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 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. 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 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: Three categories: 1. Pair-wise comparisons (Post-Hoc Comparison) 2. Comparison specified prior to performing the experiment (Planned comparison) (Planned comparison) 3. Comparison specified after observing the outcome of the experiment (Un-planned comparison) experiment (Un-planned comparison) Statistical inference procedures of pair-wise comparisons: Statistical inference procedures of pair-wise comparisons: Fisher’s least significant difference (LSD) method Fisher’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 are t means Suppose there 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. 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. 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: Conditions common to all the methods: The ANOVA model is the one way analysis of variance The ANOVA model is the one way analysis of variance The conditions required to perform the ANOVA are satisfied. The conditions required to perform the ANOVA are satisfied. The experiment is fixed-effect. The experiment is fixed-effect. Pair Comparison

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. 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: In terms of Power and the probability of making a Type I error, the tests discussed can be ordered as follows: Tukey HSD Test Student-Newman-Keuls Test Duncan Multiple Range Test Fisher 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 (LSD) Method This method builds on the equal variances t-test of the difference between two means. 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. The test statistic is improved by using MSE rather than s p 2. It is concluded that  i and  j differ (at  % significance level if |  i -  j | > LSD, where It is concluded that  i and  j differ (at  % significance level if |  i -  j | > LSD, where

Critical t for a test about equality = t  (2),

Example: Cassava yields (ton/ha) Source of variation Degrees of Freedom Sum of Square Mena Square F calculated Treatment313645,33334 Block34013,33310 Error9121,33 Total1518 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 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: The Significant Difference or the range value: where r ,p, is the Duncan’s Significant Range Value with parameters p (= range-value) and (= MSE degree-of-freedom), and experiment-wise alpha level  (=  joint ).

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

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. 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 The Significant Difference or the range value for this test is where q ,a, is the Studentized Range Statistic with parameters p (= range-value) and (= MSE degree-of-freedom), and experiment-wise alpha level  (=  joint ). where q ,a, is the Studentized Range Statistic with parameters p (= range-value) and (= MSE degree-of-freedom), and experiment-wise alpha level  (=  joint ).

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

The test procedure: Assumes equal number of observation per populations. Assumes equal number of observation per populations. Find a critical number  as follows: Find a critical number  as follows: dft = treatment degrees of freedom =degrees of freedom = dfe n g = number of observations per population  = significance level q  (dft, ) = a critical value obtained from the studentized range table Tukey HSD Procedure

Studentized Range Statistic

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 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. 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: 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:

Planned Comparisons or Contrasts Each contrast can be specified as: and it is required: Each contrast can be specified as: and it is required: A sum-of-squares can be calculated for a contrast as A sum-of-squares can be calculated for a contrast as

Planned Comparisons Each contrast has 1 degree-of-freedom, and a contrast can be tested by comparing it to the MSE for the ANOVA: Each contrast has 1 degree-of-freedom, and a contrast can be tested by comparing it to the MSE for the ANOVA:

Un-planned Comparisons or Contrasts If more than 1 contrast is tested, it is important that the contrasts all be orthogonal, that is If more than 1 contrast is tested, it is important that the contrasts all be orthogonal, that is Note that It can be tested at most t-1 orthogonal contrasts.

Contrast orthogonal examples TreatmentYields (ton/ha) Adira-419 GH-625 GH-718 Local18 The mean effect of local and high yielding varieties The mean effect of high yielding and promising lines

Orthogonal Polynomials 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 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)

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 model): A second-order model (quadratic 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 background Theoretical background Scatter diagram Scatter diagram Orthogonal polynomial test Orthogonal polynomial test Choosing order of the model

Theoretical background Can be searched from previous research or literature Can be searched from previous research or literature Examples: 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

Orthogonal Linear Contrasts for Polynomial Regression