1. Understanding and conducting
ANOVA
Understanding and conducting

2. Basics of ANOVA Null hypothesis:
All groups are coming from the same population
Differences in means are due to normal sampling fluctuations between samples drawn from the same population
If null hypothesis is true, then certain other things must be true.
 Test if that is the case (F-test)
If yes  accept null hypothesis
If no  reject null hypothesis

3. Example
Students were grouped into low-medium and high levels of motivation.
They were asked number of hours per week they spent doing homework.
The question is, whether low-medium-high motivated students spend
time doing homework that significantly differs.

4. Example
Null and Alternative Hypotheses

5. Example
Means of each group

6. Example
Under the null hypothesis, all of these are coming from the same population
Variance in means

7. Example
Between groups estimate of variance

8. Example
Within groups estimate of variance

9. F-Statistic (Fisher’s test, 1923)
Logic of the F statistic:
The two estimates would be approximately equal if samples were drawn randomly from a single population.
If the ratio of “between” samples variance to “within” samples variance is taken, it should be approximately 1.
Depending on sample size
Depending on variation of sample means among all possible samples
If the ratio of “between” samples variance is much higher than “within” samples variance, then sample means vary more than expected by chance.
This is evidence that the independent variable is associated with significant differentiation of means.

10. Understanding the F-test
Between groups
variance
Within groups
Variance due to
membership of
groups
True variance
between subjects
measurement
error + =

11. Assumptions of the F-test
Interval/ratio level measurement of the dependent variable (robust against deviations from normality)
Independence of observations (each unit of analysis is independently selected into the sample)

12. Which one is NOT a null hypothesis for ANOVA comparing the political conservatism of low, middle and high SES individuals?
Low, middle, and high ability groups do not differ from each other.
Low, middle, and high ability groups are just like simple random samples drawn from the same population.
Low, middle and high ability groups differ from each other in their political conservatism.
SES of individuals is not associated with their political conservatism.

13. How is an F test differ from a t test?
T/F - The two tests are different because their null hypotheses are different.

14. How is an F test differ from a t test?
T/F - The two tests are different because the t test compares 2 group means and the F test can compare multiple group means.

15. How is an F test differ from a t test?
T/F - The two tests differ in the way we interpret the p values.

16. How is an F test differ from a t test?
T/F - The two tests differ in the way we reason when we make inferences from a sample to a population.

17. The F test works by comparing two estimates
The F test works by comparing two estimates. Those two things are the estimates of WHAT?
Population means
Population variances
Sample means
Sample standard deviations

18. There are two different estimates of the population variance in an F test. One is called the WITHIN GROUPS estimate of the population variance. Which statement is not true about the WITHIN GROUPS variance?
WG variance is given that name because it is obtained by averaging person-to-person variance estimates for each group.
WG variance is the variance that quantifies the person-to-person variability of all persons in all groups combined.
WG variance is compared to the “between groups” estimate of the population variance for an F test.
WG variance is just a summary estimate of what the population variance is, based on each one of the sample variances.

19. One of the estimates of the population variance in an F test is called the BETWEEN GROUPS estimate of the population variance. Which statement is not true about the BETWEEN GROUPS variance?
BG variance is given that name because it is obtained by focusing on the variability of group means.
BG variance is the variance estimate that uses the SEM formula relating the population variance to the sample-to-sample variability.
BG variance is compared to the “within groups” estimate of the population variance for an F test.
BG variance is the estimate of the variance under the assumption that the alternative hypothesis that the groups are coming from different populations.

20. When will the F test be equal to 1?
When the differences between the sample means are large enough.
When the alternative hypothesis is true.
When the error variance is too large.
When the null hypothesis is true.

21. Thinking exercise
Spider web on ANOVA

22. Multifactorial Designs: Beginning Multivariate Analysis
Read G&W

23. Excel file – Multifactorial ANOVA introduction
More than one IW may influence a DV
Sex
Relationship Satisfaction
Family Type

24. Interaction effects There can be many different types of interactions
Interactions must be interpreted in the light of theory
They are symmetric mathematically
Main effects cannot be interpreted if there are interaction effects in the model
Main effects are “marginal” to the interaction effects

25. Satisfaction is influenced by family background only in one group
Intact
Disrupted

26. Excel file – Multifactorial ANOVA introduction
More than one IW may influence a DV
Sex
Relationship Satisfaction
Family Type

27. Satisfaction is influenced by family background only in one group
Intact
Disrupted

28. Excel file – Multifactorial ANOVA introduction
More than one IW may influence a DV
Sex
Relationship Satisfaction
Family Type

29. Satisfaction is influenced by family background only in one group
Intact
Disrupted

30. Excel file – Multifactorial ANOVA introduction
More than one IW may influence a DV
Sex
Relationship Satisfaction
Family Type

31. Satisfaction is influenced by family background only in one group
Intact
Disrupted

32. SPSS output: Family type effects

33. SPSS output: Sex effects

34. SPSS output: Family type and sex effects at the same time

35. SPSS output: Family type and sex effects at the same time and interdependent

36. SPSS output: Family type and sex effects at the same time and interdependent

37. Understanding the F statistics: Equal sample sizes
Factor B
… m 1 2 k n
Factor A

38. Understanding the F statistics: Equal sample sizes
Variance of the attribute
in the subjects in each group
Factor B
… m 1 2 k n
Factor A

39. Understanding the F statistics: Equal sample sizes
Factor B
… m 1 2 k n
Factor A

40. Understanding the F statistics: Equal sample sizes
Factor B
… m 1 2 k n
Factor A

41. Understanding the F statistics: Equal sample sizes
Factor B
… m 1 2 k n
Factor A

42. Reviewing multifactor F statistics
Within group variance
Factor B
… m 1 2 k n
Between groups variance of Factor A
Factor A
Between groups variance of Factors A*B
Remember: This subtracts Factor A and Factor B marginal variances
Between groups variance of Factor B

43. Language of Multifactorial ANOVA
The present research has a m*n design.
Main (marginal) effects of Factor A were significant (not significant).
Main (marginal) effects of Factor B were significant (not significant).
Interaction effects of Factor A by Factor B were significant (not significant).

44. READ AND STUDY: Example 14.2, page 429