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 investigators want to make specific

comparisons of the means and to estimate comparisons of the means and to estimate pooled errorpooled 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 2. Comparison specified prior to performing the experimentexperiment

(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 are The conditions required to perform the ANOVA are

satisfied.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 10

Error 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 values for means next to each other along the real number line, and those with 1, 2, … , a means in real number line, and those with 1, 2, … , a means in between the two means being compared.between the 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 test or the range value for this test isis

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:

gn

MSEdfedftq ),(

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:

:54

54

540

yyH

H

A

0:

:5431

5431

54310

yyyyH

H

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

1

2

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 19

GH-6 25

GH-7 18

Local 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 Examples

Treatment (Urea dosage) kg/ha

Yields (ton/ha)

50 19

100 25

150 18

200 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 Can be searched from previous research or literatureor literature

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