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Chapter 14Repeated Measures andTwo Factor Analysis of Variance
PowerPoint Lecture Slides
Essentials of Statistics for the Behavioral Sciences Seventh Edition
by Frederick J. Gravetter and Larry B. Wallnau
Chapter 14 Learning Outcomes
Concepts to review
• Independent-measures analysis of variance(Chapter 13)
• Repeated measures designs (Chapter 11)
• Individual differences
14.1 Overview
• Analysis of Variance– Evaluated mean differences of two or
groups
• Complex Analysis of Variance– Samples are related not independent
(Repeated measures ANOVA)– More than one factor is tested
(Factorial ANOVA, here Two-Factor)
14.2 Repeated Measures ANOVA
• Advantages of repeated measures designs– Individual differences among participants do
not influence outcomes– Smaller number of subjects needed to test all
the treatments• Repeated Measures ANOVA
– Compares two or more treatment conditions with the same subjects tested in all conditions
– Studies same group of subjects at two or more different times.
Hypotheses for repeated measures ANOVA
• Null hypothesis: in the population, there are no mean differences among the treatment groups
• Alternate hypothesis states that there are mean differences among the treatment groups.
...: 3210 H
H1: At least one treatment mean μ is different from another
Individual Differences in the Repeated Measures ANOVA
• F ratio based on variances– Same structure as independent measures– Variance due to individual differences is
not present
effect treatment no withexpected es)(differenc variance
treatments between es)(differenc varianceF
Individual differences
• Participant characteristics that vary from one person to another.– Not systematically present in any treatment
group or by research design
• Characteristics may influence measurements on the outcome variable– Eliminated from the numerator by the
research design– Must be removed from the denominator
statistically
Logic of repeated measures ANOVA
• Numerator of the F ratio includes– Systematic differences caused by treatments– Unsystematic differences caused by random
factors (reduced because same individuals in all treatments)
• Denominator estimates variance reasonable to expect from unsystematic factors– Effect of individual differences is removed– Residual (error) variance remains
Figure 14.1 Structure of the Repeated-Measures ANOVA
Stage One Repeated-Measures ANOVA 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– Removing the individual differences from the
denominator
– Compute SSBetween subjects and subtract it from SSWithin treatments to find SSError
Stage Two Repeated-Measures ANOVA equations
N
G
k
PSS subjectsbetween
22
subjectsbetweentreatments withinerror SSSSSS
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
treatments between
treatments betweenntstreatme between df
SSMS
error
mentstreat between
MS
MSF
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
error
treatments between
subjectsbetweentotal
treatments between
SS
SS
SSSS
SS
2
Post hoc tests with Repeated Measures ANOVA
• 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
Assumptions of the Repeated Measures ANOVA
• 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
from an ANOVA for a repeated-measures research study. How many subjects participated in the study?
Learning Check - Answer• A researcher obtains an F-ratio with df = 2, 12
from an ANOVA for a repeated-measures research study. How many subjects participated in the study?
Learning Check
• Decide if each of the following statements is True or False.
Answer
14.2 Two-Factor ANOVA
• Factorial designs– Consider more than one factor– 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
• Non-parallel lines (cross or converge) in a graph– Indicate interaction is occurring
• Typically called the A x B interaction
Figure 14.2 Graph of group means with and without interaction
Structure of the Two-Factor Analysis
• 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
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 14.3 Structure of the Two- Factor Analysis of Variance
Stage One of the Two-Factor Analysis of Variance
N
GXSStotal
22
treatment each insidetreatmentswithin SSSS
N
G
n
TSS treatmentsbetween
22
Stage 2 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 ANOVAdftotal = 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-ratio 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
Effect Size for Two-Factor ANOVA
• η2, is computed as the percentage of variability not explained by other factors.
treatments withinA
A
AxBBtotal
AA SSSS
SS
SSSSSS
SS
2
treatments withinB
B
AxBAtotal
BB SSSS
SS
SSSSSS
SS
2
treatments withinAxB
AxB
BAtotal
AxBAxB SSSS
SS
SSSSSS
SS
2
Figure 14.4 Sample means for Example 14.3
Assumptions for the Two-Factor ANOVA
• The validity of the ANOVA presented in this chapter depends on three assumptions common to other hypothesis tests1. The observations within each sample must
be independent of each other
2. The populations from which the samples are selected must be normally distributed
3. 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 _____.
Learning Check - Answer
• If a two-factor analysis of variance produces a statistically significant interaction, then you can conclude that _____.
Learning Check
• Decide if each of the following statements is True or False.
Answer
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