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{\sl Soybean Growth and Disease \hfil 05/04/2000}
{\hfil\thepage\hfil}
\pagestyle{verbose}
\begin{document}
% 1 ==================================================================
\begin{slide}{}
\begin{center}
{\Huge\blue Factors Involving Soybean Growth and Disease}\\[4ex]
\emph{\Large\mg{Hao Zhang}}\\[1.5ex]
\end{center}
\end{slide}
% 2 ==================================================================
\begin{slide}
\Ti{Outlines}
\begin{enumerate}
\item Introduction
\begin{itemize}
\item Data Description
\item Experimental Design
\item Objectives
\end{itemize}  
\item Analysis
\begin{itemize}
\item Model Selection
\item Analysis \& Interpretation
\end{itemize} 
\item Summary
\end{enumerate}
\end{slide}
% 3 ===================================================================
\begin{slide}
\begin{center}
\Ti{Introduction}
\end{center}
\begin{itemize} 
\item Experimenter: Luis Del Rio (Department of Plant Pathology)
\item Pathogen: White mold for soybeans
\item Factors
\begin{itemize}
\item Row spacing treatment ( 1 = 7.5'', 2 = 15'', 3 = 30'') 
\item Variety treatment ( 1 \& 2 )
\item Time: 1996 - 1999 (four years)
\item Location: location 1 \& location 2
\end{itemize}
\end{itemize}
\end{slide}
%4 ===================================================================
\begin{slide}
\Ti{Response Variables} 
\begin{enumerate}
\item White mold
\begin{itemize}
\item its incidence is described by the percentage of infected plants in a field
\end{itemize}
\item Yield
\item Lodging 
\begin{itemize}
\item it is scored to describe the degree of flattening status of soybeans 
\item its range is 1 to 5\\
1 = all plants stand erect, 5 = all plants lie flat
\end{itemize}
\end{enumerate}
\end{slide}
% 4 ==================================================================
\begin{slide}
\Ti{Experimental Design}
\begin{itemize}
\item Multilocation Trial
\item Design Within Each Location: split plot design with whole plots 
conducted as RCBD (randomized complete block design)
\begin{itemize}
\item The experiment was conducted using four blocks
\item Each block was divided into three equal parts; Row spacing 1, 2 and 3 
were randomly assigned to each part, or {\bf whole-plot unit}
\item Each whole plot consisted of two sub-plots, or {\bf split-plot units};
Variety 1 and 2 were randomly assigned to each split plot
\end{itemize}
\end{itemize}
\end{slide}

% 5 =======================================================================
\begin{slide}
\Ti{Objectives}
\begin{itemize}
\item to understand the treatment effects (row spacing \& variety) within
each location in every year
\item to compare treatments averaged over geographical area represented
by locations and test treatment-by-location interaction
\item to understand the overall effects of the treatments over years
\end{itemize}

Analysis will have three parts: 
\begin{itemize}
\item [(1)] analysis within each location
\item [(2)] analysis for multilocation experiments
\item [(3)] analysis over years
\end{itemize}
\end{slide}
% 6 ==========================================================================
\begin{slide}
\Ti{Analysis within each location}
\begin{itemize}
\item Model
\begin{center}
$y_{ijk} = \mu_{ij} + b_k + e_{ik} +  \epsilon_{ijk}$
\end{center}
or 
\begin{center}
$y_{ijk} = \mu + \alpha_i + b_k + e_{ik} + \beta_{j} +
 {(\alpha*\beta)}_{ij}+ \epsilon_{ijk}$
\end{center}
for $i = 1,2,3$; $j = 1,2$; and $k = 1,2,3,4$

\item Three Fixed Effects
\begin{itemize}
\item $\alpha_i$ denotes the $i$th row spacing effect (whole plot treatment)
\item $\beta_j$ denotes the $j$th variety (split plot treatment)
\item ${(\alpha*\beta)}_{ij}$ denotes the interaction effect of the $i$th row spacing 
 and the $j$th variety
\end{itemize}
\end{itemize}
\end{slide}
% 7 ==========================================================================
\begin{slide}
\Ti{Random Effects}
\begin{itemize}
\item $b_k$ is the random effect of block $k$. Assume 
 $b_k \sim i.i.d. ~  N(0, {\sigma_b}^2)$;

\item $e_{ik}$ denotes the whole plot error, the random effect of 
the $i$th row spacing in block $k$. Assume
 $e_{ik} \sim i.i.d. ~  N(0, {\sigma_e}^2)$;

\item $\epsilon_{ijk}$ denotes the split plot error, the random 
effect of the $j$th variety grown with the $i$th row spacing in the 
$k$th block. Assume $\epsilon_{ijk} \sim i.i.d. ~  N(0, {\sigma_\epsilon}^2)$.

