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- 1. MANCOVA EDUC. 303 (ADVANCED STATISTICS) GIENA L. ODICTA, PhD Course Facilitator CRISTIAN V. CAPAPAS Discussant
- 2. What is the One-Way MANCOVA? MANCOVA is a short term for Multivariate Analysis of Covariance (MAnCova). The words one and way in the name indicate that the analysis includes only one independent variable. Like all analyses of covariance, the MANCOVA is a combination of a One-Way MANOVA preceded by a regression analysis.
- 3. What is the One-Way MANCOVA? In basic terms, the MANCOVA looks at the influence of one or more independent variables on one dependent variable while removing the effect of one or more covariate factors. To do that the One-Way MANCOVA first conducts a regression of the covariate variables on the dependent variable. Thus it eliminates the influence of the covariates from the analysis. Then the residuals (the unexplained variance in the regression model) are subject to MANOVA, which tests whether the independent variable still influences the dependent variables after the influence of the covariate(s) has been removed.
- 4. What is the One-Way MANCOVA? The One-Way MANCOVA includes one independent variable, one or more dependent variables and the MANCOVA can include more than one covariate, and SPSS handles up to ten. If the One-Way MANCOVA model has more than one covariate it is possible to run the MANCOVA with contrasts and post hoc tests just like the one-way ANCOVA or the ANOVA to identify the strength of the effect of each covariate.
- 5. Assumptions of One-Way MANCOVA Test Multivariate Normality: The dependent variables should follow a multivariate normal distribution within each group. Multivariate normality ensures that the statistical inferences drawn from the analysis are robust and accurate. Homogeneity of Covariance Matrices: The covariance matrices of the dependent variables should be approximately equal across all groups. The homogeneity of covariance matrices ensures that the relationships between variables are consistent, allowing for meaningful comparisons. Homogeneity of Regression Slopes: The relationships between the independent variable and each dependent variable, as well as the covariates, should be consistent across all groups. This assumption ensures the reliability of regression slopes across groups.
- 6. Assumptions of One-Way MANCOVA Test Homogeneity of Regression Slopes: The relationships between the independent variable and each dependent variable, as well as the covariates, should be consistent across all groups. This assumption ensures the reliability of regression slopes across groups. Absence of Outliers: The dataset should be free from outliers that could disproportionately influence the results. Outliers can distort the estimation of parameters and compromise the integrity of the analysis. Linearity: The relationships between the independent variable, dependent variables, and covariates should be linear. This assumption ensures that the impact of the independent variable is consistent across the range of values.
- 7. The hypothesis of One-Way MANCOVA Test Main Effects Null Hypothesis: There are no significant differences in the combined set of dependent variables across the levels of the independent variable, after adjusting for the covariates. Alternative Hypothesis: There are significant differences in the combined set of dependent variables across the levels of the independent variable, after adjusting for the covariates.
- 8. The hypothesis of One-Way MANCOVA Test Covariate Effect Null Hypothesis: The impact of the covariate(s) on the combined set of dependent variables is not significant. Alternative Hypothesis: The impact of the covariate(s) on the combined set of dependent variables is significant.
- 9. Example of One-Way MANCOVA Test Scenario: A researcher is studying the impact of different teaching methods on student performance in two subjects: Science and English. The researcher also wants to control for the effects of prior academic achievement (as measured by the students GPA) since students enter the courses with different levels of prior academic ability. The study has three groups of students: 1. Traditional Teaching Method (Group 1) 2. Online Teaching Method (Group 2) 3. Blended Teaching Method (Group 3)
- 10. Example of One-Way MANCOVA Test The researcher has collected data on the following variables: Dependent Variables Science Test Scores English Test Scores Independent Variable (Factor) Teaching Method (3 levels: Traditional, Online, Blended) Covariate GPA (prior academic achievement)
- 11. Example of One-Way MANCOVA Test Tasks: 1. State the Research Hypothesis Null Hypothesis (Ho): There is no significant difference in the Science and English test scores between the teaching methods when controlling for GPA. Alternative Hypothesis (Ha): At least one teaching method differs significantly in the Science or English test scores when controlling for GPA.
- 12. Example of One-Way MANCOVA Test Tasks: 2. Assumptions for MANCOVA: Multivariate Normality: Are the dependent variables (Science and English test scores) normally distributed for each group? Homogeneity of Covariance Matrices: Is the covariance matrix of the Science and English test scores the same across the different groups? Linearity: Are the relationships between the dependent variables and the covariate (GPA) linear? Independence of Observations: Are the students test scores independent of each other?
- 13. Example of One-Way MANCOVA Test Tasks: 3. Data Structure: Assume the researcher collected the following data for each student in the study.
- 14. Example of One-Way MANCOVA Test Tasks: 4. Conducting the MANCOVA:
- 15. Step by Step Procedure in Running One Way MANCOVA Test in SPSS Software Tasks: 4. Conducting the MANCOVA:
- 16. Step by Step Procedure in Running One Way MANCOVA Test in SPSS Software Tasks: 4. Conducting the MANCOVA:
- 24. Results
- 25. Interpretation of Data Factor Name: The factor being studied is the "Group Teaching Method." This means that we are comparing how different teaching methods (Traditional, Online, and Blended) affect some dependent variable. Levels of the Factor: Level 1: Traditional Teaching Method (4 participants) Level 2: Online Teaching Method (4 participants) Level 3: Blended Teaching Method (4 participants) N (Sample Size): For each teaching method group, there are 4 participants. This is a small sample size for each group, totaling 12 participants across all three groups.
- 26. Results
- 27. Interpretation of Data Science Test Scores: On average, students performed best in the Blended Teaching Method (89.25), followed by the Online Teaching Method (86.75), and then the Traditional Teaching Method (81.25). The standard deviations are relatively similar across these groups, but the Blended method shows slightly more consistent results. English Test Scores: Again, students performed best in the Blended Teaching Method (82.75), followed by Online Teaching Method (79.00), and lastly the Traditional Teaching Method (73.75). The variability in English scores was lowest in the Online method (SD = 2.160).
- 28. Interpretation of Data Overall Performance: The Blended Teaching Method appears to be the most effective for both Science and English, with the highest mean scores in both subjects. The Traditional Teaching Method shows the lowest performance on both tests. In conclusion, the data suggests that the Blended Teaching Method may have a positive impact on students' test scores in both Science and English compared to the Traditional and Online methods.
- 29. Results
- 30. Interpretation of Data The Box's M test results suggest that the assumption of equal covariance matrices across the groups is not violated (since the p-value is greater than 0.05). In this case, the p-value is 0.075, which is greater than 0.05, suggesting that there is no significant difference between the covariance matrices of the groups. Therefore, you fail to reject the null hypothesis, and you can conclude that the covariance matrices are equal across the groups. Therefore, you can proceed with your analysis under the assumption that the covariance matrices are equal.
- 31. Results
- 32. Interpretation of Data Intercept and GPA both have significant effects on the dependent variables, explaining a large portion of the variance (85.9% for Intercept, 79.6% for GPA). GTM has a marginally non-significant effect, as indicated by a p-value of 0.063, which is greater than 0.05, although it explains a substantial amount of variance (41%). You would reject the null hypothesis for Intercept and GPA, meaning they have significant effects on the dependent variables, but fail to reject the null hypothesis for GTM, meaning there is insufficient evidence to say GTM significantly influences the dependent variables.
- 33. Results
- 34. Interpretation of Data 1. Science Test Score: F = 0.226 df1 = 2, df2 = 9 p-value = 0.802 Null hypothesis: Levenes test assumes the null hypothesis that the error variances (residuals) are equal across groups. p-value = 0.802: Since the p-value is greater than 0.05, you fail to reject the null hypothesis. This means that there is no significant difference in the error variances across the groups for the Science Test Score. Therefore, the assumption of homogeneity of variances (equal error variances) holds for this dependent variable.
- 35. Interpretation of Data 2. English Test Score: F = 0.427 df1 = 2, df2 = 9 p-value = 0.665 Null hypothesis: As with the Science Test Score, the null hypothesis is that the error variances are equal across the groups. p-value = 0.665: Since this p-value is also greater than 0.05, you fail to reject the null hypothesis. This means that there is no significant difference in the error variances across the groups for the English Test Score either, so the assumption of homogeneity of variances is also met for this variable.
- 36. Interpretation of Data Conclusion: For both the Science Test Score and the English Test Score, the p-values from Levene's test are both much greater than 0.05, meaning there is no significant difference in the error variances across the groups. You can therefore conclude that the assumption of equal error variances holds for both dependent variables, which is important for the validity of certain statistical tests, like ANOVA or MANOVA.
- 37. Results
- 38. Interpretation of Data 1. Corrected Model: Science Test Score: Type III Sum of Squares = 198.446 F = 29.722, p = 0.000 Partial Eta Squared = 0.918 English Test Score: Type III Sum of Squares = 203.946 F = 21.707, p = 0.000 Partial Eta Squared = 0.891
- 39. Interpretation of Data Interpretation: The Corrected Model represents the overall effect of all the independent variables (GPA, GTM, and the intercept) on the dependent variable. Both Science Test Score and English Test Score have significant models, as the p-values are both 0.000, which is less than the usual alpha level of 0.05. This means that the independent variables (GPA, GTM, etc.) together explain a significant portion of the variation in both the Science and English test scores. Partial Eta Squared shows the proportion of variance explained by the model. For Science Test Score, 0.918 means that 91.8% of the variance in Science Test Scores is explained by the independent variables. For English Test Score, 0.891 means that 89.1% of the variance in English Test Scores is explained.
