Analysis of a balanced two factor ANOVA model
- Page ID
- 204
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)