\item They are independent of each other.
\end{itemize}
\end{slide}
% 8 =========================================================================
\begin{slide}
\Ti{White Mold fitting within locations (1996)}
{\small{
\begin{table}[h]
  \begin{center}
    \begin{tabular}{l||c|c|c||c|c|c}
\hline
\hline
Cov Parm & \multicolumn{3}{c||}{location 1}&\multicolumn{3}{c}{location 2}\\
\hline
\hline
 & Estimate  & Std Error  & Pr Z & Estimate  & Std Error  & Pr Z \\
\hline
block (${\hat{\sigma}_b}^2$) & 0 & . & .& 0 & . & . \\
block*row (${\hat{\sigma}_e}^2$) & 0.8258 &0.9414 & 0.1902 & 0 & . & . \\
Residual(${\hat{\sigma_\epsilon}}^2$) & 1.8749 & 0.8838 & 0.0169 & 
2.4817 & 0.8272 & 0.0013\\
\hline
\hline
 \end{tabular}
    \caption{Covariance parameter estimates for white mold (1996)}
    \label{tab:1}
  \end{center}
\end{table}
}}
\end{slide}
% 9===============================================================
\begin{slide}
{\small{
\begin{table}[tbph]
  \begin{center}
    \begin{tabular}{l|c||c|c|c||c|c|c}
\hline
\hline
Effect & NDF &\multicolumn{3}{c||}{location 1} 
&\multicolumn{3}{c}{location 2}\\
\hline
\hline
 & & DDF & F Value & Pr $>$  F &  DDF & F Value & Pr $>$ F\\
\hline
Row & 2 & 9 & 6.10 & 0.0211 & 18 & 1.34 & 0.2877\\
Variety & 1 & 9 & 61.72 & $<$0.0001 & 18 & 4.41 & 0.0502\\
Row * Variety & 2 & 9 & 4.21 & 0.0511 & 18 & 0.92 & 0.4168\\ 
\hline
\hline
 \end{tabular}
    \caption{Type III tests of fixed results for white mold (1996)}
    \label{tab:2}
  \end{center}
\end{table}
}}
\end{slide}
% 10 ======================================================================
\begin{slide}
{\small{
\begin{table}[tbph]
  \begin{center}
    \begin{tabular}{c|c||c|c|c|c|c|c}
\hline
\hline
 &$H_0$ & Estimate & Std Error & DF & t Value & Pr $>$ t & Tukey P\\
\hline
\hline
&$\beta_1 - \beta_2 = 0$ & 4.3917 & 0.5590 & 9 & 7.86 &  $<$.0001 & $<$.0001 \\
Loc 1&$\alpha_1 - \alpha_2 = 0$ & -1.3375 & 0.9390 & 9 & -1.42 & 0.1881 & 0
.3696\\
 &$\alpha_1 - \alpha_3 = 0$& 1.9250 & 0.9390 & 9 & 2.05 & 0.0706 & 0.1519\\
& $\alpha_2 - \alpha_3 = 0$ & 3.2625 & 0.9390 & 9 & 3.47 & 0.0070 & 0.0173\\ 
\hline
Loc 2 & $\beta_1 - \beta_2 = 0$ & 1.3500 & 0.6431 & 18 & 2.10 & 0.0502 & 0.0502\\
\hline
\hline
 \end{tabular}
    \caption{Main effect hypotheses testing for white mold (1996)}
    \label{tab:3}
  \end{center}
\end{table}
}}
\end{slide}
% 11 ======================================================================
\begin{slide}
\begin{figure}[thbp]
  \begin{center}
   \includegraphics[height=2.8in,width=4.7in]{Wm96L1.conf.ps}
     \caption{Confidence intervals for white mold (location 1)}
   \label{fig:1}
  \end{center}
\end{figure}
\end{slide}
% 12==============================================================
