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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: