1. Keller: Stats for Mgmt & Econ, 7th Ed Analysis of Variance
May 8, 2018
Chapter 15
Analysis of Variance
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

2. Analysis of Variance…
Analysis of variance is a technique that allows us to compare two or more populations of interval data.
Analysis of variance is:
 an extremely powerful and widely used procedure.
 a procedure which determines whether differences exist between population means.
 a procedure which works by analyzing sample variance.

3. One-Way Analysis of Variance…
Independent samples are drawn from k populations:
Note: These populations are referred to as treatments.
It is not a requirement that n1 = n2 = … = nk.

4. One Way Analysis of Variance…
New Terminology:
x is the response variable, and its values are responses.
xij refers to the ith observation in the jth sample.
E.g. x35 is the third observation of the fifth sample.
The grand mean, , is the mean of all the observations, i.e.:
(n = n1 + n2 + … + nk)

5. One Way Analysis of Variance…
More New Terminology:
The unit that we measure is the experimental unit.
Population classification criterion is called a factor.
Each population is a factor level.

6. Example 15-1… An apple juice company has a new product featuring…
more convenience,
similar or better quality, and
lower price
when compared with existing juice products.
Which factor should an advertising campaign focus on?
Before going national, test markets are set-up in three cities, each with its own campaign, and data is recorded…
Do differences is sales exist between the test markets?

7. comma added for clarity
Example 15.1…
Terminology
x is the response variable, and its values are responses.
weekly sales is the response variable;
the actual sales figures are the responses in this example.
xij refers to the ith observation in the jth sample.
E.g. x42 is the fourth week’s sales in city #2: 717 pkgs.
x20, 3 is the last week of sales for city #3: 532 pkgs.
comma added for clarity

8. Example 15.1… The unit that we measure is the experimental unit.
Terminology
The unit that we measure is the experimental unit.
Weeks in the three cities when we recorded sales.
Population classification criterion is called a factor.
The advertising strategy is the factor we’re interested in. This is the only factor under consideration (hence the term “one way” analysis of variance).
Each population is a factor level.
In this example, there are three factor levels: convenience, quality, and price.

9. Example 15.1… The null hypothesis in this case is: H0:
IDENTIFY
The null hypothesis in this case is:
H0:
i.e. there are no differences between population means.
Our alternative hypothesis becomes:
H1: at least two means differ
OK. Now we need some test statistics…

10. sum across k treatments
Test Statistics…
Since is of interest to us, a statistic that measures the proximity of the sample means to each other would also be of interest.
Such a statistic exists, and is called the between-treatments variation. It is denoted SST, short for “sum of squares for treatments”. Its is calculated as:
grand mean
sum across k treatments
A large SST indicates large variation between sample means which supports H1.

11. Test Statistics…
SST gave us the between-treatments variation. A second statistic, SSE (Sum of Squares for Error) measures the within-treatments variation.
SSE is given by: or:
In the second formulation, it is easier to see that it provides a measure of the amount of variation we can expect from the random variable we’ve observed.

12. Example 15.1… Since: If it were the case that:
COMPUTE
Since:
If it were the case that:
then SST = 0 and our null hypothesis, H0:
would be supported.
More generally, a “small value” of SST supports the null hypothesis. The question is, how small is “small enough”?

13. Example 15.1…
COMPUTE
The following sample statistics and grand mean were computed…
Hence, the between-treatments variation, sum of squares for treatments, is:
Is SST = 57, “large enough” to indicate the population means differ?

14. Example 15.1… We calculate the sample variances as:
COMPUTE
We calculate the sample variances as:
and from these, calculate the within-treatments variation (sum of squares for error) as:
We still need a couple more quantities in order to relate SST and SSE together in a meaningful way…

15. Mean Squares… The mean square for treatments (MST) is given by:
The mean square for errors (MSE) is given by:
And the test statistic:
is F-distributed with k–1 and n–k degrees of freedom.
Aha! We must be close…

16. Example 15.1…
COMPUTE
We can calculate the mean squares treatment and mean squares error quantities as:
Giving us our F-statistic of:
Does F = 3.23 fall into a rejection region or not? How does it compare to a critical value of F?
Note these required conditions:
1. The populations tested are normally distributed.
2. The variances of all the populations tested are equal.

17. Example 15.1…
INTERPRET
Since the purpose of calculating the F-statistic is to determine whether the value of SST is large enough to reject the null hypothesis, if SST is large, F will be large.
Hence our rejection region is:
Our value for FCritical is:

18. Example 15.1…
INTERPRET
Since F = 3.23 is greater than FCritical = 3.15, we reject the null hypothesis (H0: ) in favor of the alternative hypothesis (H1: at least two population means differ).
That is: there is enough evidence to infer that the mean weekly sales differ between the three cities.
Stated another way: we are quite confident that the strategy used to advertise the product will produce different sales figures.

19. Summary of Techniques (so far)…

20. ANOVA Table…
The results of analysis of variance are usually reported in an ANOVA table…
Source of
Variation
degrees of
freedom
Sum of Squares
Mean Square
Treatments
k–1
SST
MST=SST/(k–1)
Error
n–k
SSE
MSE=SSE/(n–k)
Total
n–1
SS(Total)
F-stat=MST/MSE

21. Example 15.1… Using Excel: Tools, Data Analysis…, Anova: Single Factor
COMPUTE
Using Excel:
Tools, Data Analysis…, Anova: Single Factor
We produce the following output…
compare
“SST”
“SSE”
p-value vs. 0.05…

22. Identifying Factors…
Factors that Identify the One-Way Analysis of Variance:

23. Analysis of Variance Experimental Designs
Experimental design is one of the factors that determines which technique we use.
In the previous example we compared three populations on the basis of one factor – advertising strategy.
One-way analysis of variance is only one of many different experimental designs of the analysis of variance.

24. Analysis of Variance Experimental Designs
A multifactor experiment is one where there are two or more factors that define the treatments.
For example, if instead of just varying the advertising strategy for our new apple juice product we also varied the advertising medium (e.g. television or newspaper), then we have a two-factor analysis of variance situation.
The first factor, advertising strategy, still has three levels (convenience, quality, and price) while the second factor, advertising medium, has two levels (TV or print).

25. Independent Samples and Blocks
Similar to the ‘matched pairs experiment’, a randomized block design experiment reduces the variation within the samples, making it easier to detect differences between populations.
The term block refers to a matched group of observations from each population.
We can also perform a blocked experiment by using the same subject for each treatment in a “repeated measures” experiment.

26. Independent Samples and Blocks
The randomized block experiment is also called the two-way analysis of variance, not to be confused with the two-factor analysis of variance. To illustrate where we’re headed…
we’ll
do this
first

27. Randomized Block Analysis of Variance
The purpose of designing a randomized block experiment is to reduce the within-treatments variation to more easily detect differences between the treatment means.
In this design, we partition the total variation into three sources of variation:
SS(Total) = SST + SSB + SSE
where SSB, the sum of squares for blocks, measures the variation between the blocks.

28. Randomized Blocks…
In addition to k treatments, we introduce notation for b blocks in our experimental design…
mean of the observations of the 1st treatment
mean of the observations of the 2nd treatment

29. Sum of Squares : Randomized Block…
Squaring the ‘distance’ from the grand mean, leads to the following set of formulae…
test statistic for treatments
test statistic for blocks

30. ANOVA Table…
We can summarize this new information in an analysis of variance (ANOVA) table for the randomized block analysis of variance as follows…
Source of Variation
d.f.:
Sum of Squares
Mean Square
F Statistic
Treatments
k–1
SST
MST=SST/(k–1)
F=MST/MSE
Blocks
b–1
SSB
MSB=SSB/(b-1)
F=MSB/MSE
Error
n–k–b+1
SSE
MSE=SSE/(n–k–b+1)
Total
n–1
SS(Total)

31. Test Statistics & Rejection Regions…

32. Example 15.2…
IDENTIFY
Are there difference in the effectiveness of four new cholesterol drugs? 25 groups of men were matched according to age & weight, and the results were recorded.
The hypotheses to test in this case are:
H0:
H1: At least two means differ

33. Example 15.2… Each of the four drugs can be considered a treatment.
IDENTIFY
Each of the four drugs can be considered a treatment.
Each group) can be blocked, because they are matched by age and weight.
By setting up the experiment this way, we eliminates the variability in cholesterol reduction related to different combinations of age and weight. This helps detect differences in the mean cholesterol reduction attributed to the different drugs.

34. Example 15.2… The Data : : : : : Treatment Block
: : : : :
Block
There are b = 25 blocks, and
k = 4 treatments in this example.

35. Example 15.2… We obtain the output from Excel… COMPUTE
Tools > Data Analysis > Anova: Two Factor Without Replication
a.k.a. Randomized Block
b-1
k-1
Rows  Blocks
MSB
MST
Columns  Treatments

36. Example 15.2…
INTERPRET
The F-statistic to determine whether differences exist between the four drugs (treatments; columns) is 4.12 and is greater than FCritical=1.67. Its p-value is
Thus we reject H0 in favor of the research hypothesis: at least two means differ (i.e. there are differences between the treatments).

37. Example 15.2…
INTERPRET
The other F-statistic, (p-value = 0; also greater than FCritical=4.12) indicates that there are differences between the groups of men (blocks; rows), that is: age & weight have an impact, but our experiment design accounts for that.

38. Identifying Factors…
Factors that Identify the Randomized Block of the Analysis of Variance:

39. Two-Factor Analysis of Variance…
The original set-up for Example 15.1 examined one factor, namely the effects of the marketing strategy on sales.
Emphasis on convenience,
Emphasis on quality, or
Emphasis on price.
Suppose we introduce a second factor, that being the effects of the selected media on sales, that is:
Advertise on television, or
Advertise in newspapers.
To which factor(s) or the interaction of factors can we attribute any differences in mean sales of apple juice?

40. More Terminology…
A complete factorial experiment is an experiment in which the data for all possible combinations of the levels of the factors are gathered. This is also known as a two-way classification.
The two factors are usually labeled A & B, with the number of levels of each factor denoted by a & b respectively.
The number of observations for each combination is called a replicate, and is denoted by r. For our purposes, the number of replicates will be the same for each treatment, that is they are balanced.

41. Example 15.3… The Data Factor “A” • Strategy Factor “B” Medium
There are a = 3 levels of factor A, b = 2 levels of factor B, yielding 3 x 2 = 6 replicates, each replicate has r = 10 observations…

42. Possible Outcomes…
Fig. 15.5
This figure illustrates the case where there are differences between levels of A, but no difference between the levels of B and no interaction between A & B: 1 2 3
Levels 1 and 2 of factor B
Levels of factor A
Mean response

43. Possible Outcomes…
Fig. 15.6
This figure illustrates the case where there are differences between levels of B, but no differences between the levels of A and no interaction between A & B: 1 2 3
Level 2 of factor B
Levels of factor A
Mean response
Level 1 of factor B

44. Possible Outcomes…
Fig. 15.4
This figure illustrates the case where there are differences between levels of A, and there are differences between the levels of B, but and no interaction between A & B:
(i.e. the factors affect sales independently, which means there is no interaction) 1 2 3
Level 1 of factor B
Levels of factor A
Mean response
Level 2 of factor B

45. Possible Outcomes… This figure shows the levels of A & B interacting:1 2 3
Level 1 of factor B
Levels of factor A
Mean response
Level 2 of factor B

46. ANOVA Table… Table 15.8 Source of Variation d.f.: Sum of Squares
Mean Square
F Statistic
Factor A
a-1
SS(A)
MS(A)=SS(A)/(a-1)
F=MS(A)/MSE
Factor B
b–1
SS(B)
MS(B)=SS(B)/(b-1)
F=MS(B)/MSE
Interaction
(a-1)(b-1)
SS(AB)
MS(AB) = SS(AB)
[(a-1)(b-1)]
F=MS(AB)/MSE
Error
n–ab
SSE
MSE=SSE/(n–ab)
Total
n–1
SS(Total)

47. Two Factor ANOVA…
Test for the differences between the Levels of Factor A…
H0: The means of the a levels of Factor A are equal
H1: At least two means differ
Test statistic: F = MS(A) / MSE
Example 15.3: Are there differences in the mean sales
caused by different marketing strategies?
H0:

48. Two Factor ANOVA…
Test for the differences between the Levels of Factor B…
H0: The means of the a levels of Factor B are equal
H1: At least two means differ
Test statistic: F = MS(B) / MSE
Example 15.3: Are there differences in the mean sales
caused by different advertising media?
H0:

49. Two Factor ANOVA… Test for interaction between Factors A and B…
H0: Factors A and B do not interact to affect the mean responses.
H1: Factors A and B do interact to affect the mean responses.
Test statistic: F = MS(AB) / MSE
Example 15.3: Are there differences in the mean sales caused by interaction between marketing strategy and advertising medium??
H0:
H1: At least two means differ

50. Example 15.3… Using the data, we use Tools > Data Analysis… >
COMPUTE
Using the data, we use Tools > Data Analysis… >
Anova: Two-Factor With Replication and get:
Factor B - Media
Factor A - Mktg Strategy
Interaction of A&B
Error

51. Example 15.3…
INTERPRET
There is evidence at the 5% significance level to infer that differences in weekly sales exist between the different marketing strategies (Factor A).

