Please write up and interpret the results for the following repeated measures ANOVA, using the Activity 6.sav data set. Score_0 through Score _12 is the repeated measure (7 levels) and gender is a fixed factor. Discuss especially both main effects and the presence/absence of an interaction between the two.
All of the relevant data is given below.
|
Within-Subjects Factors
|
|
Measure: MEASURE_1
|
|
Score
|
Dependent Variable
|
|
1
|
Score_0
|
|
2
|
Score_2
|
|
3
|
Score_4
|
|
4
|
Score_6
|
|
5
|
Score_8
|
|
6
|
Score_10
|
|
7
|
Score_12
|
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Between-Subjects Factors
|
|
|
Value Label
|
N
|
|
Gender
|
F
|
Female
|
8
|
|
M
|
Male
|
4
|
|
Descriptive Statistics
|
|
|
Gender
|
Mean
|
Std. Deviation
|
N
|
|
Pre-test score
|
Female
|
28.25
|
8.172
|
8
|
|
Male
|
32.25
|
19.432
|
4
|
|
Total
|
29.58
|
12.221
|
12
|
|
Week 2 score
|
Female
|
29.75
|
6.319
|
8
|
|
Male
|
39.75
|
13.889
|
4
|
|
Total
|
33.08
|
10.113
|
12
|
|
Week 4 score
|
Female
|
33.63
|
5.181
|
8
|
|
Male
|
39.00
|
16.432
|
4
|
|
Total
|
35.42
|
9.885
|
12
|
|
Week 6 score
|
Female
|
35.88
|
6.556
|
8
|
|
Male
|
35.25
|
17.802
|
4
|
|
Total
|
35.67
|
10.671
|
12
|
|
Week 8 score
|
Female
|
39.38
|
5.370
|
8
|
|
Male
|
41.00
|
16.633
|
4
|
|
Total
|
39.92
|
9.718
|
12
|
|
Week 10 score
|
Female
|
44.88
|
5.743
|
8
|
|
Male
|
47.25
|
13.961
|
4
|
|
Total
|
45.67
|
8.690
|
12
|
|
Week 12 score
|
Female
|
48.38
|
8.518
|
8
|
|
Male
|
53.25
|
13.793
|
4
|
|
Total
|
50.00
|
10.189
|
12
|
|
Multivariate Testsa
|
|
Effect
|
Value
|
F
|
Hypothesis df
|
Error df
|
Sig.
|
|
Score
|
Pillai's Trace
|
.961
|
20.439b
|
6.000
|
5.000
|
.002
|
|
Wilks' Lambda
|
.039
|
20.439b
|
6.000
|
5.000
|
.002
|
|
Hotelling's Trace
|
24.526
|
20.439b
|
6.000
|
5.000
|
.002
|
|
Roy's Largest Root
|
24.526
|
20.439b
|
6.000
|
5.000
|
.002
|
|
Score * Gender
|
Pillai's Trace
|
.491
|
.804b
|
6.000
|
5.000
|
.607
|
|
Wilks' Lambda
|
.509
|
.804b
|
6.000
|
5.000
|
.607
|
|
Hotelling's Trace
|
.965
|
.804b
|
6.000
|
5.000
|
.607
|
|
Roy's Largest Root
|
.965
|
.804b
|
6.000
|
5.000
|
.607
|
|
a. Design: Intercept + Gender
Within Subjects Design: Score
|
|
b. Exact statistic
|
|
Mauchly's Test of Sphericitya
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|
|
Measure: MEASURE_1
|
|
|
Within Subjects Effect
|
Mauchly's W
|
Approx. Chi-Square
|
df
|
Sig.
|
Epsilonb
|
|
|
Greenhouse-Geisser
|
Huynh-Feldt
|
Lower-bound
|
|
|
Score
|
.001
|
56.876
|
20
|
.000
|
.441
|
.674
|
.167
|
|
|
Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix.
|
|
|
a. Design: Intercept + Gender
Within Subjects Design: Score
|
|
|
b. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table.
|
|
| |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Tests of Within-Subjects Effects
|
|
Measure: MEASURE_1
|
|
Source
|
Type III Sum of Squares
|
df
|
Mean Square
|
F
|
Sig.
|
|
Score
|
Sphericity Assumed
|
3246.536
|
6
|
541.089
|
20.609
|
.000
|
|
Greenhouse-Geisser
|
3246.536
|
2.646
|
1227.164
|
20.609
|
.000
|
|
Huynh-Feldt
|
3246.536
|
4.045
|
802.659
|
20.609
|
.000
|
|
Lower-bound
|
3246.536
|
1.000
|
3246.536
|
20.609
|
.001
|
|
Score * Gender
|
Sphericity Assumed
|
182.155
|
6
|
30.359
|
1.156
|
.342
|
|
Greenhouse-Geisser
|
182.155
|
2.646
|
68.853
|
1.156
|
.341
|
|
Huynh-Feldt
|
182.155
|
4.045
|
45.035
|
1.156
|
.344
|
|
Lower-bound
|
182.155
|
1.000
|
182.155
|
1.156
|
.307
|
|
Error(Score)
|
Sphericity Assumed
|
1575.321
|
60
|
26.255
|
|
|
|
Greenhouse-Geisser
|
1575.321
|
26.456
|
59.546
|
|
|
|
Huynh-Feldt
|
1575.321
|
40.447
|
38.948
|
|
|
|
Lower-bound
|
1575.321
|
10.000
|
157.532
|
|
|
|
Tests of Within-Subjects Contrasts
|
|
Measure: MEASURE_1
|
|
Source
|
Score
|
Type III Sum of Squares
|
df
|
Mean Square
|
F
|
Sig.
|
|
Score
|
Linear
|
2962.680
|
1
|
2962.680
|
46.905
|
.000
|
|
Quadratic
|
143.040
|
1
|
143.040
|
4.305
|
.065
|
|
Cubic
|
51.361
|
1
|
51.361
|
2.242
|
.165
|
|
Order 4
|
73.724
|
1
|
73.724
|
2.765
|
.127
|
|
Order 5
|
3.584
|
1
|
3.584
|
.405
|
.539
|
|
Order 6
|
12.147
|
1
|
12.147
|
4.444
|
.061
|
|
Score * Gender
|
Linear
|
25.537
|
1
|
25.537
|
.404
|
.539
|
|
Quadratic
|
21.254
|
1
|
21.254
|
.640
|
.442
|
|
Cubic
|
66.694
|
1
|
66.694
|
2.911
|
.119
|
|
Order 4
|
55.767
|
1
|
55.767
|
2.092
|
.179
|
|
Order 5
|
5.060
|
1
|
5.060
|
.572
|
.467
|
|
Order 6
|
7.841
|
1
|
7.841
|
2.869
|
.121
|
|
Error(Score)
|
Linear
|
631.638
|
10
|
63.164
|
|
|
|
Quadratic
|
332.272
|
10
|
33.227
|
|
|
|
Cubic
|
229.083
|
10
|
22.908
|
|
|
|
Order 4
|
266.594
|
10
|
26.659
|
|
|
|
Order 5
|
88.403
|
10
|
8.840
|
|
|
|
Order 6
|
27.330
|
10
|
2.733
|
|
|
|
Tests of Between-Subjects Effects
|
|
Measure: MEASURE_1
|
|
Transformed Variable: Average
|
|
Source
|
Type III Sum of Squares
|
df
|
Mean Square
|
F
|
Sig.
|
|
Intercept
|
114349.339
|
1
|
114349.339
|
188.733
|
.000
|
|
Gender
|
290.720
|
1
|
290.720
|
.480
|
.504
|
|
Error
|
6058.804
|
10
|
605.880
|
|
|
a. Is the assumption of sphericity violated? How can you tell? What does this mean in the context of interpreting the results?
|
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-bound
|
|
Score
|
.001
|
56.876
|
20
|
.000
|
.441
|
.674
|
.167
|
|
Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix.
