ANOVA

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Subject
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
HR ly ing
80
71
74
68
76
72
72
60
78
86
86
74
60
60
68
54
68
64
64
60
70
80
60
60
82
64
84
95
78
56
72
68
76
76
68
56
HR sitting
66
73
80
76
84
80
88
64
72
88
88
74
62
76
69
58
72
88
90
66
80
84
84
68
75
68
88
92
80
65
80
72
100
84
81
80
HR standing
60
86
78
72
88
88
92
70
68
96
96
80
65
72
75
60
80
64
90
70
86
86
90
72
86
72
96
91
80
76
104
92
80
92
76
88
Effects of posture
84
80
HR 76
(bpm)
72
68
64
lying
sitting
standing
Does posture affect HR?
Analysis of variance (ANOVA): Based on the null hypothesis that all 3
samples are drawn from the same population, so that the variance of each
sample will be an estimate of the same (population) variance (). Thus,
according to the null hypothesis, the variance within samples or the variance
between samples are both estimates of the population variance 
The “within samples” variance is determined by calculating, for each
sample, the sum of squares (SS) of differences between each value and
the sample mean, adding these up and dividing by the number of degrees
of freedom.
The “between samples” variance is determined by (1) calculating the mean
of all values for all samples (the “grand mean”), (2) in each sample, replace
each individual value by the mean of that sample; (3) then calculating the
sum of squares (SS) of differences between each new replaced value and
the grand mean, adding these up and dividing by the number of degrees of
freedom.
“within samples” variance
Subj e c t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36

x
(x1  x)
2
(SS)
HR ly ing
80
71
74
68
76
72
72
60
78
86
86
74
60
60
68
54
68
64
64
60
70
80
60
60
82
64
84
95
78
56
72
68
76
76
68
56
HR sitting
“between samples” variance
HR sta nding
66
73
80
76
84
80
88
64
72
88
88
74
62
76
69
58
72
88
90
66
80
84
84
68
75
68
88
92
80
65
80
72
100
84
81
80
HR ly ing
60
86
78
72
88
88
92
70
68
96
96
80
65
72
75
60
80
64
90
70
86
86
90
72
86
72
96
91
80
76
104
92
80
92
76
88
70.56
77.64
81.03
3322.89
3252.31
4336.97
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
7 0 .5 6
70.56
x
Grand mean = 76.41
 (x  76.41)
1
Total SS = 3322.89 + 3252.31 + 4336.97 = 10912.17
Variance estimate = 10912.17/105 = 103.93
1233.01
HR sta nding
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
7 7 .6 4
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
8 1 .0 3
77.64
54.49
81.03
768.40
Total SS = 1233.01 + 54.49 + 768.40 = 2055.90
No. of degrees of freedom = (36 -1) + (36 -1) + (36 -1) = 105

2
HR sitting


No. of degrees of freedom = (3-1) = 2
Variance estimate = 2055.87/2 = 1027.95
Thus, the estimate of the population variance based on the variations
between sample means (1027.95) is much greater than the estimate
based on variations of individual values about their own sample means
(103.93). If the null hypothesis were correct, one would expect the two
estimates to be similar.
The degree of discrepancy is called F, and is calculated as the ratio:
between sample variance estimate/ within sample variance estimate
In this case, F = 1029.95/103.93 = 9.89
Assuming the null hypothesis is true, what is the probability of obtaining
By chance a value of F ≥ 9.89 on (2,105) degrees of freedom?
