Comparing Means Anova F-test can be used to determine whether the expected responses at the t levels of an experimental factor.
Download ReportTranscript Comparing Means Anova F-test can be used to determine whether the expected responses at the t levels of an experimental factor.
Comparing Means
Anova
F-test can be used to determine whether the expected responses at the t levels of an experimental factor differ from each other When the null hypothesis is rejected, it may be desirable to find which mean(s) is (are) different, and at what ranking order.
In practice, it is actually not primary interest to test the null hypothesis, instead the investigators want to make specific comparisons of the means and to estimate pooled error
Means comparison
Three categories: 1. Pair-wise comparisons (Post-Hoc Comparison) 2. Comparison specified prior to performing the experiment (Planned comparison) 3. Comparison specified after observing the outcome of the experiment (Un-planned comparison) Statistical inference procedures of pair-wise comparisons: Fisher’s least significant difference (LSD) method Duncan’s Multiple Range Test (DMRT) Student Newman Keul Test (SNK) Tukey’s HSD (“Honestly Significantly Different”) Procedure
Pair Comparison
Suppose there are t means
x
1 ,
x
2 , ,
x t
An F-test has revealed that there are significant differences amongst the t means Performing an analysis to determine precisely where the differences exist.
Pair Comparison
Two means are considered different if the difference between the corresponding sample means is larger than a critical number. Then, the larger sample mean is believed to be associated with a larger population mean.
Conditions common to all the methods: The ANOVA model is the one way analysis of variance The conditions required to perform the ANOVA are satisfied.
The experiment is fixed-effect.
Comparing Pair-comparison methods
With the exception of the F-LSD test, there is no good theoretical argument that favors one pair-comparison method over the others. Professional statisticians often disagree on which method is appropriate.
In terms of Power and the probability of making a Type I error, the tests discussed can be ordered as follows: MORE Power HIGHER P[Type I Error]
Tukey HSD Test Student-Newman-Keuls Test Duncan Multiple Range Test Fisher LSD Test
Pairwise comparisons are traditionally considered as “post hoc” and not “a priori”, if one needs to categorize all comparisons into one of the two groups
Fisher Least Significant Different (LSD) Method
This method builds on the equal variances t-test of the difference between two means.
The test statistic is improved by using MSE rather than s p 2 .
It is concluded that level if | m i m i and m j differ (at a % significance m j | > LSD, where
LSD
t
a 2 ,
dfe
1
MSE
(
n i
1
n j
)
Critical t for a test about equality = t a (2),
Example: Cassava yields (ton/ha)
Source of variation Degrees of Freedom Sum of Square
Treatment Block Error 3 3 9 136 40 12
Mena Square
45,333 13,333 1,33
F calculated
34 10 Total 15 18 F-table: 3,86
Duncan’s Multiple Range Test
The Duncan Multiple Range test uses different Significant Difference values for means next to each other along the real number line, and those with 1, 2, … , a means in between the two means being compared.
The Significant Difference or the range value:
R p
r
a ,
p
,
MSE n
where r a ,p, alpha level a is the Duncan’s Significant Range Value with parameters (= range-value) and (= a joint ).
p
(= MSE degree-of-freedom), and experiment-wise
Duncan’s Multiple Range Test
MSE is the mean square error from the ANOVA table and n is the number of observations used to calculate the means being compared.
