1. One-way Analysis of Variance
Chapter 16
One-way Analysis of Variance
Fundamental Statistics for the Behavioral Sciences, 4th edition
David C. Howell
©1999 Brooks/Cole Publishing Company/ITP   

2. Chapter 16 One-way Analysis of Variance
Major Points
An example
The logic
Calculations
Unequal sample sizes
Multiple comparisons
Fisher’s LSD
Bonferroni t
Cont.

3. Chapter 16 One-way Analysis of Variance
Major Points--cont.
Assumptions of analysis of variance
Magnitude of effect
eta squared
omega squared
Review questions

4. Logic of the Analysis of Variance
Chapter 16 One-way Analysis of Variance
Logic of the Analysis of Variance
Null hypothesis: H0 Population means equal
m1 = m2 = m3 = m4
Alternative hypothesis: H1
Not all population means equal.
Cont.

5. Chapter 16 One-way Analysis of Variance
Logic--cont.
Create a measure of variability among group means
MSgroups
Create a measure of variability within groups
MSerror
Cont.

6. Chapter 16 One-way Analysis of Variance
Logic--cont.
Form ratio of MSgroups /MSerror
Ratio approximately 1 if null true
Ratio significantly larger than 1 if null false
“approximately 1” can actually be as high as 2 or 3, but not much higher

7. Epinephrine and Memory
Chapter 16 One-way Analysis of Variance
Epinephrine and Memory
Based on Introini-Collison & McGaugh (1986)
Trained mice to go left on Y maze
Injected with 0, .1, .3, or 1.0 mg/kg epinephrine
Next day trained to go right in same Y maze
dep. Var. = # trials to learn reversal
More trials indicates better retention of Day 1
Reflects epinephrine’s effect on memory
Introini-Collison, I.B., & Mcgaugh, J.L. (1986) Epinephrine modulates long-term retention of an aversively motivated discrimination. Behavioral and Neural Biology, 45,

8. Chapter 16 One-way Analysis of Variance
Grand mean = 3.78

9. Chapter 16 One-way Analysis of Variance
Calculations
Start with Sum of Squares (SS)
We need:
SStotal
SSgroups
SSerror
Compute degrees of freedom (df )
Compute mean squares and F
Cont.

10. Chapter 16 One-way Analysis of Variance
Calculations--cont.

11. Degrees of Freedom (df )
Chapter 16 One-way Analysis of Variance
Degrees of Freedom (df )
Number of “observations” free to vary
dftotal = N - 1
Variability of N observations
dfgroups = g - 1
variability of g means
dferror = g (n - 1)
n observations in each group = n - 1 df
times g groups

12. Chapter 16 One-way Analysis of Variance
Summary Table

13. Chapter 16 One-way Analysis of Variance
Conclusions
The F for groups is significant.
We would obtain an F of this size, when H0 true, less than one time out of 1000.
The difference in group means cannot be explained by random error.
The number of trials to learn reversal depends on level of epinephrine.
Cont.

14. Chapter 16 One-way Analysis of Variance
Conclusions--cont.
The injection of epinephrine following learning appears to consolidate that learning.
High doses may have negative effect.

15. Chapter 16 One-way Analysis of Variance
Unequal Sample Sizes
With one-way, no particular problem
Multiply mean deviations by appropriate ni as you go
The problem is more complex with more complex designs, as shown in next chapter.
Example from Foa, Rothbaum, Riggs, & Murdock (1991)
Foa, E.B., Rothbaum, B.O., Riggs, D.S., & Murdock, T.B. (1991). Treatment of posttraumatic stress disorder in rape victims: A comparison between cognitive-behavioral procedures and counseling. Journal of Consulting and Clinical Psychology, 59,

16. Post-Traumatic Stress Disorder
Chapter 16 One-way Analysis of Variance
Post-Traumatic Stress Disorder
Four treatment groups given psychotherapy
Stress Inoculation Therapy (SIT)
Standard techniques for handling stress
Prolonged exposure (PE)
Reviewed the event repeatedly in their mind
Cont.

