Analysis of Variance
ANOVA
1-Way
Basic Ideas:
Underlying justification: Central Limit Theorem
Primary Goal: Testing equality among means in several groups
Secondary Goal: Estimating size of effects if it can be shown that all groups are not the same.
These are accomplished by estimating variance in two ways.
Or by solving an algebraic tautology.
An example:
Percent of a Standard 50-word list heard correctly in the presence of background noise.
24 subjects with normal hearing were played standard audiology tapes of English words at low volume, with a noisy background. They repeated the words and were scored correct or incorrect in their perception of the words. The order of list presentation was randomized.
The word lists are standard audiology tools for assessing hearing. They are calibrated to be equally difficult to perceive. However, the original calibration was performed with normal-hearing subjects and no noise background. The experimenter wished to determine whether the lists were still equally difficult to understand in the presence of a noisy background.
Source: Loven, Faith. (1981). A Study of the Interlist Equivalency of the CID W-22 Word List Presented in Quiet and in Noise. Unpublished MS Thesis, University of Iowa.
Data
List 1 / List 2 / List 3 / List 428 / 20 / 24 / 26
24 / 16 / 32 / 24
32 / 38 / 20 / 22
30 / 20 / 14 / 18
34 / 34 / 32 / 24
30 / 30 / 22 / 30
36 / 30 / 20 / 22
32 / 28 / 26 / 28
48 / 42 / 26 / 30
32 / 36 / 38 / 16
32 / 32 / 30 / 18
38 / 36 / 16 / 34
32 / 28 / 36 / 32
40 / 38 / 32 / 34
28 / 36 / 38 / 32
48 / 28 / 14 / 18
34 / 34 / 26 / 20
28 / 16 / 14 / 20
40 / 34 / 38 / 40
18 / 22 / 20 / 26
20 / 20 / 14 / 14
26 / 30 / 18 / 14
36 / 20 / 22 / 30
40 / 44 / 34 / 42 / Total
Mean / 32.8 / 29.7 / 25.3 / 25.6 / 28.3
Variance / 54.9 / 64.9 / 69.2 / 60.5 / 62.4
Sorted list:
18 / 16 / 14 / 14
20 / 16 / 14 / 14
24 / 20 / 14 / 16
26 / 20 / 14 / 18
28 / 20 / 16 / 18
28 / 20 / 18 / 18
28 / 22 / 20 / 20
30 / 28 / 20 / 20
30 / 28 / 20 / 22
32 / 28 / 22 / 22
32 / 30 / 22 / 24
32 / 30 / 24 / 24
32 / 30 / 26 / 26
32 / 32 / 26 / 26
34 / 34 / 26 / 28
34 / 34 / 30 / 30
36 / 34 / 32 / 30
36 / 36 / 32 / 30
38 / 36 / 32 / 32
40 / 36 / 34 / 32
40 / 38 / 36 / 34
40 / 38 / 38 / 34
48 / 42 / 38 / 40
48 / 44 / 38 / 42 / Total
Mean / 32.8 / 29.7 / 25.3 / 25.6 / 28.3
Variance / 54.9 / 64.9 / 69.2 / 60.5 / 62.4
Grand mean / 28.3
Five number Summary:
List 1 / List 2 / List 3 / List 4* / 48 / 44 / 38 / 42
H / 35 / 34 / 31 / 30
M / 32 / 30 / 25 / 25
H / 30 / 28 / 20 / 21
* / 18 / 16 / 14 / 14
Are the differences observed among the groups statistically significant?
Variance / 54.9 / 64.9 / 69.2 / 60.5 / 62.4
Grand mean / 28.3
effects / 4.4 / 1.4 / -3.1 / -2.7
effects^2 / 19.7 / 1.8 / 9.4 / 7.4 / 9.6
BSS / 472.6 / 44.0 / 225.1 / 178.8 / 230.1
920.5
WSS / 1317 / 1558 / 1660 / 1452
5988
SS / df / MS / F
Between / 920 / 3 / 307 / 4.7
Within / 5988 / 92 / 65
Total / 6908 / 95