1. One way Analysis of Variance (ANOVA)
Comparing k Populations

2. The F test – for comparing k means
Situation
We have k normal populations
Let mi and s denote the mean and standard deviation of population i.
i = 1, 2, 3, … k.
Note: we assume that the standard deviation for each population is the same.
s1 = s2 = … = sk = s

3. We want to test
against

4. Computing Formulae:
Compute 1) 2) 3) 4) 5)

5. The data Assume we have collected data from each of k populations
Let xi1, xi2 , xi3 , … denote the ni observations from population i.
i = 1, 2, 3, … k.

6. Then 1) 2) 3)

7. Anova Table Mean Square F-ratio Between k - 1 SSBetween MSBetween
Source
d.f.
Sum of Squares
Mean Square
F-ratio
Between
k - 1
SSBetween
MSBetween
MSB /MSW
Within
N - k
SSWithin
MSWithin
Total
N - 1
SSTotal

8. Example
In the following example we are comparing weight gains resulting from the following six diets
Diet 1 - High Protein , Beef
Diet 2 - High Protein , Cereal
Diet 3 - High Protein , Pork
Diet 4 - Low protein , Beef
Diet 5 - Low protein , Cereal
Diet 6 - Low protein , Pork

10. Thus
Thus since F > we reject H0

11. Anova Table Mean Square F-ratio Between 5 4612.933 922.587 4.3**
Source
d.f.
Sum of Squares
Mean Square
F-ratio
Between 5
4.3**
(p = )
Within 54
Total 59
* - Significant at 0.05 (not 0.01)
** - Significant at 0.01

12. Equivalence of the F-test and the t-test when k = 2

13. the F-test

15. Hence

16. Factorial Experiments
Analysis of Variance

17. k Categorical independent variables A, B, C, … (the Factors) Let
Dependent variable Y
k Categorical independent variables A, B, C, … (the Factors)
Let
a = the number of categories of A
b = the number of categories of B
c = the number of categories of C
etc.

18. The Completely Randomized Design
We form the set of all treatment combinations – the set of all combinations of the k factors
Total number of treatment combinations
t = abc….
In the completely randomized design n experimental units (test animals , test plots, etc. are randomly assigned to each treatment combination.
Total number of experimental units N = nt=nabc..

19. The treatment combinations can thought to be arranged in a k-dimensional rectangular block1 2 b 1 2 A a

20. C B A

21. The Completely Randomized Design is called balanced
If the number of observations per treatment combination is unequal the design is called unbalanced. (resulting mathematically more complex analysis and computations)
If for some of the treatment combinations there are no observations the design is called incomplete. (In this case it may happen that some of the parameters - main effects and interactions - cannot be estimated.)

22. Example In this example we are examining the effect of
The level of protein A (High or Low) and the source of protein B (Beef, Cereal, or Pork) on weight gains (grams) in rats.
We have n = 10 test animals randomly assigned to k = 6 diets

23. The k = 6 diets are the 6 = 3×2 Level-Source combinations
High - Beef
High - Cereal
High - Pork
Low - Beef
Low - Cereal
Low - Pork

24. Gains in weight (grams) for rats under six diets
Table
Gains in weight (grams) for rats under six diets
differing in level of protein (High or Low) and s
ource of protein (Beef, Cereal, or Pork)
Level
of Protein High Protein Low protein
Source
of Protein Beef Cereal Pork Beef Cereal Pork
Diet
Mean
Std. Dev

25. Treatment combinations
Source of Protein
Beef
Cereal
Pork
Level of Protein
High
Diet 1
Diet 2
Diet 3
Low
Diet 4
Diet 5
Diet 6

26. Summary Table of Means Source of Protein
Level of Protein Beef Cereal Pork Overall
High
Low
Overall

27. Profiles of the response relative to a factor
A graphical representation of the effect of a factor on a reponse variable (dependent variable)

28. Profile Y for A Y Levels of A a
This could be for an individual case or averaged over a group of cases
This could be for specific level of another factor or averaged levels of another factor a … 1 2 3
Levels of A

29. Profiles of Weight Gain for Source and Level of Protein

30. Profiles of Weight Gain for Source and Level of Protein

31. Example – Four factor experiment
Four factors are studied for their effect on Y (luster of paint film). The four factors are:
1) Film Thickness - (1 or 2 mils)
2) Drying conditions (Regular or Special)
3) Length of wash (10,30,40 or 60 Minutes), and
4) Temperature of wash (92 ˚C or 100 ˚C)
Two observations of film luster (Y) are taken for each treatment combination

