ANCOVA and Factorial ANOVA: A Case Study Essay

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Updated: Mar 22nd, 2024

SPSS Assignment

Exploratory Data Analysis

Descriptives
Classroom sizeStatisticStd. Error
Math_Score10 or lessMean93.2500.81394
95% Confidence Interval for MeanLower Bound91.5464
Upper Bound94.9536
5% Trimmed Mean93.2778
Median93.0000
Variance13.250
Std. Deviation3.64005
Minimum87.00
Maximum99.00
Range12.00
Interquartile Range6.50
Skewness.002.512
Kurtosis-1.080.992
11-19Mean89.1000.72873
95% Confidence Interval for MeanLower Bound87.5747
Upper Bound90.6253
5% Trimmed Mean89.1667
Median89.5000
Variance10.621
Std. Deviation3.25900
Minimum82.00
Maximum95.00
Range13.00
Interquartile Range5.75
Skewness-.342.512
Kurtosis-.292.992
20 or moreMean85.20001.59868
95% Confidence Interval for MeanLower Bound81.8539
Upper Bound88.5461
5% Trimmed Mean85.2222
Median86.5000
Variance51.116
Std. Deviation7.14953
Minimum72.00
Maximum98.00
Range26.00
Interquartile Range11.25
Skewness-.177.512
Kurtosis-.824.992
Descriptives
GenderStatisticStd. Error
Math_ScoreFemaleMean87.16671.32707
95% Confidence Interval for MeanLower Bound84.4525
Upper Bound89.8808
5% Trimmed Mean87.3704
Median88.5000
Variance52.833
Std. Deviation7.26865
Minimum72.00
Maximum98.00
Range26.00
Interquartile Range10.50
Skewness-.272.427
Kurtosis-.698.833
MaleMean91.2000.58408
95% Confidence Interval for MeanLower Bound90.0054
Upper Bound92.3946
5% Trimmed Mean91.0556
Median91.0000
Variance10.234
Std. Deviation3.19914
Minimum86.00
Maximum99.00
Range13.00
Interquartile Range4.00
Skewness.557.427
Kurtosis.244.833

ANCOVA and Factorial ANOVA: A Case Study

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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 LabelN
Classroom size110 or less20
211-1920
320 or more20
GenderFFemale30
MMale30
Descriptive Statistics
Dependent Variable: Math_Score
Classroom sizeGenderMeanStd. DeviationN
10 or lessFemale93.80003.9384110
Male92.70003.4335010
Total93.25003.6400520
11-19Female88.50003.9791110
Male89.70002.4060110
Total89.10003.2590020
20 or moreFemale79.20004.1846310
Male91.20003.2249010
Total85.20007.1495320
TotalFemale87.16677.2686530
Male91.20003.1991430
Total89.18335.9275060
Levene’s Test of Equality of Error Variancesa
Dependent Variable: Math_Score
Fdf1df2Sig.
.822554.539
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
SourceType III Sum of SquaresdfMean SquareFSig.Partial Eta Squared
Corrected Model1381.483a5276.29721.576.000.666
Intercept477220.0171477220.01737266.639.000.999
Classroom648.2332324.11725.311.000.484
Gender244.0171244.01719.056.000.261
Classroom * Gender489.2332244.61719.102.000.414
Error691.5005412.806
Total479293.00060
Corrected Total2072.98359
a. R Squared =.666 (Adjusted R Squared =.636)
Estimates
Dependent Variable: Math_Score
Classroom sizeMeanStd. Error95% Confidence Interval
Lower BoundUpper Bound
10 or less93.250.80091.64694.854
11-1989.100.80087.49690.704
20 or more85.200.80083.59686.804
Pairwise Comparisons
Dependent Variable: Math_Score
(I) Classroom size(J) Classroom sizeMean Difference (I-J)Std. ErrorSig.b95% Confidence Interval for Differenceb
Lower BoundUpper Bound
10 or less11-194.150*1.132.0011.8816.419
20 or more8.050*1.132.0005.78110.319
11-1910 or less-4.150*1.132.001-6.419-1.881
20 or more3.900*1.132.0011.6316.169
20 or more10 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 SquaresdfMean SquareFSig.Partial Eta Squared
Contrast648.2332324.11725.311.000.484
Error691.5005412.806
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
GenderMeanStd. Error95% Confidence Interval
Lower BoundUpper Bound
Female87.167.65385.85788.477
Male91.200.65389.89092.510
Pairwise Comparisons
Dependent Variable: Math_Score
(I) Gender(J) GenderMean Difference (I-J)Std. ErrorSig.b95% Confidence Interval for Differenceb
Lower BoundUpper Bound
FemaleMale-4.033*.924.000-5.886-2.181
MaleFemale4.033*.924.0002.1815.886
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 SquaresdfMean SquareFSig.Partial Eta Squared
Contrast244.0171244.01719.056.000.261
Error691.5005412.806
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 sizeGenderMeanStd. Error95% Confidence Interval
Lower BoundUpper Bound
10 or lessFemale93.8001.13291.53196.069
Male92.7001.13290.43194.969
11-19Female88.5001.13286.23190.769
Male89.7001.13287.43191.969
20 or moreFemale79.2001.13276.93181.469
Male91.2001.13288.93193.469
Math_Score
Student-Newman-Keulsa,b
Classroom sizeNSubset
123
20 or more2085.2000
11-192089.1000
10 or less2093.2500
Sig.1.0001.0001.000
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).

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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
SourceType III Sum of SquaresdfMean SquareFSig.Partial Eta Squared
Corrected Model892.250a3297.41714.106.000.430
Intercept54416.327154416.3272580.866.000.979
Memory244.0171244.01711.573.001.171
Color648.2332324.11715.372.000.354
Error1180.7335621.085
Total479293.00060
Corrected Total2072.98359
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.

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

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