Analysis of Variance (ANOVA)

One-Way Classified Data

Fixed Effect Model:

• Assumes that the levels of the factor are the only levels of interest.
• The goal is to determine if there are significant differences between the group means.
• Model: \(Y_{ij} = \mu + \alpha_i + \epsilon_{ij}\)
  • $ Y_{ij} $: Observation from group $ i $, replicate $ j $
  • $ \mu $: Overall mean
  • $ \alpha_i $: Effect of group $ i $ (fixed effect)
  • $ \epsilon_{ij} $: Random error

Random Effect Model:

• Assumes that the levels of the factor are randomly selected from a larger population.
• The focus is on the variability among the group means.
• Model: \(Y_{ij} = \mu + \alpha_i + \epsilon_{ij}\)
  • $ \alpha_i $: Random effect with $ \alpha_i \sim N(0, \sigma^2_\alpha) $

Two-Way Classified Data (One Observation per Cell)

Fixed Effect Model:

• Both factors have fixed levels.
• Model: \(Y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} + \epsilon_{ijk}\)
  • $ \alpha_i $: Effect of factor A
  • $ \beta_j $: Effect of factor B
  • $ (\alpha\beta)_{ij} $: Interaction effect between A and B

Random Effect Model:

• Both factors are random effects.
• Model: \(Y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} + \epsilon_{ijk}\)
  • $ \alpha_i \sim N(0, \sigma^2_\alpha) $

  • $ \beta_j \sim N(0, \sigma^2_\beta) $

  • $(\alpha\beta){ij} \sim N(0, \sigma^2{\alpha\beta})$

Mixed Effect Model:

• One factor is fixed and the other is random.
• Model: \(Y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} + \epsilon_{ijk}\)
  • $ \alpha_i $: Fixed effect
  • $ \beta_j \sim N(0, \sigma^2_\beta) $

Two-Way Classified Data (m observations per cell)

Fixed Effect Model:

• Both factors have fixed levels, and multiple observations per cell.
• Model: \(Y_{ijkm} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} + \epsilon_{ijkm}\)

Random Effect Model:

• Both factors are random effects, with multiple observations per cell.
• Model: \(Y_{ijkm} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} + \epsilon_{ijkm}\)

Mixed Effect Model:

• One factor is fixed and the other is random, with multiple observations per cell.
• Model: