One-Way Analysis of Variance (ANOVA)

One-way ANOVA is the simplest form of this statistical technique. It extends the two-sample t-test to situations with more than two groups, avoiding the problem of inflated Type I error that arises from performing multiple pairwise t-tests. The ‘one-way’ or ‘one-factor’ designation indicates that the groups are defined by a single categorical variable. For example, in a study comparing the effectiveness of three different diets, ‘diet type’ is the single factor. The underlying statistical model for an observation \(y_{ij}\) (the i-th observation in the j-th group) is \(y_{ij} = \mu + \tau_j + \epsilon_{ij}\), where \(\mu\) is the overall grand mean, \(\tau_j\) is the effect of being in group j, and \(\epsilon_{ij}\) is the random error term. The analysis proceeds by calculating the F-statistic. If the F-test yields a significant result (i.e., the p-value is below a chosen significance level), it indicates that at least one group mean is different from the others. However, ANOVA does not specify which groups are different. To identify the specific differences, post-hoc tests like Tukey’s HSD or Bonferroni correction are required.