1. Statistics for Managers Using Microsoft Excel 3rd Edition
Chapter 9
Analysis of Variance
© 2002 Prentice-Hall, Inc.

2. Chapter Topics
The completely randomized design: one-factor analysis of variance
ANOVA assumptions
F test for difference in c means
The Tukey-Kramer procedure
The factorial design: two-way analysis of variance
Examine effects of factors and interaction
Kruskal-Wallis rank test for differences in c medians
© 2002 Prentice-Hall, Inc.

3. General Experimental Setting
Investigator controls one or more independent variables
Called treatment variables or factors
Each treatment factor contains two or more levels (or categories/classifications)
Observe effects on dependent variable
Response to levels of independent variable
Experimental design: the plan used to test hypothesis
© 2002 Prentice-Hall, Inc.

4. Completely Randomized Design
Experimental units (subjects) are assigned randomly to treatments
Subjects are assumed homogeneous
Only one factor or independent variable
With two or more treatment levels
Analyzed by
One-factor analysis of variance (one-way ANOVA)
© 2002 Prentice-Hall, Inc.

5. Randomized Design Example
Factor (Training Method)
Factor Levels (Treatments)
Randomly Assigned Units
Dependent Variable (Response)
21 hrs
17 hrs
31 hrs
27 hrs
25 hrs
28 hrs
29 hrs
20 hrs
22 hrs



© 2002 Prentice-Hall, Inc.

6. One-Factor Analysis of Variance F Test
Evaluate the difference among the mean responses of two or more (c ) populations
e.g.: Several types of tires, oven temperature settings
Assumptions
Samples are randomly and independently drawn
This condition must be met
Populations are normally distributed
F test is robust to moderate departure from normality
Populations have equal variances
Less sensitive to this requirement when samples are of equal size from each population
© 2002 Prentice-Hall, Inc.

7. Why ANOVA?
Could compare the means one by one using Z or t tests for difference of means
Each Z or t test contains Type I error
The total Type I error with k pairs of means is 1- (1 - a) k
e.g.: If there are 5 means and use a = .05
Must perform 10 comparisons
Type I error is 1 – (.95) 10 = .40
40% of the time you will reject the null hypothesis of equal means in favor of the alternative even when the null is true!
© 2002 Prentice-Hall, Inc.

8. Hypotheses of One-Way ANOVA
All population means are equal
No treatment effect (no variation in means among groups)
At least one population mean is different (others may be the same!)
There is treatment effect
Does not mean that all population means are different
© 2002 Prentice-Hall, Inc.

9. One-Factor ANOVA (No Treatment Effect)
The Null Hypothesis is True
© 2002 Prentice-Hall, Inc.

10. One-Factor ANOVA (Treatment Effect Present)
The Null Hypothesis is NOT True
© 2002 Prentice-Hall, Inc.

11. One-Factor ANOVA (Partition of Total Variation)
Total Variation SST
Variation Due to Treatment SSA +
Variation Due to Random Sampling SSW =
Commonly referred to as:
Sum of Squares Among
Sum of Squares Between
Sum of Squares Model
Sum of Squares Explained
Sum of Squares Treatment
Among Groups Variation
Commonly referred to as:
Sum of Squares Within
Sum of Squares Error
Sum of Squares Unexplained
Within Groups Variation
© 2002 Prentice-Hall, Inc.

12. Total Variation
© 2002 Prentice-Hall, Inc.

13. Total Variation
(continued)
© 2002 Prentice-Hall, Inc.

14. Among-Group Variation
Variation Due to Differences Among Groups.
© 2002 Prentice-Hall, Inc.

15. Among-Group Variation
(continued)
© 2002 Prentice-Hall, Inc.

16. Within-Group Variation
Summing the variation within each group and then adding over all groups.
© 2002 Prentice-Hall, Inc.

17. Within-Group Variation
(continued)
© 2002 Prentice-Hall, Inc.

18. Within-Group Variation
(continued)
For c = 2, this is the pooled-variance in the t-Test.
If more than two groups, use F Test.
For two groups, use t-Test. F Test more limited.
© 2002 Prentice-Hall, Inc.

19. One-Factor ANOVA F Test Statistic
MSA is mean squares among or between variances
MSW is mean squares within or error variances
Degrees of freedom
© 2002 Prentice-Hall, Inc.

20. One-Factor ANOVA Summary Table
Source of Variation
Degrees of Freedom
Sum of Squares
Mean Squares (Variance)
F Statistic
Among (Factor)
c – 1
SSA
MSA = SSA/(c – 1 )
MSA/MSW
Within (Error)
n – c
SSW
MSW =
SSW/(n – c )
Total
n – 1
SST = SSA + SSW
© 2002 Prentice-Hall, Inc.

21. Features of One-Factor ANOVA F Statistic
The F Statistic is the ratio of the among estimate of variance and the within estimate of variance
The ratio must always be positive
Df1 = c -1 will typically be small
Df2 = n - c will typically be large
The ratio should be closed to 1 if the null is true
© 2002 Prentice-Hall, Inc.

22. Features of One-Factor ANOVA F Statistic
(continued)
The numerator is expected to be greater than the denominator
The ratio will be larger than 1 if the null is false
© 2002 Prentice-Hall, Inc.

23. One-Factor ANOVA F Test Example
Machine1 Machine2 Machine
As production manager, you want to see if three filling machines have different mean filling times. You assign 15 similarly trained and experienced workers, five per machine, to the machines. At the .05 significance level, is there a difference in mean filling times?
© 2002 Prentice-Hall, Inc.

24. One-Factor ANOVA Example: Scatter Diagram
Machine1 Machine2 Machine
Time in Seconds 27 26 25 24 23 22 21 20 19 • • • • • • • • • • • • • • •
© 2002 Prentice-Hall, Inc.

25. One-Factor ANOVA Example Computations
Machine1 Machine2 Machine
© 2002 Prentice-Hall, Inc.

26. Mean Squares (Variance)
Summary Table
Source of Variation
Degrees of Freedom
Sum of Squares
Mean Squares (Variance)
F Statistic
Among (Factor)
3-1=2
MSA/MSW
=25.60
Within (Error)
15-3=12
.9211
Total
15-1=14
© 2002 Prentice-Hall, Inc.

27. One-Factor ANOVA Example Solution
Test Statistic:
Decision:
Conclusion:
H0: 1 = 2 = 3
H1: Not All Equal
 = .05
df1= df2 = 12
Critical Value(s):
MSA 23 .
5820 F    25 . 6
MSW .
9211
Reject at  = 0.05
 = 0.05
There is evidence that at least one  i differs from the rest. F
3.89
© 2002 Prentice-Hall, Inc.

28. Solution In EXCEL Use tools | data analysis | ANOVA: single factor
EXCEL worksheet that performs the one-factor ANOVA of the example
© 2002 Prentice-Hall, Inc.

29. The Tukey-Kramer Procedure
Tells which population means are significantly different
e.g.: 1 = 2  3
Two groups whose means may be significantly different
Post hoc (a posteriori) procedure
Done after rejection of equal means in ANOVA
Ability for pair-wise comparisons
Compare absolute mean differences with critical range
f(X)    X = 1 2 3
© 2002 Prentice-Hall, Inc.

30. The Tukey-Kramer Procedure: Example
1. Compute absolute mean differences:
Machine1 Machine2 Machine
2. Compute Critical Range:
3. All of the absolute mean differences are greater. There is a significance difference between each pair of means at 5% level of significance.
© 2002 Prentice-Hall, Inc.

31. Solution in PHStat
Use PHStat | c-sample tests | Tukey-Kramer procedure …
EXCEL worksheet that performs the Tukey-Kramer procedure for the previous example
© 2002 Prentice-Hall, Inc.

32. Two-Way ANOVA Examines the effect of
Two factors on the dependent variable
e.g.: Percent carbonation and line speed on soft drink bottling process
Interaction between the different levels of these two factors
e.g.: Does the effect of one particular percentage of carbonation depend on which level the line speed is set?
© 2002 Prentice-Hall, Inc.

33. Two-Way ANOVA Assumptions Normality Homogeneity of Variance
(continued)
Assumptions
Normality
Populations are normally distributed
Homogeneity of Variance
Populations have equal variances
Independence of Errors
Independent random samples are drawn
© 2002 Prentice-Hall, Inc.

34. Two-Way ANOVA Total Variation Partitioning+
Variation Due to Treatment A
SSA
d.f.= r-1
SSB +
Variation Due to Treatment B
d.f.= c-1
Total Variation
SSAB +
Variation Due to Interaction
SST =
d.f.= (r-1)(c-1)
d.f.= n-1
SSE
Variation Due to Random Sampling
d.f.= rc(n’-1)
© 2002 Prentice-Hall, Inc.

35. Two-Way ANOVA Total Variation Partitioning
© 2002 Prentice-Hall, Inc.

36. Total Variation
© 2002 Prentice-Hall, Inc.

37. Factor A Variation Sum of Squares Due to Factor A
= the difference among the various levels of factor A and the grand mean
© 2002 Prentice-Hall, Inc.

38. Factor B Variation Sum of Squares Due to Factor B
= the difference among the various levels of factor B and the grand mean
© 2002 Prentice-Hall, Inc.

39. Interaction Variation
Sum of Squares Due to Interaction between A and B
= the effect of the combinations of factor A and factor B
© 2002 Prentice-Hall, Inc.

