3 Factorial designs with more than two factors

Jan A. van den Brakel

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The results developed for $K\times L$ factorial designs are extended to designs with more than two factors. A more general notation for the treatment factors is introduced first. Let $A_g$ denote the $g-\text{th}$ treatment factor in the experiment with levels $a_g=1,\mathrm{...},M_g$. In the general case there are $\text{g}=\text{1,}\mathrm{...}\text{,G}$ factors included in the experiment. The population parameters observed under the $M_1{M}_2\mathrm{...}{M}_G$ treatment combinations are collected in the vector $\overline{Y}={({\overline{Y}}_{\mathrm{11...1}},\mathrm{...},{\overline{Y}}_{a_1{a}_2\mathrm{...}{a}_G},\mathrm{...},{\overline{Y}}_{M_1{M}_2\mathrm{...}{M}_G})}^t$. The index for the levels of a factor runs within each level of its preceding factor. Thus index $a_g$ runs from $a_g=1,\mathrm{...},M_g$ within each level of $a_{(g-1)}$. Hypotheses about the main effects and interactions are, as motivated in section 2.2, formulated about $\overline{Y}$ in expectation over the measurement error model.

The contrast matrices for the main effects and interactions in (2.13) are developed for the general case of a $M_1\times M_2\times \mathrm{...}\times M_G$ factorial design. Let $A=\{1,\mathrm{...},G\}$ denote the set of labels for the factors and ${\tilde{C}}_{A_g}=(j_{(M_g-1)}|-I_{(M_g-1)})$. The following three functions are defined first;

$$J_{1_g}=\{\begin{array}{cc}{j}_{M_1}^t\otimes \mathrm{...}\otimes j_{M\left(g-1\right)}^t\hfill & :g>1\hfill \\ 1\hfill & :g=1\hfill \end{array},$$

$$J_{2_g}=\{\begin{array}{cc}{j}_{M\left(g+1\right)}^t\otimes \mathrm{...}\otimes j_{M_G}^t\hfill & :g<G\hfill \\ 1\hfill & :g=G\hfill \end{array},$$

$$J_{3_{g,g\text{'}}}=\{\begin{array}{cc}{j}_{M\left(g+1\right)}^t\otimes \mathrm{...}\otimes j_{M\left(g\text{'}-1\right)}^t\hfill & :g\text{'}-g>1\hfill \\ 1\hfill & :g\text{'}=g+1\hfill \end{array}.$$

The main effect of factor $A_g$ is defined as the $M_g-1$ contrasts between the $M_g$ levels, averaged over the levels of the other $G-1$ factors and is given by:

$$C_{A_{g_1}}={({\displaystyle {\prod }_{g\in A\backslash \left\{g_1\right\}}{M}_g})}^{-1}{J}_{1_{g_1}}\otimes {\tilde{C}}_{A_{g_1}}\otimes J_{2_{g_1}},g_1=1,\dots ,G.$$

Postmultiplication of ${\tilde{C}}_{A_{g_1}}$ by $J_{2_{g_1}}$ sums over the levels of the factors $A_{(g_1+1)}\mathrm{...}{A}_G$ that are nested within each level of $A_{g_1}$.Subsequently, ${\tilde{C}}_{A_{g_1}}$ defines the $M_{g_1}-1$ contrasts between the levels of $A_{g_1}$ that are nested within each combination of the levels of $A_1\mathrm{...}{A}_{(g_1-1)}$. Premultiplication of ${\tilde{C}}_{A_{g_1}}$ by $J_{1_{g_1}}$ adds the contrast matrices ${\tilde{C}}_{A_{g_1}}$ that are nested within all combinations of the levels of $A_1\mathrm{...}{A}_{(g_1-1)}$

