Exploratory Data Analysis, Factorial ANOVA and ANCOVA in Statistics

This article covers Exploratory Data Analysis, Factorial ANOVA and ANCOVA in Statistics with solved examples and tables. It includes analysis of a dataset with variables like Gender, Classroom and Math Score, and also covers ANCOVA with a mock analysis of performance scores in training with Gender and Training Class as independent variables. Subject: Statistics, Course Code: STAT2, University: Not mentioned

Running head: STATISTICS 2
Statistics 2
Name of the student
Name of the university
Author’s note

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Table of Contents
1. Exploratory Data analysis:.......................................................................................................2
Part a)...........................................................................................................................................2
Part b)...........................................................................................................................................4
Part c)...........................................................................................................................................5
2. Factorial ANOVA (analysis of variance):................................................................................5
Part a)...........................................................................................................................................6
Part b)...........................................................................................................................................6
Part c)...........................................................................................................................................8
Part d)...........................................................................................................................................9
Part e).........................................................................................................................................11
3. Analysis of covariance (ANCOVA):.....................................................................................11
Interpretation of Mock ANCOVA output table:........................................................................16
References:....................................................................................................................................18

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1. Exploratory Data analysis:
In the given data set, mainly three variables are undertaken that are “Gender”, “Classroom”
and “Math_Score”. Among 60 participants, 30 are males and rest of 30 are females. Total 60
students attend their classrooms in three classes with 20 students in each class. There are total six
groups present in the data set according to the paired values of “Gender” and “Classroom”.
The two types of “Gender” are “Male” (M) and “Female” (F). The ordinal variable
“Classroom” is classified in three kinds. “1” refers Small classroom where not more than 10
children could be accommodated. “2” refers Medium classroom where the accommodation is for
11 to 19 students and “3” refers Large classroom where the accommodation is for more than 20
students.
Part a)
Table 1: Table of Between-Subjects Factors
Table 2: Exploratory data summary of Mathematics Score

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Figure 1: Profile plot of Estimated Marginal Means of Scores of Mathematics with respect
to Gender

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Figure 2: Profile plot of Estimated Marginal Means of Scores of Mathematics with respect
to Classroom
Part b)
The SPSS generated profile plot shows that both males and females average score is
highest in mathematics in case of small classroom. For the females, the average score in
mathematics decreases rapidly as the size of classroom increases. However, for the males, the
average score of mathematics first decreases as classroom size turns out to be small to medium
and then increases as the size of classroom becomes Large. The difference of average scores of
mathematics for male and female in small and medium classroom are not high. However, the

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difference of average scores of mathematics for male and female in large classroom is
significantly high.
From the other perspective, the average score of mathematics in small classroom is higher for
females than males. However, Males score in mathematics more than Females score in medium
and large size classroom.
Part c)
Table 3: Descriptive Statistics of Mathematics Score
The average Scores in mathematics for the accounted six groups are –
a) Mean scores of “Females” whose classroom was “small” = 93.8
b) Mean scores of “Females” whose classroom was “medium” = 88.5
c) Mean scores of “Females” whose classroom was ‘large” = 79.2

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d) Mean scores of “Males” whose classroom was “small” = 92.7
e) Mean scores of “Males” whose classroom was “medium” = 89.7
f) Mean scores of “Males” whose classroom was “large” = 91.2
2. Factorial ANOVA (analysis of variance):
Table 4: Tests of Between-Subjects Effects in Factorial ANOVA method
Part a)
According to the Factorial ANOVA table, the main effect of the factor “Gender” has p-
value 0.0 (<0.05). Hence, the effect of “Gender” is insignificant. It could be inferred that the
average scores in mathematics in both kinds of “Gender” are unequal. The main effect of gender
is found statistically significant in this analysis.
Part b)
The p-value of main effect of the factor “Classroom” has p-value 0.0 (<0.05). Hence, the
effect of “Classroom” is also insignificant. We can conclude that the average scores in

