1. Factorial Designs & Managing Violated Statistical Assumptions
Developing Study Skills and Research Methods (HL20107)
Dr James Betts

2. Lecture Outline: Factorial Research Designs Revisited
Mixed Model 2-way ANOVA
Fully independent/repeated measures 2-way ANOVA
Statistical Assumptions of ANOVA.

3. Last Week Recap
In last week’s lecture we saw two worked examples of 1-way analyses of variance
However, many experimental designs have more than one independent variable (i.e. factorial design)

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

5. 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)

6. 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

7. 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

8. 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

9. 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

10. Factorial Designs: Interpretation
210
180
150 ?
Heart Rate (beatsmin-1)
120 90 60 30
Resting
Exercise

11. Factorial Designs: Interpretation
210
180
150
Heart Rate (beatsmin-1)
120 ? 90 60 30
Resting
Exercise

12. 2-way mixed model ANOVA: Partitioning
Systematic
Variance
(resting vs exercise)
= variance between means due to exercise
Systematic
Variance
(male vs female)
= variance between means due to gender
Systematic
Variance
(Interaction)
= variance between means due IV interaction
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.

13. Procedure for computing 2-way mixed model ANOVA
Step 1: Complete the table
i.e.
-square each raw score
-total the raw scores for each subject (e.g. XT)
-square the total score for each subject (e.g. (XT)2)
-Total both columns for each group
-Total all raw scores and squared scores (e.g. X & X2).

14. Procedure for computing 2-way mixed model ANOVA
Step 2: Calculate the Grand Total correction factor
GT =
(X)2 N

15. Procedure for computing 2-way mixed model ANOVA
Step 3: Compute total Sum of Squares
SStotal= X2 - GT

16. Procedure for computing 2-way mixed model ANOVA
Step 4: Compute Exercise Effect Sum of Squares
SSex= GT
= GT 
(Xex)2
nex
(XRmale+XRfemale)2
(XEmale+XEfemale)2
nRmale+fem
nEmale+fem

17. Procedure for computing 2-way mixed model ANOVA
Step 5: Compute Gender Effect Sum of Squares
SSgen= GT
= GT 
(Xgen)2
ngen
(XRmale+XEmale)2
(XRfem+XEfem)2
nmaleR+E
nfemR+E

18. Procedure for computing 2-way mixed model ANOVA
Step 6: Compute Interaction Effect Sum of Squares
SSint= GT
= (SSex+SSgen) - GT 
(Xex+gen)2
nex+gen
(XRmale)2
(XRfem)2
(XEmale)2
(XEfem)2
nRmale
nRfem
nEmale
nEfem

19. Procedure for computing 2-way mixed model ANOVA
Step 7: Compute between subjects Sum of Squares
SSbet= SSgen- GT
= SSgen- GT
(XS)2 nk
(XT)2+(XD)2+(XH)2+(XJ)2+(XK)2+(XA)2+(XS)2+(XL)2 nk

20. Procedure for computing 2-way mixed model ANOVA
Step 8: Compute within subjects Sum of Squares
SSwit= SStotal - (SSex+SSgen+SSint+SSbet)

21. Procedure for computing 2-way mixed model ANOVA
Step 9: Determine the d.f. for each sum of squares
dftotal = (N - 1)
dfex = (k - 1)
dfgen = (r - 1)
dfint = (k - 1)(r - 1)
dfbet = r(n - 1)
dfwit = r(n - 1)(k - 1)

22. Procedure for computing 2-way mixed model ANOVA
Systematic
Variance
(resting vs exercise)
Step 10: Estimate the Variances
SSex =
dfex
Systematic
Variance
(male vs female)
SSgen =
dfgen
Systematic
Variance
(Interaction)
SSint =
dfint
Error
Variance
(within subjects)
SSwit =
dfwit
Error
Variance
(between subjects)
SSbet =
dfbet

23. Procedure for computing 2-way mixed model ANOVA
Systematic
Variance
(resting vs exercise)
Step 11: Compute F values
SSex =
dfex
Systematic
Variance
(male vs female)
SSgen =
dfgen
Systematic
Variance
(Interaction)
SSint =
dfint
Error
Variance
(within subjects)
SSwit =
dfwit
Error
Variance
(between subjects)
SSbet =
dfbet

24. Procedure for computing 2-way mixed model ANOVA
Step 12: Consult F distribution table as before
Exercise
Gender
Interaction

25. 2-way mixed model ANOVA: SPSS Output
Systematic Varianceex
SSex
Calculated Fex
dfex
SSwit
Error Variancewit
dfwit

26. 2-way mixed model ANOVA: SPSS OutputGT
Calculated Fgen
SSgen
Systematic Variancegen
dfgen
dfbet
SSbet
Error Variancebet

27. 2-way mixed model ANOVA
Systematic
Variance
(resting vs exercise)
The previous calculation and associated partitioning is an example of a 2-way mixed model ANOVA
i.e. exercise = repeated measures
gender = independent measures
Systematic
Variance
(male vs female)
Systematic
Variance
(Interaction)
Error
Variance
(within subjects)
Error
Variance
(between subjects)

28. 2-way Independent Measures ANOVA
Systematic
Variance
(resting vs exercise)
So for a fully unpaired design
e.g. males vs females
&
rest group vs exercise group
Systematic
Variance
(male vs female)
Systematic
Variance
(Interaction)
Error
Variance
(within subjects)
Error
Variance
(between subjects)

29. 2-way Independent Measures ANOVA
Systematic
Variance
(resting vs exercise)
Systematic
Variance
(male vs female)
Systematic
Variance
(Interaction)
Error
Variance
(within subjects)
Error
Variance
(between subjects)

30. 2-way Repeated Measures ANOVA
Systematic
Variance
(resting vs exercise)
…but for a fully paired design
e.g. morning vs evening
&
rest vs exercise
Systematic
Variance
(am vs pm)
Systematic
Variance
(Interaction)
Error Variance
(within subjectsexercise)
Error Variance
(within subjectstime)
Error Variance
(within subjectsinteract)

31. 2-way Repeated Measures ANOVA
Systematic
Variance
(resting vs exercise)
Systematic
Variance
(am vs pm)
Systematic
Variance
(Interaction)
Error Variance
(within subjectsexercise)
Error Variance
(within subjectstime)
Error Variance
(within subjectsinteract)

32. Summary: 2-way ANOVA
2-way (factorial) ANOVA may be appropriate whenever there are multiple IV’s to compare
We have worked through a mixed model but you should familiarise yourself with paired/unpaired procedures
You should also ensure you are aware what these effects actually look like graphically.

33. Statistical Assumptions
As with other parametric tests, ANOVA is associated with a number of statistical assumptions
When these assumptions are violated we often find that an inferential test performs poorly
We therefore need to determine not only whether an assumption has been violated but also whether that violation is sufficient to produce statistical errors.

34. e.g. ND assumption from last year
Sustained Isometric Torque (seconds)

35. e.g. ND assumption from last year

36. Assumptions of ANOVA N acquired through random sampling
Data must be of at least the interval LOM (continuous)
Independence of observations
Homogeneity of variance
All data is normally distributed

37. “ANOVA is generally robust to violations of the normality assumption, in that even when the data are non-normal, the actual Type I error rate is usually close to the nominal (i.e., desired) value.”
Maxwell & Delaney (1990) Designing Experiments & Analyzing Data: A Model Comparison Perspective, p. 109
“If the data analysis produces a statistically significant finding when no test of sphericity is conducted…you should disregard the inferential claims made by the researcher.”
Huck & Cormier (1996) Reading Statistics & Research, p. 432

38. Group A
Placebo
Supplement 1
Group B
Supplement 2
Group C

39. Plac.
Supp. 1
Supp. 2
Plac.-Supp. 1
Supp. 1-Supp. 2
Plac.-Supp. 2
Tom
2.4
3.0
3.3
-0.6
-0.3
-0.9
Dick
2.2
2.5
0.1
-0.2
Harry
1.8
1.9
-0.1
-0.4
James
1.6
1.1
1.2
0.5
0.4
Mean
2.0
2.1
2.3
SD2
0.7
0.2
0.04
0.3

40. Many texts recommend ‘Mauchley’s Test of Sphericity’
A χ2 test which, if significant, indicates a violation to sphericity
However, this is not advisable on four counts:
1.)
2.)
3.)
4.)

41. Managing Violations to Sphericity
How should we analyse aspherical data?
Option 1
Option 2
Option 3

42. Paired 1-way MANOVA: SPSS Output

43. Paired 1-way ANOVA: SPSS Output

44. Summary ANOVA and Sphericity
ANOVA is a generally robust inferential test
Unpaired data are susceptible to heterogeneity of variance only if group sizes are unequal
Paired data are susceptible to asphericity only if multiple comparisons are made
Suggested solutions for the latter include either MANOVA or epsilon corrected df depending on sample size relative to number of levels.