1
Assignment 7
FOUN 823
Directions:
(1) Please email your assignment answers in an MS Word file to the instructor
( XXXXXXXXXX) by the designated deadline. No overdue assignment will be graded
unless a prior a
angement is made due to some extenuating circumstances.
(2) For this assignment, please analyze the SPSS data file in the same assignment
folder with within-subjects ANOVA to answer various assignment questions.
(3) The alpha level (α) will be .05 for the hypothesis testing on this assignment.
(4) No partial credits will be assigned to an inco
ect ve
al conclusion supported by the
co
ect statistics or a co
ect ve
al conclusion supported by the inco
ect statistics. In
addition, no partial credits will be assigned to an inco
ect SPSS table.
(5) Please contact me ( XXXXXXXXXX) for any questions regarding what you are
equired to do for this assignment. Don’t make your own assumption when an
assignment question is unclear to you.
2
A researcher studied the change of the depressive symptoms over time among
30 patents. Please conduct a one-way within-subjects ANOVA on the depressive
symptom data in the SPSS data file to answer the assignment questions:
1. According to the Mauchly’s test results, is the sphericity assumption violated?
Please support your ve
al conclusion with the relevant statistics in APA style.
XXXXXXXXXXPoint)
2. What will be the sum of squares (SS) of Time variable? (1 Point)
3. What will be the sum of squares (SS) of E
or? (1 Point)
4. Did patients’ depressive symptoms change over time? Given no violation of the
sphericity assumption, please use the standard F test results to support you
ve
al conclusion. Given the violation of the sphericity assumption, please use
the F tests with the Greenhouse-Geisser co
ection and the Huynh-Feldt
co
ection to support your ve
al conclusion (1 Point)
5. Please copy the Profile Plots graph from the SPSS outputs then paste it to your
submitted assignment. (1 Point)
Quantitative Method III
1
Within-Subjects (Repeated
Measures) ANOVA
FOUN 823
Cherng-Jyh Yen, Ph.D.
2
Within-Subjects (Repeated
Measures) ANOVA
• Within-Subjects ANOVA
• Within-subjects independent variable
• Same participants going through all levels of the
within-subjects independent variable, such as
treatment, time, and so forth
• Repeated measures dependent variable
• Same participants being measured on the same
dependent variable at each level of the independent
variable (repeated measures)
• Main advantage over between-subjects
ANOVA: Increased statistical power
• Subjects serve as their own control
• Remove the SS related to Subject from the e
or
variance
• Decomposition of SSTotal
Between-subjects ANOVA:
SSTotal = SSA + SSS/A
Within-subjects ANOVA:
SSTotal = SSA + (SSS + SSAxS); SSAxS as the e
or variance
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Within-Subjects ANOVA
• A Numerical Example
XXXXXXXXXXTime 1 Time 2 Time 3 Time 4 Time 5
XXXXXXXXXXPretest 1) (Pretest XXXXXXXXXXPosttest XXXXXXXXXXPosttest XXXXXXXXXXPosttest 3)
Subject
XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX5 �̅�1. = 8.200
XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX5 �̅�2. = 7.000
XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX6 �̅�3. = 7.000
XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX4 �̅�4. = 5.600
XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX5 �̅�5. = 6.400
XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX7 �̅�6. = 8.000
XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX4 �̅�7. = 7.600
XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX5 �̅�8. = 6.000
�̅�.1 = 9.125 �̅�.2 = 9.000 �̅�.3 = 6.375 �̅�.4 = 5.250 �̅�.5 = 5.125
�̅�T = 6.975
Within-subjects independent variable: Time
Time 1: 1 week before the treatment
Time 2: 1 hour before the treatment
Time 3: 1 week after the treatment
Time 4: 2 weeks after the treatment
Time 5: 3 weeks after the treatment
Repeated measures dependent variable: Symptom Severity
Omnibus, unfocused research question related
to the main effect of Time: Does depressive symptom
severity change over time (i.e. differ over time points)?
Focused research question related to the
contrast of planned comparison: Does the average
post-treatment depressive symptom severity differ from the
average pre-treatment depressive symptom severity?
4
Within-Subjects ANOVA
• Sphericity Assumption
• Homogeneity of population variances and
co
elations of difference scores
• Assumption violation → Inco
ect observed
p value → Inco
ect decision of the null
hypothesis → Inco
ect ve
al conclusion
5
Within-Subjects ANOVA
• Mauchly’s test of sphericity
assumption
Observed p < ? → Reject the null hypothesis of no
violation of the sphericity assumption
Observed p > ? → Retain the null hypothesis of no
violation of the sphericity assumption
If ? = .05
Ve
al Conclusion: The alpha level was set at .05. The
esults suggested the violation of the sphericity assumption,
W (9) = .04, p = .043.
Mauchly's Test of Sphericitya
Measure: MEASURE_1
Within Subjects
Effect
Mauchly's
W
Approx.
Chi-Square df Sig.
Epsilonb
Greenhouse-
Geisser
Huynh-
Feldt
Lower-
ound
Time XXXXXXXXXX XXXXXXXXXX
Tests the null hypothesis that the e
or covariance matrix of the orthonormalized transformed dependent
variables is proportional to an identity matrix.
a. Design: Intercept
Within Subjects Design: Time
. May be used to adjust the degrees of freedom for the averaged tests of significance