52. Example 15.3…
INTERPRET
There is insufficient evidence at the 5% significance level to infer that differences in weekly sales exist between television and newspaper advertising (Factor B).

53. Example 15.3…
INTERPRET
There is not enough evidence to conclude that there is an interaction between marketing strategy and advertising medium that affects mean weekly sales (interaction of Factor A & Factor B).

54. See for yourself… Plot this data…
…as a line chart to see the effects of Factors A & B…
Television
Newspaper

55. See for yourself…
There are differences between the levels of factor A, no difference between the levels of factor B, and no interaction is apparent.

56. See for yourself…
INTERPRET
These results indicate that emphasizing quality produces the highest sales and that television and newspapers are equally effective.

57. Identifying Factors…
Independent Samples Two-Factor Analysis of Variance…

58. Summary of ANOVA… two-factor analysis of variance
one-way analysis of variance
two-way analysis of variance
a.k.a. randomized blocks

59. Multiple Comparisons…
When we conclude from the one-way analysis of variance that at least two treatment means differ (i.e. we reject the null hypothesis that H0: ), we often need to know which treatment means are responsible for these differences.
We will examine three statistical inference procedures that allow us to determine which population means differ:
• Fisher’s least significant difference (LSD) method
• Bonferroni adjustment, and
• Tukey’s multiple comparison method.

60. Multiple Comparisons…
Two means are considered different if the difference between the corresponding sample means is larger than a critical number. The general case for this is, IF
THEN we conclude and differ.
The larger sample mean is then believed to be associated with a larger population mean.

61. Fisher’s Least Significant Difference…
What is this critical number, NCritical ?
One measure is the Least Significant Difference, given by:
LSD will be the same for all pairs of means if all k sample sizes are equal. If some sample sizes differ, LSD must be calculated for each combination.

62. Back to Example 15.1…
With k=3 treatments (marketing strategy based on convenience, quality, or price), we will perform three comparisons based on the sample means:
We compare these to the Least Significant Difference we calculate as (at 5% significance):

63. Example 15.1 • Fisher’s LSD
We can compute the difference of means manually or use Excel: Tools > Data Analysis Plus > Multiple Comparisons
we conclude that only the means
for convenience and quality differ

64. Bonferroni Adjustment to LSD Method…
Fisher’s method may result in an increased probability of committing a type I error.
We can adjust Fisher’s LSD calculation by using the “Bonferroni adjustment”.
Where we used alpha ( ), say .05, previously, we now use and adjusted value for alpha:
where

65. Example 15.1 • Bonferroni’s Adjustment
Since we have k=3 treatments, C=k(k–1)/2=3(2)/2=3, hence we set our new alpha value to:
Thus, instead of using t.05/2 in our LSD calculation, we are going to use t.0167/2
Use Excel’s TINV()
function to calculate
t.0167/2=2.467

66. Example 15.1 • Bonferroni’s Adjustment
Again, we can use the Multiple Comparisons tool to do the number crunching for us:
Similar result as before…

67. Tukey’s Multiple Comparison Method…
As before, we are looking for a critical number to compare the differences of the sample means against. In this case:
Note: is a lower case Omega, not a “w”
Critical value of the Studentized range
with n–k degrees of freedom
Table 7 - Appendix B
harmonic mean of the sample sizes

68. Example 15.1 • Tukey’s Method…
Again, we can select a pair of means, manually calculate the difference between the larger and smaller mean, compare this difference to our calculated value of omega ( ), and so on for each pair of sample means. Or use Excel:
Similar result as before…
Compare

69. Which method to use?
In example 15.1, all three multiple comparison methods yielded the same results. This will not always be the case! Generally speaking…
If you have identified two or three pairwise comparisons that you wish to make before conducting the analysis of variance, use the Bonferroni method.
If you plan to compare all possible combinations, use Tukey’s comparison method.