Transcript Document 7719600

Simple ANOVA Comparing the
Means of Three or More
Groups
Chapter 9
ANOVA Terminology
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The purpose of this experiment was to compare
the effects of intensity of training (low, med,
high) on aerobic fitness (VO2).
The independent variable Intensity of Training is
called a FACTOR.
The FACTOR has 3 LEVELS (low, med, high)
The dependent variable in this experiment is
VO2
ANOVA allows for multiple comparisons while
still keeping alpha at 0.05.
Familywise or Experimentwise Error Rate
The purpose of this experiment was to compare the effects
of NUMBER OF DAYS TRAINING PER WEEK (1, 2, 3, 4,
5, 6) on STRENGTH.
The number of days training is a factor with 6 levels. We
could use multiple t-tests to compare (1 v 2, 1 v 3, 1 v 4, 1 v
5, 1 v 6; 2 v 3, 2 v 4, 2 v 5, 2 v 6; 3 v 4, 3 v 5, 3 v 6; 4 v 5, 4
v 6; 5 v 6). That would require 15 t-tests. This would
cause alpha to inflate from 0.05 to 0.26 greatly increasing
the probability of making a Type I ERROR.
ANOVA fixes this problem by doing only one test.
Assumptions of ANOVA
Sources of Variance
Between Groups
variance is the deviation
of the group means from
the Grand MEAN.
Within Groups variance
is the deviation of
individual scores from
their Group Means.
Mean Square and F Ratio
Strength Training Groups
Within Groups Deviations
Sum of Squared Within Deviations
Between Groups Deviations
Between Groups Sum of Squared
Deviations
F Statistic is a Ration of Variances
50.74
 12.69
4
47.43
 1.58
30
12.69
 8.03
1.58
Each Sum of Squares is divided by
its df to produce a Mean Square.
F ratio is the ratio of variances
F = MSb / MSe
Critical Values of F Statistic
Scheffe Post Hoc
Scheffe Results
Scheffe is less
powerful than Tukey
Tukey HSD
Tukey Results
The Group 1 vs Group 2 was only significant at 0.05 with the more
conservative Scheffe.
Critical Values of q Statistic
R2 (also called eta2) and ω2
R2 or eta2 are rough
estimates the size of
the effect.
ω2 is a more exact
test of the Effect.
Effects of Play on Stress
Scheffe Post Hoc
Tukey Post Hoc
Again, Tukey is more powerful than Scheffe.
Single Factor (One-Way) ANOVA
Enter Value Labels for Independent
Variable Group
One-Way or Single Factor ANOVA
Enter Independent and Dependent
Variables
Options Button
Post hoc Tests
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Usually you would
choose just one
Post hoc test.
Tukey is the most
powerful, which is
why it is used most
often.
Scheffe can handle
unequal group sizes,
but it is not very
powerful
One-Way ANOVA Output: Descriptives
Descriptives
s trength
N
Group 1
Group 2
Group 3
Group 4
Control
Total
7
7
7
7
7
35
Mean
5.2857
7.8571
5.0000
6.2857
4.4286
5.7714
Std. Deviation
1.11270
1.67616
.81650
1.49603
.97590
1.69923
Std. Error
.42056
.63353
.30861
.56544
.36886
.28722
95% Confidence Interval for
Mean
Lower Bound Upper Bound
4.2566
6.3148
6.3070
9.4073
4.2449
5.7551
4.9021
7.6693
3.5260
5.3311
5.1877
6.3551
Minim
One-Way ANOVA Output: Homogeneity of
Variance?
Test of Homogeneity of Variances
s trength
Levene
Statis tic
1.388
df1
df2
4
30
Sig.
.262
One-Way ANOVA Output: Summary Table
ANOVA
s trength
Between Groups
Within Groups
Total
Sum of
Squares
50.743
47.429
98.171
df
4
30
34
Mean Square
12.686
1.581
F
8.024
Sig.
.000
One-Way ANOVA Output: Means Plot
8.00
Mean of strength
7.00
6.00
5.00
4.00
Group 1
Group 2
Group 3
Group
Group 4
Control
One-Way ANOVA Output: Tukey Post Hoc
One-Way ANOVA Output: Scheffe Post Hoc
One-Way ANOVA Output: