- An ANCOVA is different from an ANOVA in
|it adds another independent variable to the equation|
|it statistically controls the dependent variable for something called a covariate|
|it keeps the independent variables constant and then examines their effects on the dependent variable|
|it removes all the variance from the dependent variable and then analyzes the effects of the independent variable/s on the dependent variable|
- The following is a list of assumptions relevant to ANOVA and ANCOVA. Which of the following statistics assumptions are UNIQUE to ANCOVA only (not something you would check for an ANOVA). There can be more than one answer that you could check off, but they must be unique to ANCOVA.
|Normality of the Dependent Variable|
|Homogeneity of regression slopes|
|Homogeneity of variance|
|Normality of Covariates|
|Unequal n in groups|
|Outliers in Dependent Variable and Covariates|
- You are interested in evaluating the effect of recent relaxation training (IV1; relax.training) and level of tension (IV2; tension) on the respondents’ scores on Beck Anxiety Inventory (DV; BAI), while controlling for the level of perceived stress (Covariate; stress). Use the attached SPSS file to conduct your data screening and analysis.
For this question, first write out the appropriate research questions and null hypotheses for the main effects and interaction (HINT: see M&V, section 5.4).
For questions 4-16, please use this dataset: Dataset for Lab 4 _Q4-16_.sav
|For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac). Paragraph Open Sans,sans-serif 10pt P 0 WORDSPOWERED BY TINY|
- Are there any outliers in this dataset? (HINT: Check univariate normality of the DV and the Covariate by looking at their z scores that you can obtain through Analyze – Descriptive Statistics – Descriptives – move over BAI and stress variables to the Variable(s) window – click on Save standardized values as variables; Consider any cases with z scores below -3 and or above +3 as outliers)
- According to the skewness numbers, do the data meet the assumption of normality? (HINT: Check univariate normality of the DV and the Covariate by looking at their total skewness and kurtosis numbers that can be obtained through Analyze – Descriptive Statistics – Descriptives – move over BAI and stress variables to the Variable(s) window – click on Options – click on Kurtosis and Skewness boxes under “Distribution”)
- Use the following standard to determine if a transformation of the data is necessary: consider a skewness value of -.75 to .75 to be normal skewness, .76 to 1.00 to be moderate positive skewness and -.76 to -1.00 to be moderate negative skewness, 1.01 to 2.00 to be substantial positive skewness and -1.01 to -2.00 to be substantial negative skewness, and 2.01 and above to be severe positive skewness and -2.01 and below to be severe negative skewness. Based on this standard, is a transformation necessary for this data?
- Was there a problem with homogeneity of regression slopes?
(Hint: See the checklist for conducting ANCOVA in your M & V text, and information on about homogeneity of regression slopes in your books and class/ lab notes. Ensure you are using the correct DV, IV and COV in your boxes)
|Yes, since the interaction between the factors and the covariate was significant, F(3, 196) = 4.26, p = .006|
|No, since the interaction between the factors and the covariate was not significant, F(3, 193) = 2.29, p = .08|
|Yes, since the interaction between the factors and the covariate was not significant, F(3, 193) = 2.29, p = .08|
|No, since the interaction between the factors and the covariate was significant, F(3, 196) = 4.26, p = .006|
|No, since the interaction between the factors and the covariate was not significant, F(1, 193) = .166, p = .68|
- Examine the data to determine if there is a problem with homogeneity of variance. Was there a problem with homogeneity of variance?
|Yes, since Levene’s test was significant, with p = .006|
|Yes, since Levene’s test was not significant, with p = .091|
|No, since Levene’s test was not significant, with p = .091|
|No, since Levene’s test was significant, with p = .006|
- Now conduct the ANCOVA. According to the results, is the factor interaction significant? (Analyze – General Linear Model – Univariate – Model – click Full factorial; in the ouput look at the F-test results for the interaction between the two factors)
|Yes, the factor interaction is significant, F(1, 195) = 24.01, p < .001, partial η² = .11.|
|No, the factor interaction is not significant, F(1, 195) = .22, p = .64, partial η² = .001.|
|Yes, the factor interaction is significant, F(1, 195) = 2.13, p = .15, partial η² = .011.|
|No, the factor interaction is not significant, F(1, 195) = 2.13, p = .15, partial η² = .011.|
- Look at the line plot of the factors. According to this line plot, do the factors (IVs) appear to have disordinal interaction?
- Are main effects for level of tension significant?
- Are main effects for history of relaxation training significant?
- Does the covariate, level of perceived stress, significantly influence the DV?
- Which of the following has the highest effect size?
|The factor interaction|
- Based on the results of your analysis, please fill in the blanks for the following Results section. Even if violations of assumptions might cause someone to not run this ANCOVA, go ahead and run it anyway and just report the problems.
A × analysis of covariance (ANCOVA) was conducted to investigate whether tension and relaxation training would significantly affect respondents’ scores on the Beck Anxiety Inventory, after controlling for the perceived level of stress. Prior to analysis, all variables were examined through SPSS software for accuracy of data entry, missing values, and outliers. No obvious data entry errors or outliers were found. There were cases with missing data in the dataset. Then data were examined for normality. This examination revealed that both continuous variables (the dependent variable and the covariate) had skewness values within the acceptable range between to , which indicated that transformations were necessary (BAI skewness = , stress skewness = ). Further, equality of groups was examined, and it was found that split between the groups for both independent variables was rather .
Next, assumptions of homogeneity of regression slopes and homogeneity of variance for ANCOVA were examined. This examination revealed that Levene’s test was , F(, ) = , p =, which indicated unequal variances among groups. Interaction between the independent variables (tension and relaxation training) and the covariate (stress) was , F(, ) = , p = Therefore, assumption of homogeneity of regression slopes was , and further analysis was acceptable.
Results of the ANCOVA revealed that the covariate (stress) did not have a significant effect on the dependent variable, F(, ) = , p = , partial η2 = . There was also no significant interaction of tension and relaxation training on the participants’ BAI scores, F(, ) = , p = , partial η² = . Main effect results revealed that respondents’ BAI scores were affected by tension alone, F(, ) = , p < , partial η2 = , and by relaxation training alone, F(, ) = , p < , partial η2 = .
Overall, results of the ANCOVA revealed that after adjusting for stress, participants with low tension (adjusted M = ) scored significantly lower on the BAI as compared to the participants with high tension (adjusted M = ). Further, participants who recently completed relaxation training (adjusted M = ) also received significantly lower BAI scores as compared to those who never had any relaxation training (adjusted M = ). In short, when controlling for stress, tension accounted for % of the variance in the respondents’ BAI scores, and relaxation training accounted for % of this variance. These results indicate that while there are significant differences, the effect sizes of these factors are small to medium.
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