1. Chapter 10: Analysis of Variance: Comparing More Than Two Means

2. Where We’ve Been
Presented methods for estimating and testing hypotheses about a single population mean
Presented methods for comparing two population means
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

3. Where We’re Going
Discuss the critical elements in the design of a sampling experiment
Investigate completely randomized, randomized block, and factorial designs
Show how to analyze data using a technique called analysis of variance
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

4. 10.1: Elements of a Designed Experiment
Elements of Designed Experiments
Factors
(possible impacting the response variable)
Quantitative Factors
(Numerical)
Qualitative Factors
(Non-numerical)
Response Variable
(or dependent variable)
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

5. 10.1: Elements of a Designed Experiment
Elements of Designed Experiments
Factors
(possible impacting the response variable)
Quantitative Factors
(Numerical)
Qualitative Factors
(Non-numerical)
Response Variable
(or dependent variable)
Factor Levels are the values of the factors
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

6. 10.1: Elements of a Designed Experiment
Treatments are the factor-level combinations
In the example above, a variety of different GPA – Hours Studied combinations could occur within each subset (Yes or No) of the Study Group factor
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

7. 10.1: Elements of a Designed Experiment
An experimental unit is the object on which the response and factors are observed or measured
In the example above, an individual student would be the experimental unit
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

8. 10.1: Elements of a Designed Experiment
In a designed experiment the analyst controls the treatments and the selection of experimental units to each treatment
In an observational experiment the analyst observes the treatment and response on a sample of experimental units
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

9. 10.1: Elements of a Designed Experiment
In a designed experiment the analyst controls the treatments and the selection of experimental units to each treatment
In an observational experiment the analyst observes the treatment and response on a sample of experimental units
The method by which the experimental units are selected determines the type of experiment
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

10. 10.1: Elements of a Designed Experiment
Population of Experimental Units
Sample of Experimental Units
Apply factor-level combinations
Treatment 1 Sample
Treatment 2 Sample
Treatment 3 Sample …
Treatment k Sample
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

11. 10.2: The Completely Randomized Design
The completely randomized design is a design in which treatments are randomly assigned to the experimental units or in which independent random samples of experimental units are selected for each treatment.
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

12. 10.2: The Completely Randomized Design
Randomly assign observations to treatments
Completely Randomized Design
Randomly assign treatments to observations
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

13. 10.2: The Completely Randomized Design
Very often the object is to determine whether the varying treatments result in different means:
H0: µ1 = µ2 = µ3 = µ4 = ··· = µk
Ha: At least two of the k treatment means differ
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

14. 10.2: The Completely Randomized Design
Testing the equity of the means involves comparing the variability among the different treatments as well as within the treatments, adjusted for degrees of freedom.
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

15. 10.2: The Completely Randomized Design
Adjusting for degrees of freedom produces comparable measures of variability
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

16. 10.2: The Completely Randomized Design
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

17. 10.2: The Completely Randomized Design
The ratio of the variability among the treatment means to that within the treatment means is an F -statistic:
with k-1 numerator and n-k denominator degrees of freedom.
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

18. 10.2: The Completely Randomized Design
If F*  1, the difference between the treatment means may be attributable to sampling error.
If F* > 1 (significantly), there is support for the alternative hypothesis that the treatments themselves produce different results.
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

19. 10.2: The Completely Randomized Design
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

20. 10.2: The Completely Randomized Design
Conditions required for a Valid ANOVA F-Test: Completely Randomized Design
The samples are randomly selected in an independent manner from the k treatment populations.
All k sampled populations have distributions that are approximately normal.
The k population variances are equal.
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

21. 10.2: The Completely Randomized Design
The USGA compares the driving distance of four brands of golf balls.
H0: µ1 = µ2 = µ3 = µ4
Ha: The mean distances differ for at least two of the brands
 = .10
Test Statistic: F = MST/MSE
Rejection region: F > 2.25 = F.10 with v1 = 3 and v2 = 36
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

22. 10.2: The Completely Randomized Design
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

23. 10.2: The Completely Randomized Design
The USGA compares the driving distance of four brands of golf balls.
H0: µ1 = µ2 = µ3 = µ4
Ha: The mean distances differ for at least two of the brands
 = .10 Test Statistic: F = MST/MSE
Rejection region: F > 2.25 = F.10 with v1 = 3 and v2 = 36
Source
Degrees of Freedom
Sum of Squares
Mean Square F
p-value
Brands 3
2,794.39
931.46
43.99
.000
Error 36
762.30
21.18
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

24. 10.2: The Completely Randomized Design
The USGA compares the driving distance of four brands of golf balls.
H0: µ1 = µ2 = µ3 = µ4
Ha: The mean distances differ for at least two of the brands
 = .10 Test Statistic: F = MST/MSE
Rejection region: F > 2.25 = F.10 with v1 = 3 and v2 = 36
Since the calculated
F > 2.25, we reject the null hypothesis.
Source
Degrees of Freedom
Sum of Squares
Mean Square F
p-value
Brands 3
2,794.39
931.46
43.99
.000
Error 36
762.30
21.18
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

25. 10.2: The Completely Randomized Design
If the conditions for ANOVA are not met, a nonparametric procedure is recommended (see Chapter 14).
If the null hypothesis is not rejected, that is not conclusive proof that the treatment means are all equal.
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

26. 10.3: Multiple Comparisons of Means
Suppose the ANOVA F-test indicates differences in the means. To determine the differences, we would compare the differences of the means.
With k treatment means, there are
c = k(k – 1)/2
pairs of means to be compared, and each would have a significance level smaller than the overall .
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

27. 10.3: Multiple Comparisons of Means
To retain the overall confidence level, various techniques are available for pair wise comparisons:
Tukey – treatment sample sizes are equal
Bonferroni - treatment sample sizes may be unequal
Scheffé – general procedure for all linear combinations of treatment means
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

28. 10.3: Multiple Comparisons of Means
Let’s go back to the four brands of golf balls in the previous example:
Rank the treatment means with an overall 95% level of confidence using Tukey’s procedure.
Estimate the highest ranked golf ball's mean driving distance.
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

29. 10.3: Multiple Comparisons of Means
Pair wise Comparisons for Four Golf ball Brands
Based on a SAS ANOVA report (see pages 506-7)
Brand Comparison
95% Confidence Interval
µA - µB
< µA - µB <
µA - µC
< µA - µC <
µA - µD
-4.08 < µA - µD < 7.00
µB - µC
< µB - µC < -3.35
µB - µD
6.2 < µB - µD < 17.28
µC - µD
15.09 < µC - µD < 26.17
Brand C outperforms each of the other brands.
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

30. 10.3: Multiple Comparisons of Means
To construct a confidence interval on Brand C, we can use the descriptive statistics from the ANOVA and a straightforward one-sample t-based confidence interval (see section 7.3):
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

31. 10.4: The Randomized Block Design
Blocks (matched sets of experimental units) are formed.
Each of the b blocks has k experimental units, one for each treatment.
One experimental unit from each block is randomly assigned to each treatment, for a total of n = bk responses.
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

32. 10.4: The Randomized Block Design
To test the equity of the means, we use the ratio MST/MSE ~ F
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

33. 10.4: The Randomized Block Design
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

34. 10.4: The Randomized Block Design
Conditions required for a valid ANOVA F – Test
The b blocks are randomly selected and all k treatments are applied (in random order) to each block.
The distribution of observations corresponding to all bk block-treatment combinations are approximately normal.
The bk block-treatment distributions have equal variances.
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

35. 10.4: The Randomized Block Design
Completely Randomized Design
Randomized Block Design
Total Sum of Squares
SS(Total)
df=n-1
Sum of Squares for Treatments
SST
df=k-1
Sum of Squares for Error
SSE
df=n-k
Sum of Squares for Blocks
SSB
df=b-1
Sum of Squares for Error
df=n-b-k+1=(b-1)(k-1)
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

36. 10.4: The Randomized Block Design
Suppose the golf balls analyzed above are analyzed again using ten real golfers instead of a machine.
Each golfer is a block
Each brand is a treatment assigned in random order to each golfer
The ten drives for each brand produce the following means:
Brand A
Brand B
Brand C
Brand D
227 yards
233.2 yards
245.3 yards
220.7 yards
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

37. 10.4: The Randomized Block Design
ANOVA Table for the Golf Ball Tests
Source df SS MS F p
Treatment (Brand) 3
2,298.7
1,099.6
54.31
.000
Block (Golfer) 9
12,073.9
1,341.5
Error 27
546.6
20.2
Total 39
15,919.2
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

38. 10.4: The Randomized Block Design
Equity of Means 95% Confidence Intervals for the Golf Balls’ Distance
µA - µB
µA - µC
µA - µD
µB - µC
µB - µD
µC - µD
(-11.9,--.4)
(-24.0, -2.6)
(.6, 12.0)
(-17.9, -6.4)
(6.7, 18.2)
(18.9, 30.3)
None of the confidence intervals contain zero, so we can be 95% certain all of the brand means differ.
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

39. 10.4: The Randomized Block Design
ANOVA Table for the Golf Ball Tests
Source df SS MS F p
Treatment (Brand) 3
2,298.7
1,099.6
54.31
.000
Block (Golfer) 9
12,073.9
1,341.5
Error 27
546.6
20.2
Total 39
15,919.2
To test for block mean differences, use the ratio of MSB to MEE
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

40. 10.5: Factorial Experiments
A complete factorial experiment is a factorial experiment in which every factor-level combination is utilized. That is, the number of treatments in the experiment equals the total number of factor-level combinations.
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

41. 10.5: Factorial Experiments
Say two factors, A and B, are involved. Results could vary among observations because …
Factor A (only) matters
Factors A and B both matter and they interact
Factor B (only) matters
Factors A and B both matter, independently
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

42. 10.5: Factorial Experiments
Overall Treatment Variability
Main Effect of Factor A
Main Effect of Factor B
Interaction between A and B
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

43. 10.5: Factorial Experiments
Stage 1
Stage 2
Total Sum of Squares
SS(Total)
df=n-1
Sum of Squares for Treatments
SST
df=ab-1
Main effect sum of squares Factor A
SS(A)
df=a-1
Main effect sum of squares Factor B
SS(B)
df = b-1
Interaction sum of squares
Factors A and B
SS(AB)
df = (a-1)(b-1)
Sum of Squares for Error
SSE
df=n-ab
Sum of Squares for Error
df= n - ab
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

44. 10.5: Factorial Experiments
Tests Conducted in Analyses of Factorial Experiments: Completely Randomized Design, r Replicates per Treatment
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

45. 10.5: Factorial Experiments
Tests Conducted in Analyses of Factorial Experiments: Completely Randomized Design, r Replicates per Treatment
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

46. 10.5: Factorial Experiments
Tests Conducted in Analyses of Factorial Experiments: Completely Randomized Design, r Replicates per Treatment
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

47. 10.5: Factorial Experiments
Tests Conducted in Analyses of Factorial Experiments: Completely Randomized Design, r Replicates per Treatment
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

48. 10.5: Factorial Experiments
Tests Conducted in Analyses of Factorial Experiments: Completely Randomized Design, r Replicates per Treatment
Conditions Required:
Response distribution for each factor-level combination is normal.
Response variance is constant for all treatments.
Random and independent samples of experimental units are associated with each treatment.
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

49. 10.5: Factorial Experiments
The four brands of golf balls are tested again, this time with a driver and a 5 iron. Each brand-club combination (eight in all) is assigned randomly to four experimental units in a sequence of swings by Iron Byron.
Are the treatment means equal?
Do the factors “brand“ and “club” interact?
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

50. 10.5: Factorial Experiments
TABLE 10.13: ANOVA Summary Table for Example 10.10
Source df SS MS F
Model 7
140.35
Brand 1
32,092.11
32,093.11
936.75
Club 3
800.74
266.91
7.79
Interaction
765.96
255.32
7.45
Error 24
822.24
34.26
Total 31
34,482.05
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

51. 10.5: Factorial Experiments
TABLE 10.13: ANOVA Summary Table for Example 10.10
Source df SS MS F
Model 7
140.35
Brand 1
32,092.11
32,093.11
936.75
Club 3
800.74
266.91
7.79
Interaction
765.96
255.32
7.45
Error 24
822.24
34.26
Total 31
34,482.05
Reject the null hypothesis
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

52. 10.5: Factorial Experiments
TABLE 10.13: ANOVA Summary Table for Example 10.10
Source df SS MS F
Model 7
140.35
Brand 1
32,092.11
32,093.11
936.75
Club 3
800.74
266.91
7.79
Interaction
765.96
255.32
7.45
Error 24
822.24
34.26
Total 31
34,482.05
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

53. 10.5: Factorial Experiments
TABLE 10.13: ANOVA Summary Table for Example 10.10
Source df SS MS F
Model 7
140.35
Brand 1
32,092.11
32,093.11
936.75
Club 3
800.74
266.91
7.79
Interaction
765.96
255.32
7.45
Error 24
822.24
34.26
Total 31
34,482.05
Reject the null hypothesis
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

54. 10.5: Factorial Experiments
Further analysis (see text) suggests that, although the factor “Club” clearly has an impact on distance, the results for “Brand “ are more ambiguous: Brand B hit with a 5 iron outdistances the others, but not when hit with the driver.
Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance