Biometrical Letters (BL template)

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\markboth{\textnormal{\small \hfill  K. Ozawa K., S. Kuriki, S. Mejza   \hfill}}{\textnormal{\small \hfill Incomplete split-block designs constructed by affine $\alpha$-resolvable designs \hfill}}


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%------------------------------------------------------------TITLE------------------------------------------------------------  
\begin{BLtitle}
Incomplete split-block designs constructed by affine $\alpha$-resolvable designs
\end{BLtitle}


%------------------------------------------------------------AUTHORS------------------------------------------------------------

\begin{BLauthor}
Kazuhiro Ozawa$^{1}$,  Shinji Kuriki$^{2}$,  Stanis{\l}aw Mejza$^{3}$
\end{BLauthor}


%------------------------------------------------------------ADRESSES ---------------------------------------------------

%
\begin{BLaffiliation}
$^1$Department of Nursing, Gifu College of Nursing, Hashima, Gifu, 501-6295, Japan, e-mail: [email protected] \\
$^2$Department of Mathematical Sciences, Graduate School of Engineering, Osaka Prefecture University, Naka-ku, Sakai, Osaka, 599-8531, Japan,\\ e-mail: [email protected] \\
$^3$Department of Mathematical and Statistical Methods, Pozna\'n University of Life Sciences, Wojska Polskiego 28, PL-60-637 Pozna\'n, Poland, \\
e-mail: [email protected] \\
\end{BLaffiliation}


%
%
%------------------------------------------------------------ABSTRACT------------------------------------------------------------
\begin{Summary}
We construct an incomplete split-block design (ISBD) by the semi-Kronecker product of two affine $\alpha$-resolvable designs for row and column treatments......
\end{Summary}
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%------------------------------------------------------------KEY_WORDS---------------------------------------------------

\begin{Key_words}
affine $\alpha$-resolvable designs, incomplete split-block designs...
\end{Key_words}



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\pagewiselinenumbers
\section{Introduction}
 We consider a two-factor experiment of split-block type with $b$ blocks, in which the first factor $A$ occurs at $v_1$ levels $A_1,A_2,\ldots,A_{v_1}$ and the second factor $B$ occurs at $v_2$ levels $B_1,B_2,\ldots,B_{v_2}$. Each block is divided into $k_1$ rows and $k_2$ columns. The levels of $A$, called row treatments, are randomly assigned to the rows, and the levels of $B$, called column treatments, are randomly assigned to the columns. Such a design is called a split-block design. Here we consider an incomplete split-block design (ISBD) such that $k_1 < v_1$ and $k_2 < v_2$.
 
 ISBDs are often used in the biological, agricultural and environmental sciences. Many authors (for example, Mejza (1987, 1998), Mejza (1992) and Hering and Mejza (1997)) have considered methods for the analysis of data and constructions for ISBDs. In particular, Mejza (1992) and Hering and Mejza (1997) considered a mixed linear model with fixed treatment effects and random block, row and column effects. The $h$th factorial treatment combination effect $\tau_h$ is defined by
\begin{eqnarray}\nonumber
 \tau_{h}=\mu+\alpha_{i}+\beta_{j}+(\alpha\beta)_{ij}
\end{eqnarray}
for $h=(i-1)v_2+j$, $i=1,2,\ldots,v_1$ and $j=1,2,\ldots,v_2$, where $\mu$ is the general mean, $\alpha_i$ denotes the main effect of level $A_i$ of the factor $A$, $\beta_j$ denotes the main effect of level $B_j$ of the factor $B$, and $(\alpha\beta)_{ij}$ denotes the interaction effect of $A_i$ and $B_j$.
\begin{figure}[h!]
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\includegraphics[scale=0.45]{figure.jpg}
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\caption{This is the first figure}
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\subsection{Subsection 1}
%
 Here $\sum_{i=1}^{v_1} \alpha_i=0$, $\sum_{j=1}^{v_2} \beta_j=0$, $\sum_{i=1}^{v_1} (\alpha\beta)_{ij}=0$ for $j=1,2,\ldots,v_2$ and $\sum_{j=1}^{v_2} (\alpha\beta)_{ij}=0$ for $i=1,2,\ldots,v_1$. This model (Hering and Mejza, 1997) results from a three-step randomization, i.e., ....... 
The stratum information matrices are
\begin{eqnarray}
\begin{array}{cl}
\mbox{\bf A}_1= & \frac{1}{k_1k_2}\mbox{\bf N}_1^{ }\mbox{\bf N}_1'-\frac{r}{v}\mbox{\bf J}_v, \quad \mbox{\bf A}_2=\frac{1}{k_2}\mbox{\bf
N}_2^{ }\mbox{\bf N}_2'-\frac{1}{k_1k_2}\mbox{\bf N}_1^{ }\mbox{\bf N}_1',
\\
 \mbox{\bf A}_3=  & \frac{1}{k_1}\mbox{\bf N}_3^{ }\mbox{\bf
N}_3'-\frac{1}{k_1k_2}\mbox{\bf N}_1^{ }\mbox{\bf N}_1' \quad {\rm and }  \\
\mbox{\bf
A}_4=  & r\mbox{\bf I}_v-\frac{1}{k_2}\mbox{\bf N}_2^{ }\mbox{\bf
N}_2'-\frac{1}{k_1}\mbox{\bf N}_3^{ }\mbox{\bf N}_3'+\frac{1}{k_1k_2}\mbox{\bf
N}_1^{ }\mbox{\bf N}_1'.
\label{ai}
\end{array}
\end{eqnarray}
\begin{table}[!ht]
\begin{center}
\caption{This is an example table }
\small
 \begin{tabular}{c c c c} 
 \hline
 Col1 & Col2 & Col2 & Col3 \\ [0.5ex] 
 \hline
 1 & 6 & 87837 & 787 \\ 
 2 & 7 & 78 & 5415 \\
 3 & 545 & 778 & 7507 \\
 4 & 545 & 18744 & 7560 \\
  5 & 88 & 788 & 6344 \\ 
 \hline
\end{tabular}
\end{center}
\end{table}