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mathematics for all the three types of classroom sizes are not equal (Source:
Academic.udayton.edu., 2018). The main effect of classroom size is also found statistically
significant in this analysis.
Table 5: Table of Estimated Marginal means of Gender
Overall, the average score of mathematics in case of males is greater than females (91.2>87.167).
Table 6: Table of Estimated Marginal means of Classroom
The students of small classroom scores is greater than the scores of students of medium
classroom (93.25>89.1). The scores in mathematics are least for the students of large classroom
(85.2).
Table 7: Table of Estimated Marginal means of interaction effect of Gender and Classroom

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Female students who get learning in small classroom generally get maximum score in
mathematics. On the other hand, female students who get learning in large classroom, score least
marks in the mathematics exam.
Table 8: Table of Post Hoc test of Multiple Comparisons of Classrooms

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Part c)
The previous two-way ANOVA table indicates p-value of interaction effect of “Gender”
and “Classroom” as 0.0 (<0.05). Therefore, the interaction of these two factors has statistical
significance. It could be concluded that the effect of types of gender and classrooms are not
same. Here, the interaction effect of these two factors is statistically significant too.
The difference average scores in mathematics of the small and large classroom students
are significantly high (±8.05) followed by small and medium classroom students (±4.15). All the
p-values (0.0, 0.002 and 0.003) of differences of mean score are less than 0.05
(Statistics.laerd.com., 2018). Therefore, we can conclude that the differences of average scores in
mathematics are significant to each other.
Table 9: Table of Homogeneous Subsets of the Mathematics Scores
The average scores of mathematics for three sizes of classrooms are undertaken here. The
table of homogeneity provides the significant p-value = 1 (>0.05). Therefore, the assertion of
absence of homogeneity in all the three types of classrooms could be rejected (Lomax and
Surman, 2007).

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Part d)
The hypotheses that are to be tested here are:
Null hypothesis (H0): The girls score higher than boys in the maths exam when small classroom
is accommodated.
Alternative hypothesis (HA): The girls score same as boys in the maths exam when small
classroom is accommodated.
Table 10: Tables for testing the equality of averages of scores of maths for the small
classroom among 10 boys and 10 girls

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The average of score of maths in small classroom is 93.8 for females and 92.7 for males. The
ANOVA table which verifies the equality of averages provides the F-statistic 0.443 with
calculated p-value = 0.514. The p-value is greater than 0.05. Therefore, we accept the null
hypothesis that girls perform better than boys do when capacity of classroom is fewer (small
classroom).
Part e)
According to the two-way factorial ANOVA table as well as the Post-hoc test and multiple
comparison test, it is evident that the both the main effects “Gender” and “Classroom” have
significant associations with score of mathematics. Gender and Classroom in collaborate have
their significant interaction effect that also takes part in the deviation of maths score (Field,
2013).
Simple main effects analysis indicated that-

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a) Males and females (2 groups) have difference in their maths score.
b) The three kinds of classrooms like small, medium and large (3 groups) have the
differences in maths score within themselves.
c) Due to significant interaction effect, for both types of genders and all types of classrooms
(6 groups) have the differences in their maths score.
3. Analysis of covariance (ANCOVA):
Analysis of covariance (ANCOVA) has features of both ANOVA and regression. An
ANCOVA model has more or one additional numerical variable known as covariates related to
the response variable. We include covariates to decrease the variance in the error terms and to
have more precise measurement of the treatment effects. Therefore, ANCOVA examines the
main and interaction effects of the factors to control the effects of the covariate.
My research area of interest is the requirement for additional professional development
training for police officers in United States. In this mock analysis, the dependent variable is
“Performance_Score” and the independent variable is “Gender” and “Training_Class”. An
ANCOVA table is going to be executed that indicates both the independent variable as well as
the significant covariate predictor of the dependent variable.
Table 12: Table of Homogeneity
Levene's Test of Equality of Error Variancesa
Dependent Variable: Performance score in training
F df1 df2 Sig.
.658 5 54 .657
Tests the null hypothesis that the error variance of the dependent variable is equal across
groups.
a. Design: Intercept + Sex + training_class + Sex * training_class