- 40. Interpretation of Data 2. Intercept: Science Test Score: Type III Sum of Squares = 108.407 F = 48.710, p = 0.000 Partial Eta Squared = 0.859 English Test Score: Type III Sum of Squares = 110.650 F = 35.331, p = 0.000 Partial Eta Squared = 0.815
- 41. Interpretation of Data Interpretation: The Intercept term tests whether there is a significant effect on the test scores regardless of the independent variables. Both tests have p = 0.000, which is highly significant. This suggests that the intercept (the baseline effect) plays a large role in explaining the test scores. The Partial Eta Squared values (0.859 for Science and 0.815 for English) indicate that the intercept alone accounts for a substantial portion of the variance in both test scores (85.9% for Science and 81.5% for English).
- 42. Interpretation of Data 3. GPA: Science Test Score: Type III Sum of Squares = 64.446 F = 28.957, p = 0.001 Partial Eta Squared = 0.784 English Test Score: Type III Sum of Squares = 40.446 F = 12.915, p = 0.007 Partial Eta Squared = 0.617
- 43. Interpretation of Data Interpretation: GPA has a significant effect on both the Science and English Test Scores, as both p-values are less than 0.05. For Science Test Scores, the F-value of 28.957 indicates that GPA is strongly associated with Science scores, explaining 78.4% of the variance (Partial Eta Squared = 0.784). For English Test Scores, GPA also has a significant effect, but the effect is smaller, explaining 61.7% of the variance in English scores.
- 44. Interpretation of Data 4. GTM: Science Test Score: Type III Sum of Squares = 12.481 F = 2.804, p = 0.119 Partial Eta Squared = 0.412 English Test Score: Type III Sum of Squares = 25.522 F = 4.075, p = 0.060 Partial Eta Squared = 0.505
- 45. Interpretation of Data Interpretation: GTM has no significant effect on Science Test Scores (p = 0.119, greater than 0.05), meaning you fail to reject the null hypothesis for GTM in predicting Science scores. For English Test Scores, the effect of GTM is marginally significant (p = 0.060, which is slightly above 0.05). This suggests that GTM may have a potential effect on English scores, but the evidence is not strong enough to confirm significance at the 0.05 level. The Partial Eta Squared values indicate the proportion of variance explained by GTM. For Science, it explains 41.2% of the variance, and for English, it explains 50.5%. While these are moderate effects, they are not statistically significant for Science and only marginally so for English.
- 46. Interpretation of Data 5. Error: This section provides the Error sum of squares, degrees of freedom, and mean squares. The error term represents the variation in the dependent variables that is not explained by the model (including GPA and GTM). For Science Test Score, the error variance is 17.804 with 8 degrees of freedom. For English Test Score, the error variance is 25.054 with 8 degrees of freedom.
- 47. Interpretation of Data 5. Error: This section provides the Error sum of squares, degrees of freedom, and mean squares. The error term represents the variation in the dependent variables that is not explained by the model (including GPA and GTM). For Science Test Score, the error variance is 17.804 with 8 degrees of freedom. For English Test Score, the error variance is 25.054 with 8 degrees of freedom.
- 48. Interpretation of Data Summary: Significant effects: The Intercept and GPA have significant effects on both Science and English Test Scores. The Corrected Model (which includes all predictors) also explains a significant portion of the variance in both Science and English Test Scores. Marginal effect: GTM shows a marginal effect on English Test Scores but has no significant effect on Science Test Scores. Effect size: The Partial Eta Squared values suggest that the independent variables (GPA, GTM) explain a large proportion of the variance in both test scores, especially GPA, which has the highest effect sizes.
- 49. Interpretation of Data In conclusion, GPA is the most significant and influential factor affecting both Science and English Test Scores, while GTM has a weaker and less significant influence.
- 50. Results
- 51. Results
- 52. Take Home Problem Set Activity Scenario: A researcher is investigating the impact of three different types of exercise programs (Strength Training, Aerobic Exercise, and Yoga) on physical fitness, measured by two outcomes: cardiovascular endurance and flexibility. Additionally, the researcher is interested in controlling for the participants' baseline body mass index (BMI), as it is believed that BMI may influence the fitness outcomes.
- 53. Take Home Problem Set Activity Data for Analysis: You are given the following data for 60 participants: Note: Each group has 20 participants.