\begin{slide}
\begin{figure}[htbp]
  \begin{center}
   \includegraphics[height=2.8in,width=4.7in]{Wm96L1.inter.ps}
     \caption{(1) Interaction for loc 1 (2) Confidence intervals for loc 2}
   \label{fig:2}
  \end{center}
\end{figure}
\end{slide}
% 13 ==================================================================
\begin{slide}
\Ti{Yield fitting within locations (1996)}
{\small{
\begin{table}[tbph]
  \begin{center}
    \begin{tabular}{l|c||c|c|c||c|c|c}
\hline
\hline
Effect & NDF &\multicolumn{3}{c||}{location 1} 
&\multicolumn{3}{c}{location 2}\\
\hline
\hline
 & & DDF & F Value & Pr $>$  F &  DDF & F Value & Pr $>$ F\\
\hline
Row & 2 & 15 & 0.08 & 0.9275 & 6 & 9.17 & 0.0150\\
Variety & 1 & 15 & 0.33 & 0.5768 & 9 & 2.18 & 0.1741\\
Row * Variety & 2 & 15 & 21.41 & $<$.0001 & 9 & 1.64 & 0.2474\\ 
\hline
\hline
 \end{tabular}
    \caption{Type III tests of fixed results for yield (1996)}
    \label{tab:4}
  \end{center}
\end{table}
}}
\end{slide}
% 14 ============================================================
\begin{slide}
{\small{
\begin{table}[tbph]
  \begin{center}
    \begin{tabular}{c||c|c|c|c|c|c}
      \hline
      \hline
      $H_0$ & Estimate & Std Error & DF & t Value & Pr $>$ t & Tukey P\\
      \hline
      \hline
      $\alpha_1 - \alpha_2 = 0$ & -599.37 & 989.80 & 6 & -0.61 & 0.5670 & 0.8225\\
      $\alpha_1 - \alpha_3 = 0$& 3335.12 & 989.80 & 6 & 3.37 & 0.0151 & 0.0347\\
      $\alpha_2 - \alpha_3 = 0$ & 3934.50 & 989.80 & 6 & 3.98 & 0.0073 &0.0172\\
      \hline
      quadratic in variety 1 &-1904.45 & 760.59 & 8.19 & -2.50 & 0.0361\\
      quadratic in variety 2 & -839.66 & 760.59 & 8.19 & -1.10 & 0.3010\\
      \hline
    \end{tabular}
    \caption{related hypothesis testing for yield (location 2)}
    \label{tab:5}
  \end{center}
\end{table}
}}
\end{slide}
% 15 ==================================================================
\begin{slide}
\begin{figure}[thbp]
  \begin{center}
    \includegraphics[height=2.8in,width=4.7in]{Yield96.inter.ps}
    \caption{Interaction plots for yield within locations(1996)}
    \label{fig:3}
  \end{center}
\end{figure}
\end{slide}
%16 ================================================================
\begin{slide}
\Ti{Lodging fitting within locations (1996)}
{\small{
\begin{table}[thbp]
    \begin{center}
    \begin{tabular}{l|c||c|c|c||c|c|c}
      \hline
      \hline
Effect & NDF &\multicolumn{3}{c||}{location 1} 
&\multicolumn{3}{c}{location 2}\\
\hline
\hline
& & DDF & F Value & Pr $>$  F &  DDF & F Value & Pr $>$ F\\
\hline
Variety & 1 & 9 & 139.64 & $<$0.0001 & 18 & 10.80 & 0.0041\\
Row & 2 & 9 & 6.58 & 0.0174 & 18 & 0.90 & 0.4241\\
Row*Variety & 2 & 9 & 4.23 & 0.0508 & 18 & 0.90 & 0.4241\\ 
\hline