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The homogeneity testing with the help of Levene’s Test (equality of error variances)
shows that F (5, 54) = 0.658 with the significant p-value 0.657. The p-value is greater than 0.05.
Therefore, it could be said that average of performance scores in training period considering both
Sex and training class in all the six groups are not homogeneous (Source: Sheffield.ac.uk., 2018).
Table 13: Tests of Between-Subjects Effects of Scores of Mathematics
Tests of Between-Subjects Effects
Dependent Variable: Performance score in training
Source Type III Sum of
Squares
df Mean Square F Sig.
Corrected Model 279.133a 5 55.827 .317 .901
Intercept 317699.267 1 317699.267 1803.667 .000
Sex 160.067 1 160.067 .909 .345
training_class 110.533 2 55.267 .314 .732
Sex * training_class 8.533 2 4.267 .024 .976
Error 9511.600 54 176.141
Total 327490.000 60
Corrected Total 9790.733 59
a. R Squared = .029 (Adjusted R Squared = -.061)
The ANCOVA table as output tells us that the p-value of “Sex” is 0 and “Training_class”
is 0.732. Hence, these Sex factor differs in effectiveness to the score of performance in training
however the factor training class do not differ in effectiveness to the performance score. The
interaction effect of Gender and Classroom has significant p-value 0.976, which is greater than
0.05. The joint interaction effect of these factors does not create differences to the performance
score.

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It could be inferred that the gender, classroom and their interaction effect do not create
differences to the score of training performance.
We can reject the null hypothesis (H0): RL2 = RS2. Hence, it could be inferred that the interactions
contribute to the model and homogeneity of the regression slope is unsustainable.
Table 14: Table of Estimated Marginal Means of Mathematics Scores with respect to
Gender
Sex
Dependent Variable: Performance score in training
Sex Mean Std. Error 95% Confidence Interval
Lower Bound Upper Bound
F 71.133 2.423 66.275 75.991
M 74.400 2.423 69.542 79.258
Table 15: Pair wise Comparisons of estimated Marginal Means of Mathematics Scores with
respect to Gender
Pairwise Comparisons
Dependent Variable: Performance score in training
(I) Sex (J) Sex Mean
Difference (I-J)
Std. Error Sig.a 95% Confidence Interval for
Differencea
Lower Bound Upper Bound
F M -3.267 3.427 .345 -10.137 3.604
M F 3.267 3.427 .345 -3.604 10.137
Based on estimated marginal means
a. Adjustment for multiple comparisons: Bonferroni.

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The difference of average scores of training performance between both types of genders
is -3.267. Hence, according to the p-value (0.345>0.005), the mean difference is insignificant.
Therefore, the average performance score for males and females are not different with 95%
probability.
Table 16: Univariate test of estimated marginal means of Mathematics Scores with respect
to Gender
Univariate Tests
Dependent Variable: Performance score in training
Sum of Squares df Mean Square F Sig.
Contrast 160.067 1 160.067 .909 .345
Error 9511.600 54 176.141
The F tests the effect of Sex. This test is based on the linearly independent pairwise
comparisons among the estimated marginal means.
The significant p-value of the F-statistic (0.909) is 0.345. Therefore, the mean scores of
training performance of males and average scores of training performance of females are not
different from each other. The main effect to gender is found insignificant.
Table 17: Table of Estimated Marginal Means of Mathematics Scores with respect to
Classroom
Estimates
Dependent Variable: Performance score in training
training_class Mean Std. Error 95% Confidence Interval
Lower Bound Upper Bound
1.00 74.300 2.968 68.350 80.250

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2.00 73.000 2.968 67.050 78.950
3.00 71.000 2.968 65.050 76.950
Table 18: Pair wise Comparisons of estimated Marginal Means of Mathematics Scores with
respect to Classroom
Pairwise Comparisons
Dependent Variable: Performance score in training
(I)
training_class
(J)
training_class
Mean
Difference
(I-J)
Std.
Error
Sig.a 95% Confidence Interval
for Differencea
Lower
Bound
Upper
Bound
1.00 2.00 1.300 4.197 1.000 -9.070 11.670
3.00 3.300 4.197 1.000 -7.070 13.670
2.00 1.00 -1.300 4.197 1.000 -11.670 9.070
3.00 2.000 4.197 1.000 -8.370 12.370
3.00 1.00 -3.300 4.197 1.000 -13.670 7.070
2.00 -2.000 4.197 1.000 -12.370 8.370
Based on estimated marginal means
a. Adjustment for multiple comparisons: Bonferroni.
The difference of average scores of training performance between all types of training
classes has the p-value (1>0.05). Hence, the mean difference is insignificant. Therefore, the
average performance score for different training classes are same with 95% probability.
Table 19: Univariate test of estimated marginal means of Mathematics Scores with respect
to Classroom

