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Comparing Comparing MeansMeans
AnovaAnova F-test can be used to determine whether the F-test can be used to determine whether the
expected responses at the t levels of an expected responses at the t levels of an experimental factor differ from each otherexperimental factor differ from each other
When the null hypothesis is rejected, it may be When the null hypothesis is rejected, it may be desirable to find which mean(s) is (are) desirable to find which mean(s) is (are) different, and at what ranking order.different, and at what ranking order.
In practice, it is actually not primary interest to In practice, it is actually not primary interest to test the null hypothesis, instead the test the null hypothesis, instead the
investigators want to make specific comparisons investigators want to make specific comparisons of the means and to estimate pooled errorof the means and to estimate pooled error
Means comparisonMeans comparison Three categories:Three categories:
1. Pair-wise comparisons (Post-Hoc Comparison)1. Pair-wise comparisons (Post-Hoc Comparison)2. Comparison specified prior to performing the experiment2. Comparison specified prior to performing the experiment
(Planned comparison)(Planned comparison)3. Comparison specified after observing the outcome of the3. Comparison specified after observing the outcome of the
experiment (Un-planned comparison)experiment (Un-planned comparison) Statistical inference procedures of pair-wise Statistical inference procedures of pair-wise
comparisons:comparisons: Fisher’s least significant difference (LSD) methodFisher’s least significant difference (LSD) method Duncan’s Multiple Range Test (DMRT)Duncan’s Multiple Range Test (DMRT) Student Newman Keul Test (SNK)Student Newman Keul Test (SNK) Tukey’s Tukey’s HSD (“Honestly Significantly Different”)
Procedure
Suppose Suppose there are t means are t means
An F-test has revealed that there are significant differences amongst the t means
Performing an analysis to determine precisely where the differences exist.
txxx ,,2,1
Pair ComparisonPair Comparison
Two means are considered different if the Two means are considered different if the difference between the corresponding sample difference between the corresponding sample means is larger than a critical number. Then, means is larger than a critical number. Then, the larger sample mean is believed to be the larger sample mean is believed to be associated with a larger population mean.associated with a larger population mean.
Conditions common to all the methods:Conditions common to all the methods: The ANOVA model is the one way analysis of The ANOVA model is the one way analysis of
variancevariance The conditions required to perform the ANOVA The conditions required to perform the ANOVA
are satisfied.are satisfied. The experiment is fixed-effect.The experiment is fixed-effect.
Pair Comparison
Comparing Pair-comparison methods With the exception of the F-With the exception of the F-LSDLSD test, there is no test, there is no
good theoretical argument that favors one pair-good theoretical argument that favors one pair-comparison method over the others. comparison method over the others. Professional statisticians often disagree on which Professional statisticians often disagree on which method is appropriate.method is appropriate.
In terms of Power and the probability of making In terms of Power and the probability of making a a Type IType I error, the tests discussed can be error, the tests discussed can be ordered as follows:ordered as follows:
Tukey HSD Test Student-Newman-Keuls Test Duncan Multiple Range TestFisher LSD Test
MORE Power HIGHER P[Type I Error]
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 Fisher Least Significant Different (LSD) Method(LSD) Method
This method builds on the equal variances t-This method builds on the equal variances t-test of the difference between two means.test of the difference between two means.
The test statistic is improved by using MSE The test statistic is improved by using MSE rather than srather than spp
22.. It is concluded that It is concluded that ii and and jj differ (at differ (at % %
significance level if |significance level if |ii - - jj| > LSD, where| > LSD, where
)11(,2ji
dfe nnMSEtLSD
Critical t for a test about equality = t(2),
Example:Example:Cassava yields (ton/ha)Cassava yields (ton/ha)
Source of variation
Degrees of Freedom
Sum of Square
Mena Square
F calculated
Treatment
3 136 45,333 34
Block 3 40 13,333 10Error 9 12 1,33
Total 15 18
F-table: 3,86
Duncan’s Multiple Range TestDuncan’s Multiple Range Test The Duncan Multiple Range test uses different The Duncan Multiple Range test uses different Significant Significant
DifferenceDifference values for means next to each other along the real values for means next to each other along the real number line, and those with 1, 2, … , a means in between the number line, and those with 1, 2, … , a means in between the two means being compared.two means being compared.
