1. Chapter 13 Comparing Groups: Analysis of Variance Methods
Learn ….
How to use Statistical inference
To Compare Several Population
Means

2. How Can We Compare Several Means?
Section 13.1
How Can We Compare Several Means?
One-Way ANOVA

3. Analysis of Variance
The analysis of variance method compares means of several groups
Let g denote the number of groups
Each group has a corresponding population of subjects
The means of the response variable for the g populations are denoted by µ1, µ2, … µg

4. Hypotheses and Assumptions for the ANOVA Test
The analysis of variance is a significance test of the null hypothesis of equal population means:
H0: µ1 = µ2 = …= µg
The alternative hypothesis is:
Ha: At least two of the population means are unequal

5. Hypotheses and Assumptions for the ANOVA Test
The assumptions for the ANOVA test comparing population means are as follows:
The population distributions of the response variable for the g groups are normal with the same standard deviation for each group

6. Hypotheses and Assumptions for the ANOVA Test
Randomization:
In a survey sample, independent random samples are selected from the g populations
In an experiment, subjects are randomly assigned separately to the g groups

7. Example: How Long Will You Tolerate Being Put on Hold?
An airline has a toll-free telephone number for reservations
The airline hopes a caller remains on hold until the call is answered, so as not to lose a potential customer

8. Example: How Long Will You Tolerate Being Put on Hold?
The airline recently conducted a randomized experiment to analyze whether callers would remain on hold longer, on the average, if they heard:
An advertisement about the airline and its current promotion
Muzak (“elevator music”)
Classical music

9. Example: How Long Will You Tolerate Being Put on Hold?
The company randomly selected one out of every 1000 calls in a week
For each call, they randomly selected one of the three recorded messages
They measured the number of minutes that the caller stayed on hold before hanging up (these calls were purposely not answered)

10. Example: How Long Will You Tolerate Being Put on Hold?

11. Example: How Long Will You Tolerate Being Put on Hold?
Denote the holding time means for the populations that these three random samples represent by:
µ1 = mean for the advertisement
µ2 = mean for the Muzak
µ3 = mean for the classical music

12. Example: How Long Will You Tolerate Being Put on Hold?
The hypotheses for the ANOVA test are:
H0: µ1=µ2=µ3
Ha: at least two of the population means are different

13. Example: How Long Will You Tolerate Being Put on Hold?
Here is a display of the sample means:

14. Example: How Long Will You Tolerate Being Put on Hold?
As you can see from the graph on the previous page, the sample means are quite different
This alone is not sufficient evidence to enable us to reject H0

15. Variability between Groups and within Groups
The ANOVA method is used to compare population means
It is called analysis of variance because it uses evidence about two types of variability

16. Variability Between Groups and Within Groups
Two examples of data sets with equal means but unequal variability:

17. Variability Between Groups and Within Groups
Which case do you think gives stronger evidence against H0:µ1=µ2=µ3?
What is the difference between the data in these two cases?

18. Variability Between Groups and Within Groups
In both cases the variability between pairs of means is the same
In ‘Case b’ the variability within each sample is much smaller than in ‘Case a.’
The fact that ‘Case b’ has less variability within each sample gives stronger evidence against H0

19. ANOVA F-Test Statistic
The analysis of variance (ANOVA) F-test statistic is:
The larger the variability between groups relative to the variability within groups, the larger the F test statistic tends to be

20. ANOVA F-Test Statistic
The test statistic for comparing means has the F-distribution
The larger the F-test statistic value, the stronger the evidence against H0

21. ANOVA F-test for Comparing Population Means of Several Groups
Assumptions:
Independent random samples
Normal population distributions with equal standard deviations

22. ANOVA F-test for Comparing Population Means of Several Groups
Hypotheses:
H0:µ1=µ2= … =µg
Ha:at least two of the population means are different

23. ANOVA F-test for Comparing Population Means of Several Groups
Test statistic:
F- sampling distribution has df1= g -1, df2 = N – g = (total sample size – no. of groups)

24. ANOVA F-test for Comparing Population Means of Several Groups
P-value: Right-tail probability above the observed F- value
Conclusion: If decision is needed, reject if P-value ≤ significance level (such as 0.05)

25. The Variance Estimates and the ANOVA Table
Let σ denote the standard deviation for each of the g population distributions
One assumption for the ANOVA F-test is that each population has the same standard deviation, σ

26. The Variance Estimates and the ANOVA Table
The F-test statistic is the ratio of two estimates of σ2, the population variance for each group
The estimate of σ2 in the denominator of the F-test statistic uses the variability within each group
The estimate of σ2 in the numerator of the F-test statistic uses the variability between each sample mean and the overall mean for all the data

27. The Variance Estimates and the ANOVA Table
Computer software displays the two estimates of σ2 in the ANOVA table
The MS column contains the two estimates, which are called mean squares
The ratio of the two mean squares is the F- test statistic
This F- statistic has a P-value

28. Example: ANOVA for Customers’ Telephone Holding Times
This example is a continuation of a previous example in which an airline conducted a randomized experiment to analyze whether callers would remain on hold longer, on the average, if they heard:
An advertisement about the airline and its current promotion
Muzak (“elevator music”)
Classical music

