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One-Way ANOVA
Introduction to Analysis of Variance (ANOVA)
What is ANOVA?
ANOVA is short for ANalysis Of VAriance Used with 3 or more groups to test for MEAN
DIFFS. E.g., caffeine study with 3 groups:
No caffeine Mild dose Jolt group
Level is value, kind or amount of IV Treatment Group is people who get specific
treatment or level of IV Treatment Effect is size of difference in means
Rationale for ANOVA (1)
We have at least 3 means to test, e.g., H0: 1 = 2 = 3.
Could take them 2 at a time, but really want to test all 3 (or more) at once.
Instead of using a mean difference, we can use the variance of the group means about the grand mean over all groups.
Logic is just the same as for the t-test. Compare the observed variance among means (observed difference in means in the t-test) to what we would expect to get by chance.
Rationale for ANOVA (2)
Suppose we drew 3 samples from the same population. Our results might look like this:
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4
3
2
1
0
100-10-20
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Raw Scores (X)
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Three Samples from the Same Population
Mean 1
Mean 2
Mean 3
Standard Dev Group 3
Note that the means from the 3 groups are not exactly the same, but they are close, so the variance among means will be small.
Rationale for ANOVA (3)
Suppose we sample people from 3 different populations. Our results might look like this:
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4
3
2
1
0
Three Samples from 3 Diffferent Populations
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Three Samples from 3 Diffferent Populations
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Three Samples from 3 Diffferent Populations
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Raw Scores (X)
Three Samples from 3 Diffferent Populations
Mean 1
Mean 2Mean 3
SD Group 1
Note that the sample means are far away from one another, so the variance among means will be large.
Rationale for ANOVA (4)Suppose we complete a study and find the following results (either graph). How would we know or decide whether there is a real effect or not?
100-10-20
4
3
2
1
0
100-10-20
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Raw Scores (X)
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Three Samples from the Same Population
Mean 1
Mean 2
Mean 3
Standard Dev Group 3
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4
3
2
1
0
Three Samples from 3 Diffferent Populations
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Three Samples from 3 Diffferent Populations
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Three Samples from 3 Diffferent Populations
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Raw Scores (X)
Three Samples from 3 Diffferent Populations
Mean 1
Mean 2Mean 3
SD Group 1
To decide, we can compare our observed variance in means to what we would expect to get on the basis of chance given no true difference in means.
Review
When would we use a t-test versus 1-way ANOVA?
In ANOVA, what happens to the variance in means (between cells) if the treatment effect is large?
Rationale for ANOVA We can break the total variance in a study into meaningful pieces that correspond to treatment effects and error. That’s why we call this Analysis of Variance.
Definitions of Terms Used in ANOVA:
GX The Grand Mean, taken over all observations.
AX
1AX
The mean of any level of a treatment.
The mean of a specific level (1 in this case) of a treatment.
iX The observation or raw data for the ith person.
The ANOVA ModelA treatment effect is the difference between the overall, grand mean, and the mean of a cell (treatment level).
GA XXEffectIV
Error is the difference between a score and a cell (treatment level) mean.
Ai XXError
The ANOVA Model:
)()( AiGAGi XXXXXX
An individual’s score is The grand
mean+
A treatment or IV effect + Error
The ANOVA Model
)()( AiGAGi XXXXXX The grand
meanA treatment or IV effect
Error
40
30
20
10
0
Fre
quen
cy
ANOVA Data by Treatment LevelANOVA Data by Treatment Level
ANOVA Data by Treatment Level
Grand Mean
Treatment Mean
Error
IV Effect
The graph shows the terms in the equation. There are three cells or levels in this study. The IV effect and error for the highest scoring cell is shown.
ANOVA CalculationsSums of squares (squared deviations from the mean) tell the story of variance. The simple ANOVA designs have 3 sums of squares.
2)( Gitot XXSSThe total sum of squares comes from the distance of all the scores from the grand mean. This is the total; it’s all you have.
2)( AiW XXSS The within-group or within-cell sum of squares comes from the distance of the observations to the cell means. This indicates error.
2)( GAAB XXNSS The between-cells or between-groups sum of squares tells of the distance of the cell means from the grand mean. This indicates IV effects.WBTOT SSSSSS
Computational Example: Caffeine on Test Scores
G1: Control G2: Mild G3: Jolt
Test Scores
75=79-4 80=84-4 70=74-4
77=79-2 82=84-2 72=74-2
79=79+0 84=84+0 74=74+0
81=79+2 86=84+2 76=74+2
83=79+4 88=84+4 78=74+4
Means
79 84 74
SDs (N-1)
3.16 3.16 3.16
G1 75 79 16
Control 77 79 4
M=79 79 79 0
SD=3.16 81 79 4
83 79 16
G2 80 79 1
M=84 82 79 9
SD=3.16 84 79 25
86 79 49
88 79 81
G3 70 79 81
M=74 72 79 49
SD=3.16 74 79 25
76 79 9
78 79 1
Sum 370
GXiX 2)( Gi XX
Total Sum of Squares
2)( Gitot XXSS
In the total sum of squares, we are finding the squared distance from the Grand Mean. If we took the average, we would have a variance.
