For the results of an ANOVA to be considered valid, several key assumptions about the data must be met. These are: (1) Independence of observations, meaning the errors are uncorrelated. (2) Normality, where the residuals for each group are approximately normally distributed. (3) Homoscedasticity, or homogeneity of variances, meaning the variance of residuals is equal across all groups.
These assumptions relate to the residuals (the differences between observed values and the group means), not the raw data itself. Independence is the most critical assumption and is typically ensured by proper experimental design and random sampling; violations can lead to severely biased results. Normality means the distribution of residuals within each group should follow a bell curve. ANOVA is considered relatively robust to moderate violations of this assumption, especially with large and balanced sample sizes, due to the Central Limit Theorem. Homoscedasticity ([latex]\sigma_1^2 = \sigma_2^2 = \dots = \sigma_k^2[/latex]) means the spread or scatter of data points around their group mean should be similar for all groups. Significant violation of this assumption (heteroscedasticity) can increase the rate of Type I errors. Statisticians have developed diagnostic tools to check these assumptions. For example, Q-Q plots can assess normality, and Levene’s test or Bartlett’s test can check for homogeneity of variances. If assumptions are severely violated, researchers may need to transform the data or use alternative statistical methods that do not rely on these assumptions.
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