40. Random Error Sum of Squares Error
= the differences among the observations within each cell and the corresponding cell means
© 2002 Prentice-Hall, Inc.

41. Two-Way ANOVA: The F Test Statistic
F Test for Factor A Main Effect
H0: 1 .= 2 . = ••• = r .
H1: Not all i . are equal
Reject if F > FU
F Test for Factor B Main Effect
H0: 1 = . 2 = ••• =  c
H1: Not all . j are equal
Reject if F > FU
F Test for Interaction Effect
H0: ij = 0 (for all i and j)
H1: ij  0
Reject if F > FU
© 2002 Prentice-Hall, Inc.

42. Two-Way ANOVA Summary Table
Source of Variation
Degrees of Freedom
Sum of Squares
Mean Squares
F Statistic
Factor A (Row)
r – 1
SSA
MSA = SSA/(r – 1)
MSA/ MSE
Factor B (Column)
c – 1
SSB
MSB =
SSB/(c – 1)
MSB/ MSE
AB (Interaction)
(r – 1)(c – 1)
SSAB
MSAB = SSAB/ [(r – 1)(c – 1)]
MSAB/ MSE
Error
rc (n’ – 1)
SSE
MSE =
SSE/[rc (n’ – 1)]
Total
rc n’ – 1
SST
© 2002 Prentice-Hall, Inc.

43. Features of Two-Way ANOVA F Test
Degrees of freedom always add up
rcn’-1=rc(n’-1)+(c-1)+(r-1)+(c-1)(r-1)
Total=error+column+row+interaction
The denominator of the F Test is always the same but the numerator is different.
The sums of squares always add up
© 2002 Prentice-Hall, Inc.

44. Kruskal-Wallis Rank Test for c Medians
Extension of Wilcoxon rank-sum test
Tests the equality of more than 2 (c) population medians
Distribution-free test procedure
Used to analyze completely randomized experimental designs
Use 2 distribution to approximate if each sample group size nj > 5
df = c – 1
© 2002 Prentice-Hall, Inc.

45. Kruskal-Wallis Rank Test
Assumptions
Independent random samples are drawn
Continuous dependent variable
Data may be ranked both within and among samples
Populations have same variability
Populations have same shape
Robust with regard to last two conditions
Use F test in completely randomized designs and when the more stringent assumptions hold
© 2002 Prentice-Hall, Inc.

46. Kruskal-Wallis Rank Test Procedure
Obtain ranks
In event of tie, each of the tied values gets their average rank
Add the ranks for data from each of the c groups
Square to obtain tj2
© 2002 Prentice-Hall, Inc.

47. Kruskal-Wallis Rank Test Procedure
(continued)
Compute test statistic
Number of observation in j –th sample
H may be approximated by chi-square distribution with df = c –1 when each nj >5
© 2002 Prentice-Hall, Inc.

48. Kruskal-Wallis Rank Test Procedure
(continued)
Critical value for a given a
Upper tail
Decision rule
Reject H0: M1 = M2 = ••• = mc if test statistic H >
Otherwise do not reject H0
© 2002 Prentice-Hall, Inc.

49. Kruksal-Wallis Rank Test: Example
Machine1 Machine2 Machine
As production manager, you want to see if three filling machines have different median filling times. You assign 15 similarly trained & experienced workers, five per machine, to the machines. At the .05 significance level, is there a difference in median filling times?
© 2002 Prentice-Hall, Inc.

50. Example Solution: Step 1 Obtaining a Ranking
Raw Data
Ranks
Machine1 Machine2 Machine
Machine1 Machine2 Machine 65 38 17
© 2002 Prentice-Hall, Inc.

51. Example Solution: Step 2 Test Statistic Computation
© 2002 Prentice-Hall, Inc.

52. Kruskal-Wallis Test Example Solution
H0: M1 = M2 = M3
H1: Not all equal
 = .05
df = c - 1 = = 2
Critical Value(s):
Test Statistic:
H = 11.58
Decision:
Reject at
 = .05
Conclusion:
 = .05
There is evidence that population medians are not all equal.
© 2002 Prentice-Hall, Inc.

53. Kruskal-Wallis Test in PHStat
PHStat | c-sample tests | Kruskal-Wallis rank sum test …
Example solution in excel spreadsheet
© 2002 Prentice-Hall, Inc.

54. Chapter Summary
Described the completely randomized design: one-factor analysis of variance
ANOVA assumptions
F test for difference in c means
The Tukey-Kramer procedure
Described the factorial design: two-way analysis of variance
Examine effects of factors and interaction
Discussed Kruskal-Wallis rank test for differences in c medians
© 2002 Prentice-Hall, Inc.