The The Significant DifferenceSignificant Difference or the range value: or the range value:
where rwhere r,p,,p, is the is the Duncan’s Significant Range ValueDuncan’s Significant Range Value with with parameters parameters pp (= range-value) and (= range-value) and (= MSE degree-of-freedom), and (= MSE degree-of-freedom), and experiment-wise alpha level experiment-wise alpha level (= (= jointjoint).).
nMSErR pp ,,
Duncan’s Multiple Range TestDuncan’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 Multiple Range Statistic -- ALPHA = 0.05
2 3 4 5 6 7 8 9 10 20 50 1001 18.00 18.00 18.00 18.00 18.00 18.00 18.00 18.00 18.00 18.00 18.00 18.002 6.09 6.09 6.09 6.09 6.09 6.09 6.09 6.09 6.09 6.09 6.09 6.093 4.50 4.50 4.50 4.50 4.50 4.50 4.50 4.50 4.50 4.50 4.50 4.504 3.93 4.01 4.02 4.02 4.02 4.02 4.02 4.02 4.02 4.02 4.02 4.025 3.64 3.74 3.79 3.83 3.83 3.83 3.83 3.83 3.83 3.83 3.83 3.836 3.46 3.58 3.64 3.68 3.68 3.68 3.68 3.68 3.68 3.68 3.68 3.687 3.35 3.47 3.54 3.58 3.60 3.61 3.61 3.61 3.61 3.61 3.61 3.618 3.26 3.39 3.47 3.52 3.55 3.56 3.56 3.56 3.56 3.56 3.56 3.569 3.20 3.34 3.41 3.47 3.50 3.52 3.52 3.52 3.52 3.52 3.52 3.52
10 3.15 3.30 3.37 3.43 3.46 3.47 3.47 3.47 3.47 3.48 3.48 3.4811 3.11 3.27 3.35 3.39 3.43 3.44 3.45 3.46 3.46 3.48 3.48 3.4812 3.08 3.23 3.33 3.36 3.40 3.42 3.44 3.44 3.46 3.48 3.48 3.4813 3.06 3.21 3.30 3.35 3.38 3.41 3.42 3.44 3.45 3.47 3.47 3.4714 3.03 3.18 3.27 3.33 3.37 3.39 3.41 3.42 3.44 3.47 3.47 3.4715 3.01 3.16 3.25 3.31 3.36 3.38 3.40 3.42 3.43 3.47 3.47 3.4716 3.00 3.15 3.23 3.30 3.34 3.37 3.39 3.41 3.43 3.47 3.47 3.4717 2.98 3.13 3.22 3.28 3.33 3.36 3.38 3.40 3.42 3.47 3.47 3.4718 2.97 3.12 3.21 3.27 3.32 3.35 3.37 3.39 3.41 3.47 3.47 3.4719 2.98 3.11 3.19 3.26 3.31 3.35 3.37 3.39 3.41 3.47 3.47 3.4720 2.95 3.10 3.18 3.25 3.30 3.34 3.36 3.38 3.40 3.47 3.47 3.4730 2.89 3.04 3.12 3.20 3.25 3.29 3.32 3.35 3.37 3.47 3.47 3.4740 2.86 3.01 3.10 3.17 3.22 3.27 3.30 3.33 3.35 3.47 3.47 3.4760 2.83 2.98 3.08 3.14 3.20 3.24 3.28 3.31 3.33 3.47 3.48 3.48
100 2.80 2.95 3.05 3.12 3.18 3.22 3.26 3.29 3.32 3.47 3.53 3.53inf 2.77 2.92 3.02 3.09 3.15 3.19 3.23 3.26 3.29 3.47 3.61 3.67
Degrees of freedom
p
Student-Newman-Keuls Test Similar to the Duncan Multiple Range test, the Student-Similar to the Duncan Multiple Range test, the Student-
Newman-Keuls Test uses different Newman-Keuls Test uses different Significant DifferenceSignificant Difference values for means next to each other, and those with 1, 2, values for means next to each other, and those with 1, 2, … , a means in between the two means being compared.… , a means in between the two means being compared.
The The Significant DifferenceSignificant Difference or the range value for this or the range value for this test istest is
where qwhere q,a,,a, is the is the Studentized Range StatisticStudentized Range Statistic with with parameters parameters pp (= range-value) and (= range-value) and (= MSE (= MSE degree-of-freedom), and experiment-wise alpha degree-of-freedom), and experiment-wise alpha level level (= (= jointjoint).).
nMSEqK pp ,,
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 StatisticCritical Points for the Studentized Range Statistic -- ALPHA = 0.05
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 205 3.64 4.60 5.22 5.67 6.03 6.33 6.58 6.80 6.99 7.17 7.32 7.47 7.60 7.72 7.83 7.93 8.03 8.12 8.216 3.46 4.34 4.90 5.30 5.63 5.90 6.12 6.32 6.49 6.65 6.79 6.92 7.03 7.14 7.24 7.34 7.43 7.51 7.597 3.34 4.16 4.68 5.06 5.36 5.61 5.82 6.00 6.16 6.30 6.43 6.55 6.66 6.76 6.85 6.94 7.02 7.10 7.178 3.26 4.04 4.53 4.89 5.17 5.40 5.60 5.77 5.92 6.05 6.18 6.29 6.39 6.48 6.57 6.65 6.73 6.80 6.879 3.20 3.95 4.41 4.76 5.02 5.24 5.43 5.59 5.74 5.87 5.98 6.09 6.19 6.28 6.36 6.44 6.51 6.58 6.64
10 3.15 3.88 4.33 4.65 4.91 5.12 5.30 5.46 5.60 5.72 5.83 5.93 6.03 6.11 6.19 6.27 6.34 6.40 6.4711 3.11 3.82 4.26 4.57 4.82 5.03 5.20 5.35 5.49 5.61 5.71 5.81 5.90 5.98 6.06 6.13 6.20 6.27 6.3312 3.08 3.77 4.20 4.51 4.75 4.95 5.12 5.27 5.39 5.51 5.61 5.71 5.80 5.88 5.95 6.02 6.09 6.15 6.2113 3.06 3.73 4.15 4.45 4.69 4.88 5.05 5.19 5.32 5.43 5.53 5.63 5.71 5.79 5.86 5.93 5.99 6.05 6.1114 3.03 3.70 4.11 4.41 4.64 4.83 4.99 5.13 5.25 5.36 5.46 5.55 5.64 5.71 5.79 5.85 5.91 5.97 6.0315 3.01 3.67 4.08 4.37 4.59 4.78 4.94 5.08 5.20 5.31 5.40 5.49 5.57 5.65 5.72 5.78 5.85 5.90 5.9616 3.00 3.65 4.05 4.33 4.56 4.74 4.90 5.03 5.15 5.26 5.35 5.44 5.52 5.59 5.66 5.73 5.79 5.84 5.9017 2.98 3.63 4.02 4.30 4.52 4.70 4.86 4.99 5.11 5.21 5.31 5.39 5.47 5.54 5.61 5.67 5.73 5.79 5.8418 2.97 3.61 4.00 4.28 4.49 4.67 4.82 4.96 5.07 5.17 5.27 5.35 5.43 5.50 5.57 5.63 5.69 5.74 5.7919 2.96 3.59 3.98 4.25 4.47 4.65 4.79 4.92 5.04 5.14 5.23 5.31 5.39 5.46 5.53 5.59 5.65 5.70 5.7520 2.95 3.58 3.96 4.23 4.45 4.62 4.77 4.90 5.01 5.11 5.20 5.26 5.36 5.43 4.49 5.55 5.61 5.66 5.7124 2.92 3.53 3.90 4.17 4.37 4.54 4.68 4.81 4.92 5.01 5.10 5.16 5.25 5.32 5.38 5.44 5.49 5.55 5.5930 2.89 3.49 3.85 4.10 4.30 4.46 4.60 4.72 4.82 4.92 5.00 5.08 5.15 5.21 5.27 5.33 5.38 5.43 5.4740 2.86 3.44 3.79 4.04 4.23 4.39 4.52 4.63 4.73 4.82 4.90 4.98 5.04 5.11 5.16 5.22 5.27 5.31 5.3660 2.83 3.40 3.74 3.98 4.16 4.31 4.44 4.55 4.65 4.73 4.81 4.88 4.94 5.00 5.06 5.11 5.15 5.20 5.24
120 2.80 3.36 3.68 3.92 4.10 4.24 4.36 4.47 4.56 4.64 4.71 4.78 4.84 4.90 4.95 5.00 5.04 5.09 5.13inf 2.77 3.31 3.63 3.86 4.03 4.17 4.29 4.39 4.47 4.55 4.62 4.68 4.74 4.80 4.85 4.89 4.93 4.97 5.01
pDegrees of freedom
The test procedure: The test procedure: Assumes equal number of observation per Assumes equal number of observation per
populations.populations. Find a critical number Find a critical number as follows: as follows:
gnMSEdfedftq ),(
dft = treatment degrees of freedom =degrees of freedom = dfeng = number of observations per population = significance levelq(dft,) = a critical value obtained from the studentized range table
Tukey HSD Procedure
Studentized Range Studentized Range StatisticStatistic
Critical Points for the Studentized Range Statistic -- ALPHA = 0.05
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 205 3.64 4.60 5.22 5.67 6.03 6.33 6.58 6.80 6.99 7.17 7.32 7.47 7.60 7.72 7.83 7.93 8.03 8.12 8.216 3.46 4.34 4.90 5.30 5.63 5.90 6.12 6.32 6.49 6.65 6.79 6.92 7.03 7.14 7.24 7.34 7.43 7.51 7.597 3.34 4.16 4.68 5.06 5.36 5.61 5.82 6.00 6.16 6.30 6.43 6.55 6.66 6.76 6.85 6.94 7.02 7.10 7.178 3.26 4.04 4.53 4.89 5.17 5.40 5.60 5.77 5.92 6.05 6.18 6.29 6.39 6.48 6.57 6.65 6.73 6.80 6.879 3.20 3.95 4.41 4.76 5.02 5.24 5.43 5.59 5.74 5.87 5.98 6.09 6.19 6.28 6.36 6.44 6.51 6.58 6.64
10 3.15 3.88 4.33 4.65 4.91 5.12 5.30 5.46 5.60 5.72 5.83 5.93 6.03 6.11 6.19 6.27 6.34 6.40 6.4711 3.11 3.82 4.26 4.57 4.82 5.03 5.20 5.35 5.49 5.61 5.71 5.81 5.90 5.98 6.06 6.13 6.20 6.27 6.3312 3.08 3.77 4.20 4.51 4.75 4.95 5.12 5.27 5.39 5.51 5.61 5.71 5.80 5.88 5.95 6.02 6.09 6.15 6.2113 3.06 3.73 4.15 4.45 4.69 4.88 5.05 5.19 5.32 5.43 5.53 5.63 5.71 5.79 5.86 5.93 5.99 6.05 6.1114 3.03 3.70 4.11 4.41 4.64 4.83 4.99 5.13 5.25 5.36 5.46 5.55 5.64 5.71 5.79 5.85 5.91 5.97 6.0315 3.01 3.67 4.08 4.37 4.59 4.78 4.94 5.08 5.20 5.31 5.40 5.49 5.57 5.65 5.72 5.78 5.85 5.90 5.9616 3.00 3.65 4.05 4.33 4.56 4.74 4.90 5.03 5.15 5.26 5.35 5.44 5.52 5.59 5.66 5.73 5.79 5.84 5.9017 2.98 3.63 4.02 4.30 4.52 4.70 4.86 4.99 5.11 5.21 5.31 5.39 5.47 5.54 5.61 5.67 5.73 5.79 5.8418 2.97 3.61 4.00 4.28 4.49 4.67 4.82 4.96 5.07 5.17 5.27 5.35 5.43 5.50 5.57 5.63 5.69 5.74 5.7919 2.96 3.59 3.98 4.25 4.47 4.65 4.79 4.92 5.04 5.14 