# Factorial Anova for the Levels of Music Liking

## Introduction

This paper provides a factorial analysis of variance (factorial ANOVA) of the data contained in the Fugazi.sav file, which accompanies the book by Field (2013, p. 541). The assumptions for the factorial ANOVA are stated and tested; the null and alternative hypotheses are provided; the contents of the SPSS syntax file is supplied; and the results of the ANOVA are supplied in tables and reported as a text according to the guidelines of American Psychological Association ([APA], 2010). All the SPSS output is provided in the Appendices.

## Review of Chapter 13 (Field, 2013, pp. 507-542)

Field (2013) provides guidelines for conducting a factorial ANOVA. The author describes the general theory behind the factorial ANOVA, the types of factorial designs, and provides and analyzes an example of a study which can be conducted using the factorial ANOVA. Further, the model sum of squares is discussed; the main effects of the independent variables and the interaction effect are explained. After that, the instructions for conducting the test in SPSS are given, and the ways of interpreting the output are offered. Calculating effect sizes is also touched upon. A number of tasks are provided at the end of the chapter.

## Review of APA (2010) Guidelines

APA (2010) provides general guidelines for reporting results of studies, ANOVAs in particular (pp. 46-47). The results section should include the type of test, a general description of the sample, and the results of the study, such as the degrees of freedom, the F values, the *p* values, the effect sizes; if the F is significant, it is possible to include the results of post-hoc tests.

## Underlying Assumptions for Factorial ANOVA

The assumptions for a factorial ANOVA are as follows (Warner, 2013, pp. 506-507):

- the scores of the dependent variable are quantitative;
- the distribution of the scores of the dependent variable is close to normal;
- the values of the dependent variable are achieved via independent observations, and the independent variables are independent of one another;
- the variances of values of the dependent variable are homogenous in different groups;
- there are no extreme outliers.

## Testing the Assumptions

The assumption (1) is met because the dependent variable is measured as an interval. The assumption (3) is met because there were no repeated measurements, and the independent variables (age of respondents and type of music) are mutually independent.

To test the assumption (2), it is possible to examine the histogram for the *liking* variable:

Unfortunately, the distribution is not normal. However “factorial ANOVA is fairly robust to violations of the normality assumption… unless the numbers of cases in the cells are very small and/or unequal” (Warner, 2013, p. 507). Thus, it may still be possible to conduct the factorial ANOVA.

The histogram can also be utilized to test the assumption (5) and confirm that no extreme outliers are present.

To test (4), the homogeneity of variances, it is possible to use the Levene test:

The Levene’s test of equality of error variances yielded F (5, 84) = 1.189, but *p*=.322 is more than α=.05, which indicates that there are no significant differences between the variances of the values of the dependent variable in the groups. Thus, the assumption (4) is met.

## Ways of Addressing the Violations of Assumptions

It is possible to state that if the assumption (1) was not met, it would be pointless to run ANOVA until the scores are quantitative. The violation of (3) could be dealt with by not taking into account the repeated measurements. If one wishes to address the violation of the assumption of the normality (2), or the violation of the assumption of homogeneity of variances across groups (4), it is possible to apply transformations to the data, such as raising each value of the dependent variable to the same power (“PROPHET StatGuide,” n.d.). The presence of extreme outliers, that is, the violation of (5), can be handled by removing the outliers from the data set.

### Null and Alternative Hypotheses

H_{0,music}: M_{1,music}=M_{2,music}=M_{2,music}

H_{0,age}: M_{1,age}=M_{2,age}

H_{0,music×age}: there are no significant differences between the means of the *music×age* groups.

H_{A,music}: there is a significant difference between the means of the *music* groups.

H_{A,age}: M_{1,age}≠M_{2,age}

H_{0,music*age}: there are significant differences between the means of the *music*age* groups.

### Syntax File

The contents of the Syntax file can be found in Appendix 1.

### SPSS Output

All of the SPSS output can be found in Appendix 2.

### Results Tables

The results tables can be found in Appendix 3.

## Results

A 2×3 factorial ANOVA was conducted by using SPSS in order to identify whether the type of music (1=Fugazi; 2=ABBA; 3=Barf Grooks), the age of respondents (1=older, 40+; 2=young, 0-40), and the interaction between the type of music and the age of respondents could predict the liking rating of music. It was expected that the combination of age=2 and music=1 would yield higher scores in the liking rating, whereas the combination of age=1 and music=3 would also yield higher scores in the liking rating (Field, 2013, p. 541). The number of participants in each age group was 45; each group was equally split into three subgroups and assigned to listen to a different type of music–either 1, or 2, or 3. Thus, the number of respondents in each of the six music*age groups was 15.

Preliminary data screening was carried out to determine whether there are violations of the assumptions of the factorial ANOVA; the histogram showed that the distribution of the liking rating variable was non-normal, but no transformations were utilized. Simultaneously, the Levene test displayed no significant violation of the assumption of the homogenous variances among the six groups.

The analysis revealed that there was a significant difference in the means of liking for the type of music; the main effect yielded F (2, 84) = 105.620 at *p*<.001. Thus, for Fugazi, M=-4.83, SD=74.23; for ABBA, M=62.03, SD=18.35; for Barf Grooks, M=1.40, SD=77.41. The effect size as estimated by partial η^{2} was.715, which is a very large effect size. The observed power was 1.0, as calculated at α=.05.

No significant difference in the means of liking for the age was detected; the main effect yielded F (1, 84) =.002 at *p*=.966.

There was also a significant difference in the means of liking for the interaction music*age, F (2, 84) = 400.977 at *p*<.001. The effect size as estimated by partial η^{2} was.905, which is a very large effect size. The observed power was 1.0, as calculated at α=.05.

Post-hoc tests (Bonferroni correction; see Appendix 3) showed that there was a significant difference in liking for Fugazi and Abba (mean difference=-66.87, p<.001), Abba and Barf Grooks (mean difference=60.63, p<.001); no significant difference was found for Fugazi and Barf Grooks (p=.671).

## Adjusting Power

In order to achieve the.80 power for the interaction music*age, it is possible to decrease the sample size (because the observed power is stated to be 1.0, it is needed to decrease it). In general, it is also possible to increase α (for instance, take.10 instead of.05) to decrease the effect size. To find out the necessary sample size to obtain the power of.80, it is possible to use a table such as the one provided by Bausell and Li (2002, p. 263); see the Appendix 4.

## References

American Psychological Association. (2010). *Publication manual of the American Psychological Association* (6th ed.). Washington, DC: American Psychological Association.

Bausell, R. B., & Li, Y.-F. (2002). *Power analysis for experimental research: A practical guide for the biological, medical and social sciences*. Cambridge, UK: Cambridge University Press.

Field, A. (2013). *Discovering statistics using IBM SPSS statistics: And sex and drugs and rock’n’roll* (4th ed.). Thousand Oaks, CA: Sage Publications.

*PROPHET StatGuide: Possible alternatives if your data violate two-factor factorial ANOVA assumptions*. (n.d.). Web.

Warner, R. M. (2013). *Applied statistics: From bivariate through multivariate techniques* (2nd ed.). Thousand Oaks, CA: SAGE Publications.