e-Statistics

Analysis of Variance

  1. Firstly the column of response values is specified.
  2. Secondly the column indicating ``treatments'' is specified.
  3. Another column indicating ``blocks'' is selected.
We then proceed to compute the analysis of variance table (AOV table) which summarizes the degree of freedom (df), the sum of squares (SS), and mean squares (MS).

Various statistics in the AOV table for randomized block design are described as in the following.

Source Degree of freedom $ SS$ Mean square F-statistic
Treatment $ k-1$ $ SS_{\mbox{tr}}$ $ \displaystyle MS_{\mbox{tr}} = \frac{SS_{\mbox{tr}}}{k-1}$ $ \displaystyle F = \frac{MS_{\mbox{tr}}}{MS_{\mbox{error}}}$
Block $ b-1$ $ SS_{\mbox{bl}}$ $ \displaystyle MS_{\mbox{bl}} = \frac{SS_{\mbox{bl}}}{b-1}$ $ \displaystyle F = \frac{MS_{\mbox{bl}}}{MS_{\mbox{error}}}$
Error $ kbn-k-b+1$ $ SS_{\mbox{error}}$ $ \displaystyle
MS_{\mbox{error}} = \frac{SS_{\mbox{error}}}{n-k-b+1}$  
Total within blocks $ kbn-1$ $ SS_{\mbox{total}}$    

  1. $ \displaystyle\bar{X}_{i \cdot\cdot} = \frac{1}{bn} \sum_{j=1}^{b}\sum_{l=1}^n X_{ijl}$ is called the treatment sample mean for each treatment $ i = 1,\ldots,k$.
  2. $ \displaystyle\bar{X}_{\cdot j \cdot} = \frac{1}{kn} \sum_{i=1}^{k}\sum_{l=1}^n X_{ijl}$ is the block sample mean for each block $ j = 1,\ldots,b$.
  3. $ \displaystyle\bar{X}_{\cdot\cdot\cdot}
= \frac{1}{kbn} \sum_{i=1}^k\sum_{j=1}^{b}\sum_{l=1}^n X_{ijl}$ is the overall sample mean.
  4. $ SS_{\mbox{tr}} = bn\displaystyle\sum_{i=1}^k (\bar{X}_{i \cdot\cdot} - \bar{X}_{\cdot\cdot\cdot})^2$ is the treatment sum of sqaures.
  5. $ SS_{\mbox{bl}} = kn\displaystyle\sum_{j=1}^b (\bar{X}_{\cdot j \cdot} - \bar{X}_{\cdot\cdot\cdot})^2$ is the block sum of sqaures,
  6. $ SS_{\mbox{error}} = \displaystyle\sum_{i=1}^k\sum_{j=1}^{b}\sum_{l=1}^n
(X_{i...
...r{X}_{i \cdot\cdot} - \bar{X}_{\cdot j \cdot}
+ \bar{X}_{\cdot\cdot\cdot})^2$ is the error sum of sqaures.
  7. $ SS_{\mbox{total}} = \displaystyle\sum_{i=1}^k \: \sum_{j=1}^b\sum_{l=1}^n
(X_{ijl} - \bar{X}_{\cdot\cdot\cdot})^2$ is the total sum of sqaures. Together we obtain the algebraic identity

    $\displaystyle SS_{\mbox{total}}
= SS_{\mbox{tr}} + SS_{\mbox{bl}} + SS_{\mbox{error}} .
$