# Analysis of a balanced two factor ANOVA model

### 1  Analysis of a balanced two factor ANOVA model

$$\alpha$$

Model:

$$Y_{ijk}=\mu_{\cdot\cdot}+\alpha_i+\beta_j+(\alpha\beta)_{ij}+\epsilon_{ijk}, \qquad i=1,\cdots,a; ~~j=1,\cdots,b; ~k=1,\cdots,n, (1)$$

$$\label{eq:two_factor_ANOVA} Y_{ijk}=\mu_{\cdot\cdot}+\alpha_i+\beta_j+(\alpha\beta)_{ij}+\epsilon_{ijk}, \qquad i=1,\cdots,a; ~~j=1,\cdots,b; ~k=1,\cdots,n,$$

where

• $$\mu..$$, $$\alpha_i$$'s $$\beta_j$$'s and ($$alpha\beta)_{ij}$$'s are unknown parameters (fixed effects) subject to identifiability constraints:

$$\sum_{i=1}^a \alpha_i=0, \sum_{j=1}^b \beta_j=0 (2)$$

$$\sum_{i=1}^a (\alpha\beta)_{ij}=0, ~~j=1,\cdots,b; \sum_{j=1}^b (\alpha\beta)_{ij}=0, ~~ i=1,\cdots,a . (3)$$

• Distributional assumption : $$\epsilon_{ijk}$$ are i.i.d.   (independently and identically distributed) as N(0, $$\sigma^2$$).
• In another word, $$Y_{ijk}$$'s are independent random variables with normal distribution with

$$\mu_{ij} := \mathbb{E}(Y_{ijk}) = \mu_{\cdot\cdot}+\alpha_i+\beta_j+(\alpha\beta)_{ij},$$

and Var($$Y_{ijk}$$) = $$\sigma^2$$, where $$\alpha_i$$'s, $$\beta_j$$'s and $$(\alpha\beta)_{ij}$$'s are subject to the identifiability constnraints (2) and (3).

#### 1.1 Point estimates of the population means

We estimate the population means by the corresponding sample means.

$$\overline{Y}_{ij\cdot}&=&\frac{1}{n}\sum_{k=1}^n Y_{ijk} &\longrightarrow& \mu_{ij}=\mu_{\cdot\cdot}+\alpha_i+\beta_j+(\alpha\beta)_{ij}\\ \overline{Y}_{i\cdot\cdot}&=&\frac{1}{bn}\sum_{j=1}^b\sum_{k=1}^nY_{ijk} &\longrightarrow& \mu_{i\cdot}=\mu_{\cdot\cdot}+\alpha_i\\ \overline{Y}_{\cdot j\cdot}&=&\frac{1}{an}\sum_{i=1}^a\sum_{k=1}^nY_{ijk} &\longrightarrow& \mu_{\cdot j}=\mu_{\cdot\cdot}+\beta_j\\ \overline{Y}_{\cdots}&=&\frac{1}{abn}\sum_{i=1}^a\sum_{j=1}^b\sum_{k=1}^nY_{ijk} &\longrightarrow& \mu_{\cdot\cdot}$$
The effects (main effects and interaction effects) can be estimated accordingly.

$$\widehat{\alpha}_i & := \overline{Y}_{i\cdot\cdot}-\overline{Y}_{\cdots} & \longrightarrow \alpha_i=\mu_{i\cdot}-\mu_{\cdot\cdot}\\ \widehat{\beta}_j & := \overline{Y}_{\cdot j\cdot}-\overline{Y}_{\cdots} & \longrightarrow \beta_j=\mu_{\cdot j}-\mu_{\cdot\cdot} \\ \widehat{(\alpha\beta)}_{ij} & :=\overline{Y}_{ij\cdot}-\overline{Y}_{\cdots}-(\overline{Y}_{i\cdot\cdot}-\overline{Y}_{\cdots})-(\overline{Y}_{\cdot j\cdot}-\overline{Y}_{\cdots}) & \\ & =\overline{Y}_{ij\cdot}-\overline{Y}_{i\cdot\cdot}-\overline{Y}_{\cdot j\cdot}+\overline{Y}_{\cdots} & \longrightarrow (\alpha\beta)_{ij}=\mu_{ij}-\alpha_i-\beta_j+\mu_{\cdot\cdot}$$

#### 1.2 ANOVA decomposition of sum squares

Basic decomposition:

$$SSTO = SSTR + SSE.$$

where

$$SSTO &=& \sum_{i=1}^a\sum_{j=1}^b \sum_{k=1}^n (Y_{ijk} - \overline{Y}_{\cdots})^2 \\ SSTR &=& n \sum_{i=1}^a \sum_{j=1}^b (\overline{Y}_{ij\cdot} - \overline{Y}_{\cdots})^2\\ SSE &=& \sum_{i=1}^a\sum_{j=1}^b \sum_{k=1}^n (Y_{ijk} - \overline{Y}_{ij\cdot})^2$$

### Contributors

• Yingwen Li (UCD)
• Debashis Paul (UCD)