29. Example: ANOVA for Customers’ Telephone Holding Times
Denote the holding time means for the populations that these three random samples represent by:
µ1 = mean for the advertisement
µ2 = mean for the Muzak
µ3 = mean for the classical music

30. Example: ANOVA for Customers’ Telephone Holding Times
The hypotheses for the ANOVA test are:
H0:µ1=µ2=µ3
Ha:at least two of the population means are different

31. Example: ANOVA for Customers’ Telephone Holding Times

32. Example: ANOVA for Customers’ Telephone Holding Times
Since P-value < 0.05, there is sufficient evidence to reject H0:µ1=µ2=µ3
We conclude that a difference exists among the three types of messages in the population mean amount of time that customers are willing to remain on hold

33. Assumptions for the ANOVA F-Test and the Effects of Violating Them
Population distributions are normal
Moderate violations of the normality assumption are not serious
These distributions have the same standard deviation, σ
Moderate violations are also not serious

34. Assumptions for the ANOVA F-Test and the Effects of Violating Them
You can construct box plots or dot plots for the sample data distributions to check for extreme violations of normality
Misleading results may occur with the F-test if the distributions are highly skewed and the sample size N is small

35. Assumptions for the ANOVA F-Test and the Effects of Violating Them
Misleading results may also occur with the F-test if there are relatively large differences among the standard deviations (the largest sample standard deviation being more than double the smallest one)

36. State the null hypothesis
The 1998 General Social Survey asked subjects how many friend they have. Is this associated with the respondent’s astrological sign (the 13 symbols of the Zodiac)? The ANOVA table for the data reports F=0.61
State the null hypothesis
The population means for all 12 Zodiac signs are the same.
At least two population means are different.

37. State the alternative hypothesis
The 1998 General Social Survey asked subjects how many friend they have. Is this associated with the respondent’s astrological sign (the 13 symbols of the Zodiac)? The ANOVA table for the data reports F=0.61
State the alternative hypothesis
The population means for all 12 Zodiac signs are the same.
At least two population means are different.

38. The 1998 General Social Survey asked subjects how many friend they have. Is this associated with the respondent’s astrological sign (the 13 symbols of the Zodiac)? The ANOVA table for the data reports F=0.61
Based on what you know about the F distribution would you guess that the test value of 0.61 provides strong evidence against the null hypothesis? No
Yes

39. The 1998 General Social Survey asked subjects how many friend they have. Is this associated with the respondent’s astrological sign (the 13 symbols of the Zodiac)? The ANOVA table for the data reports F=0.61
The P-value associated with the F-statistic is At a significance level of 0.05, what is the correct decision?
Reject Ho
Fail to Reject Ho
Reject Ha
Fail to Reject Ha

40. How Should We Follow Up an ANOVA F-Test?
Section 13.2
How Should We Follow Up an ANOVA F-Test?

41. Follow Up to an ANOVA F-Test
When an analysis of variance F-test has a small P-value, the test does not specify which means are different or how different they are
We can estimate differences between population means with confidence intervals

42. Confidence Intervals Comparing Pairs of Means
For two groups i and j, with sample means yi and yj having sample sizes ni and nj, the 95% confidence interval for µi - µj is:
The t-score has df = N – g

43. Confidence Intervals Comparing Pairs of Means
Some software refers to this confidence method as the Fisher method
When the confidence interval does not contain 0, we can infer that the population means are different
The interval shows just how different the means may be

44. Example: Number of Good Friends and Happiness
A recent GSS study asked: “About how many good friends do you have?”
The study also asked each respondent to indicate whether they were ‘very happy,’ ‘pretty happy,’ or ‘not too happy’

45. Example: Number of Good Friends and Happiness
Let the response variable y = number of good friends
Let the categorical explanatory variable x = happiness level

46. Example: Number of Good Friends and Happiness

47. Example: Number of Good Friends and Happiness
Construct a 95% CI to compare the population mean number of good friends for the two categories: ‘very happy’ and ‘pretty happy.’
95% CI formula:

48. Example: Number of Good Friends and Happiness
First, use the output to find s:
Use software or a table to find the t-value of 1.963

49. Example: Number of Good Friends and Happiness
Next calculate the interval:
Since the CI contains only positive numbers, this suggest that, on average, people who are very happy have more good friends than people who are pretty happy

50. Controlling Overall Confidence with Many Confidence Intervals
The confidence interval method just discussed is mainly used when g is small or when only a few comparisons are of main interest
The confidence level of 0.95 applies to any particular confidence interval that we construct

51. Controlling Overall Confidence with Many Confidence Intervals
How can we construct the intervals so that the 95% confidence extends to the entire set of intervals rather than to each single interval?
Methods that control the probability that all confidence intervals will contain the true differences in means are called multiple comparison methods

52. Controlling Overall Confidence with Many Confidence Intervals
The method that we will use is called the Tukey method
It is designed to give overall confidence level very close to the desired value (such as 0.95)
This method is available in most software packages