5.23 5.31 5.39 5.46 5.53 5.59 5.65 5.70 5.7520 2.95 3.58 3.96 4.23 4.45 4.62 4.77 4.90 5.01 5.11 5.20 5.26 5.36 5.43 4.49 5.55 5.61 5.66 5.7124 2.92 3.53 3.90 4.17 4.37 4.54 4.68 4.81 4.92 5.01 5.10 5.16 5.25 5.32 5.38 5.44 5.49 5.55 5.5930 2.89 3.49 3.85 4.10 4.30 4.46 4.60 4.72 4.82 4.92 5.00 5.08 5.15 5.21 5.27 5.33 5.38 5.43 5.4740 2.86 3.44 3.79 4.04 4.23 4.39 4.52 4.63 4.73 4.82 4.90 4.98 5.04 5.11 5.16 5.22 5.27 5.31 5.3660 2.83 3.40 3.74 3.98 4.16 4.31 4.44 4.55 4.65 4.73 4.81 4.88 4.94 5.00 5.06 5.11 5.15 5.20 5.24
120 2.80 3.36 3.68 3.92 4.10 4.24 4.36 4.47 4.56 4.64 4.71 4.78 4.84 4.90 4.95 5.00 5.04 5.09 5.13inf 2.77 3.31 3.63 3.86 4.03 4.17 4.29 4.39 4.47 4.55 4.62 4.68 4.74 4.80 4.85 4.89 4.93 4.97 5.01
tDegrees of freedom
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”
Scheffe
Planned Comparisons or ContrastsPlanned Comparisons or Contrasts In some cases, an experimenter may know ahead In some cases, an experimenter may know ahead
of time that it is of interest to compare two of time that it is of interest to compare two different means, or groups of means.different means, or groups of means.
An effective way to do this is to use An effective way to do this is to use contrastscontrasts or or planned comparisonsplanned comparisons. These represent . These represent specific hypotheses in terms of the treatment specific hypotheses in terms of the treatment means such as:means such as:
0::
5454
540
yyHH
A
0::
54315431
54310
yyyyHH
A
Planned Comparisons or ContrastsPlanned Comparisons or Contrasts Each contrast can be specified as:Each contrast can be specified as:
and it is required:and it is required:
A sum-of-squares can be calculated for a contrast A sum-of-squares can be calculated for a contrast asas
t
iii ycC
1
01
t
iic
t
i i
t
i iiC
cn
ycss
12
2
1
Planned ComparisonsPlanned Comparisons Each contrast has 1 degree-of-freedom, and a Each contrast has 1 degree-of-freedom, and a
contrast can be tested by comparing it to the contrast can be tested by comparing it to the MSEMSE for the ANOVA: for the ANOVA:
),1(1
dfeFdfeSSESSc
Un-planned Comparisons or Un-planned Comparisons or ContrastsContrasts
If more than 1 contrast is tested, it is important If more than 1 contrast is tested, it is important that the contrasts all be that the contrasts all be orthogonalorthogonal, that is , that is
Note that It can be tested at most t-1 orthogonal contrasts.
