Chapter 13 Power Points for Essentials of Statistics for the Behavioral Sciences, Gravetter & Wallnau, 8th ed

1. Chapter 13
Repeated-Measures and
Two-Factor Analysis of Variance
PowerPoint Lecture Slides
Essentials of Statistics for the
Behavioral Sciences
Eighth Edition
by Frederick J. Gravetter and Larry B. Wallnau

2. Chapter 13 Learning Outcomes
• Understand logic of repeated-measures
ANOVA study1
• Compute repeated-measures ANOVA to
evaluate mean differences for single-factor
repeated-measures study
2
• Measure effect size, perform post hoc tests
and evaluate assumptions required for
single-factor repeated-measures ANOVA
3

3. • Measure effect size, interpret results and
articulate assumptions for two-factor ANOVA
Ch 13 Learning Outcomes
(continued)
• Understand logic of two-factor study and matrix
of group means4
• Describe main effects and interactions from
pattern of group means in two-factor ANOVA5
• Compute two-factor ANOVA to evaluate means
for two-factor independent-measures study6
7

4. Tools You Will Need
• Independent-Measures Analysis of Variance
(Chapter 12)
• Repeated-Measures Designs (Chapter 11)
• Individual Differences

5. 13.1 Overview
• Analysis of Variance
– Evaluated mean differences for two or more
groups
– Limited to one independent variable (IV)
• Complex Analysis of Variance
– Samples are related; not independent
(Repeated-measures ANOVA)
– Two independent variables are manipulated
(Factorial ANOVA; only Two-Factor in this text)

6. 13.2 Repeated-Measures ANOVA
• Independent-measures ANOVA uses multiple
participant samples to test the treatments
• Participant samples may not be identical
• If groups are different, what was responsible?
– Treatment differences?
– Participant group differences?
• Repeated-measures solves this problem by
testing all treatments using one sample of
participants

7. Repeated-Measures ANOVA
• Repeated-Measures ANOVA used to evaluate
mean differences in two general situations
– In an experiment, compare two or more
manipulated treatment conditions using the same
participants in all conditions
– In a nonexperimental study, compare a group of
participants at two or more different times
• Before therapy; After therapy; 6-month follow-up
• Compare vocabulary at age 3, 4 and 5

8. Repeated-Measures ANOVA
Hypotheses
• Null hypothesis: in the population there are no
mean differences among the treatment groups
• Alternate hypothesis: there is one (or more)
mean differences among the treatment groups
...: 3210  H
H1: At least one treatment
mean μ differs from another

9. General structure of the
ANOVA F-Ratio
• F ratio based on variances
– Numerator measures treatment mean differences
– Denominator measures treatment mean
differences when there is no treatment effect
– Large F-ratio  greater treatment differences
than would be expected with no treatment effects
effecttreatmentnowithexpectedes)(differencvariance
treatmentsbetweenes)(differencvariance
F 

10. Individual differences
• Participant characteristics may vary
considerably from one person to another
• Participant characteristics can influence
measurements (Dependent Variable)
• Repeated measures design allows control of
the effects of participant characteristics
– Eliminated from the numerator by the research
design
– Must be removed from the denominator
statistically

11. Structure of the F-Ratio for
Repeated-Measures ANOVA
ally)mathematicremovedsdifferencel(individua
effecttreatmentnowithexpectedes)(differencvariance
s)differenceindividual(without
eatmentsbetween tres)(differencvariance
F 
The biggest change between independent-
measures ANOVA and repeated-measures ANOVA
is the addition of a process to mathematically
remove the individual differences variance
component from the denominator of the F-ratio

12. Repeated-Measures ANOVA
Logic
• Numerator of the F ratio includes
– Systematic differences caused by treatments
– Unsystematic differences caused by random
factors are reduced because the same individuals
are in all treatments
• Denominator estimates variance reasonable
to expect from unsystematic factors
– Effect of individual differences is removed
– Residual (error) variance remains

13. Figure 13.1 Structure of the
Repeated-Measures ANOVA

14. Repeated-Measures ANOVA
Stage One Equations
N
G
XSStotal
2
2
 
 treatmenteachinsidetreatmentswithin SSSS
N
G
n
T
SS treatmentsbetween
22
 

15. Two Stages of the Repeated-
Measures ANOVA
• First stage
– Identical to independent samples ANOVA
– Compute SStotal, SSbetween treatments and
SSwithin treatments
• Second stage
– Done to remove the individual differences from
the denominator
– Compute SSbetween subjects and subtract it from
SSwithin treatments to find SSerror (also called residual)

16. Repeated-Measures ANOVA
Stage Two Equations
N
G
k
P
SS subjectsbetween
22
_  
bjectsbetween_suatmentswithin tre SSSSSSerror 

17. Degrees of freedom for
Repeated-Measures ANOVA
dftotal = N – 1
dfwithin treatments = Σdfinside each treatment
dfbetween treatments = k – 1
dfbetween subjects = n – 1
dferror = dfwithin treatments – dfbetween subjects

18. Mean squares and F-ratio for
Repeated-Measures ANOVA
error
error
error
df
SS
MS 
treatmentsbetween
treatmentsbetween
treatmentsbetween
df
SS
MS
_
_
_ 
error
mentstreatbetween
MS
MS
F 

19. F-Ratio General Structure for
Repeated-Measures ANOVA
)(
)(
sdifferenceindividualwithout
sdifferenceicunsystemat
sdifferenceindividualwithout
sdifferenceicunsystemateffectstreatment
F



20. Effect size for the
Repeated-Measures ANOVA
• Percentage of variance explained by the
treatment differences
• Partial η2 is percentage of variability that has not
already been explained by other factors
or
subjectsbetweentotal
eatmentsbetween tr2
SS SS
SS


errorSSSS
SS


eatmentsbetween tr
eatmentsbetween tr2


21. In the Literature
• Report a summary of descriptive statistics (at
least means and standard deviations)
• Report a concise statement of the ANOVA
results
– E.g., F (3, 18) = 16.72, p<.01, η2 = .859

22. Repeated Measures ANOVA
post hoc tests (posttests)
• Significant F indicates that H0 (“all populations
means are equal”) is wrong in some way
• Use post hoc test to determine exactly where
significant differences exist among more than
two treatment means
– Tukey’s HSD and Scheffé can be used
– Substitute SSerror and dferror in the formulas

23. Repeated-Measures ANOVA
Assumptions
• The observations within each treatment
condition must be independent
• The population distribution within each
treatment must be normal
• The variances of the population distribution
for each treatment should be equivalent

24. Learning Check
• A researcher obtains an F-ratio with df = 2, 12
in a repeated-measures study ANOVA. How
many subjects participated in the study?
• 15A
• 14B
• 13C
• 7D

25. Learning Check - Answer
• A researcher obtains an F-ratio with df = 2, 12
in a repeated-measures study ANOVA. How
many subjects participated in the study?
• 15A
• 14B
• 13C
• 7D

26. Learning Check
• Decide if each of the following statements
is True or False
• For the repeated-measures ANOVA,
degrees of freedom for SSerror could be
written as [(N–k) – (n–1)]
T/F
• The first stage of the repeated-
measures ANOVA is the same as the
independent-measures ANOVA
T/F

27. Learning Check - Answer
• dferror = dfw/i treatments – dfbetwn subjects
• Within treatments df = N-k;
between subjects df = n-1
True
• After the first stage analysis, the
second stage analysis adjusts for
individual differences
True

28. Repeated-Measures ANOVA
Advantages and Disadvantages
• Advantages of repeated-measures designs
– Individual differences among participants do not
influence outcomes
– Smaller number of participants needed to test all
the treatments
• Disadvantages of repeated-measures designs
– Some (unknown) factor other than the treatment
may cause participant’s scores to change
– Practice or experience may affect scores
independently of the actual treatment effect

29. 13.3 Two-Factor ANOVA
• Both independent variables and quasi-
independent variables may be employed as
factors in Two-Factor ANOVA
• An independent variable (factor) is
manipulated in an experiment
• A quasi-independent variable (factor) is not
manipulated but defines the groups of scores
in a nonexperimental study

30. 13.3 Two-Factor ANOVA
• Factorial designs
– Consider more than one factor
• We will study two-factor designs only
• Also limited to situations with equal n’s in each group
– Joint impact of factors is considered
• Three hypotheses tested by three F-ratios
– Each tested with same basic F-ratio structure
effecttreatmentnowithexpectedes)(differencvariance
treatmentsbetweenes)(differencvariance
F 

31. Main Effects
• Mean differences among levels of one factor
– Differences are tested for statistical significance
– Each factor is evaluated independently of the
other factor(s) in the study
21
21
:
:
1
0
AA
AA
H
H




21
21
:
:
1
0
BB
BB
H
H





32. Interactions Between Factors
• The mean differences between individuals
treatment conditions, or cells, are different
from what would be predicted from the
overall main effects of the factors
• H0: There is no interaction between
Factors A and B
• H1: There is an interaction between
Factors A and B

33. Interpreting Interactions
• Dependence of factors
– The effect of one factor depends on the level or
value of the other
– Sometimes called “non-additive” effects because
the main effects do not “add” together predictably
• Non-parallel lines (cross, converge or diverge)
in a graph indicate interaction is occurring
• Typically called the A x B interaction

34. Figure 13.2 Group Means Graphed
without (a) and with (b) Interaction

35. Structure of the Two-Factor
Analysis of Variance
• Three distinct tests
– Main effect of Factor A
– Main effect of Factor B
– Interaction of A and B
• A separate F test is conducted for each
• Results of one are independent of the others
effecttreatmentnoisthereifexpectedsdifferencemeanvariance
treatmentsbetweensdifferencemeanvariance
F
)(
)(


36. Two Stages of the Two-Factor
Analysis of Variance
• First stage
– Identical to independent samples ANOVA
– Compute SStotal, SSbetween treatments and
SSwithin treatments
• Second stage
– Partition the SSbetween treatments into three separate
components: differences attributable to Factor A;
to Factor B; and to the AxB interaction

37. Figure 13.3 Structure of the
Two-Factor Analysis of Variance

38. Stage One of the Two-Factor
Analysis of Variance
N
G
XSStotal
2
2
 
 menteach treatinsideSSSS treatmentswithin
N
G
n
T
SS treatmentsbetween
22
 

39. Stage Two of the Two Factor
Analysis of Variance
• This stage determines the numerators for the
three F-ratios by partitioning SSbetween treatments
N
G
n
T
SS
row
row
A
22
  N
G
n
T
SS
col
col
B
22
 
BAtreatmentsbetweenAxB SSSSSSSS 

40. Degrees of freedom for
Two-Factor ANOVA
dftotal = N – 1
dfwithin treatments = Σdfinside each treatment
dfbetween treatments = k – 1
dfA = (number of rows) – 1
dfB = (number of columns)– 1
dferror = dfwithin treatments – dfbetween subjects

41. Mean squares and F-ratios for
the Two-Factor ANOVA
reatmentstwithin
reatmentstwithin
reatmentstwithin
df
SS
MS 
AxB
AxB
AxB
B
B
B
A
A
A
df
SS
MS
df
SS
MS
df
SS
MS 
within
AxB
AxB
within
B
B
within
A
A
MS
MS
F
MS
MS
F
MS
MS
F 

42. Two-Factor ANOVA
Summary Table Example
Source SS df MS F
Between treatments 200 3
Factor A 40 1 40 4
Factor B 60 1 60 *6
A x B 100 1 100 **10
Within Treatments 300 20 10
Total 500 23
F.05 (1, 20) = 4.35*
F.01 (1, 20) = 8.10**
(N = 24; n = 6)

43. Two-Factor ANOVA Effect Size
• η2, is computed to show the percentage of
variability not explained by other factors
treatmentswithinA
A
AxBBtotal
A
A
SSSS
SS
SSSSSS
SS



2

treatmentswithinB
B
AxBAtotal
B
B
SSSS
SS
SSSSSS
SS
_
2




treatmentswithinAxB
AxB
BAtotal
AxB
AxB
SSSS
SS
SSSSSS
SS



2


44. In the Literature
• Report mean and standard deviations (usually
in a table or graph due to the complexity of
the design)
• Report results of hypothesis test for all three
terms (A & B main effects; A x B interaction)
• For each term include F, df, p-value & η2
• E.g., F (1, 20) = 6.33, p<.05, η2 = .478

45. Interpreting the Results
• Focus on the overall pattern of results
• Significant interactions require particular
attention because even if you understand the
main effects, interactions go beyond what
main effects alone can explain.
• Extensive practice is typically required to be
able to clearly articulate results which include
a significant interaction

46. Figure 13.4
Sample means for Example 13.4

47. Two-Factor ANOVA
Assumptions
• The validity of the ANOVA presented in this
chapter depends on three assumptions
common to other hypothesis tests
– The observations within each sample must be
independent of each other
– The populations from which the samples are
selected must be normally distributed
– The populations from which the samples are
selected must have equal variances
(homogeneity of variance)

48. Learning Check
• If a two-factor analysis of variance produces a
statistically significant interaction, then you
can conclude that _____
• either the main effect for factor A or the main effect
for factor B is also significantA
• neither the main effect for factor A nor the main
effect for factor B is significantB
• both the man effect for factor A and the main effect
for factor B are significantC
• the significance of the main effects is not related to
the significance of the interactionD

49. Learning Check - Answer
• If a two-factor analysis of variance produces a
statistically significant interaction, then you
can conclude that _____
• either the main effect for factor A or the main effect
for factor B is also significantA
• neither the main effect for factor A nor the main
effect for factor B is significantB
• both the man effect for factor A and the main effect
for factor B are significantC
• the significance of the main effects is not related
to the significance of the interactionD

50. Learning Check
• Decide if each of the following statements
is True or False
• Two separate single-factor ANOVAs provide
exactly the same information that is obtained
from a two-factor analysis of variance
T/F
• A disadvantage of combining 2 factors in an
experiment is that you cannot determine how
either factor would affect participants’ scores
if it were examined in an experiment by itself
T/F

51. Learning Check - Answers
• Main effects in Two-Factor ANOVA are
identical to results of two One-Way
ANOVAs; but Two-Factor ANOVA
provides Interaction results too!
False
• The two-factor ANOVA allows you to
determine the effect of one variable
controlling for the effect of the other
False

52. Figure 13.5 Independent-
Measures Two-Factor Formulas

53. Figure 13.6 Example 13.1 SPSS
Output for Repeated-Measures

54. Figure 13.7 Example 13.4 SPSS
Output for Two-Factor ANOVA

55. Any
Questions
?
Concepts
?
Equations?