Refreshing the Basics of ANOVA

Analysis of Variance, or ANOVA allows us to discern if there are any statistically significant differences between the means of three or more independent groups. While t-tests are used to compare two means, ANOVA comes into play when you want to compare more than two.

The ANOVA Equation and Its Interpretation

At its core, ANOVA is represented by the following equation:

\[ Y_{ij} = \mu + \alpha_i + \varepsilon_{ij} \]

Where:

  • \(Y_{ij}\) represents the observed value,

  • \(\mu\) is the overall mean,

  • \(\alpha_i\) is the effect of the ith group, and

  • \(\varepsilon{ij}\) is the residual or error.

To illustrate this, let’s consider a scenario where we are studying the effect of different diets on the weight of marine creatures. ANOVA helps us determine whether the differences in weight are simply due to random variation, or whether certain diets actually lead to a change in weight. The equation breaks down the observed weight (\(Y_{ij}\)) into the overall mean weight (\(\mu\)), the effect of each diet (\(\alpha_i\)), and the residual variation not explained by the diet (\(\varepsilon{ij}\)).

The Assumptions of ANOVA

Like any statistical test, ANOVA operates under certain assumptions. Meeting these assumptions allows for a robust analysis of your data. The three main assumptions are:

  1. Independence: Each sample should be selected independently from the others. In our marine science example, if we’re analyzing growth rates of coral under different light conditions, one coral’s growth should not affect another’s.

  2. Normality: The residuals, or errors, should be normally distributed. This means that the differences between the observed values and the values predicted by the model should follow a normal distribution.

  3. Homogeneity of Variances: The variances within each group should be roughly equal. This means that the variability in weight should be similar across all diet groups.

review

What statistical test do we use to test normality?

What statistical test can you use to test homogeneity of variances?

What plots can you use to check normality and homogeneity of variances?

Applying ANOVA in R

applying ANOVA in R is straightforward. Let’s say we have a dataset named coral_growth, with the variables GrowthRate and LightCondition. To perform an ANOVA to test if light condition affects growth rate, you can use the aov() function:

coral_growth <- data.frame(
  GrowthRate = c(10, 12, 15, 8, 9, 11, 14, 13, 16),
  LightCondition = factor(c("Low", "Low", "Low", "Medium", "Medium", "Medium", "High", "High", "High")))

# Performing ANOVA in R
anova_result <- aov(GrowthRate ~ LightCondition, data = coral_growth)

# Viewing the summary of the ANOVA
summary(anova_result)

The aov() function fits the ANOVA model, and the summary() function provides detailed results, including the F-statistic and the p-value, which helps determine if the differences between group means are statistically significant.