Answer: P = 0.00012
Therefore the null hypothesis is rejected
ANOVA tests can be performed on Excel. The results will look like the
following:
A nova: Single F ac tor
SU M M A RY
Groups
H R lying
H R s itting
H R s tanding
Count
36
36
36
A NO V A
Source of Variation
Between G roups
Within G roups
SS
2 0 5 5 .9
1 0 9 1 2 .2
T otal
1 2 9 6 8 .1
Sum
2540
2795
2917
df
2
105
107
Average
Variance
7 0 .6
9 4 .9
7 7 .6
9 2 .9
8 1 .0
1 2 3 .9
MS
1 0 2 7 .9 5
1 0 3 .9 3
F
9 .8 9
P-value
0 .0 0 0 1 2
F crit
3 .0 8
In the data set above, the gender of the subject was also recorded, as follows:
HR ly ing
80
71
74
68
76
72
72
60
78
86
86
74
60
60
68
54
68
64
64
60
70
80
60
60
82
64
84
95
78
56
72
68
76
76
68
56
HR sitting
66
73
80
76
84
80
88
64
72
88
88
74
62
76
69
58
72
88
90
66
80
84
84
68
75
68
88
92
80
65
80
72
100
84
81
80
HR sta ndingGende r
6 0 fe male
8 6 fe male
7 8 fe male
7 2 fe male
8 8 fe male
8 8 fe male
9 2 fe male
7 0 fe male
6 8 fe male
9 6 fe male
9 6 fe male
8 0 fe male
6 5 fe male
7 2 fe male
7 5 fe male
6 0 fe male
8 0 fe male
6 4 fe male
9 0 male
7 0 male
8 6 male
8 6 male
9 0 male
7 2 male
8 6 male
7 2 male
9 6 male
9 1 male
8 0 male
7 6 male
1 0 4 male
9 2 male
8 0 male
9 2 male
7 6 male
8 8 male
Effects of posture
92
88
84
girls
boys
80
76
72
68
64
1)
2)
3)
lying
sitting
standing
Does posture affect HR?
Is there a difference between males and
females?
Are the effects of posture and gender on
HR additive or interactive?
In this case, there are two factors that affect the variable (posture and gender). To answer
the previous questions, an ANOVA (two factor with replication) is required. The results
are as follows:
A nov a: T wo-F ac tor With Replic ation
S U M M A RY
H R ly ing
H R s itting
H R s tanding
T otal
female
C ount
S um
A verage
V arianc e
18
1271
7 0 .6 1
7 9 .4 3
18
1358
7 5 .4 4
8 9 .6 7
18
1390
7 7 .2 2
1 3 9 .0 1
54
4019
7 4 .4 3
1 0 6 .7 8
18
1269
7 0 .5 0
1 1 6 .0 3
18
1437
7 9 .8 3
9 1 .4 4
18
1527
8 4 .8 3
8 5 .4 4
54
4233
7 8 .3 9
1 2 9 .9 0
36
2540
7 0 .5 6
9 4 .9 4
36
2795
7 7 .6 4
9 2 .9 2
36
2917
8 1 .0 3
1 2 3 .9 1
1
2
2
102
MS
4 2 4 .0 4
1 0 2 7 .9 5
1 3 5 .4 0
1 0 0 .1 7
male
C ount
S um
A verage
V arianc e
Total
C ount
S um
A verage
V arianc e
A NO V A
Source of Variation
S ample
C olumns
I nterac tion
Within
T otal
SS
4 2 4 .0 4
2 0 5 5 .9 1
2 7 0 .8 0
1 0 2 1 7 .3 3
1 2 9 6 8 .0 7
df
107
F
4 .2 3
1 0 .2 6
1 .3 5
P-value
0 .0 4 2 1 9
0 .0 0 0 0 9
0 .2 6 3 4 0
F crit
3 .9 3
3 .0 9
3 .0 9
Imagine that we have a slightly different set of data, where the differences between
the genders are more exaggerated, as follows:
Ge nde r
female
female
female
female
female
female
female
female
female
female
female
female
female
female
female
female
female
female
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
HR ly ing HR sitting HR standing
80
71
74
68
76
72
72
60
78
86
86
74
60
60
68
54
68
64
64
60
70
80
60
60
82
64
84
95
78
56
72
68
76
76
68
56
67
74
81
77
85
81
89
65
73
89
89
75
63
77
70
59
73
89
94
69
83
87
87
71
78
71
92
96
83
68
83
75
104
87
84
83
62
89
80
74
91
91
95
72
70
99
99
82
67
74
77
62
82
66
97
76
93
93
97
78
93
78
104
98
86
82
112
99
86
99
82
95
Effects of posture
92
88
84
girls
boys
80
76
72
68
64
1)
2)
3)
lying
sitting
standing
Does posture affect HR?
Is there a difference between males and
females?
Are the effects of posture and gender on
HR additive or interactive?
