Running head: ANCOVA AND FACTORIAL ANCOVA
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Section 3 - Activity 6 -- waltzekcSTAT8028-6
NORTHCENTRAL UNIVERSITY
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Learner: Waltzek, Chris
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STAT8028-6
Dr. William Forrester
Business Statistics
Assignment Number 6
Learner Comments: Hello Dr. Forrester and thank you for your helpful / insightful feedback.
C.W.
ANCOVA AND FACTORIAL ANCOVA
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ANCOVA and Factorial ANCOVA
Chris G. Waltzek
Northcentral University
ANCOVA AND FACTORIAL ANCOVA
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Abstract
Exploratory data analysis (EDA) is performed on all the variables in the Activity 6.sav data set,
in this brief paper. The results are examined by group with the appropriate graphs. A brief
analysis of the data is provided. The descriptive statistics for the sample are presented. A
factorial ANOVA is performed using Activity 6.sav data set. The main effect of gender and
classroom size is examined. Post hoc tests are included. Interaction between the two variables is
examined. The researcher’s hypothesis, that girls would do better than boys in classrooms with
fewer students, is confirmed. My area of research is restated. One mock independent variable
and two mock dependent variables are identified and a mock ANCOVA is performed. The
hypothetical output is included.
ANCOVA AND FACTORIAL ANCOVA
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ANCOVA & Factorial ANOVA
Exploratory Data Analysis
Figure 1.1 illustrates the three classroom sizes and the resulting math scores further delineated
with blue / green labels for females / males respectively. When the classroom size is 10 or less
the female test score mean is better than that of males. However, as the classroom size increases
the female test scores decline as well as relative to that of male test scores. The exploratory data
analysis supports the researcher’s assertion that classroom size has a significant impact upon
math scores.
Figure 1.1. Math Scores - Classroom size: 10 or Less
Figure 1.1. Created with IBM SPSS Statistics Version 19. Copyright 1989 by IBM Company.
Next the mean / standard deviation for each classroom size is explored by gender. Table 1.1
further corroborates the researcher’s assumption that female math scores suffer as the classroom
size increases, particularly when compared to that of male test scores.
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Table 1.1 Classroom Size / Math Scores by
Gender
Descriptive Statistics
Dependent Variable: Math_Score
Classroom size Gender
10 or less
Female
Male
Total
11-19
Female
Male
Total
20 or more
Female
Male
Total
Total
Female
Male
Total
Mean
93.8000
92.7000
93.2500
88.5000
89.7000
89.1000
79.2000
91.2000
85.2000
87.1667
91.2000
89.1833
Std.
Deviation
3.93841
3.43350
3.64005
3.97911
2.40601
3.25900
4.18463
3.22490
7.14953
7.26865
3.19914
5.92750
N
10
10
20
10
10
20
10
10
20
30
30
60
Note: Created with IBM SPSS Statistics Version 19. Copyright 1989 by IBM Company.
Factorial ANOVA
It is essential to determine whether or not the male and female math scores differ
significantly. Table 1.2 reveals the t- score, p < .05, indicating that male / female mean test
scores are satisfactorily dissimilar.
The ANOVA output in Table 1.2 shows that the covariate p < .05 significantly predicts the
dependent variable. Thus the math scores are influenced by the classroom size. I created a
scatterplot with the covariate and dependent variable. The interpolation lines in Figure 1.2
further illustrate how female math scores decline as class size increases. As long as the
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classroom size remains small, 10 or less there is not a noticeable effect. But as the classroom
size increases the blue female scores decline indicating a significant drop.
Table 1.2. Factorial ANOVA Output
Tests of Between-Subjects Effects
Dependent Variable: Math_Score
Type III Sum
Mean
Source
of Squares
df
Square
a
F
Sig.
Corrected Model
1381.483
5
276.297
21.576
Intercept
477220.017
1 477220.017 37266.639
Classroom
648.233
2
324.117
25.311
Gender
244.017
1
244.017
19.056
Classroom *
489.233
2
244.617
19.102
Gender
Error
691.500
54
12.806
Total
479293.000
60
Corrected Total
2072.983
59
a. R Squared = .666 (Adjusted R Squared = .636)
Note: Created with IBM SPSS Statistics Version 19. Copyright 1989 by IBM
Company.
.000
.000
.000
.000
.000
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Figure 1.2. Math Score / Classroom Size
Figure 1.2. Created with IBM SPSS Statistics Version 19. Copyright 1989 by IBM Company.
Table 1.3 reveals that Levene’s test is not significant (F(5, 54) = 1.21, p > .05) indicating that the
assumption of homogeneity of variance is satisfied.
Table 1.3. Levene's Test of Equality of
Error Variances
Dependent Variable: Math_Score
F
df1
df2
Sig.
.822
5
54
.539
Tests the null hypothesis that the error
variance of the dependent variable is equal
across groups.
a. Design: Intercept + Classroom + Gender
+ Classroom * Gender
Note: Created with IBM SPSS Statistics
Version 19. Copyright 1989 by IBM
Company.
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Main Effect
In order to determine the effect sizes of the factorial ANOVA variables, gender, classroom
size and gender * classroom size, equations 1.1 - 1.3 (Field, 2009) are utilized.
1.1.
1.2.
1.3.
The main effect of gender: (F (1, 54) = 19.06, p < .001, w = .11) indicates that the gender is
significant, with a small effect. Even when the classroom size covariate is held constant, gender
has a significant impact on test scores. To better understand the impact of gender on math
scores, Figure 1.3 illustrates the gender effect without the classroom size component. Clearly
gender is a factor of math scores.
ANCOVA AND FACTORIAL ANCOVA
Figure 1.3. Impact of Gender on Math Scores
Figure 1.3. Created with IBM SPSS Statistics Version 19. Copyright 1989 by IBM Company.
Post Hoc Test
The Bonferroni post hoc test in Table 1.4 is significant p < .05 confirming the finding that
female student math scores suffer as class size increases. However, the post hoc test does not
take into account the interaction between gender and classroom size (Field, 2009).
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Table 1.4. Post Hoc Test
Multiple Comparisons
Dependent Variable: Math_Score
(I) Classroom (J) Classroom
size
size
Bonferr 10 or less
11-19
oni
20 or more
11-19
10 or less
20 or more
20 or more
10 or less
11-19
Mean
Difference Std.
(I-J)
Error
*
4.1500 1.1316
2
8.0500* 1.1316
2
-4.1500* 1.1316
2
3.9000* 1.1316
2
*
-8.0500 1.1316
2
-3.9000* 1.1316
2
Sig.
.002
95% Confidence
Interval
Lower
Upper
Bound
Bound
1.3539
6.9461
.000
5.2539
10.8461
.002
-6.9461
-1.3539
.003
1.1039
6.6961
.000
-10.8461
-5.2539
.003
-6.6961
-1.1039
Based on observed means.
The error term is Mean Square (Error) = 12.806.
*.The mean difference is significant at the .05 level.
Note: Created with IBM SPSS Statistics Version 19. Copyright 1989 by IBM Company.
Classroom Size Effect
The classroom size is also significant (F (2, 54) = 25.31, p < .001, w = .31) with a medium
large effect on math scores. The negative relationship between the two variables signifies that as
class size increases math scores decline. To better understand the impact of classroom size on
math scores, Figure 1.4 illustrates the effect of classroom size without the gender component.
Clearly classroom size is a significant contributor to math scores.
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Figure 1.4. Impact of Classroom Size on Math Scores
Figure 1.4. Created with IBM SPSS Statistics Version 19. Copyright 1989 by IBM Company.
Interaction Effect: Classroom Size / Gender
Table 1.2 shows that the gender * classroom size interaction variable resulted in (F (2, 54) =
19.10, p < .001, w =.22 (medium effect). Clearly as class size increases to 20 or more, female
scores drop abruptly. The finding is further substantiated by the Bonferroni post hoc test in
Table 1.4. In addition, Table 1.5 and Figure 1.5 illustrate the dramatic drop off in female math
scores when class size increases to 20 or more.
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Table 1.5. Classroom Size x Gender
Dependent Variable: Math_Score
Classroom size Gender
10 or less
Female
Male
11-19
Female
Male
20 or more
Female
Male
Mean
93.800
92.700
88.500
89.700
79.200
Std.
Error
1.132
1.132
1.132
1.132
1.132
91.200
1.132
95% Confidence Interval
Lower
Upper
Bound
Bound
91.531
96.069
90.431
94.969
86.231
90.769
87.431
91.969
76.931
81.469
88.931
93.469
Note: Created with IBM SPSS Statistics Version 19. Copyright 1989 by IBM Company.
