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Analysis of a balanced two factor ANOVA model

1  Analysis of a balanced two factor ANOVA model

$$\alpha$$

Model:

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

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

 

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)