The results are as follows. Note that the P-values are less
A nova: T wo-F ac tor With Replic ation
SU M M A RY
H R lying
H R s itting
H R s tanding
T otal
female
C ount
Sum
A verage
V arianc e
18
1 2 7 1 .0 0
7 0 .6 1
7 9 .4 3
18
18
1 3 7 8 .3 7 1 4 3 1 .7 0
7 6 .5 8
7 9 .5 4
9 2 .3 8
1 4 7 .4 7
54
4 0 8 1 .0 7
7 5 .5 8
1 1 6 .4 6
18
1 2 6 9 .0 0
7 0 .5 0
1 1 6 .0 3
18
18
1 4 9 4 .4 8 1 6 4 9 .1 6
8 3 .0 3
9 1 .6 2
9 8 .9 0
9 9 .6 6
54
4 4 1 2 .6 4
8 1 .7 2
1 7 7 .5 3
36
2 5 4 0 .0 0
7 0 .5 6
9 4 .9 4
36
36
2 8 7 2 .8 5 3 0 8 0 .8 6
7 9 .8 0
8 5 .5 8
1 0 3 .6 1
1 5 7 .5 7
male
C ount
Sum
A verage
V arianc e
Total
C ount
Sum
A verage
V arianc e
A NO V A
Source of Variation
Sample
C olumns
I nterac tion
Within
T otal
SS
df
1 0 1 7 .9 5
4 1 3 5 .0 6
6 7 0 .2 3
1 0 7 7 5 .8 7
1
2
2
102
1 6 5 9 9 .1 1
107
MS
1 0 1 7 .9 5
2 0 6 7 .5 3
3 3 5 .1 1
1 0 5 .6 5
F
9 .6 4
1 9 .5 7
3 .1 7
P-value
0 .0 0 2 5
0 .0 0 0 0 0 0 0 6
0 .0 4 6 1
F crit
3 .9 3
3 .0 9
3 .0 9
Imagine a quite different data set, as follows:
Gender
female
female
female
female
female
female
female
female
female
female
female
female
female
female
female
female
female
female
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
HR ly ing HR sitting HR standing
80
71
74
68
76
72
72
60
78
86
86
74
60
60
68
54
68
64
90
70
86
86
90
72
86
72
96
91
80
76
104
92
80
92
76
88
66
73
80
76
84
80
88
64
72
88
88
74
62
76
69
58
72
88
94
69
83
87
87
71
78
71
92
96
83
68
83
75
104
87
84
83
60
86
78
72
88
88
92
70
68
96
96
80
65
72
75
60
80
64
64
60
70
80
60
60
82
64
84
95
78
56
72
68
76
76
68
56
Effects of posture
92
girls
88
boys
84
80
76
72
68
64
1)
2)
3)
lying
sitting
standing
Does posture affect HR?
Is there a difference between males and
females?
Are the effects of posture and gender on
HR additive or interactive?
The results are as follows:
A nova: T wo-F ac tor With Replic ation
S U M M A RY
H R lying
H R s itting
H R s tanding
T otal
female
C ount
S um
A verage
V arianc e
18
1271
7 0 .6 1
7 9 .4 3
18
1358
7 5 .4 4
8 9 .6 7
18
1390
7 7 .2 2
1 3 9 .0 1
54
4019
7 4 .4 3
1 0 6 .7 8
18
1527
8 4 .8 3
8 5 .4 4
18
1 4 9 4 .4 8
8 3 .0 3
9 8 .9 0
18
54
1 2 6 9 4 2 9 0 .4 8
7 0 .5 0 7 9 .4 5 3 3 3
1 1 6 .0 3 1 3 7 .7 3 7 8
36
2798
7 7 .7 2
1 3 2 .0 9
36
2 8 5 2 .4 8
7 9 .2 4
1 0 6 .3 8
36
2659
7 3 .8 6
1 3 5 .4 9
male
C ount
S um
A verage
V arianc e
Total
C ount
S um
A verage
V arianc e
A NO V A
Source of Variation
SS
S ample
6 8 2 .4 2
C olumns
5 5 3 .0 0
I nterac tion
2 0 6 2 .1 3
Within
1 0 3 4 4 .1 8
T otal
1 3 6 4 1 .7 2 6
df
MS
1
2
2
102
107
6 8 2 .4 2
2 7 6 .5 0
1 0 3 1 .0 6
1 0 1 .4 1
F
6 .7 3
2 .7 3
1 0 .1 7
P-value
0 .0 1 0 8 8
0 .0 7 0 2 2
0 .0 0 0 0 9
F crit
3 .9 3
3 .0 9
3 .0 9
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