Analysis of a balanced two factor ANOVA model

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1 Analysis of a balanced two factor ANOVA model
1. 1.1 Point estimates of the population means
2. 1.2 ANOVA decomposition of sum squares
Contributors

1 Analysis of a balanced two factor ANOVA model

$$\alpha$$

Model:

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

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

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)