\subsection{Subsection 2}
%
$v=v_1v_2$, $\mbox{\bf N}_1$, $\mbox{\bf N}_2$ and $\mbox{\bf N}_3$ are the incidence matrices for treatment combinations vs. blocks, treatment combinations vs. rows and treatment combinations vs. columns, respectively, $\mbox{\bf I}_v$ is the identity matrix of order $v$ and $\mbox{\bf J}_v$ is a $v \times v$ matrix whose elements are all unity. 


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\section{The method of construction of ISBDs}
%
 Firstly, we need the semi-Kronecker product of two matrices, which will be used to construct ISBDs (see Mejza et al., 2001). 
Suppose that two matrices $\mbox{\bf C}$ and $\mbox{\bf D}$ are divided into the same number $t$ of submatrices as follows:
\begin{eqnarray}\nonumber
 \mbox{\bf C}=[\mbox{\bf C}_1:\mbox{\bf C}_2:\cdots:\mbox{\bf C}_t] \quad\mbox{and}\quad \mbox{\bf D}=[\mbox{\bf D}_1:\mbox{\bf D}_2 :\cdots:\mbox{\bf D}_t].
\end{eqnarray}

\noindent
{\bf Example 1.} 
 We consider an affine $2$-RD(9,4,6) and an affine $3$-RD(16,6,12) with $\beta_1=3$, $\beta_2=4$, $b_1=6$, $b_2=8$, $t=2$, $q_{11}=3$, $q_{12}=4$, $q_{21}=8$, $q_{22}=9$ and the incidence matrices $\mbox{\bf N}_A=(\mbox{\bf N}_{A1}:\mbox{\bf N}_{A2})$ and $\mbox{\bf N}_B=(\mbox{\bf N}_{B1}:\mbox{\bf N}_{B2})$, where
\begin{eqnarray}\nonumber
\mbox{\bf N}_{A1}=\left(
\begin{array}{ccc}
 1 & 0 & 1 \\
 1 & 0 & 1 \\
 1 & 0 & 1 \\
 1 & 1 & 0 \\
 1 & 1 & 0 \\
 1 & 1 & 0 \\
 0 & 1 & 1 \\
 0 & 1 & 1 \\
 0 & 1 & 1 \\
\end{array}
\right), \quad \quad 
\mbox{\bf N}_{A2}=\left(
\begin{array}{ccc}
 1 & 0 & 1 \\
 1 & 1 & 0 \\
 0 & 1 & 1 \\
 1 & 0 & 1 \\
 1 & 1 & 0 \\
 0 & 1 & 1 \\
 1 & 0 & 1 \\
 1 & 1 & 0 \\
 0 & 1 & 1 \\
\end{array}
\right), 
\end{eqnarray}

\noindent
The ISBD ${\tilde {\cal D}}$ constructed by the semi-Kronecker product has $24$ blocks with 6 rows and 12 columns in each block.
%, and has the $18 \times 4 \times 6=432$ observations.
The ISBD can be expressed in the following way:


\begin{table}[!ht]
\begin{center}
\caption{This is an example table }
\small
\begin{tabular}{l@{\hspace{1.5cm}}l@{\hspace{1.5cm}}l}
 \hline
$\beta _{0}$ & 182569.8 &  $\left( 7356.856\right) $ \\ 
$\beta _{1}$ & 92.26604 & $\left( 106.6340\right) $ \\ 
$\beta _{2}$ & 182708.7 & $\left( 89.78595\right) $  \\ 
$\beta _{3}$ & 90.25090 & $\left( 1.300387\right) $ \\
$\beta _{4}$ & 0.998389  & $\left( 0.000868\right) $ \\ 
\hline
\end{tabular}
\end{center}
\end{table}

%4----------------------------    Acknowledgements  --------------------------------
\begin{Acknowledgements}
The research was partially supported by ......
\end{Acknowledgements}





%------------------------------------------------------------References------------------------------------------------------------
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\harvarditem[]{}{}{} Granger C. W. J. (1969): Investigating causal relations by
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\harvarditem[]{}{}{} Kleinberg S. (2012): Causality, Probability, and Time. Cambridge
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\harvarditem[]{}{}{} Maddala G.S., Lahiri K. (2009): Diagnostic Checking, Model
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\harvarditem[]{}{}{} Phillips P.C.B. (1986): Understanding spurious regressions in
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\harvarditem[]{}{}{} Shiffrin R.M. (2016): Drawing causal inference from big data.
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America 113: 7308--7309.

\end{BLReferences}
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