12 ANOVA: Analysis of Variance

Used to compare the means of a criterion (dependent) variable between two or more groups defined by the predictor (independent variable). The test computes an F value indicative of the ratio between the variation between groups to the variation within groups to determine if the observed differences between groups on the criterion variable represent differences between populations from which the samples are drawn (alternative hypothesis, H₁). Or, if the observed differences are due purely to chance (null hypothesis, H₀).

The analysis below will use the same data set used for the Independent Samples t-Test. Therefore, the experiment and description is the same.

12.1 Assumptions

Sample data is normally distributed:
- Check frequency distributions, q-q plots;
- If the data is not normally distributed, proceed, but with caution. The F test is robust to departures from normality because the chances of Type I error are not increased by deviations from normality;
The groups are homogeneous (have equal variances):
- Check homogeneity of variances (Levene’s test);
- If NOT homogeneous:
  - For between-subjects variables, do not worry. F is robust and violations are not likely to increase the chance of Type I error significantly;
  - For within-subjects variables, worry, as the results can suggest false significant effects;
Independence of observations seems a reasonable assumptions since each participant was measured individually, independent of the others.

12.1.1 Testing Normality

Normality is tested similarly to the Analysis of normality presented in The t-Test chapter. Please follow the guidance provided in that section.

The analysis of normality presented in the t-Test chapter, using both visual and computation methods, suggests that despite some departures from normal distribution, the data follows a normal distribution. Therefore, the assumption of normality is verified.

12.1.2 Homogeneity of Variances

Let’s use Levene’s test to look at the homogeneity of variances.

Levene's Test for Homogeneity of Variance (center = median)
      Df F value Pr(>F)
group  1       0   0.96
      98

The test shows (p > 0.05) that the null hypothesis (H₀) that error variances of the criterion variable (Height) are equal across groups (Females & Males) should be accepted.

12.2 The ANOVA Test

            Df Sum Sq Mean Sq F value Pr(>F)    
Gender       1    906     906     116 <2e-16 ***
Residuals   98    766       8                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

A significance value of p < .001 suggests that the null hypothesis can be rejected safely and the alternative hypothesis accepted. For reporting the output should be converted to the appropriate format for the publication venue. As an example, the American Psychological Association Manual of Style recommends the following table format (Table 12.1).

Table 12.1: ANOVA results in APA format

	Df	Sum of Squares	Mean Square	F	p (sig)
Gender	1	906.0	906.0	115.9	<.000
Residuals	98	766.2	7.8