1. Chapter 10
Analysis of Variance

2. Chapter 10 - Chapter Outcomes
After studying the material in this chapter, you should be able to:
Recognize the applications that call for the use of analysis of variance.
Understand the logic of analysis of variance.
Be aware of several different analysis of variance designs and understand when to use each one.
Perform a single factor hypothesis test using analysis of variance manually and with the aid of Excel or Minitab software.

3. Chapter 10 - Chapter Outcomes (continued)
After studying the material in this chapter, you should be able to:
Conduct and interpret post-analysis of variance pairwise comparisons procedures.
Recognize when randomized block analysis of variance is useful and be able to perform the randomized block analysis.
Perform two factor analysis of variance tests with replications using Excel or Minitab and interpret the output.

4. One-Way Analysis of Variance
One-way analysis of variance is a design in which independent samples are obtained from k levels of a single factor for the purpose of testing whether the k levels have equal means.

5. One-Way Analysis of Variance
A factor refers to a quantity under examination in an experiment as a possible cause of variation in the response variable.

6. One-Way Analysis of Variance
Levels refer to the categories, measurements, or strata of a factor of interest in the current experiment.

7. One-Way Analysis of Variance
Null hypothesis in an ANOVA experiment:
Alternative hypothesis in an ANOVA experiment:

8. One-Way Analysis of Variance
COMPLETELY RANDOMIZED DESIGN
An experiment is completely randomized if it consists of the independent random selection of observations representing each level of one factor.

9. One-Way Analysis of Variance
BALANCED DESIGN
An experiment is said to have a balanced design if the factor levels have equal sample sizes.

10. One-Way Analysis of Variance
The ANOVA test is based on three assumptions:
All populations are normally distributed.
The population variances are equal.
The observations are independent, meaning that any one individual value is not dependent on the value of any other observation.

11. One-Way Analysis of Variance
Total Variation
The total variation in the data refers to the aggregate dispersion of the individual data values across the various factor levels.

12. One-Way Analysis of Variance
Within-Sample Variation
The dispersion that exists among the data values within a particular factor level is called the within-sample variation.

13. One-Way Analysis of Variance
Between-Sample Variation
The dispersion among the factor sample means is called the between-sample variation.

14. One-Way Analysis of Variance
PARTITIONED SUM OF SQUARES
where:
TSS = Total Sum of Squares
SSB = Sum of Squares Between
SSW = Sum of Squares Within

15. One-Way Analysis of Variance
Null hypothesis in an ANOVA experiment:
Alternative hypothesis in an ANOVA experiment:

16. One-Way Analysis of Variance
F-Max test for Equal Variances
where:
s2max = Largest sample variance
s2min = Smallest sample variance

17. One-Way Analysis of Variance
TOTAL SUM OF SQUARES
where:
TSS = Total sum of squares
k = Number of populations (levels)
ni = Sample size from population i
xij = jth measurement from population i
= Grand Mean (mean of all the data values)

18. One-Way Analysis of Variance
SUM OF SQUARES BETWEEN
where:
SSB = Sum of Squares Between Samples
k = Number of populations (levels)
ni = Sample size from population i
= Sample mean from population i
= Grand Mean

19. One-Way Analysis of Variance
SUM OF SQUARES WITHIN
SSW = TSS - SSB or
where:
SSW = Sum of Squares Within Samples
k = Number of populations
ni = Sample size from population i
= Sample mean from population i
xij = jth measurement from population i

20. One-Way ANOVA Table (Table 10-3)
Source of SS df MS
F Ratio
Variation
Between
SSB
k - 1
SSB
Samples
K-1
MSB
MSW
Within
N - k
SSW
N-k
SSW
Samples
Total
TSS
N - 1
k = Number of populations
N = Sum of the sample sizes from all populations
df = Degrees of freedom

21. One-Way ANOVA Table (Table 10-3)

22. Example of a One-Way ANOVA Table (Table 10-4)
Source of SS df MS
F Ratio
Variation
7.333
Between
7.333 22 3
Samples
Within
198.88 = 28
Samples
Total
220.88 31

23. One-Way Analysis of Variance (Figure 10-5)
H0: 1= 2 = 3 = 4
HA: At least two population means are different
 = 0.05
Degrees of freedom:
D1 = k - 1 = = 3
D2 = N - k = = 28
Rejection Region F
Since F=  F= 2. 95, do not reject H0

24. The Tukey-Kramer Procedure
The Tukey-Kramer procedure is a method for testing which populations have different means, after the one-way ANOVA null hypothesis has been rejected.

25. The Tukey-Kramer Procedure and One-Way ANOVA
The experiment-wide error rate is the proportion of experiments in which at least one of the set of confidence intervals constructed does not contain the true value of the population parameter being estimated.