\hline
\end{tabular}
\caption{Type III tests of fixed results for lodging (1996)}
    \label{tab:6}
  \end{center}
\end{table}
}}
\end{slide}
% 17 ======================================================================
\begin{slide}
{\small{
\begin{table}[htbp]
  \begin{center}
    \begin{tabular}{c|c||c|c|c|c|c|c}
      \hline
      \hline
      &$H_0$ & Estimate & Std Error & DF & t Value & Pr $>$ t & Tukey P\\
      \hline
      \hline
      &$\beta_1 - \beta_2 = 0$ & 2.6667 & 0.2257 & 9 & 11.82 & $<$.0001 & $<$.0001 \\
      Loc 1&$\alpha_1 - \alpha_2 = 0$ & -0.1250 & 0.3005 & 9 & -0.42 & 0.6871 & 0.9099\\
      &$\alpha_1 - \alpha_3 = 0$& 0.8750 & 0.3005 & 9 & 2.91 & 0.0172 & 0.0415\\
      & $\alpha_2 - \alpha_3 = 0$ & 1.000 & 0.3005 & 9 & 3.33 & 0.0088 & 0.0217\\ 
      \hline
      Loc 2 & $\beta_1 - \beta_2 = 0$ & 0.5000 & 0.1521 & 18 & 3.29 & 0.0041 & 0.0041\\
      \hline
      \hline
    \end{tabular}
    \caption{Main effect hypothesis testing for lodging (1996)}
    \label{tab:7}
  \end{center}
\end{table}
}}
\end{slide}
% 18 ======================================================================
\begin{slide}
\begin{figure}[thbp]
  \begin{center}
    \includegraphics[height=2.8in, width=4.6in]{Lodge96.inter.ps}
    \caption{Interaction plots for lodging within locations(1996)}
    \label{fig:4}
  \end{center}
\end{figure}
\end{slide}
% 19 =====================================================================
\begin{slide}
\Ti{Analysis for multilocation experiment}
\begin{itemize}
\item Model
\begin{center}
    $y_{ijkl} = \mu + \tau_l + \alpha_i + {b(\tau)}_{lk} + 
  {e(\tau)}_{lik} + \beta_{j} + {(\alpha*\beta)}_{ij} +
 {(\tau*\alpha)}_{li} + 
    {(\tau*\alpha*\beta)}_{lij} +  \epsilon_{ijkl}$
\end{center}

\item $\tau_l$ denotes the main effect of the $l$th location; 
${(\tau*\alpha)}_{li}$ and ${(\tau*\alpha*\beta)}_{lij}$ 
denotes the interaction effects of location $l$ with other
treatments

\item random effects
\begin{itemize}
\item ${b(\tau)}_{lk}$ is the random effect of block $k$ within location $l$
\item ${e(\tau)}_{lik}$ is the random effect of  row spacing $i$ 
in block $k$ within location $l$
\item $\epsilon_{ijkl}$ denotes the random effect of the $j$th variety 
grown with the $i$th row spacing in the $k$th block within location $l$
\end{itemize}
\end{itemize}
\end{slide}
% 20 =================================================================
\begin{slide}
\Ti{White mold fitting in 1996 (combined locations)}
{\small{
\begin{table}[hbtp]
    \begin{center}
    \begin{tabular}{l||c|c|c|c}
      \hline
      \hline
      Cov Parm  & Estimate  & Std Error & Z value & Pr Z \\
      \hline
      block(location) (${\hat{\sigma}_b}^2$) & 0 & . & . & . \\