Figure 1.5. Impact of Classroom Size & Gender on Math Scores
Figure 1.5. Created with IBM SPSS Statistics Version 19. Copyright 1989 by IBM Company.
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Therefore it is safe to assume that there is a significant interaction between gender and
classroom size. To properly assess the relationship between gender and classroom size Figure
1.1 reveals that math scores vary significantly for men and women as classroom size increases,
i.e. the difference between the blue and green bars changes significantly for men and women
(Field, 2009). There appears to be ample evidence in support of the researcher’s initial
hypothesis.
ANCOVA Research Applications
Research - Dependent / Independent Variables
My area of interest involves adjusting the CAPM model with a trend component, resulting in
the CAPMT. The dependent variables are total portfolio return and the S&P 500 return. The
independent variable is the market trend.
Mock ANCOVA
According to Field (2009) the covariate (portfolio return) must be autonomous from the
independent variable (trend). Field suggests using the t- test, ANOVA or the ANCOVA. If it is
determined that the means do not differ significantly then the covariate may be used in the
model. The main effect of the trend is not significant, F (1, 58) = 1.05, p > .05. The means do
not differ significantly and it is safe to use the covariate. Since there was a preexisting
hypothesis, post hoc tests are not performed.
Levene’s test in Table 1.6 is not significant (F (2, 27) = 4.62, p > .05) indicating that the
assumption of homogeneity of variance is satisfied.
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Table 1.6. Mock Levene's Test of
Equality of Error Variances
Dependent Variable: Libido
F
df1
df2
Sig.
4.618
2
27
.29
Tests the null hypothesis that the error
variance of the dependent variable is
equal across groups.
Note: Created with IBM SPSS Statistics
Version 19. Copyright 1989 by IBM
Company.
Since each group contains equal numbers of participants the effect size is computed using
omega squared (w²) in equation 1.4.
1.4.
The main effect is determined using data from Table 1.7. The trend predictor is significant: (F
(1, 26) = 4.14, p < .05, w = .37 (medium-large effect size)) indicating that the market trend has a
significant impact on portfolio returns. The S&P 500 covariate has a significant positive
relationship and a large effect size (F (1, 26) = 4.21, p < .05, w = .68) indicating that as the
general market increases, portfolio returns rise substantially. The dependent variable has a
positive relationship with both covariates indicating that as either increases, expected portfolio
returns are enhanced.
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Table 1.7. Mock ANCOVA Output
Tests of Between-Subjects Effects
Dependent Variable: Portfolio Return
Type III Sum
Mean
Source
of Squares
df
Square
F
Sig.
Corrected
51.058
3
10.640
3.500
.030
Model
Intercept
76.069
1
76.069
25.020
.000
Trend
20.185
1
12.593
4.142
.027
S&P_500
12.056
1
14.325
4.210
.035
Port_Return
15.255
1
13.359
4.310
.048
Trend *
15.321
1
13.356
4.320
.049
S&P_500
Error
49.047
26
3.040
Total
683.000
31
Corrected
100.105
29
Total
Note: Created with IBM SPSS Statistics Version 19. Copyright 1989 by IBM
Company.
Variable Interaction Effect
The independent variable and covariate, trend * S&P 500 resulted with (F (1, 26) = 4.32, p <
.05, w = .45 (large effect size)). Therefore it is safe to assume that there is a significant
interaction between the trend and S&P 500. Post hoc tests were not administered because a
preexisting hypothesis was applied.
Main Effect Findings
Judging by the significant values p < .05, which according to Kazmier (2003) is most popular
due to the ease of calculation, the market trend and the S&P 500 covariate significantly predicted
the dependent variable. The amount of variation accounted for in the model is 51 units; the
ANCOVA AND FACTORIAL ANCOVA
market trend comprised 20, the S&P return variable included 12 units and the interaction
variable accounted for 15 units. The covariate reduced the unexplained variance to 49 units.
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References
Field, A. (2009). Discovering statistics using SPSS. London, UK:
SAGE Publications Ltd. Retrieved from
http://www.coursesmart.com/9781847879073/
Kazmier, L. J. (2003). Schaum’s outline of theory and
problems of business statistics. New York,
NY: McGraw-Hill. Retrieved from
http://site.ebrary.com.proxy1.ncu.edu/lib/ncent/docDetail.action?
docID=10051516&