53. Example: Multiple Comparison Intervals for Number of Good Friends

54. What if There Are Two Factors?
Section 13.3
What if There Are Two Factors?
Two-Way ANOVA

55. ANOVA One-way ANOVA is a bivariate method:
It has a quantitative response variable
It has one categorical explanatory variable
Two-way ANOVA is a multivariate method:
It has two categorical explanatory variables

56. A recent study at Iowa State University:
Example: How Does Corn Yield Depend on Amounts of Fertilizer and Manure?
A recent study at Iowa State University:
A field was portioned into 20 equal-size plots
Each plot was planted with the same amount of corn seed
The goal was to study how the yield of corn later harvested depended on the levels of use of nitrogen-based fertilizer and manure
Each factor (fertilizer and manure) was measured in a binary manner

57. What are the four treatments you can compare with this experiment?
Example: How Does Corn Yield Depend on Amounts of Fertilizer and Manure?
What are the four treatments you can compare with this experiment?

58. Inference about Effects in Two-Way ANOVA
In two-way ANOVA, a null hypothesis states that the population means are the same in each category of one factor, at each fixed level of the other factor

59. Example: How Does Corn Yield Depend on Amounts of Fertilizer and Manure?
We could test:
H0: Mean corn yield is equal for plots at the low and high levels of fertilizer, for each fixed level of manure

60. Example: How Does Corn Yield Depend on Amounts of Fertilizer and Manure?
We could also test:
H0: Mean corn yield is equal for plots at the low and high levels of manure, for each fixed level of fertilizer

61. Example: How Does Corn Yield Depend on Amounts of Fertilizer and Manure?
The effect of individual factors tested with the two null hypotheses (the previous two pages) are called main effects

62. Assumptions for the Two-way ANOVA F-test
The population distribution for each group is normal
The population standard deviations are identical
The data result from a random sample or randomized experiment

63. F-test Statistics in Two-way ANOVA
For testing the main effect for a factor, the test statistic is the ratio of mean squares:

64. F-test Statistics in Two-way ANOVA
When the null hypothesis of equal population means for the factor is true, the F-test statistic values tend to fluctuate around 1
When it is false, they tend to be larger
The P-value is the right-tail probability above the observed F-value

65. Example: Testing the Main Effects for Corn Yield
Data and sample statistics for each group:

66. Example: Testing the Main Effects for Corn Yield
Output from Two-way ANOVA:

67. Example: Testing the Main Effects for Corn Yield
First consider the hypothesis:
H0: Mean corn yield is equal for plots at the low and high levels of fertilizer, for each fixed level of manure

68. Example: Testing the Main Effects for Corn Yield
From the output, you can obtain the F-test statistic of 6.33 with its corresponding P-value of 0.022
The small P-value indicates strong evidence that the mean corn yield depends on fertilizer level

69. Example: Testing the Main Effects for Corn Yield
Next consider the hypothesis:
H0: Mean corn yield is equal for plots at the low and high levels of manure, for each fixed level of fertilizer

70. Example: Testing the Main Effects for Corn Yield
From the output, you can obtain the F-test statistic of 6.88 with its corresponding P-value of 0.018
The small P-value indicates strong evidence that the mean corn yield depends on manure level

71. Exploring Interaction between Factors in Two-Way ANOVA
No interaction between two factors means that the effect of either factor on the response variable is the same at each category of the other factor

72. Exploring Interaction between Factors in Two-Way ANOVA

73. Exploring Interaction between Factors in Two-Way ANOVA
A graph showing interaction:

74. Testing for Interaction
In conducting a two-way ANOVA, before testing the main effects, it is customary to test a third null hypothesis stating that their is no interaction between the factors in their effects on the response

75. Testing for Interaction
The test statistic providing the sample evidence of interaction is:
When H0 is false, the F-statistic tends to be large

76. Example: Testing for Interaction with Corn Yield Data
ANOVA table for a model that allows interaction:

77. Example: Testing for Interaction with Corn Yield Data
The test statistic for H0: no interaction is
F = 1.10 with a corresponding P-value
of 0.311
There is not much evidence of interaction
We would not reject H0 at the usual significance levels, such as 0.05

78. Check Interaction before Main Effects
In practice, in two-way ANOVA, you should first test the hypothesis of no interaction

79. Check Interaction before Main Effects
If the evidence of interaction is not strong (that is, if the P-value is not small), then test the main effects hypotheses and/or construct confidence intervals for those effects

80. Check Interaction before Main Effects
If important evidence of interaction exists, plot and compare the cell means for a factor separately at each category of the other factor

81. An experiment randomly assigns 100 subjects suffering from high cholesterol to one of four groups: low-dose Lipitor, high-dose Lipitor, low-dose Pravachol and high-dose Pravachol. After three months of treatment, the change in cholesterol level is measured.
What is the response variable?
Cholesterol level
Drug dosage
Drug type

82. An experiment randomly assigns 100 subjects suffering from high cholesterol to one of four groups: low-dose Lipitor, high-dose Lipitor, low-dose Pravachol and high-dose Pravachol. After three months of treatment, the change in cholesterol level is measured.
What are the factors?
Cholesterol level and drug type
Drug dosage and cholesterol level
Drug type and drug dosage