|
|
a. Design: Intercept + Gender
Within Subjects Design: Score
|
|
b. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table.
|
The above table depicts the results of Mauchly's Test of Spheriicty which tests for one of the assumptions of the ANOVA with repeated measures, namely, sphericity (homogeneity of covariance). This particular table is important for viewing as this assumption is commonly violated. In this case, since p-value is less than .05, I conclude that there are significant differences between the variance of difference. Therefore, the condition of sphericity has not been met.
b. Is there a main effect of gender? Is so, explain the effect. Use post hoc tests when necessary (or explain why they are not required in this specific case).
In this case, there is no main effect of gender since gender has a p-value of .504 which means that this is not significant at the 5% level. Also, in this case, the effect is not significant so there is no need for a post hoc test. Moreover, if the effect was significant, then we would not be able to perform the post hoc test since we only have two categories. Post hoc can be run if there are more than two classifications.
c. Is there a main effect tie (i.e. an increase in scores from Week 0 to Week 12)? If so, explain the effect. Use post hoc tests when necessary (or explain why they are not required in this specific case). Examine the output carefully and give as much detail as possible in your findings.
|
Tests of Within-Subjects Effects
|
|
Measure: MEASURE_1
|
|
Source
|
Type III Sum of Squares
|
df
|
Mean Square
|
F
|
Sig.
|
|
Score
|
Sphericity Assumed
|
3246.536
|
6
|
541.089
|
20.609
|
.000
|
|
Greenhouse-Geisser
|
3246.536
|
2.646
|
1227.164
|
20.609
|
.000
|
|
Huynh-Feldt
|
3246.536
|
4.045
|
802.659
|
20.609
|
.000
|
|
Lower-bound
|
3246.536
|
1.000
|
3246.536
|
20.609
|
.001
|
|
Score * Gender
|
Sphericity Assumed
|
182.155
|
6
|
30.359
|
1.156
|
.342
|
|
Greenhouse-Geisser
|
182.155
|
2.646
|
68.853
|
1.156
|
.341
|
|
Huynh-Feldt
|
182.155
|
4.045
|
45.035
|
1.156
|
.344
|
|
Lower-bound
|
182.155
|
1.000
|
182.155
|
1.156
|
.307
|
|
Error(Score)
|
Sphericity Assumed
|
1575.321
|
60
|
26.255
|
|
|
|
Greenhouse-Geisser
|
1575.321
|
26.456
|
59.546
|
|
|
|
Huynh-Feldt
|
1575.321
|
40.447
|
38.948
|
|
|
|
Lower-bound
|
1575.321
|
10.000
|
157.532
|
|
|
|
Pairwise Comparisons
|
|
Measure:MEASURE_1
|
|
(I) SCORE
|
(J) SCORE
|
Mean Difference (I-J)
|
Std. Error
|
Sig.a
|
95% Confidence Interval for Differencea
|
|
Lower Bound
|
Upper Bound
|
|
|
1
|
|
2
|
-4.500
|
3.383
|
.213
|
-12.039
|
3.039
|
|
3
|
-6.062*
|
2.412
|
.031
|
-11.437
|
-.688
|
|
4
|
-5.312*
|
2.117
|
.031
|
-10.029
|
-.596
|
|
5
|
-9.937*
|
2.693
|
.004
|
-15.938
|
-3.937
|
|
6
|
-15.813*
|
2.683
|
.000
|
-21.791
|
-9.834
|
|
7
|
-20.563*
|
3.524
|
.000
|
-28.415
|
-12.710
|
|
2
|
|
1
|
4.500
|
3.383
|
.213
|
-3.039
|
12.039
|
|
3
|
-1.562
|
1.542
|
.335
|
-4.998
|
1.873
|
|
4
|
-.812
|
2.324
|
.734
|
-5.990
|
4.365
|
|
5
|
-5.437*
|
1.914
|
.018
|
-9.701
|
-1.174
|
|
6
|
-11.313*
|
2.063
|
.000
|
-15.909
|
-6.716
|
|
7
|
-16.063*
|
3.154
|
.000
|
-23.089
|
-9.036
|
|
3
|
|
1
|
6.062*
|
2.412
|
.031
|
.688
|
11.437
|
|
2
|
1.562
|
1.542
|
.335
|
-1.873
|
4.998
|
|
4
|
.750
|
1.097
|
.510
|
-1.693
|
3.193
|
|
5
|
-3.875*
|
1.058
|
.004
|
-6.233
|
-1.517
|
|
6
|
-9.750*
|
1.336
|
.000
|
-12.726
|
-6.774
|
|
7
|
-14.500*
|
2.739
|
.000
|
-20.603
|
-8.397
|
|
4
|
|
1
|
5.312*
|
2.117
|
.031
|
.596
|
10.029
|
|
2
|
.812
|
2.324
|
.734
|
-4.365
|
5.990
|
|
3
|
-.750
|
1.097
|
.510
|
-3.193
|
1.693
|
|
5
|
-4.625*
|
1.019
|
.001
|
-6.895
|
-2.355
|
|
6
|
-10.500*
|
1.202
|
.000
|
-13.177
|
-7.823
|
|
7
|
-15.250*
|
2.711
|
.000
|
-21.291
|
-9.209
|
|
5
|
|
1
|
9.937*
|
2.693
|
.004
|
3.937
|
15.938
|
|
2
|
5.437*
|
1.914
|
.018
|
1.174
|
9.701
|
|
3
|
3.875*
|
1.058
|
.004
|
1.517
|
6.233
|
|
4
|
4.625*
|
1.019
|
.001
|
2.355
|
6.895
|
|
6
|
-5.875*
|
.716
|
.000
|
-7.471
|
-4.279
|
|
7
|
-10.625*
|
2.065
|
.000
|
-15.226
|
-6.024
|
|
6
|
|
1
|
15.813*
|
2.683
|
.000
|
9.834
|
21.791
|
|
2
|
11.313*
|
2.063
|
.000
|
6.716
|
15.909
|
|
3
|
9.750*
|
1.336
|
.000
|
6.774
|
12.726
|
|
4
|
10.500*
|
1.202
|
.000
|
7.823
|
13.177
|
|
5
|
5.875*
|
.716
|
.000
|
4.279
|
7.471
|
|
7
|
-4.750*
|
1.705
|
.019
|
-8.548
|
-.952
|
|
7
|
|
1
|
20.563*
|
3.524
|
.000
|
12.710
|
28.415
|
|
2
|
16.063*
|
3.154
|
.000
|
9.036
|
23.089
|
|
3
|
14.500*
|
2.739
|
.000
|
8.397
|
20.603
|
|
4
|
15.250*
|
2.711
|
.000
|
9.209
|
21.291
|
|
5
|
10.625*
|
2.065
|
.000
|
6.024
|
15.226
|
|
6
|
4.750*
|
1.705
|
.019
|
.952
|
8.548
|
|
Based on estimated marginal means
|
|
a. Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments).
|
|
*. The mean difference is significant at the .05 level.
|
The mean effect of score is significant at 5% level of significance. From the table, I am able to ascertain the F-value for the score factor, its associated significance level, and the effect size (Partial Eta Squared). Because my data violated the assumption of sphericity, I examine the values in the Greenhouse-Geisser row (if sphericity had not been violated, I would have looked under the Sphericity Assumed row). Thus, I can report that when using an ANOVA with repeated measures with a Greenhouse-Geiseer correction, the mean scores for weeks were statistically significantly different (F(2.646,60) = 20.609, p < 0.0005.
In addition, in looking at the above Paired Comparisons Table, I recognize the labels associated with score in the experiment from the Within-Subject Factors Table. This is a table which gives the significance level of differences between the individual time points. It can be seen that there was a significant difference in scores in training from pre to week 12. The p-values indicate the significant differences between the groups.