3.8 Two-Factor Studies

The RBD is a one-factor study, the interest is on one factor the treatments. The blocks are used because of not having enough homogeneous experimental units.
Two-factors, where factor \(A\) has \(a-\)levels, factor \(B\) has \(b-\)levels.

Then we have \(ab\) treatments.

The Model

\[Y_{ijl}=\mu _{ij}+\varepsilon _{ijl},\quad \varepsilon _{ijl}\thicksim ^{iid}(0,\sigma ^2)\] \[i=1,2,\ldots ,a\quad j=1,2,\ldots ,b\quad l=1,2,\ldots ,n\] Total number of observations \(abn\)

\[\widehat {\mu }_{ij}=\overline {Y}_{ij}.\quad \sum ^a_{i=1}\sum ^b_{j=1}\sum ^n_{l=1}\big (Y_{ijl}-\mu _{ij}\big )^2\] \[\text {SSE} = \sum \sum \sum \big (Y_{ijl}-\overline {Y}_{ij}.\big )^2\]

• \(df=ab(n-1)=\) total number of observations \(-\) parameters.
• \(Y_{ijl}= l^{\text {th}}\) observation under \(i^{\text {th}}\) level of factor \(A\) and \(j^{\text {th}}\) level of factor \(B\).
• \(\mu _{ij}=\) the expected mean of the treatment corresponding to the \(i^{\text {th}}\) level of factor \(A\) and \(j^{\text {th}}\) level of factor \(B\).
• \(\overline {Y}_{ij}.=\frac {1}{n}\sum \limits ^n_{l=1}Y_{ijl}=\) sample mean for the \(i^{\text {th}}\) level of factor \(A\) and \(j^{\text {th}}\) level of factor \(B\).
• \(\overline {Y}_i..=\frac {1}{nb}\sum \limits ^b_{j=1}\sum \limits ^n_{l=1}Y_{ijl}=\) Sample mean for the \(i^{\text {th}}\) level for factor \(A\).
• \(\overline {Y}._j.=\frac {1}{na}\sum \limits ^a_{i=1}\sum \limits ^n_{l=1}Y_{ijl}= \) Sample mean for the \(j^{\text {th}}\) level of factor \(B\).
• \(\overline {Y}...=\frac {1}{nab}\sum \limits ^a_{i=1}\sum \limits ^b_{j=1}\sum \limits ^n_{l=1}Y_{ijl}=\) Sample mean
  

Let \(\displaystyle {Q(Y)=\sum ^a_{i=1}\sum ^b_{j=1}\sum ^n_{l=1}\big (Y_{ijl}-\mu _{ij}\big )^2}\) the least squares estimators for \(\mu _{ij'}\)s are \(\overline {Y}_{ij'}\)s.

\[e_{ijl}=Y_{ijl}-\overline {Y}_{ij}.\]

• \(\text {SST}=\displaystyle {\sum ^a_{i=1}\sum ^b_{j=1}\sum ^n_{l=1}\big (Y_{ijl}-\overline {Y}...\big )^2}\quad \) Corrected total sum of squares.
• \(\text {SSTrt} = \displaystyle {n\sum ^a_{i=1}\sum ^b_{j=1}\big (\overline {Y}_{ij}.-\overline {Y}...\big )^2}\quad \) Treatment sum of squares.
• \(\text {SSE} = \displaystyle {\sum ^a_{i=1} \sum ^b_{j=1} \sum ^n_{l=1} \big (Y_{ijl}-\overline {Y}_{ij}.\big )^2}\quad \) Error sum of squares.
  

\[\underbrace {\text {SST}}_{abn-1} = \underbrace {\text {SSTrt}}_{ab-1}+\underbrace {\text {SSE}}_{ab(n-1)}\] \[Y_{ijl}-\overline {Y}... = \overline {Y}_{ij}.-\overline {Y}...+Y_{ijl}-\overline {Y}_{ij}.\]

Under \(H_0:\) the means are all equal (all the treatments are the same) \[H_0:\mu _{11}=\mu _{12}=\mu _{21}=\cdots \mu _{ab}\] \[\frac {\text {MSTrt}}{\text {MSE}}\thicksim f_{ab-1,ab(n-1)}\]

Reject \(H_0\) if \(\displaystyle {\frac {\text {MSTrt}}{\text {MSE}}\geq f^{\alpha }_{ab-1,ab(n-1)}}\)

\[Y_{ijl} = \mu + \alpha _i +\beta _j + (\alpha \beta )_{ij} + \varepsilon _{ijl}\] \[\mu = \overline {Y}... \quad \alpha _i=\overline {Y}_i..-\overline {Y}... \quad \beta _j= \overline {Y}.j.-\overline {Y}...\quad (\alpha \beta )_{ij}= \underbrace {\overline {Y}_{ij}.-\overline {Y}_i..-\overline {Y}._j.-\overline {Y}...}_{\text {interaction}}\] \[\varepsilon _{ijl}=Y_{ijl}-\overline {Y}_{ij}.\] \[\text {SST} = \underbrace {\text {SSA} + \text {SSB} + SS(AB)}_{\text {SSTrt}}+ \text {SSE}\] \[Y_{ijl} = \mu _{il}+\varepsilon _{ijl},\quad \varepsilon _{ijl}\thicksim ^{iid}N(0,\sigma ^2)\] \[i=1,2,\ldots , a\qquad j=1,2,\ldots ,b\qquad l=1,2,\ldots , n\] \begin {align*} Y & = XB +\varepsilon \\ Y_{ab\times 1} & = \begin {pmatrix} Y_{111} & Y_{121} & Y_{122} & \cdots & Y_{abn}\\ \end {pmatrix}^t\\ \varepsilon _{ab\times 1} & = \begin {pmatrix} e_{111}& \cdots & \cdots & \\ \end {pmatrix}^t\\ B_{ab\times 1} & = \begin {pmatrix} \mu _{11} & \cdots & \mu _{ab}\\ \end {pmatrix}^t \end {align*}

\[\underbrace {X}_{abn\times ab}= \begin {pmatrix} \mu _{11} & \mu _{12} & \mu _{13} & \cdots & \mu _{ab}\\ 1_n & 0 & 0 & & 0\\ 0 & 1_n & 0 & & 0\\ 0 & 0 & 1_n &\cdots & 0\\ \vdots & \vdots & \vdots & &\vdots \\ 0 & 0 & 0 & \cdots & 1_n\\ \end {pmatrix} \] \[Y_{ijl}=\mu + \alpha _i + \beta _j + (\alpha \beta )_{ij} + \varepsilon _{ijl}\qquad i=1,2,\ldots ,a\] \[\sum \alpha _i=\sum \beta _j =\sum _i (\alpha \beta )_{ij}=\sum _j(\alpha \beta )_{ij}=0\]

\[Y_{ijl} -\overline {Y}... = \overline {Y}_i..-\overline {Y}...+\overline {Y}._j.-\overline {Y}...+\overline {Y}_{ij}.-\overline {Y}_i..-\overline {Y}._j.+\overline {Y}...+Y_{ijl}-\overline {Y}_{ij}.\quad *\]

squaring both sides of \(*\) and summing we get

\[SST = \underbrace {SSA + SSB + SS(AB)}_{SSTrt}+SSE\] \begin {align*} SSTrt & = n\sum ^a_{i=1}\sum ^b_{j=1}\big (\overline {Y}_{ij}.-\overline {Y}...\big )^2\\ & = \underbrace {bn\sum ^a_{i=1}\big (\overline {Y}_i..-\overline {Y}...\big )^2}_{SSA}+\underbrace {an\sum ^b_{j=1}\big (\overline {Y}.j.-\overline {Y}...\big )^2}_{SSB}+ \underbrace {n\sum ^a_{i=1}\sum ^b_{j=1}\big (\overline {Y}_{ij}.-\overline {Y}_i..-\overline {Y}._j.+\overline {Y}...\big )^2}_{SS(AB)} \end {align*}

Computation Formula

\(\displaystyle {Y_{i}...=\sum ^b_{j=1}\sum ^n_{l=1}Y_{ijl}\qquad Y...=\sum ^a_{i=1}\sum ^b_{j=1}\sum ^n_{l=1}Y_{ijl}}\)

\(SSA = \displaystyle {bn\sum ^a_{i=1}\big (\overline {Y}_i..-\overline {Y}...\big )^2=\sum ^a_{i=1}\frac {Y^2_{i}..}{bn}-\frac {Y^2...}{abn}}\)

\(SSB = \displaystyle {an\sum ^b_{j=1}\big (\overline {Y}._j.-\overline {Y}...\big )^2=\frac {\sum ^b_{j=1}Y^2._j.}{an}-\frac {Y^2...}{abn}}\)

\(SST = \displaystyle {\sum ^a\sum ^b\sum ^n\big (Y_{ijl}-\overline {Y}...\big )^2=\sum \sum \sum Y^2_{ijl}-\frac {Y^2...}{abn}}\)

\(SSE=\displaystyle {\sum \sum \sum \big (Y_{ijl}-\overline {Y}_{ij}.\big )^2=\sum \sum \sum Y^2_{ijl}-\frac {\sum \sum Y^2_{ij}.}{n}}\)

\[SS(AB)=SST-SS(A)-SS(B)-SSE\]