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
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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
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
Transformations can help stabilize variances and improve normality
Common transformations: logarithmic, square root, reciprocal
Logarithmic: l o g ( x ) log(x) l o g ( x ) or l o g ( x + 1 ) log(x+1) l o g ( x + 1 ) for data with zero values
Square root: x \sqrt{x} x for data with a Poisson distribution
Reciprocal: 1 x \frac{1}{x} x 1 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