2)( Gitot XXSS
Scores on the Dependent Variable by Group
0.5
0.4
0.3
0.1
0.0
Rel
ativ
e F
requ
ency
Low High
Grand Mean
G1 75 79 16
Control 77 79 4
M=79 79 79 0
SD=3.16 81 79 4
83 79 16
G2 80 84 16
M=84 82 84 4
SD=3.16 84 84 0
86 84 4
88 84 16
G3 70 74 16
M=74 72 74 4
SD=3.16 74 74 0
76 74 4
78 74 16
Sum 120
Within Sum of Squares
iXAX 2)( Ai XX
2)( AiW XXSS
Within sum of squares refers to the variance within cells. That is, the difference between scores and their cell means. SSW estimates error.
Scores on the Dependent Variable by Group
0.5
0.4
0.3
0.1
0.0
Rel
ativ
e F
requ
ency
Low High
Cell or Treatment Mean
2)( AiW XXSS
G1 79 79 0
Control 79 79 0
M=79 79 79 0
SD=3.16 79 79 0
79 79 0
G2 84 79 25
M=84 84 79 25
SD=3.16 84 79 25
84 79 25
84 79 25
G3 74 79 25
M=74 74 79 25
SD=3.16 74 79 25
74 79 25
74 79 25
Sum 250
Between Sum of Squares
2)( GAAB XXNSS
AX GX 2)( GA XX
The between sum of squares relates the Cell Means to the Grand Mean. This is related to the variance of the means.
Scores on the Dependent Variable by Group
0.5
0.4
0.3
0.1
0.0
Rel
ativ
e F
requ
ency
Low High
Cell Mean
Grand Mean
Cell MeanCell Mean
2)( GAAB XXNSS
ANOVA Source Table (1)
Source SS df MS F
Between Groups
250 k-1=2 SS/df
250/2=
125 =MSB
F = MSB/MSW = 125/10
=12.5
Within Groups
120 N-k=
15-3=12
120/12 = 10 =
MSW
Total 370 N-1=14
ANOVA Source Table (2)
df – Degrees of freedom. Divide the sum of squares by degrees of freedom to get
MS, Mean Squares, which are population variance estimates.
F is the ratio of two mean squares. F is another distribution like z and t. There are tables of F used for significance testing.
The F Distribution
F Table – Critical ValuesNumerator df: dfB
dfW 1 2 3 4 5
5 5%
1%
6.61
16.3
5.79
13.3
5.41
12.1
5.19
11.4
5.05
11.0
10 5%
1%
4.96
10.0
4.10
7.56
3.71
6.55
3.48
5.99
3.33
5.64
12 5%
1%
4.75
9.33
3.89
6.94
3.49
5.95
3.26
5.41
3.11
5.06
14 5%
1%
4.60
8.86
3.74
6.51
3.34
5.56
3.11
5.04
2.96
4.70
Review
What are critical values of a statistics (e.g., critical values of F)?
What are degrees of freedom? What are mean squares? What does MSW tell us?
Review 6 Steps
1. Set alpha (.05).
2. State Null & Alternative
H0:
H1: not all are =.
3. Calculate test statistic: F=12.5
4. Determine critical value F.05(2,12) = 3.89
5. Decision rule: If test statistic > critical value, reject H0.
6. Decision: Test is significant (12.5>3.89). Means in population are different.
321
Post Hoc Tests
If the t-test is significant, you have a difference in population means.
If the F-test is significant, you have a difference in population means. But you don’t know where.
With 3 means, could be A=B>C or A>B>C or A>B=C.
We need a test to tell which means are different. Lots available, we will use 1.
Tukey HSD (1)
HSD means honestly significant difference.
A
W
N
MSqHSD
is the Type I error rate (.05).
q Is a value from a table of the studentized range statistic based on alpha, dfW (12 in our example) and k, the number of groups (3 in our example).
WMS Is the mean square within groups (10).
AN Is the number of people in each group (5).
33.55
1077.305. HSD
From table
MSW
ANResult for our example.
Use with equal sample size per cell.
Tukey HSD (2)
To see which means are significantly different, we compare the observed differences among our means to the critical value of the Tukey test.
The differences are:1-2 is 79-84 = -5 (say 5 to be positive).1-3 is 79-74 = 52-3 is 84-74 = 10. Because 10 is larger than 5.33, this result is significant (2 is different than 3). The other differences are not significant. Review 6 steps.
Review
What is a post hoc test? What is its use? Describe the HSD test. What does HSD
stand for?
Test
Another name for mean square is _________.
1. standard deviation
2. sum of squares
3. treatment level
4. variance
Test
When do we use post hoc tests? a. after a significant overall F test b. after a nonsignificant overall F test c. in place of an overall F test d. when we want to determine the impact of
different factors