e-Statistics

Analysis of Variance

The column of response values is specified.
The column identified with “treatments” is selected.
Another column identified with “blocks” is selected.

It proceeds 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 AOV table for randomized block design are described as follows:

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

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

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

The pairwise contrast tests the null hypothesis $\alpha_i - \alpha_j = 0$ for treatment, and $\beta_i - \beta_j = 0$ for block. It provides post hoc information about the difference among treatment and block. The point estimate $\theta_{ij}$ of contrast is obtained by $\bar{X}_{i\cdot\cdot} - \bar{X}_{j\cdot\cdot}$ for treatment, and $\bar{X}_{\cdot i\cdot} - \bar{X}_{\cdot j\cdot}$ for block. The standard error $\sigma_{ij}$ are calculated from $\hat{\sigma}_{ij}^2 = MS_{\mbox{error}}\times\left(\frac{2}{bn}\right)$ for treatment, and $\hat{\sigma}_{ij}^2 = MS_{\mbox{error}}\times\left(\frac{2}{kn}\right)$ for block. It should be noted that pairwise t-test does not take into account the simultaneity of multiple testing, and that it may give erroneous results.