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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)