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Univariate Tests
Dependent Variable: Performance score in training
Sum of Squares df Mean Square F Sig.
Contrast 110.533 2 55.267 .314 .732
Error 9511.600 54 176.141
The F tests the effect of training_class. This test is based on the linearly independent pairwise
comparisons among the estimated marginal means.
The p-value of the F-statistic is 0.732, which is greater than 0.05. Hence, the average
scores of training performance for the three different training classes are unequal. The main
effect of classroom is also insignificant.
Table 20: Table of Post Hoc tests
Multiple Comparisons
Dependent Variable: Performance score in training
Tukey HSD
(I)
training_class
(J)
training_class
Mean
Difference
(I-J)
Std.
Error
Sig. 95% Confidence Interval
Lower
Bound
Upper
Bound
1.00 2.00 1.3000 4.19691 .949 -8.8145 11.4145
3.00 3.3000 4.19691 .713 -6.8145 13.4145
2.00 1.00 -1.3000 4.19691 .949 -11.4145 8.8145
3.00 2.0000 4.19691 .883 -8.1145 12.1145
3.00 1.00 -3.3000 4.19691 .713 -13.4145 6.8145
2.00 -2.0000 4.19691 .883 -12.1145 8.1145
Based on observed means.
The error term is Mean Square(Error) = 176.141.

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The table of post hoc test indicates that all the significant p-values are greater than 0.05.
Therefore, the main effects and joint interaction effects individually do not put significant effect
on performance score at additional professional development training.
Interpretation of Mock ANCOVA output table:
The two-way ANCOVA output table refers that the main effects of Sex and Training
class size are significant to the performance scores. The joint interaction effect of these two
independent factors in ANCOVA is also statistically significant. Both the two factors cannot help
to make differences in the performance score.
Due to the independent factors, scores of mathematics in the six groups are not found to
be homogeneous. The ANOVA table provides that the value of R2 = 0.029 and Adjusted R2 = -
0.61. Therefore, a positive and insignificant association (due to regression) between dependent
variable and independent variables is found in this ANCOVA table. Bonforroni interaction effect
is utilised to test the equality of means of the six groups at 5% level of significance. From both
the two-way ANOVA and two-way ANCOVA tables, we had the same interpretation that the
average scores in mathematics in all the six groups are unequal. The significant effect of gender
and classroom are the reason behind it.
The male police officers would be equally efficient as female police officers would. The
performance average score of males would be equal as females for additional professional
development training. Another dependent factor of the topic is the difficulty level of the
professional development training. The week linear relevance of dependent and independent
variables indicates that Sex effect as well as the difficulty level of development training program
would not jointly perceive the additional professional development and performance score
significantly. The fixed effect of both factors and random effect covariance of both the factors
might not fulfill the requirement of additional professional development by training.

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References:
Academic.udayton.edu. (2018). Using SPSS for Factorial, Between-Subjects Analysis of
Variance. [online] Available at:
http://academic.udayton.edu/gregelvers/psy216/spss/2wayanovabs.htm.
Field, A. (2013). Discovering statistics using IBM SPSS statistics. sage.
Lomax, R. G., & Surman, S. H. (2007). FACTORIAL ANOVA IN SPSS. Real Data Analysis,
185.
Sheffield.ac.uk. (2018). Cite a Website - Cite This For Me. [online] Available at:
https://www.sheffield.ac.uk/polopoly_fs/1.531229!/file/MASH_ANCOVA_SPSS.pdf.
Statistics.laerd.com. (2018). Two-way ANOVA Output and Interpretation in SPSS Statistics -
Including Simple Main Effects | Laerd Statistics. [online] Available at:
https://statistics.laerd.com/spss-tutorials/two-way-anova-using-spss-statistics-2.php.