1. Comparing Means

2. 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

3. 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
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

4. Suppose there are t means
Pair Comparison
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.

5. 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.

6. 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

7. 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 sp2.
It is concluded that mi and mj differ (at a% significance level if |mi - mj| > LSD, where

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

9. Example: protein source
ANOVA Table of CRD
Source of Sum of Degrees of Mean
Variation Squares Freedom Squares Fcalc.
Treatments
Error
Total

10. 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:
where ra,p,n is the Duncan’s Significant Range Value with parameters p (= range-value) and n (= MSE degree-of-freedom), and experiment-wise alpha level a (= ajoint).

11. 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 …

12. Significant Ranges for Duncan’s Multiple Range Test

13. 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
where qa,a,n is the Studentized Range Statistic with parameters p (= range-value) and n (= MSE degree-of-freedom), and experiment-wise alpha level a (= ajoint).

14. 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 …

15. Studentized Range Statistic

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

17. Studentized Range Statistic

18. Considerable controversy:
Scheffe
There are many multiple (post hoc) comparison procedures
Considerable controversy:
“I have not included the multiple comparison methods of D.B. Duncan because I have been unable to understand their justification”

19. 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:

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

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

22. Contrast Examples Treatment Protein sources A Catfish B Tilapia C Tuna
The mean effect of freshwater and ocean fish
(m1+m2)/2 = (m3)
The mean effect of catfish and tilapia fish
m1 = m2

23. 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 (rj)
Usually, only the linear and quadratic contrasts are of interest

24. 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:

25. Polynomial Models in One Variable
A second-order model (quadratic model):

26. A second-order model
(quadratic model)

27. 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.

28. Choosing order of the model
Theoretical background
Scatter diagram
Orthogonal polynomial test

29. Scatter Diagram

30. Scatter Diagram

31. Orthogonal Linear Contrasts for Polynomial Regression

32. Orthogonal Linear Contrasts for Polynomial Regression