32. The data is tabulated below:
Regular Dry Special Dry
Minutes 92 C 100 C 92C 100 C
1-mil Thickness
2-mil Thickness

33. Definition:
A factor is said to not affect the response if the profile of the factor is horizontal for all combinations of levels of the other factors:
No change in the response when you change the levels of the factor (true for all combinations of levels of the other factors)
Otherwise the factor is said to affect the response:

34. Profile Y for A – A affects the response
Levels of B a … 1 2 3
Levels of A

35. Profile Y for A – no affect on the response
Levels of B a … 1 2 3
Levels of A

36. Definition:
Two (or more) factors are said to interact if changes in the response when you change the level of one factor depend on the level(s) of the other factor(s).
Profiles of the factor for different levels of the other factor(s) are not parallel
Otherwise the factors are said to be additive .
Profiles of the factor for different levels of the other factor(s) are parallel.

37. Interacting factors A and BY
Levels of B a … 1 2 3
Levels of A

38. Additive factors A and BY
Levels of B a … 1 2 3
Levels of A

39. If two (or more) factors interact each factor effects the response.
If two (or more) factors are additive it still remains to be determined if the factors affect the response
In factorial experiments we are interested in determining
which factors effect the response and
which groups of factors interact .

40. The testing in factorial experiments
Test first the higher order interactions.
If an interaction is present there is no need to test lower order interactions or main effects involving those factors. All factors in the interaction affect the response and they interact
The testing continues with for lower order interactions and main effects for factors which have not yet been determined to affect the response.

41. Models for factorial Experiments

42. The Single Factor Experiment
Situation
We have t = a treatment combinations
Let mi and s denote the mean and standard deviation of observations from treatment i.
i = 1, 2, 3, … a.
Note: we assume that the standard deviation for each population is the same.
s1 = s2 = … = sa = s

43. The data Assume we have collected data for each of the a treatments
Let yi1, yi2 , yi3 , … , yin denote the n observations for treatment i.
i = 1, 2, 3, … a.

44. The model Note: where has N(0,s2) distribution (overall mean effect)
(Effect of Factor A)
Note:
by their definition.

45. Model 1:
yij (i = 1, … , a; j = 1, …, n) are independent Normal with mean mi and variance s2.
Model 2:
where eij (i = 1, … , a; j = 1, …, n) are independent Normal with mean 0 and variance s2.
Model 3:
where eij (i = 1, … , a; j = 1, …, n) are independent Normal with mean 0 and variance s2 and

46. The Two Factor Experiment
Situation
We have t = ab treatment combinations
Let mij and s denote the mean and standard deviation of observations from the treatment combination when A = i and B = j.
i = 1, 2, 3, … a, j = 1, 2, 3, … b.

47. The data
Assume we have collected data (n observations) for each of the t = ab treatment combinations.
Let yij1, yij2 , yij3 , … , yijn denote the n observations for treatment combination - A = i, B = j.
i = 1, 2, 3, … a, j = 1, 2, 3, … b.

48. The model
Note:
where
has N(0,s2) distribution
and

49. The model Note: where has N(0,s2) distribution Note:
by their definition.

50. Main effects Interaction Effect Error Mean Model :
where eijk (i = 1, … , a; j = 1, …, b ; k = 1, …, n) are independent Normal with mean 0 and variance s2 and

51. Maximum Likelihood Estimates
where eijk (i = 1, … , a; j = 1, …, b ; k = 1, …, n) are independent Normal with mean 0 and variance s2 and

52. This is not an unbiased estimator of s2 (usually the case when estimating variance.)
The unbiased estimator results when we divide by ab(n -1) instead of abn

53. The unbiased estimator of s2 is
where

54. Testing for Interaction:
We want to test:
H0: (ab)ij = 0 for all i and j, against
HA: (ab)ij ≠ 0 for at least one i and j.
The test statistic
where

55. We reject
H0: (ab)ij = 0 for all i and j, If

56. Testing for the Main Effect of A:
We want to test:
H0: ai = 0 for all i, against
HA: ai ≠ 0 for at least one i.
The test statistic
where

57. We reject
H0: ai = 0 for all i, If

58. Testing for the Main Effect of B:
We want to test:
H0: bj = 0 for all j, against
HA: bj ≠ 0 for at least one j.
The test statistic
where

59. We reject
H0: bj = 0 for all j, If

60. The ANOVA Table Source S.S. d.f. MS =SS/df F A SSA a - 1 MSA
MSA / MSError B
SSB
b - 1
MSB
MSB / MSError AB
SSAB
(a - 1)(b - 1)
MSAB
MSAB/ MSError
Error
SSError
ab(n - 1)
MSError
Total
SSTotal
abn - 1

61. Computing Formulae