The range-value is: 2 if the two means being compared are adjacent 3 if one mean separates the two means being compared 4 if two means separate the two means being compared …
Significant Ranges for Duncan’s Multiple Range Test
Critical Points for Duncan's M ultiple Range Statistic -- ALPH A = 0.05
D e gre e s of fre e dom
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 30 40 60 100 inf 2
18.00
6.09
4.50
3.93
3.64
3.46
3.35
3.26
3.20
3.15
3.11
3.08
3.06
3.03
3.01
3.00
2.98
2.97
2.98
2.95
2.89
2.86
2.83
2.80
2.77
3
18.00
6.09
4.50
4.01
3.74
3.58
3.47
3.39
3.34
3.30
3.27
3.23
3.21
3.18
3.16
3.15
3.13
3.12
3.11
3.10
3.04
3.01
2.98
2.95
2.92
4
18.00
6.09
4.50
4.02
3.79
3.64
3.54
3.47
3.41
3.37
3.35
3.33
3.30
3.27
3.25
3.23
3.22
3.21
3.19
3.18
3.12
3.10
3.08
3.05
3.02
5
18.00
6.09
4.50
4.02
3.83
3.68
3.58
3.52
3.47
3.43
3.39
3.36
3.35
3.33
3.31
3.30
3.28
3.27
3.26
3.25
3.20
3.17
3.14
3.12
3.09
6
18.00
6.09
4.50
4.02
3.83
3.68
3.60
3.55
3.50
3.46
3.43
3.40
3.38
3.37
3.36
3.34
3.33
3.32
3.31
3.30
3.25
3.22
3.20
3.18
3.15
7
18.00
p
6.09
4.50
4.02
3.83
3.68
3.61
3.56
3.52
3.47
3.44
3.42
3.41
3.39
3.38
3.37
3.36
3.35
3.35
3.34
3.29
3.27
3.24
3.22
3.19
8
18.00
6.09
4.50
4.02
3.83
3.68
3.61
3.56
3.52
3.47
3.45
3.44
3.42
3.41
3.40
3.39
3.38
3.37
3.37
3.36
3.32
3.30
3.28
3.26
3.23
9
18.00
6.09
4.50
4.02
3.83
3.68
3.61
3.56
3.52
3.47
3.46
3.44
3.44
3.42
3.42
3.41
3.40
3.39
3.39
3.38
3.35
3.33
3.31
3.29
3.26
10
18.00
6.09
4.50
4.02
3.83
3.68
3.61
3.56
3.52
3.47
3.46
3.46
3.45
3.44
3.43
3.43
3.42
3.41
3.41
3.40
3.37
3.35
3.33
3.32
3.29
20
18.00
6.09
4.50
4.02
3.83
3.68
3.61
3.56
3.52
3.48
3.48
3.48
3.47
3.47
3.47
3.47
3.47
3.47
3.47
3.47
3.47
3.47
3.47
3.47
3.47
50
18.00
6.09
4.50
4.02
3.83
3.68
3.61
3.56
3.52
3.48
3.48
3.48
3.47
3.47
3.47
3.47
3.47
3.47
3.47
3.47
3.47
3.47
3.48
3.53
3.61
100
18.00
6.09
4.50
4.02
3.83
3.68
3.61
3.56
3.52
3.48
3.48
3.48
3.47
3.47
3.47
3.47
3.47
3.47
3.47
3.47
3.47
3.47
3.48
3.53
3.67
Student-Newman-Keuls Test
Similar to the Duncan Multiple Range test, the Student Newman-Keuls Test uses different Significant Difference values for means next to each other, and those with 1, 2, … , a means in between the two means being compared.
The Significant Difference or the range value for this test is
K p
q
a ,
p
,
MSE n
where q a ,a, is the Studentized Range Statistic with parameters range-value) and (= MSE degree-of-freedom), and experiment-wise alpha level a (= a
joint
).
p (=
Student-Newman-Keuls Test
MSE is the mean square error from the ANOVA table and n is the number of observations used to calculate the means being compared.
