Chapter 13Repeated-Measures andTwo-Factor Analysis of Variance
PowerPoint Lecture Slides
Essentials of Statistics for the Behavioral Sciences Eighth Edition
by Frederick J. Gravetter and Larry B. Wallnau
Chapter 13 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
• 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
Tools You Will Need
• Independent-Measures Analysis of Variance(Chapter 12)
• Repeated-Measures Designs (Chapter 11)• Individual Differences
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)
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
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
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
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
effect treatment no withexpected es)(differenc variance
treatments between es)(differenc varianceF
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
Structure of the F-Ratio for Repeated-Measures ANOVA
ally)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
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
Figure 13.1 Structure of the Repeated-Measures ANOVA
Repeated-Measures ANOVA Stage One Equations
N
GXSStotal
22
treatment each insidetreatmentswithin SSSS
N
G
n
TSS treatmentsbetween
22
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)
Repeated-Measures ANOVAStage Two Equations
N
G
k
PSS subjectsbetween
22
_
bjectsbetween_suatmentswithin tre SSSSSSerror
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
Mean squares and F-ratio for Repeated-Measures ANOVA
error
errorerror df
SSMS
treatmentsbetween
treatmentsbetweentreatmentsbetween df
SSMS
_
__
error
mentstreat between
MS
MSF
F-Ratio General Structure for Repeated-Measures ANOVA
)(
)(
sdifferenceindividualwithout
sdifferenceicunsystemat
sdifferenceindividualwithout
sdifferenceicunsystemateffectstreatment
F
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
subjectsbetween total
eatmentsbetween tr2
SS SS
SS
errorSSSS
SS
eatmentsbetween tr
eatmentsbetween tr2
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
Repeated Measures ANOVApost 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
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
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?
•15
A
•14
B
•13
C
•7D
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?
•15
A
•14
B
•13
C
•7D
Learning Check
• Decide if each of the following statements is True or False
•For the repeated-measures ANOVA, degrees of freedom for SSer
ror 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
Learning Check - Answer
•dferr
or = dfw/
i
treat
ments – dfbe
twn
subjec
ts•Wit
hin treatments df = N-k; between subjects df = n-1
True
•After the first stage analysis, the second stage analysis adjusts for individual differences
True
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
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
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
effect treatment no withexpected es)(differenc variance
treatments between es)(differenc varianceF
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
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
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
Figure 13.2 Group Means Graphed without (a) and with (b) Interaction
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
treatmentsbetweensdifferencemeanvarianceF
)(
)(
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
Figure 13.3 Structure of the Two-Factor Analysis of Variance
Stage One of the Two-Factor Analysis of Variance
N
GXSStotal
22
menteach treat insideSSSS treatmentswithin
N
G
n
TSS treatmentsbetween
22
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
TSS
row
rowA
22
N
G
n
TSS
col
colB
22
BAtreatments betweenAxB SSSSSSSS
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
Mean squares and F-ratios for the Two-Factor ANOVA
reatmentst within
reatmentst withinreatmentst within df
SSMS
AxB
AxBAxB
B
BB
A
AA df
SSMS
df
SSMS
df
SSMS
within
AxBAxB
within
BB
within
AA MS
MSF
MS
MSF
MS
MSF
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)
Two-Factor ANOVA Effect Size
• η2, is computed to show the percentage of variability not explained by other factors
treatments withinA
A
AxBBtotal
AA SSSS
SS
SSSSSS
SS
2
treatmentswithinB
B
AxBAtotal
BB SSSS
SS
SSSSSS
SS
_
2
treatments withinAxB
AxB
BAtotal
AxBAxB SSSS
SS
SSSSSS
SS
2
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
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
Figure 13.4Sample means for Example 13.4
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)
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 significant
A
•neither the main effect for factor A nor the main effect for factor B is significant
B
•both the man effect for factor A and the main effect for factor B are significant
C
•the significance of the main effects is not related to the significance of the interaction
D
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 significant
A
•neither the main effect for factor A nor the main effect for factor B is significant
B
•both the man effect for factor A and the main effect for factor B are significant
C
•the significance of the main effects is not related
to the significance of the interaction
D
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
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
Figure 13.5 Independent-Measures Two-Factor Formulas
Figure 13.6 Example 13.1 SPSS Output for Repeated-Measures
Figure 13.7 Example 13.4 SPSS Output for Two-Factor ANOVA
AnyQuestions
?
Concepts?
Equations?