The interaction between $A_{g_1}$ and $A_{g_2}$ is defined as the $M_{g_2}-1$ contrasts of factor $A_{g_2}$ between the $M_{g_1}-1$ contrasts of $A_{g_1}$ averaged over the levels of the other $G-2$ factors and is given by:

$$\begin{array}{l}{C}_{A_{g_1}{A}_{g_2}}={({\displaystyle {\prod }_{g\in A\backslash \left\{g_1,g_2\right\}}{M}_g})}^{-1}{J}_{1_{g_1}}\otimes {\tilde{C}}_{A_{g_1}}\otimes J_{3_{g_1,g_2}}\otimes {\tilde{C}}_{A_{g_2}}\otimes J_{2_{g_2}},\\ g_1=1,\dots ,G-1,g_2=1,\dots ,G,g_1<g_2.\end{array}$$

Postmultiplication of ${\tilde{C}}_{A_{g_2}}$ by $J_{2_{g_2}}$ adds the levels of the factors $A_{(g_2+1)}\mathrm{...}{A}_G$ that are nested within each level of $A_{g_2}$. ${\tilde{C}}_{A_{g_2}}$ defines the contrasts of the main effect of factor $A_{g_2}$ which are nested within each combination of the levels of $A_1\mathrm{...}{A}_{(g_2-1)}$. Postmultiplication of ${\tilde{C}}_{A_{g_1}}$ by $J_{3_{g_1,g_2}}$ sums the contrast matrices ${\tilde{C}}_{A_{g_2}}$ over the levels of $A_{(g_1+1)}\mathrm{...}{A}_{(g_2-1)}$ that are nested within each combination of the levels of $A_1\mathrm{...}{A}_{g_1}$. Premultiplication of $J_{3_{g_1,g_2}}\otimes {\tilde{C}}_{A_{g_2}}\otimes J_{2_{g_2}}$ with ${\tilde{C}}_{A_{g_1}}$ defines the contrasts of the interactions between $A_{g_1}$ and $A_{g_2}$, within each combination of the levels of $A_1\mathrm{...}{A}_{(g_1-1)}$. Finally, premultiplication of ${\tilde{C}}_{A_{g_1}}$ by $J_{1_{g_1}}$ sums the contrasts of the interactions between $A_{g_1}$ and $A_{g_2}$ over the levels of $A_1\mathrm{...}{A}_{(g_1-1)}$

The interaction between $A_{g_1}$, $A_{g_2}$ and $A_{g_3}$ is defined as the $M_{g_3}-1$ contrasts of factor $A_{g_3}$ between the interactions of $A_{g_1}$ and $A_{g_2}$, averaged over the levels of the other $G-3$ factors. This process expands in a similar way to higher order interactions, which results in the following definitions of the higher order interactions:

$$\begin{array}{cc}\hfill C_{A_{g_1}{A}_{g_2}{A}_{g_3}}& ={\left({\displaystyle {\prod }_{g\in A\backslash \left\{g_1,g_2,g_3\right\}}{M}_g}\right)}^{-1}{J}_{1_{g_1}}\otimes {\tilde{C}}_{A_{g_1}}\otimes J_{3_{g_1,g_2}}\otimes {\tilde{C}}_{A_{g_2}}\otimes J_{3_{g_2,g_3}}\otimes {\tilde{C}}_{A_{g_3}}\otimes J_{2_{g_3}},\hfill \\ & g_1=1,\dots ,G-2,g_2=2,\dots ,G-1,g_3=3,\dots ,G,g_1<g_2<g_3,\hfill \end{array}$$

$$\begin{array}{cc}\hfill C_{A_{g_1}{A}_{g_2}{A}_{g_3}{A}_{g_4}}& ={\left({\displaystyle {\prod }_{g\in A\backslash \left\{g_1,g_2,g_3,g_4\right\}}{M}_g}\right)}^{-1}{J}_{1_{g_1}}\otimes {\tilde{C}}_{A_{g_1}}\otimes J_{3_{g_1,g_2}}\otimes {\tilde{C}}_{A_{g_2}}\otimes J_{3_{g_2,g_3}}\otimes {\tilde{C}}_{A_{g_3}}\otimes J_{3_{g_3,g_4}}\otimes {\tilde{C}}_{A_{g_4}}\otimes J_{2_{g_3}},\hfill \\ & g_1=1,\dots ,G-3,g_2=2,\dots ,G-2,g_3=3,\dots ,G-1,\hfill \\ & g_4=4,\dots ,G,g_1<g_2<g_3<g_4,\hfill \\ & ⋮\hfill \\ \hfill C_{A_1{A}_2{A}_3\dots A_G}& ={\tilde{C}}_{A_{g_1}}\otimes {\tilde{C}}_{A_{g_2}}\otimes {\tilde{C}}_{A_{g_3}}\otimes \dots \otimes {\tilde{C}}_{A_{g_4}}\hfill \end{array}$$

