1. Orthogonal Linear Contrasts
This is a technique for partitioning ANOVA sum of squares into individual degrees of freedom

2. Definition
Let x1, x2, ... , xp denote p numerical quantities computed from the data.
These could be statistics or the raw observations.
A linear combination of x1, x2, ... , xp is defined to be a quantity ,L ,computed in the following manner:
L = c1x1+ c2x cpxp
where the coefficients c1, c2, ... , cp are predetermined numerical values:

3. Definition If the coefficients c1, c2, ... , cp satisfy:
Then the linear combination
L = c1x1+ c2x cpxp
is called a linear contrast.

4. Examples 1.
A linear combination
A linear contrast 2.
A linear contrast
L = x1 - 4 x2+ 6x3 - 4 x4 + x5
= (1)x1+ (-4)x2+ (6)x3 + (-4)x4 + (1)x5

5. Definition
Let A = a1x1+ a2x apxp and B= b1x1+ b2x bpxp be two linear contrasts of the quantities x1, x2, ... , xp. Then A and B are c called Orthogonal Linear Contrasts if in addition to:
a1+ a ap = 0 and
b1+ b bp = 0,
it is also true that:
a1b1+ a2b apbp = 0. .

6. Example
Let
Note:

7. Definition Let A = a1x1+ a2x2+ ... + apxp, B= b1x1+ b2x2+ ... + bpxp ,
..., and
L= l1x1+ l2x lpxp
be a set linear contrasts of the quantities x1, x2, ... , xp.
Then the set is called a set of Mutually Orthogonal Linear Contrasts if each linear contrast in the set is orthogonal to any other linear contrast..

8. Theorem:
The maximum number of linear contrasts in a set of Mutually Orthogonal Linear Contrasts of the quantities x1, x2, ... , xp is p - 1.
p - 1 is called the degrees of freedom (d.f.) for comparing quantities x1, x2, ... , xp .

9. Comments
Linear contrasts are making comparisons amongst the p values x1, x2, ... , xp
Orthogonal Linear Contrasts are making independent comparisons amongst the p values x1, x2, ... , xp .
The number of independent comparisons amongst the p values x1, x2, ... , xp is p – 1.

10. Definition denotes a linear contrast of the p means
If each mean, , is calculated from n observations then:
The Sum of Squares for testing the Linear Contrast L, is defined to be:

11. the degrees of freedom (df) for testing the Linear Contrast L, is defined to be
the F-ratio for testing the Linear Contrast L, is defined to be:

12. Theorem:
Let L1, L2, ... , Lp-1 denote p-1 mutually orthogonal Linear contrasts for comparing the p means . Then the Sum of Squares for comparing the p means based on p – 1 degrees of freedom , SSBetween, satisfies:

13. Comment
Defining a set of Orthogonal Linear Contrasts for comparing the p means
allows the researcher to "break apart" the Sum of Squares for comparing the p means, SSBetween, and make individual tests of each the Linear Contrast.

14. The Diet-Weight Gain example
The sum of Squares for comparing the 6 means is given in the Anova Table:

15. Five mutually orthogonal contrasts are given below (together with a description of the purpose of these contrasts) :
(A comparison of the High protein diets with Low protein diets)
(A comparison of the Beef source of protein with the Pork source of protein)

16. (A comparison of the Meat (Beef - Pork) source of protein with the Cereal source of protein)
(A comparison representing interaction between Level of protein and Source of protein for the Meat source of Protein)
(A comparison representing interaction between Level of protein with the Cereal source of Protein)

17. The Anova Table for Testing these contrasts is given below:
The Mutually Orthogonal contrasts that are eventually selected should be determine prior to observing the data and should be determined by the objectives of the experiment

18. Another Five mutually orthogonal contrasts are given below (together with a description of the purpose of these contrasts) :
(A comparison of the High protein diets with Low protein diets)
(A comparison of the Beef source of protein with the Pork source of protein)

19. (A comparison of the high and low protein diets for the Beef source of protein)
(A comparison of the high and low protein diets for the Cereal source of protein)
(A comparison of the high and low protein diets for the Pork source of protein)

20. The Anova Table for Testing these contrasts is given below:

21. Orthogonal Linear Contrasts
Polynomial Regression

22. Orthogonal Linear Contrasts for Polynomial Regression

23. Orthogonal Linear Contrasts for Polynomial Regression

24. Example
In this example we are measuring the “Life” of an electronic component and how it depends on the temperature on activation

25. The Anova Table L = 25.00 Q2 = -45.00 C = 0.00 Q4 = 30.00
Source SS df MS F Treat Linear Quadratic Cubic Quartic Error Total

26. The Anova Tables for Determining degree of polynomial
Testing for effect of the factor

27. Testing for departure from Linear

28. Testing for departure from Quadratic

30. Multiple Comparison Tests
Post-hoc Tests
Multiple Comparison Tests

31. Suppose we have p means
An F-test has revealed that there are significant differences amongst the p means
We want to perform an analysis to determine precisely where the differences exist.

32. Tukey’s Multiple Comparison Test

33. Let
denote the standard error of each
Tukey's Critical Differences
Two means are declared significant if they differ by more than this amount.
= the tabled value for Tukey’s studentized range p = no. of means, n = df for Error

34. Scheffe’s Multiple Comparison Test

35. Scheffe's Critical Differences (for Linear contrasts)
A linear contrast is declared significant if it exceeds this amount.
= the tabled value for F distribution (p -1 = df for comparing p means, n = df for Error)

36. Scheffe's Critical Differences
(for comparing two means)
Two means are declared significant if they differ by more than this amount.