Checking assumptions in a one-way ANOVA (or any statistical test) is crucial to ensure the validity and reliability of the results. Assumptions are the conditions that must be satisfied for the statistical test to be accurate and unbiased. If these assumptions are violated, the results may be misleading, leading to incorrect conclusions.
1. Independence
Independence of observations is largely a matter of the experimental design and the way data are collected. Easiest way to check for independence is to plot the residuals against run order or time period. We should look for a lack of any discernible pattern or trend; ideally, the points should appear randomly scattered around the horizontal axis, indicating that the residuals are not correlated with time and are considered independent.
If the residuals cluster together in certain time periods, it suggests non-independence and potential issues with the model. A gradual increase or decrease (upward or downward trend) in the residuals over time indicates a pattern that violates the independence assumption. If the residuals repeat a cyclical pattern over time, it suggests non-independence.
To further confirm independence, you can perform autocorrelation analysis on the residuals, which will provide statistical measures of the correlation between residuals at different time lags.
If you identify patterns in the time-ordered residual plot, it may be necessary to re-evaluate your model and potentially add additional variables to account for the observed time-related dependencies. For confirming indepedence, we use the Durbin Watson test.
2. Homoscedasticity (constant variance)
We can plot the residuals vs fitted to visually check for constant variance. The key thing to check for is a random scatter of points around the horizontal line at zero, indicating that the relationship between the variables is linear and the errors (residuals) have constant variance across the range of fitted values; any patterns or trends in the scatter suggest potential issues with the model, like non-linearity or unequal error variances. For constant variance, we use Bartlett test or Levene‘s test as a confirmatory test.
3. Normality
The residuals (differences between observed and predicted values) should be normally distributed. This assumption is important for the accuracy of the F-test in ANOVA. We use Q-Q plot for visual test.
If the residuals are normally distributed, the points should fall roughly along a straight diagonal line (the 45-degree line or the “line of equality”). Significant deviations from the straight line indicate departures from normality.
If the points form an S-shaped curve, it suggests the presence of heavy tails in the distribution (i.e., more outliers than expected in a normal distribution).
A convex or concave pattern indicates skewness. Upward curvature suggests a right-skewed distribution, while downward curvature indicates a left-skewed distribution.
Points that are far from the line indicate potential outliers or deviations from normality.
Checking for normality using a Q-Q plot helps verify one of the key assumptions of ANOVA. If the residuals are not normally distributed, the results of the ANOVA may not be reliable, and you might need to consider data transformations or non-parametric methods.
Visual test is followed by a confirmatory test such as Anderson Darling test or Shapiro-Wilk test that assesses whether the residuals are normally distributed. A p-value greater than 0.05 indicates that the residuals are normally distributed.
When conducting a one-way ANOVA, it’s essential to check the assumptions of independence, normality, and homogeneity of variances to ensure the validity of the results. Visual tests (e.g., scatterplots, Q-Q plots, histograms) and confirmatory tests (e.g., Shapiro-Wilk, Levene’s) are used to assess these assumptions. Focusing on the residuals allows us to isolate the effects being studied and ensure that the model assumptions are met.