01
t
iiidc
Contrast orthogonal examples
Treatment
Yields (ton/ha)
Adira-4 19GH-6 25GH-7 18Local 18
The mean effect of local and high yielding varietiesThe mean effect of high yielding and promising lines
Orthogonal Polynomials
Restrictive assumptions:Require quantitative factorsEqual spacing of factor levels (d)Equal numbers of observations at each cell (rj)Usually, only the linear and quadratic contrasts are of interest
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.
Polynomial ExamplesTreatment (Urea dosage) kg/ha
Yields (ton/ha)
50 19100 25150 18200 18
26
Orthogonal Polynomial The linear regression model y = X
+ is a general model for fitting any relationship that is linear in the unknown parameter .
Polynomial regression model:
Polynomial Models in One Variable
A second-order model (quadratic A second-order model (quadratic model):model):
A second-order model (quadratic model)
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.
Polynomial Models
30
Theoretical backgroundTheoretical background Scatter diagramScatter diagram Orthogonal polynomial testOrthogonal polynomial test
Choosing order of the model
Theoretical background Can be searched from previous research or Can be searched from previous research or
literatureliterature Examples:Examples:1. The relationship between dosages of nitrogen
application and yield (The law of diminishing return)
2. The relationship between pesticide application and pest mortality (Linear model/probit analysis)
3. The relationship between population density and yield (Exponential model/Cob-Douglass Model)
Scatter DiagramScatter Diagram
0
5
10
15
20
25
-2,5 -2 -1,5 -1 -0,5 0 0,5 1 1,5 2 2,5
Mean
Treatment
Scatter DiagramScatter Diagram
0
5
10
15
20
25
-2,5 -2 -1,5 -1 -0,5 0 0,5 1 1,5 2 2,5
Mean
Treatrment
Orthogonal Linear Contrasts for Orthogonal Linear Contrasts for Polynomial RegressionPolynomial Regression
t Polynomial 1 2 3 4 5 6 7 8 9 10 ∑ai2
3 Linear -1 0 1 2 Quadratic 1 -2 1 6 4 Linear -3 -1 1 3 20 Quadratic 1 -1 -1 1 4 Cubic -1 3 -3 1 20 5 Linear -2 -1 0 1 2 10 Quadratic 2 -1 -2 -1 2 14 Cubic -1 2 0 -2 1 10 Quartic 1 -4 6 -4 1 70 6 Linear -5 -3 -1 1 3 5 70 Quadratic 5 -1 -4 -4 -1 5 84 Cubic -5 7 4 -4 -7 5 180 Quartic 1 -3 2 2 -3 1 28 7 Linear -3 -2 -1 0 1 2 3 28 Quadratic 5 0 -3 -4 -3 0 5 84 Cubic -1 1 1 0 -1 -1 1 6 Quartic 3 -7 1 6 1 -7 3 154
Orthogonal Linear Contrasts for Orthogonal Linear Contrasts for Polynomial RegressionPolynomial Regression
t Polynomial 1 2 3 4 5 6 7 8 9 10 ai2
8 Linear -7 -5 -3 -1 1 3 5 7 168 Quadratic 7 1 -3 -5 -5 -3 1 7 168 Cubic -7 5 7 3 -3 -7 -5 7 264 Quartic 7 -13 -3 9 9 -3 -13 7 616 Quintic -7 23 -17 -15 15 17 -23 7 2184 9 Linear -4 -3 -2 -1 0 1 2 3 4 20 Quadratic 28 7 -8 -17 -20 -17 -8 7 28 2772 Cubic -14 7 13 9 0 -9 -13 -7 14 990 Quartic 14 -21 -11 9 18 9 -11 -21 14 2002 Quintic -4 11 -4 -9 0 9 4 -11 4 468 10 Linear -9 -7 -5 -3 -1 1 3 5 7 9 330 Quadratic 6 2 -1 -3 -4 -4 -3 -1 2 6 132 Cubic -42 14 35 31 12 -12 -31 -35 -14 42 8580 Quartic 18 -22 -17 3 18 18 3 -17 -22 18 2860 Quintic -6 14 -1 -11 -6 6 11 1 -14 6 780