07. Repeated-Measures and Two-Factor Analysis of Variance.pdf

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This is a material slide for applied statistic course, in the topic Repeated Measures and Twi Factor Anova.

Hope will help you understand applied statistics.

Very relevan for information system courses.

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

2. Learning Outcomes
•Understand logic of repeated-measures
ANOVA study
1
•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
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 study
6
7

4. 7.1 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)

5. 7.2 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

6. 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 nonexperimentalstudy, 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

7. Repeated-Measures ANOVA
Hypotheses

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
H
1: At least one treatment
mean μdiffers from another

9. General structure of the
ANOVA F-Ratio
•Fratio 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 effectseffect treatment no withexpected es)(differenc variance
treatments between es)(differenc variance
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 ANOVAally)mathematic removed sdifference l(individua
effect treatmentno with expected es)(differenc variance
s)difference individual(without
eatmentsbetween tr es)(differenc variance
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 Fratio 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 7.1 Structure of the
Repeated-Measures ANOVA

14. Repeated-Measures ANOVA
Stage One EquationsN
G
XSS
total
2
2
 

 treatment each insidetreatmentswithin SSSS N
G
n
T
SS
treatmentsbetween
22


15. Two Stages of the Repeated-
Measures ANOVA
•First stage
–Identical to independent samples ANOVA
–Compute SS
total, SS
betweentreatments and
SS
withintreatments
•Second stage
–Done to remove the individual differences from
the denominator
–Compute SS
betweensubjectsand subtract it from
SS
withintreatments to find SS
error(also called residual)

16. Repeated-Measures ANOVA
Stage Two EquationsN
G
k
P
SS
subjectsbetween
22
_  bjectsbetween_suatmentswithin tre SSSSSS
error 

17. Degrees of freedom for
Repeated-Measures ANOVA
df
total
= N–1
df
withintreatments
= Σdf
insideeach treatment
df
betweentreatments
=k–1
df
betweensubjects
= n–1
df
error
= df
withintreatments
–df
betweensubjects

18. Mean squares and F-ratio for
Repeated-Measures ANOVAerror
error
error
df
SS
MS  treatmentsbetween
treatmentsbetween
treatmentsbetween
df
SS
MS
_
_
_  error
mentstreat between
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 η
2
is percentage of variability that has not
already been explained by other factors
orsubjectsbetween total
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, η
2
= .859

22. Repeated Measures ANOVA
post hoc tests (posttests)
•Significant F indicates that H
0(“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’sHSDand Scheffécan be used
–Substitute SS
errorand df
errorin 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. 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

25. 7.3 Two-Factor ANOVA
•Both independent variables and quasi-
independent variables may be employed as
factors in Two-Factor ANOVA
•An independent variable (factor) is
manipulatedin an experiment
•A quasi-independent variable (factor) is not
manipulatedbut defines the groups of scores
in a nonexperimentalstudy

26. 7.3 Two-Factor ANOVA
•Factorial designs
–Consider more than one factor
•We will study two-factor designs only
•Also limited to situations with equal n’sin each group
–Joint impact of factors is considered
•Three hypotheses tested by three F-ratios
–Each tested with same basic F-ratio structureeffect treatment no withexpected es)(differenc variance
treatments between es)(differenc variance
F

27. 7.3 Two-Factor ANOVA

28. 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 study21
21
:
:
1
0
AA
AA
H
H



 21
21
:
:
1
0
BB
BB
H
H





29. 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
•H
0: There is no interaction between
Factors Aand B
•H
1: There is an interaction between
Factors Aand B

30. 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

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

32. Combination of significant and/or
nonsignificant main effects and interactions

33. Independence of Main Effects and
Interactions

34. Independence of Main Effects and
Interactions

35. Structure of the Two-Factor
Analysis of Variance
•Three distinct tests
–Main effect of Factor A
–Main effect of Factor B
–Interaction of Aand B
•A separate Ftest is conducted for each
•Results of one are independentof the otherseffecttreatmentnoisthereifexpectedsdifferencemeanvariance
treatmentsbetweensdifferencemeanvariance
F
)(
)(


36. Example: Hypothetical Data
17

37. Two Stages of the Two-Factor
Analysis of Variance
•First stage
–Identical to independent samples ANOVA
–Compute SS
total, SS
betweentreatments and
SS
withintreatments
•Second stage
–Partition the SS
betweentreatments into three separate
components: differences attributable to Factor A;
to Factor B; and to the AxBinteraction

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

39. Stage One of the Two-Factor
Analysis of VarianceN
G
XSS
total
2
2
 
 menteach treat insideSSSS
treatmentswithin N
G
n
T
SS
treatmentsbetween
22


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

41. Degrees of freedom for
Two-Factor ANOVA
df
total
= N–1
df
withintreatments
= Σdf
insideeach treatment
df
betweentreatments
=k–1
df
A
= (number of rows)–1
df
B
= (number of columns)–1
df
AxB
= df
betweentreatments
–df
A
–df
B

42. Mean squares and F-ratios for
the Two-Factor ANOVAreatmentst within
reatmentst within
reatmentst within
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 

43. 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)

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

45. 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& Bmain effects; A x B interaction)
•For each term include F, df, p-value & η
2
•E.g., F(1, 20) = 6.33, p<.05, η
2
= .478

46. 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

47. Figure 7.4
Sample means for Example 7.4

48. 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)

49. Figure 7.5 Independent-
Measures Two-Factor Formulas

50. Figure 7.6 Example 7.1 SPSS
Output for Repeated-Measures

51. Figure 7.7 Example 7.4 SPSS
Output for Two-Factor ANOVA