1.3 The Randomised Block Design
The one-way layout treats every source of variation other than the treatments as error. Often something else is known: plots lie in fields of different fertility, patients come from different age groups, measurements are made by different technicians. Ignoring such a factor inflates the error mean square and makes the treatment test insensitive.
A block is a group of units expected to be alike. By comparing treatments within blocks, the block-to-block variation is removed from the error term instead of hiding in it. The decomposition below is the same exercise as in Section 1.2 with one more term, and the payoff is visible in the error degrees of freedom: fewer of them, but a smaller mean square, and usually a sharper test.
\[Y_{ij} = \mu + \beta _i + \tau _j + \varepsilon _{ij}\] No replication, \[\quad i=1,2,\ldots ,b \qquad \varepsilon _{ij} \thicksim N(0,\sigma ^2)\qquad \sum ^b_{i=0} \beta _i=0\qquad \sum ^k_{j=0}\overline {Y}=0\]
- \(\mu = \) overall mean
- \(\beta _i = \) effect of the \(i^{\text {th}}\) block
- \(\tau _j = \) effect of the \(j^{\text {th}}\) treatment.
\begin {align*} Y_{ij} & = \mu + \beta _i + \tau _j + \varepsilon _{ij}\\ Y_{ij} & = \overline {Y}.. + \overline {Y}_i. - \overline {Y}.. + \overline {Y}._j-\overline {Y}.. + Y_{ij}-\overline {Y}_i. + \overline {Y}..\\ Y_{ij} - \overline {Y}.. & = \big (\overline {Y}_i.-\overline {Y}..\big ) + \big (\overline {Y}._j-\overline {Y}..\big ) + \big (Y_{ij}-\overline {Y}_i.-\overline {Y}._j+\overline {Y}..\big ) \end {align*}
squaring both sides and add, we get
\[\sum ^k_{j=1}\sum ^b_{i=1}\big (Y_{ij}-\overline {Y}..\big )^2 = \sum ^b_{i=1}k\big (\overline {Y}_i.-\overline {Y}..\big )^2 + \sum ^k_{j=1} b\big (\overline {Y}._j-\overline {Y}..\big )^2 + \sum ^k_{j=1}\sum ^b_{i=1}\big (Y_{ij}-\overline {Y}._j-\overline {Y}_i.+\overline {Y}..\big )^2\]
\[SST = SSB + SSTr(t) + SSE\]
\[Y_{ij} = \mu + \tau _j + \varepsilon _{ij}\,, \qquad i=1,2,\ldots ,b\quad j=1,2,\ldots ,k\] one - way balance
\[Y_{ij} = \mu + \beta _i + \tau _j + \varepsilon _{ij}\,,\qquad i=1,2,\ldots ,b\quad j=1,2,\ldots ,k\]
\begin {align*} SST & = SSB + SSTrt + SSE\\ & = (b-1) + (K-1) + (b-1)(K-1)\\ \end {align*}
\[Y_{ij} = \mu _j + \varepsilon _{ij}\quad \Big |\quad Y_{ij} = \mu + \tau _j + \varepsilon _{ij}\] \begin {align*} \text {C.Is};&\quad \overline {Y}._j\quad , \quad \overline {Y}._j-\overline {Y}._{j'}\, , \quad j\neq j'\\ & C_j \overline {Y}._j + C_{j'} \overline {Y}._{j'}\\ \end {align*}
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