17. Post-Traumatic Stress Disorder--cont.
Chapter 16 One-way Analysis of Variance
Post-Traumatic Stress Disorder--cont.
Supportive counseling (SC)
Standard counseling
Waiting List Control (WL)
No treatment

18. Chapter 16 One-way Analysis of Variance
SIT = Stress Inoculation Therapy
PE = Prolonged Exposure
SC = Supportive Counseling
WL = Waiting List Control
Grand mean =

19. Tentative Conclusions
Chapter 16 One-way Analysis of Variance
Tentative Conclusions
Fewer symptoms with SIT and PE than with other two
Also considerable variability within treatment groups
Is variability among means just a reflection of variability of individuals?

20. Chapter 16 One-way Analysis of Variance
Calculations
Almost the same as earlier
Note differences
We multiply by nj as we go along.
MSerror is now a weighted average.
Cont.

21. Chapter 16 One-way Analysis of Variance
Calculations--cont.

22. Chapter 16 One-way Analysis of Variance
Summary Table
F.05(3,41) = 2.84

23. Chapter 16 One-way Analysis of Variance
Conclusions
F is significant at a = .05
The population means are not all equal
Some therapies lead to greater improvement than others.
SIT appears to be most effective.

24. Chapter 16 One-way Analysis of Variance
Multiple Comparisons
Significant F only shows that not all groups are equal
We want to know what groups are different.
Such procedures are designed to control familywise error rate.
Familywise error rate defined
Contrast with per comparison error rate

25. Chapter 16 One-way Analysis of Variance
More on Error Rates
Most tests reduce significance level (a) for each t test.
The more tests we run the more likely we are to make Type I error.
Good reason to hold down number of tests

26. Fisher’s LSD Procedure
Chapter 16 One-way Analysis of Variance
Fisher’s LSD Procedure
Requires significant overall F, or no tests
Run standard t tests between pairs of groups.
Often we replace s 2j or pooled estimate with MSerror from overall analysis
It is really just a pooled error term, but with more degrees of freedom--pooled across all treatment groups.

27. Chapter 16 One-way Analysis of Variance
Bonferroni t Test
Run t tests between pairs of groups, as usual
Hold down number of t tests
Reject if t exceeds critical value in Bonferroni table
Works by using a more strict value of a for each comparison
Cont.

28. Chapter 16 One-way Analysis of Variance
Bonferroni t--cont.
Critical value of a for each test set at .05/c, where c = number of tests run
Assuming familywise a = .05
e. g. with 3 tests, each t must be significant at .05/3 = level.
With computer printout, just make sure calculated probability < .05/c
Necessary table is in the book

29. Chapter 16 One-way Analysis of Variance
Assumptions
Assume:
Observations normally distributed within each population
Population variances are equal
Homogeneity of variance or homoscedasticity
Observations are independent
Cont.

30. Chapter 16 One-way Analysis of Variance
Assumptions--cont.
Analysis of variance is generally robust to first two
A robust test is one that is not greatly affected by violations of assumptions.

31. Chapter 16 One-way Analysis of Variance
Magnitude of Effect
Eta squared (h2)
Easy to calculate
Somewhat biased on the high side
Formula
See slide #33
Percent of variation in the data that can be attributed to treatment differences
Cont.

32. Magnitude of Effect--cont.
Chapter 16 One-way Analysis of Variance
Magnitude of Effect--cont.
Omega squared (w2)
Much less biased than h2
Not as intuitive
We adjust both numerator and denominator with MSerror
Formula on next slide

33. Chapter 16 One-way Analysis of Variance
h2 and w2 for Foa, et al.
h2 = .18: 18% of variability in symptoms can be accounted for by treatment
w2 = .12: This is a less biased estimate, and note that it is 33% smaller.

34. Chapter 16 One-way Analysis of Variance
Review Questions
Why is it called the analysis of “variance” and not the analysis of “means?”
What do we compare to create our test?
Why would large values of F lead to rejection of H0?
What do we do differently with unequal n ’s?
Cont.

35. Review Questions--cont.
Chapter 16 One-way Analysis of Variance
Review Questions--cont.
What is the per comparison error rate?
Why would the familywise error rate generally be larger than .05 unless it is controlled?
Most instructors hate Fisher’s LSD. Can you guess why?
Why should they not hate it?
Cont.

36. Review Questions--cont.
Chapter 16 One-way Analysis of Variance
Review Questions--cont.
How does the Bonferroni test work?
What assumptions does the analysis of variance require?
What does “robust test” mean?
What do we mean by “magnitude of effect?”
What do you know if h2 is .60?