ANOVA assumptions are crucial for valid results. Normality, homogeneity of variance, and independence must be checked. Violations can lead to incorrect conclusions, so it's important to assess these assumptions using visual and formal methods.

Diagnostic tests help evaluate ANOVA assumptions. Residual plots and formal tests like Levene's and Shapiro-Wilk are used. If violations occur, data transformations or robust methods can address issues, ensuring reliable analysis and interpretation of results.

Assumptions

Normality and Its Assessment

• Normality assumes the residuals (differences between observed and predicted values) are normally distributed
• Violations of normality can lead to inaccurate p-values and confidence intervals
• Assess normality visually using Q-Q plots or histograms of residuals
  • Q-Q plots compare the distribution of residuals to a theoretical normal distribution
  • Histograms should show a bell-shaped curve for normally distributed residuals
• Formally test normality using the Shapiro-Wilk test
  • Null hypothesis: residuals are normally distributed
  • P-value < 0.05 suggests a significant departure from normality

Homogeneity of Variance and Independence

• Homogeneity of variance (homoscedasticity) assumes equal variances across groups
  • Violations (heteroscedasticity) can affect the validity of F-tests and lead to incorrect conclusions
  • Assess homogeneity visually using residual plots (residuals vs. fitted values)
    • Patterns or increasing/decreasing spread indicate heteroscedasticity
  • Formally test homogeneity using Levene's test
    • Null hypothesis: variances are equal across groups
    • P-value < 0.05 suggests significant differences in variances
• Independence of observations assumes that observations within and between groups are not related
  • Violations can occur due to repeated measures, clustering, or spatial/temporal correlation
  • Assess independence by examining the study design and data collection process
  • Violations may require alternative models (repeated measures ANOVA, mixed models)

Diagnostic Tests

Residual Plots for Assessing Assumptions

• Residual plots are graphical tools for assessing ANOVA assumptions
• Residuals vs. Fitted plot
  • Assess homogeneity of variance
  • Look for patterns, increasing/decreasing spread, or outliers
• Normal Q-Q plot
  • Assess normality of residuals
  • Compare residuals to a theoretical normal distribution
  • Deviations from a straight line indicate non-normality
• Scale-Location plot
  • Assess homogeneity of variance
  • Look for patterns or increasing/decreasing spread
• Residuals vs. Leverage plot
  • Identify influential observations
  • Points with high leverage and large residuals may have a strong influence on the model

Formal Tests for Assumptions

• Levene's test for homogeneity of variance
  • Null hypothesis: variances are equal across groups
  • P-value < 0.05 suggests significant differences in variances
  • Robust to non-normality, but sensitive to large sample sizes
• Shapiro-Wilk test for normality
  • Null hypothesis: residuals are normally distributed
  • P-value < 0.05 suggests a significant departure from normality
  • More powerful than visual assessment, but sensitive to large sample sizes
  • Alternative: Anderson-Darling test

Addressing Violations

Data Transformations

• Transformations can help stabilize variances and improve normality
• Common transformations: logarithmic, square root, reciprocal
  • Logarithmic: $log(x)$ or $log(x+1)$ for data with zero values
  • Square root: $\sqrt{x}$ for data with a Poisson distribution
  • Reciprocal: $\frac{1}{x}$ for data with a strong right skew
• Choose a transformation based on the nature of the data and the severity of the violation
• Interpret results on the transformed scale or back-transform for interpretation

Robust ANOVA Methods and Non-Parametric Alternatives

• Robust ANOVA methods are less sensitive to violations of assumptions
  • Welch's ANOVA: does not assume equal variances
  • Trimmed means ANOVA: robust to non-normality and outliers
  • Bootstrapping: resampling method to obtain robust confidence intervals and p-values
• Non-parametric alternatives do not rely on distributional assumptions
  • Kruskal-Wallis test: rank-based test for comparing medians across groups
  • Friedman test: rank-based test for repeated measures designs
  • Permutation tests: resampling method to obtain exact p-values
• Consider the trade-offs between robustness and power when selecting an alternative method