1. Analysis of Variance (ANOVA)
Developing Study Skills and Research Methods (HL20107)
Dr James Betts

2. Lecture Outline: Multiple Comparisons and Type I Errors
1-way ANOVA for Unpaired data
1-way ANOVA for Paired Data
Factorial Research Designs.

3. Males
Females

4. Placebo
Lucozade

5. Placebo
Placebo
Lucozade
Lucozade
Gatorade
Gatorade
Powerade
Powerade

6. Placebo
Placebo
Lucozade
Lucozade
Gatorade
Gatorade
Powerade
Powerade

7. What is Analysis of Variance?
ANOVA is an inferential test designed for use with 3 or more data sets
t-tests are just a form of ANOVA for 2 groups
ANOVA only interested in establishing the existence of a statistical differences, not their direction (last slide)
Based upon an F value (R. A. Fisher) which reflects the ratio between systematic and random/error variance…

8. Dependent Variable
Independent
Variable
Total Variance
between means
Systematic
Variance
Extraneous/
Confounding
(Error) Variables
Error
Variance

9. Group A
Group B
Group C

10. Group A
Group B
Group C

11. Procedure for computing 1-way ANOVA for independent samples
Step 1: Complete the table
i.e.
-square each raw score
-total the raw scores for each group
-total the squared scores for each group.

12. Procedure for computing 1-way ANOVA for independent samples
Step 2: Calculate the Grand Total correction factor
GT = =
(X)2 N
(XA+XB+XC)2 N

13. Procedure for computing 1-way ANOVA for independent samples
Step 3: Compute total Sum of Squares
SStotal= X2 - GT
= (XA2+XB2+XC2) - GT

14. Procedure for computing 1-way ANOVA for independent samples
Step 4: Compute between groups Sum of Squares
SSbet= GT
= GT 
(X)2 n
(XA)2
(XB)2
(XC)2 nA nB nC

15. Procedure for computing 1-way ANOVA for independent samples
Step 5: Compute within groups Sum of Squares
SSwit= SStotal - SSbet

16. Procedure for computing 1-way ANOVA for independent samples
Step 6: Determine the d.f. for each sum of squares
dftotal= (N - 1)
dfbet= (k - 1)
dfwit= (N - k)

17. Procedure for computing 1-way ANOVA for independent samples
Step 7/8: Estimate the Variances & Compute F =
Systematic
Variance
(between means)
SSbet
dfbet
Error
Variance
(within means)
SSwit
dfwit

18. Procedure for computing 1-way ANOVA for independent samples
Step 9: Consult F distribution table
-d1 is your df for the numerator (i.e. systematic variance)
-d2 is your df for the denominator (i.e. error variance)

19. Independent 1-way ANOVA: SPSS Output

20. Group A
Trial 1
Group B
Trial 2
Group C
Trial 3

21. Procedure for computing 1-way ANOVA for paired samples
Step 1: Complete the table
i.e.
-square each raw score
-total the raw scores for each trial & subject
-total the squared scores for each trial & subject.

22. Procedure for computing 1-way ANOVA for paired samples
Step 2: Calculate the Grand Total correction factor
GT = =
= = 54.6
(X)2 N
(X1+X2+X3)2 N
( )2
…so GT just as with unpaired data 12

23. Procedure for computing 1-way ANOVA for paired samples
Step 3: Compute total Sum of Squares
SStotal= X2 - GT
= (X12+X22+X32) - GT

24. Procedure for computing 1-way ANOVA for paired samples
Step 4: Compute between trials Sum of Squares
SSbetT= GT
= GT 
(XT)2 nT
(X1)2
(X2)2
(X3)2 n1 n2 n3

25. Procedure for computing 1-way ANOVA for paired samples
Step 5: Compute between subjects Sum of Squares
SSbetS= GT
= GT 
(XS)2 nT
(XT)2
(XD)2
(XH)2
(XJ)2 nT nD nH nJ

26. Procedure for computing 1-way ANOVA for paired samples
Step 6: Compute interaction Sum of Squares
SSint= SStotal - (SSbetT + SSbetS)

27. Procedure for computing 1-way ANOVA for paired samples
Step 7: Determine the d.f. for each sum of squares
dftotal= (N - 1)
dfbetT= (k - 1)
dfbetS= (r - 1)
dfint= (r-1)(k-1) = dfbetT x dfbetS

28. Procedure for computing 1-way ANOVA for paired samples
Step 8/9: Estimate the Variances & Compute F values =
Systematic
Variance
(between trials IV)
SSbetT
dfbetT
Systematic
Variance
(between subjects)
SSbetS
dfbetS
SSint
Error
Variance
dfint

29. Procedure for computing 1-way ANOVA for paired samples
Step 10: Consult F distribution table as before

30. Paired 1-way ANOVA: SPSS Output

31. Introduction to 2-way ANOVA
Next week we will continue to work through some examples of 2-way ANOVA (i.e. factorial designs)
However, you will come across 2-way ANOVA in this week’s lab class so there are a few terms & concepts that you should be aware of in advance...

32. Factorial Designs: Technical Terms
Levels
Main Effect
Interaction Effect

33. Factorial Designs: Multiple IV’s
Hypothesis:
The HR response to exercise is mediated by gender
We now have three questions to answer: 1) 2) 3)

34. Main Effect of Exercise Exercise*Gender Interaction
Factorial Designs: Interpretation
210
Main Effect of Exercise
Not significant
180
150
Heart Rate (beatsmin-1)
120
Main Effect of Gender
Not significant 90 60
Exercise*Gender Interaction
Not significant 30
Resting
Exercise

35. Main Effect of Exercise Exercise*Gender Interaction
Factorial Designs: Interpretation
210
Main Effect of Exercise
Significant
180
150
Heart Rate (beatsmin-1)
120
Main Effect of Gender
Not significant 90 60
Exercise*Gender Interaction
Not significant 30
Resting
Exercise

36. Main Effect of Exercise Exercise*Gender Interaction
Factorial Designs: Interpretation
210
Main Effect of Exercise
Not significant
180
150
Heart Rate (beatsmin-1)
120
Main Effect of Gender
Significant 90 60
Exercise*Gender Interaction
Not significant 30
Resting
Exercise

37. Main Effect of Exercise Exercise*Gender Interaction
Factorial Designs: Interpretation
210
Main Effect of Exercise
Not significant
180
150
Heart Rate (beatsmin-1)
120
Main Effect of Gender
Not Significant 90 60
Exercise*Gender Interaction
Significant 30
Resting
Exercise

38. Main Effect of Exercise Exercise*Gender Interaction
Factorial Designs: Interpretation
210
Main Effect of Exercise
180
150 ?
Heart Rate (beatsmin-1)
120
Main Effect of Gender 90 60
Exercise*Gender Interaction 30
Resting
Exercise

39. Main Effect of Exercise Exercise*Gender Interaction
Factorial Designs: Interpretation
210
Main Effect of Exercise
180
150
Heart Rate (beatsmin-1)
120 ?
Main Effect of Gender 90 60
Exercise*Gender Interaction 30
Resting
Exercise

40. 2-way mixed model ANOVA: Partitioning
Systematic
Variance
(resting vs exercise)
= variance between means due to
Systematic
Variance
(male vs female)
= variance between means due to
Systematic
Variance
(Interaction)
= variance between means due to
Error
Variance
(within subjects)
= uncontrolled factors plus random changes within individuals for rest vs exercise
Error
Variance
(between subjects)
= uncontrolled factors and within group differences for males vs females.

41. So for a fully unpaired design
2-way mixed model ANOVA
Systematic
Variance
(resting vs exercise)
So for a fully unpaired design
e.g. males vs females
&
rest group vs exercise group
…between subject variance (i.e. SD) has a negative impact upon all contrasts
Systematic
Variance
(male vs female)
Systematic
Variance
(Interaction)
Error
Variance
(within subjects)
Error
Variance
(between subjects)

42. Refer back to this ‘partitioning’ in your lab class
2-way mixed model ANOVA
Systematic
Variance
(resting vs exercise)
…but for a fully paired design
e.g. morning vs evening
&
rest vs exercise
…between subject variance (i.e. SD) can be removed from all contrasts.
Systematic
Variance
(am vs pm)
Systematic
Variance
(Interaction)
Error Variance
(within subjectsexercise)
Error Variance
(within subjectstime)
Error Variance
(within subjectsinteract)
Refer back to this ‘partitioning’ in your lab class