Analysis of Variance (ANOVA)

Lecture 26

March 30, 2017

Four Stages of Statistics

• Data Collection �

• Displaying and Summarizing Data �

• Probability �

• Inference• One Quantitative �

• One Categorical �

• One Categorical and One Quantitative• Matched Pairs Test �

• Difference of Two Means Test �

• ANOVA and Multiple Comparisons

• Two Categorical

• Two Quantitative

F-Distribution

• F-Distribution: continuous probability distribution with the following properties:

• Unimodal and right-skewed

• Always non-negative

• Two parameters for degrees of freedom• One for numerator, one for denominator

• Changes shape depending on degrees of freedom

Examples of F-Distribution

Example of F-Table Example #1: F-Table

• Question: What is the F-statistic with 6 df in the numerator, 11 df in the denominator, and 5% of the area in the upper tail?

• Answer: ____________________

_______

Motivation: ANOVA

• Scenario: Compare average math SAT scores for 5 students in three different majors: computer science, economics, and history.

• Question: Does it appear all the means are equal or is there some difference?

• Answer:• Means appear ________________

_____________________

• But…____________ exists withineach group

• Need inferential technique that can handle _______________

Analysis of Variance (ANOVA)

• Analysis of Variance (ANOVA): statistical technique used to compare the means of three or more populations

• Different from difference of two means test

• Use two sources of variability to compare means• Between group

• Within group

Types of Variation

• Between Group Variation: measures the amount of variation between the means of the individual groups

• “How different are the sample means from one another?”

• Within Group Variation: measures the amount of variation that exists in the samples

• “How different are the observations from each other within the individual samples?”

Comparing Types of Variation

Small Between Group• Means ____________________

Large Within Group• Observations within

groups ____________________

Large Between Group• Means _____________________

Small Within Group• Observations within

groups ____________________

ANOVA: Hypotheses and Conditions

• Used For:• Comparing if the means of three or more

independent groups are equal or if some difference exists between a pair of means

• Hypotheses: • ��: �� = �� = ⋯ = �• ��: At least two means are not equal.

• Conditions:• Data comes from normally distributed populations

• Variances of populations approximately equal

• Sampled observations independent

: Number of groups

being compared

Shortcut: Assume all of the conditions for ANOVA hold.

Example #2: ANOVA

• Scenario: Compare average math SAT scores for 5 students in three different majors: computer science, economics, and history.

• Question: Are the mean math SAT scores the same in all three majors using � = .05?

• Hypothesis Test:1. Hypotheses:

• ��: _____________________________________________________

• ��: _____________________________________________________

2. Conditions: _____________________________________________

ANOVA: Types of Variation

• Between Group Variation:

• Within Group Variation:

��� = �� �̅� − �̿� + �� �̅� − �̿

� +⋯+ � �̅ − �̿�

Sample

Size Sample

Mean

Grand Mean: mean of all observations

�̿ =���̅� + ���̅� +⋯+ ��̅

�� + �� +⋯+ �

��� = �� − 1 ��� + �� − 1 ��

� +⋯+ � − 1 ��

Sample

SizeSample

Variance

Example #2: ANOVA

• Scenario: Compare average math SAT scores for 5 students in three different majors: computer science, economics, and history.

• Question: Are the mean math SAT scores the same in all three majors?

• Data:Comp. Sci. Economics History

680 600 400

800 680 480

660 550 650

750 730 540

710 640 570

Example #2: ANOVA

• Scenario: Compare average math SAT scores for 5 students in three different majors: computer science, economics, and history.

• Question: Are the mean math SAT scores the same in all three majors using � = .05?

• Statistics:

Statistic Comp. Sci. Economics History

Mean 720 640 530

Std. Dev. 56.12 69.64 90.83

Sample Size 5 5 5

Example #2: ANOVA

• Grand Mean:

�̿ = ___________________________________________________

• Between Group Variation:

• Within Group Variation:

��� = ____________________________________________

= ____________________________________________

= ____________

��� = ___________________________________________

= ___________________________________________

= ____________

ANOVA: Test Statistic

• Mean Squared Treatment

��� =���

− 1• Mean Squared Error

��� =���

� −

• Test Statistic:

� =���

���

• Follows F-distribution with − 1 df in numerator and � − df in denominator

Example #2: ANOVA

• Scenario: Compare average math SAT scores for 5 students in three different majors: computer science, economics, and history.

• Question: Are the mean math SAT scores the same in all three majors using � = .05?

• Hypothesis Test:3. Test Statistic: Need:

• Degrees of freedom• Numerator: ______________________________

• Denominator: ______________________________

• Between group variation: ____________________

• Within group variation: ____________________

Example #2: ANOVA

• Scenario: Compare average math SAT scores for 5 students in three different majors: computer science, economics, and history.

• Question: Are the mean math SAT scores the same in all three majors using � = .05?

• Hypothesis Test:3. Test Statistic:

� = _________________________________________________

4. Critical Value: ____________________

ANOVA: Decision and Conclusion

• Decision: Reject �� for large values of the test statistic

• Implies between group variation (difference between means) is large relative to within group variation.

• Conclusion: Either…• No difference between any means

• At least two means differ

• Question: Are the mean math SAT scores the same in all three majors using � = .05?

• Hypothesis Test:5. Decision: _____________________ (_______________________)

6. Conclusion: ____________________________________________ __________________________________________________________

Example #2: ANOVA

Test Statistic:

_______________

Rejection

Region

Drawback of ANOVA

• When we reject ��, we only conclude “at least two means are not equal”

• Problem: Many different ways of rejecting ��• �� ≠ ��, �� = ��, �� = ��• �� = ��, �� ≠ ��, �� = ��• �� = ��, �� = ��, �� ≠ ��• �� ≠ ��, �� ≠ ��, �� = ��• �� ≠ ��, �� = ��, �� ≠ ��• �� = ��, �� ≠ ��, �� ≠ ��• �� ≠ �� ≠ ��

Looking Ahead: ________________________ (next class) will tell

us which of these scenarios is true.

_______________________ not equal

_______________________ not equal

________________ are equal

Example #3: Role of Mean

• Scenario: Boxplots could have same spreads but different sample means.

• Question: What impact does having more spread out means have on rejecting ��?

• Answer: _____________________________________________• Between group variation

______________________

• Within group variation______________________

• Test statistic ______________

Example #4: Role of Standard Deviation

• Scenario: Boxplots could have means of 30, 40, and 50 on left or right depending on std. devs.

• Question: What impact does having smaller standard deviations have on rejecting ��?

• Answer: _____________________________________________• Between group variation

______________________

• Within group variation______________________

• Test statistic ______________

Summary

• ANOVA: used to compare means of three or more independent populations

• Compare between group and within groupvariation to calculate test statistic

• Test Statistic: � =��

��!with − 1 and � − df

• Conclusion: either all means are equal or at least two means differ

• Drawback: Cannot tell which means differ (yet)