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ANCOVA and Factorial ANOVA: A Case Study Essay

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SPSS Assignment

Exploratory Data Analysis

Descriptives
Classroom size Statistic Std. Error
Math_Score 10 or less Mean 93.2500 .81394
95% Confidence Interval for Mean Lower Bound 91.5464
Upper Bound 94.9536
5% Trimmed Mean 93.2778
Median 93.0000
Variance 13.250
Std. Deviation 3.64005
Minimum 87.00
Maximum 99.00
Range 12.00
Interquartile Range 6.50
Skewness .002 .512
Kurtosis -1.080 .992
11-19 Mean 89.1000 .72873
95% Confidence Interval for Mean Lower Bound 87.5747
Upper Bound 90.6253
5% Trimmed Mean 89.1667
Median 89.5000
Variance 10.621
Std. Deviation 3.25900
Minimum 82.00
Maximum 95.00
Range 13.00
Interquartile Range 5.75
Skewness -.342 .512
Kurtosis -.292 .992
20 or more Mean 85.2000 1.59868
95% Confidence Interval for Mean Lower Bound 81.8539
Upper Bound 88.5461
5% Trimmed Mean 85.2222
Median 86.5000
Variance 51.116
Std. Deviation 7.14953
Minimum 72.00
Maximum 98.00
Range 26.00
Interquartile Range 11.25
Skewness -.177 .512
Kurtosis -.824 .992
Descriptives
Gender Statistic Std. Error
Math_Score Female Mean 87.1667 1.32707
95% Confidence Interval for Mean Lower Bound 84.4525
Upper Bound 89.8808
5% Trimmed Mean 87.3704
Median 88.5000
Variance 52.833
Std. Deviation 7.26865
Minimum 72.00
Maximum 98.00
Range 26.00
Interquartile Range 10.50
Skewness -.272 .427
Kurtosis -.698 .833
Male Mean 91.2000 .58408
95% Confidence Interval for Mean Lower Bound 90.0054
Upper Bound 92.3946
5% Trimmed Mean 91.0556
Median 91.0000
Variance 10.234
Std. Deviation 3.19914
Minimum 86.00
Maximum 99.00
Range 13.00
Interquartile Range 4.00
Skewness .557 .427
Kurtosis .244 .833
ANCOVA and Factorial ANOVA: A Case Study ANCOVA and Factorial ANOVA: A Case Study

The mean math score for the small classroom was 93.25 with a standard deviation (SD) of 3.64 and a standard error of 0.81. The mean math score for the medium classroom was 89.10 with a standard deviation of 3.26 and a standard error of 0.73. On the other hand, the mean math score for the large classroom was 85.20 with a standard deviation of 7.15 and a standard error of 1.59. The bar graph showed that the large classroom had the lowest math score.

The mean math score for females was 87.17 with a standard deviation of 7.27 and a standard error of 1.33. Conversely, the mean math score for males was 91.20 with a standard deviation of 3.19 and a standard error of 0.0.58. The bar graph showed that male scores were higher than female scores.

