3.9 Three-Way Analysis Of Variance

In a three-way model there are three factors of interest \[Y_{ijlm}=\mu _{ijl}+\varepsilon _{ijlm}\] \[i=1,2,\ldots ,a\qquad j=1,2,\ldots ,b\qquad l=1,2,\ldots ,k\qquad m=1,2,\ldots , n\] \[ \varepsilon _{ijlm}\thicksim ^{iid}(0,\sigma ^2)\]

There are \(abk\) treatments.

Factor \(A\), has \(a-\) levels
Factor \(B\), has \(b-\) levels
Factor \(C\), has \(k-\) levels

\[Y_{ijlm}=\mu + \alpha _i + \beta _j + \tau _l + (\alpha \beta )_{ij} + (\alpha \tau )_{il} + (\beta \tau )_{jl}+(\alpha \beta \tau )_{ijl}+\varepsilon _{ijlm}\] \[\sum \alpha _i=\sum \beta _j=\sum (\alpha \beta )_{ij}=\sum (\alpha \tau )_{il}=\sum (\beta \tau )_{jl}=\sum (\alpha \beta \tau )_{ijl}=0\]

• \(\mu =\)
• \(\alpha _i=\)
• \(\beta _j=\)
• \(\tau _l=\)
• \((\alpha \beta )_{ij}=\)
• \((\alpha \tau )_{il}=\)
• \((\beta \tau )_{jl}=\)
• \((\alpha \beta \tau )_{ijl}=\) interaction of factor \(A\) at the \(i^{\text {th}}\) level with factor \(B\) at the \(j^{\text {th}}\) level and with factor \(C\) at the \(l^{\text {th}}\) level.

\begin {align*} Y_{ijlm}-\overline {Y}.... = & \underbrace {\overline {Y}_i...-\overline {Y}....}_{\hat {\alpha }_i} + \underbrace {\overline {Y}._j..-\overline {Y}....}_{\hat {\beta }_j} + \underbrace {\overline {Y}..l.-\overline {Y}....}_{\hat {\tau }} + \underbrace {\overline {Y}_{ij}..-\overline {Y}_i...-\overline {Y}._j..+\overline {Y}....}_{(\hat {\alpha }\hat {b})_{ij}}\\ & + \underbrace {\overline {Y}_i._l.-\overline {Y}_i...-\overline {Y}.._l.+\overline {Y}....}_{(\alpha \hat {\tau })_{il}}+\underbrace {\overline {Y}._{jl}.-\overline {Y}._j..-\overline {Y}.._l.-\overline {Y}....}_{(\beta \tau )_{jl}}\\ & + \underbrace {\overline {Y}_{ijl}.-\overline {Y}_i...-\overline {Y}._j..-\overline {Y}.._l.-\overline {Y}....}_{(\alpha \beta \tau )_{ijl}} + \underbrace {Y_{ijlm}-\overline {Y}_{ijl}.}_{\hat {\varepsilon }_{ijlm}} \end {align*}

Squaring both sides of the identity above and adding, all the cross-product terms vanish, leaving \begin {align*} SST = & SSA + SSB + SSC + SSAB + SSAC + SSBC + SS(ABC) + SSE\\ abkn-1 = & (a-1) + (b-1) + (k-1) + (a-1)(b-1) + (a-1)(k-1) + (b-1)(k-1)\\ & + (a-1)(b-1)(k-1)+ abk(n-1)\\ \end {align*}

\(SST = \displaystyle {\sum _i\sum _j\sum _l\sum _m\big (Y_{ijlm}-\overline {Y}....\big )^2}\)

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

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

\(SSC = \displaystyle {abn\sum ^k_{l=1}\big (\overline {Y}.._l.-\overline {Y}....\big )^2}\)

\(SS(AB) = \displaystyle {kn\sum ^a_{i=1}\sum ^b_{j=1}\big (\overline {Y}_{ij}..-\overline {Y}_i...-\overline {Y}._j..+\overline {Y}....\big )^2}\)

\(SS(AC) = \displaystyle {bn\sum ^a_{i=1}\sum ^k_{l=1}\big (\overline {Y}_i._l.-\overline {Y}_i...-\overline {Y}.._l.+\overline {Y}....\big )^2}\)

\(SS(BC) =\displaystyle {an\sum ^b_{j=1}\sum ^k_{l=1}\big (\overline {Y}._{jl}.-\overline {Y}._j..-\overline {Y}.._l.-\overline {Y}....\big )^2}\)

\(SS(ABC) = \displaystyle {n\sum _i\sum _j\sum _l\big (\overline {Y}_{ijl}.-\overline {Y}_i...-\overline {Y}._j..-\overline {Y}.._l.+\overline {Y}....\big )}\)

\(SSE = \displaystyle {\sum _i\sum _j\sum _l\sum _m\big (Y_{ijlm}-\overline {Y}_{ijl}.\big )^2}\)