26  ANOVA

ANOVA, which stands for Analysis of Variance, is a statistical technique used to determine if there are any statistically significant differences between the means of three or more independent (unrelated) groups. It tests the hypothesis that the means of several groups are equal, and it does this by comparing the variance (spread) of scores among the groups to the variance within each group. The primary goal of ANOVA is to uncover whether there is a difference among group means, rather than determining which specific groups are different from each other.

26.1 Types of ANOVA

  1. One-Way ANOVA: Also known as single-factor ANOVA, it assesses the impact of a single factor (independent variable) on a continuous outcome variable. It compares the means across two or more groups. For example, testing the effect of different diets on weight loss.

  2. Two-Way ANOVA: This extends the one-way by not only looking at the impact of one, but two factors simultaneously on a continuous outcome. It can also evaluate the interaction effect between the two factors. For example, studying the effect of diet and exercise on weight loss.

  3. Repeated Measures ANOVA: Used when the same subjects are used for each treatment (e.g., measuring student performance at different times of the year).

  4. Multivariate Analysis of Variance (MANOVA): MANOVA is an extension of ANOVA when there are two or more dependent variables.

26.2 Assumptions of ANOVA

ANOVA relies on several assumptions about the data:

  • Independence of Cases: The groups compared must be composed of different individuals, with no individual being in more than one group.
  • Normality: The distribution of the residuals (differences between observed and predicted values) should follow a normal distribution.
  • Homogeneity of Variances: The variance among the groups should be approximately equal. This can be tested using Levene’s Test or Bartlett’s Test.

26.3 ANOVA Formula

The basic formula for ANOVA is centered around the calculation of two types of variances: within-group variance and between-group variance. The F-statistic is calculated by dividing the variance between the groups by the variance within the groups:

\[F = \frac{\text{Variance between groups}}{\text{Variance within groups}}\]

26.3.1 Steps to Conduct ANOVA

  1. State the Hypothesis:

    • Null hypothesis (H0): The means of the different groups are equal.
    • Alternative hypothesis (Ha): At least one group mean is different from the others.

  2. Calculate ANOVA: Determine the F-statistic using the ANOVA formula, which involves calculating the between-group variance and the within-group variance.

  3. Compare to Critical Value: Compare the calculated F-value to a critical value obtained from an F-distribution table, considering the degrees of freedom for the numerator (between-group variance) and the denominator (within-group variance) and the significance level (alpha, usually set at 0.05).

  4. Make a Decision: If the F-value is greater than the critical value, reject the null hypothesis. This indicates that there are significant differences between the means of the groups.

26.3.2 Post-hoc Tests

If the ANOVA indicates significant differences, post-hoc tests like Tukey’s HSD, Bonferroni, or Dunnett’s can be used to identify exactly which groups differ from each other.