26. The Tukey-Kramer Procedure and One-Way ANOVA
TUKEY-KRAMER CRITICAL RANGE
where:
q = Value from standardized range table with k and N - k degrees of freedom for the desired level of .
MSW = Mean Square Within
ni and nj = Sample sizes from populations (levels) i and j, respectively.

27. Randomized Complete Block ANOVA
A treatment is a combination of one level of each factor in an experiment associated with each observed value of the response variable.

28. Randomized Complete Block ANOVA
SUM OF SQUARES PARTITIONING - RANDOMIZED COMPLETE BLOCK DESIGN
where:
TSS = Total sum of squares
SSB = Sum of squares between factor levels
SSBL= Sum of squares between blocks
SSW = Sum of squares within levels

29. Randomized Complete Block ANOVA
SUM OF SQUARES FOR BLOCKING
where:
k = Number of levels for the factor
n = Number of blocks
= The mean of the jth block
= Grand Mean

30. Randomized Complete Block ANOVA
SUM OF SQUARES WITHIN

31. Randomized Complete Block ANOVA
The randomized block design requires the following assumptions:
The populations are normally distributed.
The populations have equal variances.
The observations are independent.

32. Randomized Block ANOVA Table (Table 10-7)
Source of SS df MS
F Ratio
Variation
Between
MSBL
b- 1
SSBL
MSBL
MSW
Blocks
Between
MSB
SSB
k - 1
MSB
Samples
MSW
Within
SSW
(k - 1)(b - 1)
MSW
Samples
Total
TSS
N - 1
k = Number of levels
b = Number of blocks
df = Degrees of freedom and N = Combined sample size

33. Randomized Block ANOVA Table (From Table 10-7)
where:

34. Randomized Block ANOVA (Figure 10-12)
H0: 1= 2 = 3
HA: At least two population means are different
 = 0.05
Degrees of freedom:
D1 = k - 1 = = 2, D2 = (n - 1)(k - 1)
= (4)(2) = 8
Rejection Region F
Since F=8.544 > F= , reject H0

35. Fisher’s Least Significant Difference (LSD) Test
FISHER’S LEAST SIGNIFICANT DIFFERENCE FOR COMPLETE BLOCK DESIGN
where:
t/2 = Upper-tailed value from Student’s t- distribution for /2 and (k -1)(n - 1) degrees of freedom
MSW = Mean square within from ANOVA table
n = Number of blocks
k = Number of levels

36. Two-Factor Analysis of Variance
Two-factor ANOVA is a technique used to analyze two factors in an analysis of variance framework.

37. Two-Factor Analysis of Variance (Figure 10-13)
SSA
Factor A
SST
SSB
Factor B
Interaction Between A and B
SSAB
SSE
Inherent Variation (Error)

38. Two-Factor Analysis of Variance
The necessary assumptions for the two factor ANOVA are:
The population values for each combination of pairwise factor levels are normally distributed.
The variances for each population are equal.
The samples are independent.
The observations are independent.

39. Randomized Block ANOVA Table (Table 10-9)
Source of SS df MS
F Ratio
Variation
MSA
Factor A
SSA
a- 1
MSA
MSE
MSB
Factor B
SSB
b - 1
MSB
MSE AB
Interaction
MSAB
(a - 1)(b - 1)
MSAB
SSAB
MSE
Error
SSE
N - ab
MSE
Total
TSS
N - 1
a = Number of levels of factor A
b = Number of levels of factor B
N = Total number of observations in all cells

40. Randomized Block ANOVA Table (From Table 10-9)
where:

41. Two Factor ANOVA Equations
Total Sum of Squares:
Sum of Squares Factor A:
Sum of Squares Factor B:
Sum of Squares Interaction Between A and B:
Sum of Squares Error:

42. Two Factor ANOVA Equations
where:
a = Number of levels of factor A
b = Number of levels of factor B
n’ = Number of replications in each cell

43. Differences Between Factor Level Mean Values: No Interaction
Mean Response
Factor B Level 1
Factor B Level 4
Factor B Level 3
Factor B Level 2 1 2 3
Factor A Levels

44. Differences Between Factor Level Mean Values: Interaction Present
Factor B Level 1
Factor B Level 2
Mean Response
Factor B Level 3
Factor B Level 4 1 2 3
Factor A Levels

45. A Caution About Interaction
When conducting hypothesis tests for a two factor ANOVA:
Test for interaction.
If interaction is present conduct a one-way ANOVA to test the levels of one of the factors using only one level of the other factor.
If no interaction is found, test factor A and factor B.

46. Key Terms Balanced Design Between-Sample Variation
Completely Randomized Design
Experiment-Wide Error Rate
Factor
Levels
One-Way Analysis of Variance
Total Variation
Treatment
Within-Sample Variation