatment) would decrease F.
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Problem #2: Post Hoc Explanation
hours
Student-Newman-Keuls
commute
45 min commute
60 min commute
30 min commute
0 min commute
Sig.
a
N
5
5
5
5
Subset for alpha = .05
1
2
1.20
1.60
2.40
2.40
3.40
.077
.068
Means for groups in homogeneous s ubsets are displayed.
a. Us es Harmonic Mean Sample Size = 5.000.
Dr. Sinn, PSYC301, The joy of 1-way ANOVA
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Problem #2: Post Hoc Explanation
hours
Student-Newman-Keuls
commute
45 min commute
60 min commute
30 min commute
0 min commute
Sig.
a
N
5
5
5
5
Subset for alpha = .05
1
2
1.20
1.60
2.40
2.40
3.40
.077
.068
Means for groups in homogeneous s ubsets are displayed.
a. Us es Harmonic Mean Sample Size = 5.000.
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Problem #2:
The hypothesis was supported. People commuting
0 minutes participated significantly more (M=3.4
hours) than people commuting 45 (M=1.2) or 60
minutes (M=1.6), F (3,16) = 7.256, p≤.05.
Commuting accounted for a large amount of
variance in community involvement, η2 = .5764.
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Follow-up Questions
Q1: Variance within group? MSwg = .650
Q2: Variance between groups? MSbg=4.717
Q3: Replacing 30 minute commuting scores with
1,4,1,4,3 would increase the within group
variability.
Q4: Increasing sampling error would decrease F.
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Review Partitioning
Study: Does alcohol
affect reaction time?
No
Alcohol
10
2 Beers
4 Beers
15
20
20
25
15
15
30
30
10
20
40
14
23
26
μna=??
μ2b=??
Identify the treatment
effect in this case.
Explain how sampling
error might arise.
Sample Means
μ4b=?? Population Means
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One-Way ANOVA
Part 2!!
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Review Partitioning
Study: Does alcohol
affect reaction time?
No
Alcohol
2 Beers
What accounts for variability
within groups?
4 Beers
10
15
20
20
25
15
15
30
30
10
20
40
What accounts for variability
between groups?
What’s the Formula for F?
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Review Partitioning
Study: Does alcohol
affect reaction time?
No
Alcohol
2 Beers
4 Beers
10
15
20
20
25
15
15
30
30
10
20
40
If the alcohol content of the
beers is not held constant, what
happens?
a. error increases
b. error decreases
c. treatment effect increases
d. treatment effect decreases
If the alcohol content of the beers
is not held constant, what
happens to F?
a. increases
b. decreases
c. neither
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Hypothesis Testing Steps
1.
Comparison: cf. three sample means.
2.
Hypothesis: Ho: μ1= μ2 = μ3
3.
Set-up: α= .05 , dfbg=K-1=3-1=2, dfwg=NT-K=12-3=9,
Fcrit = 4.26
Ha: Not all μ’s equal
• now do one-way ANOVA on SPSS
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SPSS Output - Charts
Descriptives
ALCOHOL
ALCOHOL
a
St udent-Newman-Keuls
N
no alcohol
2 beers
4 beers
Total
4
4
4
12
Mean
13.75
22.50
26.25
20.83
Std. Deviation
4.787
6.455
11.087
9.003
Std. Error
2.394
3.227
5.543
2.599
GROUP
no alcohol
2 beers
4 beers
Sig.
N
4
4
4
Subset for
alpha = .05
1
13.75
22.50
26.25
.118
Means for groups in homogeneous subs et
a. Us es Harmonic Mean Sample Size =
ANOVA
ALCOHOL
Between Groups
W ithin Groups
Total
Sum of
Squares
329.167
562.500
891.667
df
2
9
11
Mean
Square
164.583
62.500
F
2.633
Dr. Sinn, PSYC301, The joy of 1-way ANOVA
Sig.
.126
34
SPSS Output - Graphs
28
26
24
22
20
18
16
14
12
no alcohol
GROUP
Mean +- 2 SE ALCOHOL
40
30
20
10
0
N=
2 beers
4 beers
4
4
no alcohol 2 beers
4
4 beers
GROUP
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Hypothesis Testing Steps
1.
Comparison: cf. three sample means.
2.
Hypothesis: Ho: μ1= μ2 = μ3
3.
Set-up: α= .05 , dfbg=K-1=3-1=2, dfwg=NT-K=12-3=9,
Fcrit = 4.26
4.
Fobt = 2.633
5.
Retain Ho.
Ha: Not all μ’s equal
The hypothesis was not supported. The reaction times
following no alcohol (M=13.75), two beers (M=22.50),
and four beers (M=26.25) did not differ significantly,
F(2,9) = 2.633, n.s..
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Numb. of Words Recalled: Dataset A
4
8
12
Bet. Group Varib:
5
9
10
4
9
11
MSbg: _______
With. Group Varib:
5
8
12
L M H
L M H
MSwg: _______
MS BG
F

MSW G
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Numb. of Words Recalled: Dataset B
8
4
10
Bet. Group Varib:
9
5
12
9
5
11
MSbg: _______
With. Group Varib:
8
4
12
L M H
L M H
MSwg: _______
MS BG
F

MSW G
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Numb. of Words Recalled: Dataset C
7
3
9
Bet. Group Varib:
10
6
13
7
6
10
MSbg: _______
With. Group Varib:
10
3
13
L M H
L M H
MSwg: _______
MS BG
F

MSW G
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Numb. of Words Recalled: Dataset D
7
6
7
Bet. Group Varib:
10
8
7
7
6
12
MSbg: _______
With. Group Varib:
10
10
12
L M H
L M H
MSwg: _______
MS BG
F

MSW G
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Numb. of Words Recalled: Dataset E
7
6
7
10
8
7
7
6
12
10
10
12
7
6
7
10
8
7
7
6
12
10
10
12
Bet. Group Varib:
L M H
MSbg: _______
With. Group Varib:
L M H
MSwg: _______
MS BG
F

MSW G
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Numb. of Words Recalled: Dataset F
Bet. Group Varib:
L M H
MSbg: _______
With. Group Varib:
L M H
MSwg: _______
MS BG
F

MSW G
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