# 70 Analysis of Variance (ANOVA)

As discussed above, ANOVAs are used to make comparisons across three or more groups of a dependent variable(s) with one or more independent variables. ANOVA is appropriate whenever you want to test differences between the means of an interval-ratio level dependent variable across three or more categories of an independent variable. There are two techniques for doing ANOVA: one-way ANOVA and two-way ANOVA.

## One-Way ANOVA

The One-Way ANOVA compares the means of two or more independent groups in order to determine whether there is statistical evidence that the associated population means are significantly different. This technique might be useful for analyzing field studies, experiment and quasi-experiment data (Kansas State Universities Library, 2022). It must be noted that while both the One-Way ANOVA and the Independent Samples T-Test can be used to compare the means for two groups, only the One-Way ANOVA can compare the means across three or more groups.

## Two-way ANOVA

The Two-way ANOVA is similar to the one-way ANOVA except that it allows you to consider two independent variables (instead of one) while comparing the means of three or more groups of data. Before running an ANOVA, it is important to ensure that assumptions are met. Box 10.5.4.3 highlights some key assumptions.

**Box 10.6 – Some Assumptions about ANOVA**

- Your dependent variable should be measured at the continuous level (i.e., they are interval or ratio variables).
- Your two independent variables should each consist of two or more nominal or ordinal, independent groups.

Ex. Gender (2 groups: male or female), ethnicity (3 groups: Caucasian, African American and Hispanic)

- You should have independence of observations, which means that there is no relationship between the observations in each group or between the groups themselves.
- There should be no significant outliers.
- Your dependent variable should be approximately normally distributed for each combination of the groups of the two independent variables.

See UBC’s research commons for guidance on how to run an ANOVA and other procedures in SPSS https://researchcommons.library.ubc.ca/introduction-to-spss-for-statistical-analysis/

## Presenting ANOVA Results

SPSS (and most other statistical programs) presents two output tables for ANOVA results: descriptives and ANOVA. From the descriptives, you will need to record N, Mean, Std. Deviation and Std. Error. From the ANOVA table, you will need to record df, F and Sig.

ANOVAs are reported like the t test, but there are two degrees-of-freedom numbers to report. First report the between-groups degrees of freedom, then report the within-groups degrees, separated by a common. After that report the F statistic (rounded off to two decimal places) and the significance level. In addition, note the following:

- If there were significant differences, state the means and standard deviations for each group.
- If Statistically significant, you might need to dig deeper to state which group is significantly different from which. In SPSS or the statistical program you are using, simply run a post-test. Post tests (e.g. Tukey Ad hoc Post test) can be used to indicate which group is significantly different from which [see https://www.statology.org/anova-post-hoc-tests/ for further discussions and examples of post-tests]

## One-Way ANOVA

An one way analysis of variance showed that the effect of international status on grades was significant, F(3,155) = 9.94, p = .007. Post hoc analyses using the Tukey post hoc criterion for significance indicated that the average grade was significantly lower for international students (M = 70.2, SD = 2.16) than in the other two groups (regional domestic and local domestic) combined (M = 73.2, SD = 4.56), F(3, 155) = 9.37, p = .033.

OR

There was not a statistically significant difference between groups as demonstrated by one-way ANOVA (F(3,188) = .179, p = .910).

PS**Note that if the results are not significant, you should not do a post hoc analysis.

## Two-Way or Multiple Factor ANOVA

Your finding narrative should include the following:

- Identify that you are reporting on a two-way ANOVA and the variables of concern
- State the significant level (e.g. .05 level)
- Identify the effect of each of variables e.g. Independent variable #1 yield an F ratio of F (df, df)=__, p=__)
- Highlight the statistics (M and SD) for each of the attributes for Independent Variable #1 and Independent variable #2

Box 10.7 – Examples

Students’ grades in Sociology 222 were subjected to a two-way analysis of variance considering gender (females, non-females) and study status (part-time, full-time). All effects were statistically significant at the .05 significance level. The main effect of gender yielded an F ratio of F(1, 24) = 44.4, p < .001, indicating that the mean grade was significantly greater for females (M = 4.78, SD = 1.99) than for non-females (M = 2.17, SD = 1.25). The main effect of study status yielded an F ratio of F(1, 24) = 25.4, p < .01, indicating that the mean grade was significantly higher among part-time students (M = 5.49, SD = 2.25) than full-time students (M = 0.88, SD = 1.21). The interaction effect was non-significant, F(1, 24) = 1.22, p > .05.

## Chi Square

To report chi-square results in your paper, you need to identify and report on the following four values from your output: degrees of freedom and sample size in parentheses, the Pearson chi-square value (rounded to two decimal places), and the significance level In APA, chi square results are reported using the following format:

X2 (degrees of freedom, N = sample size) = chi-square statistic value, p = p value.

Let us assume that you conducted a chi-square to determine the relationship between gender and whether or not students pass Sociology 222. The first thing you would want to do is identify the four values e.g., df =2, N=1525, chi square statistic= 11.6, p=.0071

Next, you would report your chi-square results and interpret it as follows:

A chi-square test of independence was performed to examine the relation between gender and whether or not students pass SOCI 200. The relation between these variables was significant, X2 (2, N = 1525) = 11.6, p = .0071. This indicates that students who identified as males were more likely to fail than students who identified with other genders.

Remember, you must always interpret the results, i.e., state what the results mean.

**Box 10.8 – Reporting Chi Square Results**

Here are some general guidelines for reporting chi-square results:

- It is okay to include the crosstab results in your paper. However, please ensure that cross table tables follow the standard APA format: Independent variables in column and dependent variables in the row. Include numbers (and percentages in parentheses) in each cell. If the format of IV in columns and DV in rows is used, percentage the IV.
- If you include a cross table, interpret the results.
- You can cite the exact p values (e.g. p =.0013) or you note if the p value is less than .001 e.g. p < .001.
- It is important to state your hypothesis before reporting your results.
- The calculated chi-square should be stated at two decimal places
- To assess the strength of the association, we compute phi (for 2 rows x 2 columns tables i.e. 2 rows and 2 columns). For larger than 2 rows X 2 columns tables, we compute Cramer’s V. Below are some general rules of thumb to determine the strength of the relationships:
- 0.00 to 0.10 weak relationship
- 0.11 to 0.30 moderate relationship
- Greater than 0.30 strong relationship

See Healey, J. F. (2009). *Statistics: A Tool for Social Research* (Eight Edition). Wadsworth Cengage Learning.

# References

Kent State University Libraries. (2017, May 15). *SPSS tutorials*. https://libguides.library.kent.edu/SPSS

Healey, J. F. (2009). *Statistics: A Tool for Social Research* (Eight Edition). Wadsworth Cengage Learning.