The range-value is: 2 if the two means being compared are adjacent 3 if one mean separates the two means being compared 4 if two means separate the two means being compared …
Studentized Range Statistic
Critical Points for the Stude ntize d Range Statistic -- ALPH A = 0.05
D e gre e s of fre e dom
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 24 30 40 60 120 inf
3.59
3.58
3.53
3.49
3.44
3.40
3.36
3.31
3
4.60
4.34
4.16
4.04
3.95
3.88
3.82
3.77
3.73
3.70
3.67
3.65
3.63
3.61
2.96
2.95
2.92
2.89
2.86
2.83
2.80
2.77
2
3.64
3.46
3.34
3.26
3.20
3.15
3.11
3.08
3.06
3.03
3.01
3.00
2.98
2.97
3.98
3.96
3.90
3.85
3.79
3.74
3.68
3.63
4
5.22
4.90
4.68
4.53
4.41
4.33
4.26
4.20
4.57
4.51
4.15
4.45
4.11 4.41
4.08
4.05
4.02
4.00
4.37
4.33
4.30
4.28
5
5.67
5.30
5.06
4.89
4.76
4.65
4.25
4.23
4.17
4.10
4.04
3.98
3.92
3.86
4.47
4.45
4.37
4.30
4.23
4.16
4.10
4.03
6
6.03
5.63
5.36
5.17
5.02
4.91
4.82
4.75
4.69
4.64
4.59
4.56
4.52
4.49
4.79
4.77
4.68
4.60
4.52
4.44
4.36
4.29
8
6.58
6.12
5.82
5.60
5.43
5.30
5.20
5.12
5.05
4.99
4.94
4.90
4.86
4.82
4.65
4.62
4.54
4.46
4.39
4.31
4.24
4.17
7
6.33
5.90
5.61
5.40
5.24
5.12
5.03
4.95
4.88
4.83
4.78
4.74
4.70
4.67
5.04
5.01
4.92
4.82
4.73
4.65
4.56
4.47
10
6.99
6.49
6.16
5.92
5.74
5.60
5.49
5.39
5.32
5.25
5.20
5.15
5.11
5.07
4.92
4.90
4.81
4.72
4.63
4.55
4.47
4.39
9
6.80
6.32
6.00
5.77
5.59
5.46
5.35
5.27
5.19
5.13
5.08
5.03
4.99
4.96
5.21
5.17
5.14
5.11
5.01
4.92
4.82
4.73
4.64
4.55
p 11
7.17
6.65
6.30
6.05
5.87
5.72
5.61
5.51
5.43
5.36
5.31
5.26
5.23
5.20
5.10
5.00
4.90
4.81
4.71
4.62
12
7.32
6.79
6.43
6.18
5.98
5.83
5.71
5.61
5.53
5.46
5.40
5.35
5.31
5.27
5.31
5.26
5.16
5.08
4.98
4.88
4.78
4.68
13
7.47
6.92
6.55
6.29
6.09
5.93
5.81
5.71
5.90
5.80
5.63 5.71
5.55
5.64
5.49
5.44
5.39
5.35
5.57
5.52
5.47
5.43
14
7.60
7.03
6.66
6.39
6.19
6.03
5.39
5.36
5.25
5.15
5.04
4.94
4.84
4.74
5.46
5.43
5.32
5.21
5.11
5.00
4.90
4.80
15
7.72
7.14
6.76
6.48
6.28
6.11
5.98
5.88
5.79
5.71
5.65
5.59
5.54
5.50
5.53
4.49
5.38
5.27
5.16
5.06
4.95
4.85
16
7.83
7.24
6.85
6.57
6.36
6.19
6.06
5.95
5.86
5.79
5.72
5.66
5.61
5.57
5.59
5.55
5.44
5.33
5.22
5.11
5.00
4.89
17
7.93
7.34
6.94
6.65
6.44
6.27
6.13
6.02
5.93
5.85
5.78
5.73
5.67
5.63
5.65
5.61
5.49
5.38
5.27
5.15
5.04
4.93
18
8.03
7.43
7.02
6.73
6.51
6.34
6.20
6.09
5.99
5.91
5.85
5.79
5.73
5.69
5.70
5.66
5.55
5.43
5.31
5.20
5.09
4.97
19
8.12
7.51
7.10
6.80
6.58
6.40
6.27
6.15
6.05
5.97
5.90
5.84
5.79
5.74
5.75
5.71
5.59
5.47
5.36
5.24
5.13
5.01
20
8.21
7.59
7.17
6.87
6.64
6.47
6.33
6.21
6.11
6.03
5.96
5.90
5.84
5.79
Tukey HSD Procedure
The test procedure: Assumes equal number of observation per populations.