The number of rows of each contrast matrix coincides with the number of contrasts that define the various main effects and interactions. The number of columns of these matrices equals $M_1{M}_2\mathrm{...}{M}_G$.

These contrast matrices are inserted in (2.13) to define the various hypotheses about the main effects and interactions between the G treatment factors. The sampling units in the initial sample are randomly divided over all possible treatment combinations according to a CRD or an RBD, resulting in $M_1{M}_2\mathrm{...}{M}_G$ different subsamples. Let $n_{a_1\mathrm{...}{a}_G}$ denote the number of sampling units assigned to treatment combination $a_1\mathrm{...}{a}_G$ in subsample $s_{a_1\mathrm{...}{a}_G}$ and $n_{+\mathrm{...}+}$ the size of the initial sample. In the case of a CRD, the first order inclusion probabilities for the units in subsample $s_{a_1\mathrm{...}{a}_G}$ are now given by ${\pi }_i^{*}={\pi }_i(n_{a_1\mathrm{...}{a}_G}/n_{+\mathrm{...}+})$. In the case of an RBD, the first order inclusion probabilities for the units in subsample $s_{a_1\mathrm{...}{a}_G}$ are given by ${\pi }_i^{*}={\pi }_i(n_{b{a}_1\mathrm{...}{a}_G}/n_{b+\mathrm{...}+})$ where $n_{b{a}_1\mathrm{...}{a}_G}$ denotes the number of sampling units assigned to treatment combination $a_1\mathrm{...}{a}_G$ in block $b$ and $n_{b+\mathrm{...}+}$ the total number of sampling units in block $b$

Now ${\widehat{\overline{Y}}}_{a_1\mathrm{...}{a}_G;greg}$ denotes the GREG estimator for ${\overline{Y}}_{a_1\mathrm{...}{a}_G}$ based on the observations obtained in subsample $s_{a_1\mathrm{...}{a}_G}$ and is defined analogously to expression (2.18). These $M_1{M}_2\mathrm{...}{M}_G$ GREG estimators are collected in the vector ${\widehat{\overline{Y}}}_{GREG}={({\widehat{\overline{Y}}}_{\mathrm{1...1};greg},\mathrm{...},{\widehat{\overline{Y}}}_{M_1\mathrm{...}{M}_G;greg})}^t$ and is an approximately design-unbiased estimator for $\overline{Y}$ and ${\text{E}}_m\overline{Y}$. Design-based estimators for the covariance matrices of the contrasts between the elements of ${\widehat{\overline{Y}}}_{GREG}$ are defined by (2.25), where the diagonal elements of $\widehat{D}$ are defined analogously to expression (2.26) in the case of a CRD or (2.27) in the case of an RBD.

Finally hypotheses about main effects and interactions are tested with the Wald statistic (2.28), which is asymptotically distributed as a chi-squared random variable where the number of degrees of freedom equals the number of contrasts specified in the various hypotheses. As an example, the contrast matrices of the main effects and interactions in a factorial design with four factors are given in Table 3.1.

Table 3.1
Contrasts in a $M_1\times M_2\times M_3\times M_4$ factorial design
Table summary
This table displays contrasts in an $M_1\times M_2\times M_3\times M_4$ factorial design. The information is grouped by Contrast matrix and Number of contrasts (appearing as column headers).