Factorial ANOVA

Between-Subjects Factors
Value Label N
Classroom size 1 10 or less 20
2 11-19 20
3 20 or more 20
Gender F Female 30
M Male 30
Descriptive Statistics
Dependent Variable: Math_Score
Classroom size Gender Mean Std. Deviation N
10 or less Female 93.8000 3.93841 10
Male 92.7000 3.43350 10
Total 93.2500 3.64005 20
11-19 Female 88.5000 3.97911 10
Male 89.7000 2.40601 10
Total 89.1000 3.25900 20
20 or more Female 79.2000 4.18463 10
Male 91.2000 3.22490 10
Total 85.2000 7.14953 20
Total Female 87.1667 7.26865 30
Male 91.2000 3.19914 30
Total 89.1833 5.92750 60
Levene’s Test of Equality of Error Variancesa
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
Tests of Between-Subjects Effects
Dependent Variable: Math_Score
Source Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared
Corrected Model 1381.483a 5 276.297 21.576 .000 .666
Intercept 477220.017 1 477220.017 37266.639 .000 .999
Classroom 648.233 2 324.117 25.311 .000 .484
Gender 244.017 1 244.017 19.056 .000 .261
Classroom * Gender 489.233 2 244.617 19.102 .000 .414
Error 691.500 54 12.806
Total 479293.000 60
Corrected Total 2072.983 59
a. R Squared =.666 (Adjusted R Squared =.636)
Estimates
Dependent Variable: Math_Score
Classroom size Mean Std. Error 95% Confidence Interval
Lower Bound Upper Bound
10 or less 93.250 .800 91.646 94.854
11-19 89.100 .800 87.496 90.704
20 or more 85.200 .800 83.596 86.804
Pairwise Comparisons
Dependent Variable: Math_Score
(I) Classroom size (J) Classroom size Mean Difference (I-J) Std. Error Sig.b 95% Confidence Interval for Differenceb
Lower Bound Upper Bound
10 or less 11-19 4.150* 1.132 .001 1.881 6.419
20 or more 8.050* 1.132 .000 5.781 10.319
11-19 10 or less -4.150* 1.132 .001 -6.419 -1.881
20 or more 3.900* 1.132 .001 1.631 6.169
20 or more 10 or less -8.050* 1.132 .000 -10.319 -5.781
11-19 -3.900* 1.132 .001 -6.169 -1.631
Based on estimated marginal means
*. The mean difference is significant at the.05 level.
b. Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments).
Univariate Tests
Dependent Variable: Math_Score
Sum of Squares df Mean Square F Sig. Partial Eta Squared
Contrast 648.233 2 324.117 25.311 .000 .484
Error 691.500 54 12.806
The F tests the effect of Classroom size. This test is based on the linearly independent pairwise comparisons among the estimated marginal means.
Estimates
Dependent Variable: Math_Score
Gender Mean Std. Error 95% Confidence Interval
Lower Bound Upper Bound
Female 87.167 .653 85.857 88.477
Male 91.200 .653 89.890 92.510
Pairwise Comparisons
Dependent Variable: Math_Score
(I) Gender (J) Gender Mean Difference (I-J) Std. Error Sig.b 95% Confidence Interval for Differenceb
Lower Bound Upper Bound
Female Male -4.033* .924 .000 -5.886 -2.181
Male Female 4.033* .924 .000 2.181 5.886
Based on estimated marginal means
*. The mean difference is significant at the.05 level.
b. Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments).
Univariate Tests
Dependent Variable: Math_Score
Sum of Squares df Mean Square F Sig. Partial Eta Squared
Contrast 244.017 1 244.017 19.056 .000 .261
Error 691.500 54 12.806
The F tests the effect of Gender. This test is based on the linearly independent pairwise comparisons among the estimated marginal means.
3. Classroom size * Gender
Dependent Variable: Math_Score
Classroom size Gender Mean Std. Error 95% Confidence Interval
Lower Bound Upper Bound
10 or less Female 93.800 1.132 91.531 96.069
Male 92.700 1.132 90.431 94.969
11-19 Female 88.500 1.132 86.231 90.769
Male 89.700 1.132 87.431 91.969
20 or more Female 79.200 1.132 76.931 81.469
Male 91.200 1.132 88.931 93.469
Math_Score
Student-Newman-Keulsa,b
Classroom size N Subset
1 2 3
20 or more 20 85.2000
11-19 20 89.1000
10 or less 20 93.2500
Sig. 1.000 1.000 1.000
Means for groups in homogeneous subsets are displayed.
Based on observed means.
The error term is Mean Square(Error) = 12.806.
a. Uses Harmonic Mean Sample Size = 20.000.
b. Alpha =.05.

There is a significant main effect of gender F(1, 54) = 19.056, p < 0.05. Post hoc tests were not required in this case because there were fewer than three groups.

There is a main effect of classroom size F(2, 54) = 25.311, p < 0.05, indicating a significant difference between small classroom (M = 93.25, SD = 3.64), medium classroom (M = 89.10, SD = 3.26), and large classroom (M = 81.20, SD = 7.14). The Least Significant Difference (LSD) post hoc test showed that the difference between small and medium classroom was significant (p = 0.001). Similarly, significant differences were observed between small and large classroom (p <0.05) as well as between medium and large classroom (p = 0.001).

There is an interaction between the two variables as shown by the SNK post hoc test. The large classroom (20 or more) is significantly different from the medium classroom (11 to 19), which is also significantly different from the small classroom (10 or less).

There is support for the researcher’s hypothesis regarding girls’ better performance in classrooms with fewer students. In the interaction means between classroom size and gender, females in small classrooms have a higher math score of 93.8 (SE=1.132) compared to males with a mean of 92.7 (SE=1.132).

A factorial ANOVA is a statistical analysis that is done to compare the means of two or more independent variables, which break up the sample into a minimum of four categories (Fox, 2015). A factorial ANOVA was conducted to compare the main effects of classroom size and gender and the interaction effect between classroom size and gender on math scores. Classroom sizes included small (10 or less), medium (11 to 19), and large (20 or more), whereas gender consisted of two categories (male and female). A total of 20 math scores for students in each category were compared. All effects were statistically significant at the 0.05 significance level. The main effect for classroom size generated an F ratio of F(2, 54) = 25.311, p < 0.05, indicating a significant difference between small classroom (M = 93.25, SD = 3.64), medium classroom (M = 89.10, SD = 3.26) and large classroom (M = 81.20, SD = 7.14).

The impact of classroom size had a moderate effect of 48.4%. The main effect for gender yielded an F ratio of F(1, 54) = 19.056, p < 0.05, indicating that the effect for gender was significant, male (M = 91.20, SD = 3.19) and female (M = 87.17, SD = 7.27). The main impact of gender had a small effect of 26.1%. The interaction effect was significant, F(2, 54) = 19.102, p < 0.05 with a moderate effect of 41.4%. These findings showed that classroom size has an effect on the math scores of elementary-age children. Smaller classrooms are associated with higher math scores than large classrooms. In addition, female students are likely to perform better than their male counterparts in small classrooms. Therefore, classroom size and gender have a significant influence on the math scores of elementary-age children.

Applying Analytical Strategies to an Area of Research Interest

The research area of interest is to determine the effect of color on emotions.

The independent variable is color. On the other hand, two dependent variables are emotions and memory.

Mock ANCOVA Output Table

Tests of Between-Subjects Effects
Dependent Variable: Emotion
Source Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared
Corrected Model 892.250a 3 297.417 14.106 .000 .430
Intercept 54416.327 1 54416.327 2580.866 .000 .979
Memory 244.017 1 244.017 11.573 .001 .171
Color 648.233 2 324.117 15.372 .000 .354
Error 1180.733 56 21.085
Total 479293.000 60
Corrected Total 2072.983 59
a. R Squared =.430 (Adjusted R Squared =.400)

Analysis of covariance (ANCOVA) is a statistical analysis method that compares the means of two or more independent categories on a given dependent variable (Brace, Snelgar, & Kemp, 2016). One of the independent variables is considered a covariate or confounding factor that could influence the dependent variable. ANCOVA was done to compare the effect of color on emotion while adjusting for memory. There was a significant difference in the range of emotions [F(2, 56) = 15.372, p < 0.001] elicited by different colors whilst adjusting for memory. The impact of color on memory had a small effect of 17.1%, whereas the effect of color on emotions had a small to medium effect of 35.4%.

References

Brace, N., Snelgar, R., & Kemp, R. (2016). SPSS for psychologists: And everybody else (6th ed.). New York, NY: Palgrave Macmillan.

Fox, J. (2015). Applied regression analysis and generalized linear models (3rd ed.). Thousand Oaks, CA: Sage Publications.

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