Find a critical number w as follows: w
q
a (
dft
,
dfe
)
MSE n g
dft = treatment degrees of freedom =degrees of freedom = dfe n g a = number of observations per population = significance level q a (dft, ) = a critical value obtained from the studentized range table
Studentized Range Statistic
Critical Points for the Stude ntize d Range Statistic -- ALPH A = 0.05
D e gre e s of fre e dom
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 24 30 40 60 120 inf
3.59
3.58
3.53
3.49
3.44
3.40
3.36
3.31
3
4.60
4.34
4.16
4.04
3.95
3.88
3.82
3.77
3.73
3.70
3.67
3.65
3.63
3.61
2.96
2.95
2.92
2.89
2.86
2.83
2.80
2.77
2
3.64
3.46
3.34
3.26
3.20
3.15
3.11
3.08
3.06
3.03
3.01
3.00
2.98
2.97
3.98
3.96
3.90
3.85
3.79
3.74
3.68
3.63
4
5.22
4.90
4.68
4.53
4.41
4.33
4.26
4.20
4.57
4.51
4.15
4.45
4.11 4.41
4.08
4.05
4.02
4.00
4.37
4.33
4.30
4.28
5
5.67
5.30
5.06
4.89
4.76
4.65
4.25
4.23
4.17
4.10
4.04
3.98
3.92
3.86
4.47
4.45
4.37
4.30
4.23
4.16
4.10
4.03
6
6.03
5.63
5.36
5.17
5.02
4.91
4.82
4.75
4.69
4.64
4.59
4.56
4.52
4.49
4.79
4.77
4.68
4.60
4.52
4.44
4.36
4.29
8
6.58
6.12
5.82
5.60
5.43
5.30
5.20
5.12
5.05
4.99
4.94
4.90
4.86
4.82
4.65
4.62
4.54
4.46
4.39
4.31
4.24
4.17
7
6.33
5.90
5.61
5.40
5.24
5.12
5.03
4.95
4.88
4.83
4.78
4.74
4.70
4.67
5.04
5.01
4.92
4.82
4.73
4.65
4.56
4.47
10
6.99
6.49
6.16
5.92
5.74
5.60
5.49
5.39
5.32
5.25
5.20
5.15
5.11
5.07
4.92
4.90
4.81
4.72
4.63
4.55
4.47
4.39
9
6.80
6.32
6.00
5.77
5.59
5.46
5.35
5.27
5.19
5.13
5.08
5.03
4.99
4.96
5.21
5.17
5.14
5.11
5.01
4.92
4.82
4.73
4.64
4.55
t 11
7.17
6.65
6.30
6.05
5.87
5.72
5.61
5.51
5.43
5.36
5.31
5.26
5.23
5.20
5.10
5.00
4.90
4.81
4.71
4.62
12
7.32
6.79
6.43
6.18
5.98
5.83
5.71
5.61
5.53
5.46
5.40
5.35
5.31
5.27
5.31
5.26
5.16
5.08
4.98
4.88
4.78
4.68
13
7.47
6.92
6.55
6.29
6.09
5.93
5.81
5.71
5.90
5.80
5.63 5.71
5.55
5.64
5.49
5.44
5.39
5.35
5.57
5.52
5.47
5.43
14
7.60
7.03
6.66
6.39
6.19
6.03
5.39
5.36
5.25
5.15
5.04
4.94
4.84
4.74
5.46
5.43
5.32
5.21
5.11
5.00
4.90
4.80
15
7.72
7.14
6.76
6.48
6.28
6.11
5.98
5.88
5.79
5.71
5.65
5.59
5.54
5.50
5.53
4.49
5.38
5.27
5.16
5.06
4.95
4.85
16
7.83
7.24
6.85
6.57
6.36
6.19
6.06
5.95
5.86
5.79
5.72
5.66
5.61
5.57
5.59
5.55
5.44
5.33
5.22
5.11
5.00
4.89
17
7.93
7.34
6.94
6.65
6.44
6.27
6.13
6.02
5.93
5.85
5.78
5.73
5.67
5.63
5.65
5.61
5.49
5.38
5.27
5.15
5.04
4.93
18
8.03
7.43
7.02
6.73
6.51
6.34
6.20
6.09
5.99
5.91
5.85
5.79
5.73
5.69
5.70
5.66
5.55
5.43
5.31
5.20
5.09
4.97
19
8.12
7.51
7.10
6.80
6.58
6.40
6.27
6.15
6.05
5.97
5.90
5.84
5.79
5.74
5.75
5.71
5.59
5.47
5.36
5.24
5.13
5.01
20
8.21
7.59
7.17
6.87
6.64
6.47
6.33
6.21
6.11
6.03
5.96
5.90
5.84
5.79
Scheffe
There are many multiple (post hoc) comparison procedures
Considerable controversy:
“I have not included the multiple comparison methods of Duncan because I have been unable to understand their justification”
Planned Comparisons or Contrasts
In some cases, an experimenter may know ahead of time that it is of interest to compare two different means, or groups of means.
An effective way to do this is to use
contrasts
or
planned comparisons
. These represent specific hypotheses in terms of the treatment means such as:
H
0
H A
: : m 4 m 4 m 5 m 5
H
0
H A
: : m 1 m 1 m 3 m 3 m m 4 4 m 5 m 5
y
4
y
5 0
y
1
y
3
y
4
y
5 0
Planned Comparisons or Contrasts
Each contrast can be specified as:
C
i t
1
c i y i
and it is required:
i t
1
c i
0 A sum-of-squares can be calculated for a contrast as
ss C
n t i
1
c i i t
1
y c i
2
i
2
Planned Comparisons
Each contrast has 1 degree-of-freedom, and a contrast can be tested by comparing it to the MSE for the ANOVA:
SS c
1
SSE dfe
F
( 1 ,
dfe
)
Un-planned Comparisons or Contrasts
If more than 1 contrast is tested, it is important that the contrasts all be
orthogonal
, that is
i t
1
c i d i
0 Note that It can be tested at most t-1 orthogonal contrasts.
Contrast orthogonal examples
Treatment Adira-4 GH-6 GH-7 Local Yields (ton/ha) 19 25 18 18 The mean effect of local and high yielding varieties The mean effect of high yielding and promising lines
Orthogonal Polynomials
Special sets of coefficients that test for bends but manage to remain uncorrelated with one another.
Sometimes orthogonal polynomials can be used to analyze experimental data to test for curves. Restrictive assumptions: Require quantitative factors Equal spacing of factor levels (d) Equal numbers of observations at each cell (r j ) Usually, only the linear and quadratic contrasts are of interest
Polynomial Examples
Treatment (Urea dosage) kg/ha 50 100 150 200 Yields (ton/ha) 19 25 18 18
Orthogonal Polynomial
The linear regression model y = X linear in the unknown parameter .
+ is a general model for fitting any relationship that is Polynomial regression model: 26
Polynomial Models in One Variable
A second-order model (quadratic model):
A second-order model (quadratic model)
Polynomial Models
Polynomial models are useful in situations where the analyst knows that curvilinear effects are present in the true response function.
Polynomial models are also useful as approximating functions to unknown and possible very complex nonlinear relationship.
Polynomial model is the Taylor series expansion of the unknown function.
Choosing order of the model
Theoretical background Scatter diagram Orthogonal polynomial test 30
Theoretical background
Can be searched from previous research or literature 1.
2.
3.
Examples: The relationship between dosages of nitrogen application and yield (The law of diminishing return) The relationship between pesticide application and pest mortality (Linear model/probit analysis) The relationship between population density and yield (Exponential model/Cob-Douglass Model)
Scatter Diagram
25 20 15 10 5 0 -2,5 -2 -1,5 -1 -0,5 0
Treatment
0,5 1 1,5 2 2,5
Scatter Diagram
15 10 25 20 5 0 -2,5 -2 -1,5 -1 -0,5
Treatrment
0 0,5 1 1,5 2 2,5
Orthogonal Linear Contrasts for Polynomial Regression
t Polynomial 1 3 4 Linear Quadratic Linear Quadratic Cubic -1 1 -3 1 -1 2 0 -2 -1 -1 3 3 1 1 1 -1 -3 4 3 1 1 5 5 Linear Quadratic Cubic Quartic -2 2 -1 1 -1 -1 2 -4 6 Linear Quadratic Cubic Quartic -5 5 -5 1 -3 -1 7 -3 0 -2 0 6 1 -1 -2 -4 2 2 1 1 -1 -4 4 2 1 -4 -4 2 3 -1 -7 -3 6 5 5 5 1 7 Linear Quadratic Cubic Quartic -3 5 -1 3 -2 0 1 -7 -1 -3 1 1 0 -4 0 6 1 -3 -1 1 2 0 -1 -7 7 3 5 1 3 8 9 10 ∑ai2 2 6 20 4 20 10 14 10 70 70 84 180 28 28 84 6 154
Orthogonal Linear Contrasts for Polynomial Regression
t Polynomial 1 2 3 4 5 6 7 8 9 10 8 Linear Quadratic Cubic Quartic Quintic -7 7 -7 7 -7 -5 1 -3 -3 5 7 -13 -3 -1 -5 3 9 1 -5 -3 9 23 -17 -15 15 3 -3 -7 5 1 -5 -3 -13 17 -23 7 7 7 7 7 9 Linear -4 Quadratic 28 Cubic Quartic Quintic -14 -4 -3 7 7 11 -2 -1 0 1 2 -8 -17 -20 -17 -8 13 14 -21 -11 -4 9 9 -9 0 18 0 -9 -13 -7 9 9 -11 -21 14 4 3 7 -11 4 28 14 4 10 Linear Quadratic Cubic Quartic Quintic -9 -7 -5 -3 -1 1 3 5 7 9 6 2 -42 14 -1 35 -3 31 -4 -4 -3 -1 2 6 12 -12 -31 -35 -14 42 18 -22 -17 -6 14 3 18 -1 -11 -6 18 6 3 11 -17 -22 18 1 -14 6 ai2 168 168 264 616 2184 20 2772 990 2002 468 330 132 8580 2860 780