Contrast matrix	Number of contrasts (degrees of freedom)
$C_{A_1}=1/\left(M_2{M}_3{M}_4\right){\tilde{C}}_{A_1}\otimes j_{M_2}^t\otimes j_{M_3}^t\otimes j_{M_4}^t$	$M_1-1$
$C_{A_2}=1/\left(M_1{M}_3{M}_4\right)j_{M_1}^t\otimes {\tilde{C}}_{A_2}\otimes j_{M_3}^t\otimes j_{M_4}^t$	$M_2-1$
$C_{A_3}=1/\left(M_1{M}_2{M}_4\right)j_{M_1}^t\otimes j_{M_2}^t\otimes {\tilde{C}}_{A_3}\otimes j_{M_4}^t$	$M_3-1$
$C_{A_4}=1/\left(M_1{M}_2{M}_3\right)j_{M_1}^t\otimes j_{M_2}^t\otimes j_{M_3}^t\otimes {\tilde{C}}_{A_4}$	$M_4-1$
$C_{A_1{A}_2}=1/\left(M_3{M}_4\right){\tilde{C}}_{A_1}\otimes {\tilde{C}}_{A_2}\otimes j_{M_3}^t\otimes j_{M_4}^t$	$\left(M_1-1\right)\left(M_2-1\right)$
$C_{A_1{A}_3}=1/\left(M_2{M}_4\right){\tilde{C}}_{A_1}\otimes j_{M_2}^t\otimes {\tilde{C}}_{A_3}\otimes j_{M_4}^t$	$\left(M_1-1\right)\left(M_3-1\right)$
$C_{A_1{A}_4}=1/\left(M_2{M}_3\right){\tilde{C}}_{A_1}\otimes j_{M_2}^t\otimes j_{M_3}^t\otimes {\tilde{C}}_{A_4}$	$\left(M_1-1\right)\left(M_4-1\right)$
$C_{A_2{A}_3}=1/\left(M_1{M}_4\right)j_{M_1}^t\otimes {\tilde{C}}_{A_2}\otimes {\tilde{C}}_{A_3}\otimes j_{M_4}^t$	$\left(M_2-1\right)\left(M_3-1\right)$
$C_{A_2{A}_4}=1/\left(M_1{M}_3\right)j_{M_1}^t\otimes {\tilde{C}}_{A_2}\otimes j_{M_3}^t\otimes {\tilde{C}}_{A_4}$	$\left(M_2-1\right)\left(M_4-1\right)$
$C_{A_3{A}_4}=1/\left(M_1{M}_2\right)j_{M_1}^t\otimes j_{M_2}^t\otimes {\tilde{C}}_{A_3}\otimes {\tilde{C}}_{A_4}$	$\left(M_3-1\right)\left(M_4-1\right)$
$C_{A_1{A}_2{A}_3}=1/\left(M_4\right){\tilde{C}}_{A_1}\otimes {\tilde{C}}_{A_2}\otimes {\tilde{C}}_{A_3}\otimes j_{M_4}^t$	$\left(M_1-1\right)\left(M_2-1\right)\left(M_3-1\right)$
$C_{A_1{A}_2{A}_4}=1/\left(M_3\right){\tilde{C}}_{A_1}\otimes {\tilde{C}}_{A_2}\otimes j_{M_3}^t\otimes {\tilde{C}}_{A_4}$	$\left(M_1-1\right)\left(M_2-1\right)\left(M_4-1\right)$
$C_{A_1{A}_3{A}_4}=1/\left(M_2\right){\tilde{C}}_{A_1}\otimes j_{M_2}^t\otimes {\tilde{C}}_{A_3}\otimes {\tilde{C}}_{A_4}$	$\left(M_1-1\right)\left(M_3-1\right)\left(M_4-1\right)$
$C_{A_2{A}_3{A}_4}=1/\left(M_1\right)j_{M_1}^t\otimes {\tilde{C}}_{A_2}\otimes {\tilde{C}}_{A_3}\otimes {\tilde{C}}_{A_4}$	$\left(M_2-1\right)\left(M_3-1\right)\left(M_4-1\right)$
$C_{A_1{A}_2{A}_3{A}_4}={\tilde{C}}_{A_1}\otimes {\tilde{C}}_{A_2}\otimes {\tilde{C}}_{A_3}\otimes {\tilde{C}}_{A_4}$	$\left(M_1-1\right)\left(M_2-1\right)\left(M_3-1\right)\left(M_4-1\right)$

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