Running head: ANCOVA AND FACTORIAL ANCOVA 1

Section 3 - Activity 6 -- waltzekcSTAT8028-6NORTHCENTRAL UNIVERSITYASSIGNMENT COVER SHEET

Learner: Waltzek, Chris

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Business Statistics Assignment Number 6

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ANCOVA AND FACTORIAL ANCOVA 2

ANCOVA and Factorial ANCOVA

Chris G. Waltzek

Northcentral University

ANCOVA AND FACTORIAL ANCOVA 3

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 4

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 StatisticsDependent Variable: Math_Score

Classroom size Gender MeanStd.

Deviation N10 or less Female 93.8000 3.93841 10

Male 92.7000 3.43350 10Total 93.2500 3.64005 20

11-19 Female 88.5000 3.97911 10Male 89.7000 2.40601 10Total 89.1000 3.25900 20

20 or more Female 79.2000 4.18463 10Male 91.2000 3.22490 10Total 85.2000 7.14953 20

Total Female 87.1667 7.26865 30Male 91.2000 3.19914 30Total 89.1833 5.92750 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

ANCOVA AND FACTORIAL ANCOVA 6

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 OutputTests of Between-Subjects Effects

Dependent Variable: Math_Score

SourceType III Sum

of Squares df Mean Square F Sig.Corrected Model 1381.483a 5 276.297 21.576 .000Intercept 477220.017 1 477220.017 37266.639 .000Classroom 648.233 2 324.117 25.311 .000Gender 244.017 1 244.017 19.056 .000Classroom * Gender

489.233 2 244.617 19.102 .000

Error 691.500 54 12.806Total 479293.000 60Corrected Total 2072.983 59a. R Squared = .666 (Adjusted R Squared = .636)Note: Created with IBM SPSS Statistics Version 19. Copyright 1989 by IBM Company.

ANCOVA AND FACTORIAL ANCOVA 7

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_ScoreF 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 * GenderNote: Created with IBM SPSS Statistics Version 19. Copyright 1989 by IBM Company.

ANCOVA AND FACTORIAL ANCOVA 8

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.

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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 ComparisonsDependent Variable: Math_Score

(I) Classroom size

(J) Classroom size

Mean Difference

(I-J)Std.

Error Sig.

95% Confidence Interval

Lower Bound

Upper Bound

Bonferroni

10 or less 11-19 4.1500* 1.13162

.002 1.3539 6.9461

20 or more 8.0500* 1.13162

.000 5.2539 10.8461

11-19 10 or less -4.1500* 1.13162

.002 -6.9461 -1.3539

20 or more 3.9000* 1.13162

.003 1.1039 6.6961

20 or more 10 or less -8.0500* 1.13162

.000 -10.8461 -5.2539

11-19 -3.9000* 1.13162

.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 Mean Std. Error

95% Confidence IntervalLower Bound

Upper Bound

10 or less Female 93.800 1.132 91.531 96.069Male 92.700 1.132 90.431 94.969

11-19 Female 88.500 1.132 86.231 90.769Male 89.700 1.132 87.431 91.969

20 or more Female 79.200 1.132 76.931 81.469Male 91.200 1.132 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 VariancesDependent 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.

ANCOVA AND FACTORIAL ANCOVA 15

Table 1.7. Mock ANCOVA Output Tests of Between-Subjects Effects

Dependent Variable: Portfolio Return

SourceType III Sum

of Squares df Mean Square F Sig.Corrected Model

51.058 3 10.640 3.500 .030

Intercept 76.069 1 76.069 25.020 .000Trend 20.185 1 12.593 4.142 .027S&P_500Port_ReturnTrend * S&P_500

12.05615.25515.321

111

14.32513.35913.356

4.2104.3104.320

.035

.048

.049

Error 49.047 26 3.040Total 683.000 31Corrected Total 100.105 29Note: 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

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:

ANCOVA AND FACTORIAL ANCOVA 17

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&