      block*row(location) (${\hat{\sigma}_e}^2$) & 0.3292 & 0.6157 & 0.53 & 0.29
64 \\
      Residual(${\hat{\sigma_\epsilon}}^2$) & 2.2620 & 0.7540 & 3.00 & 0.0013 \\
      \hline
      \hline
    \end{tabular}
    \caption{Covariance parameter estimates for white mold (1996)}
    \label{tab:8}
  \end{center}
\end{table}
}}
\end{slide}
% 21 ============================================================
\begin{slide}
{\small{
\begin{table}[tbph]
  \begin{center}
    \begin{tabular}{l|c|c|c|c}
      \hline
      \hline
      Effect & Num Df & Den Df & F Value & Pr $>$  F \\
      \hline
      loction & 1 & 18  & 29.49 & $<$.0001\\
      row & 2 & 18 & 5.91 & 0.0106\\
      variety & 1 & 18 & 43.72 & $<$.0001\\
      row*variety & 2 & 18 & 3.35 & 0.0578\\
      location*row & 2 & 18 & 2.59 & 0.1024\\
      location*row*variety & 3 & 18 & 4.85 & 0.0120\\
      \hline
      \hline
    \end{tabular}
    \caption{Type III tests of fixed results for white mold (1996)}
    \label{tab:9}
  \end{center}
\end{table}
}}
\end{slide}
%22  =============================================================
\begin{slide}
\begin{figure}[htbp]
  \begin{center}
    \includegraphics[height=2.8in, width=4.6in]{Y96wm.inter.ps}
    \caption{Interaction plot for white mold in 1996 (digit 1 and 2 denote varieties)}
    \label{fig:5}
  \end{center}
\end{figure}
\end{slide}
% 23 ===================================================================
\begin{slide}
{\small{
\begin{table}[tbph]
  \begin{center}
    \begin{tabular}{c||c|c|c|c|c|c}
      \hline
      \hline
      $H_0$ & Estimate & Std Error & DF & t Value & Pr $>$ t & Tukey P\\
      \hline
      \hline
      $\tau_1 - \tau_2 = 0$ & 2.6792 & 0.4933 & 18 & 5.43 &  $<$.0001 & $<$.0001\\
      $\beta_1 - \beta_2 = 0$ & 2.8708 & 0.4342 & 18 & 6.61 &  $<$.0001 & $<$.00\\
      $\alpha_1 - \alpha_2 = 0$ & -0.3375 & 0.6042 & 18 & -0.56 & 0.5833 & 1.000\\
      $\alpha_1 - \alpha_3 = 0$& 1.6063 & 0.6042 & 18 & 2.66 & 0.0160 & 0.0404\\
      $\alpha_2 - \alpha_3 = 0$ & 1.9438 & 0.6042 & 18 & 3.22 & 0.0048 & 0.0126\\ 
      \hline
      \hline
    \end{tabular}
    \caption{Main effect hypothesis testing for white mold (1996)}
    \label{tab:10}
  \end{center}
\end{table}
}}
\end{slide}
% 24================================================================
\begin{slide}
\Ti{Yield fitting in 1996 (combined locations)}
{\small{
\begin{table}[tbph]
  \begin{center}
    \begin{tabular}{l|c|c|c|c}
      \hline
      \hline
      Effect & Num Df & Den Df & F Value & Pr $>$  F \\
      \hline
      location & 1 & 6 & 3.47 & 0.1119\\
      row & 2 & 12 & 7.64 & 0.0072\\
      variety & 1 & 18 & 0.17 & 0.6878\\
      row*variety & 2 & 18 & 8.77 & 0.0022\\
      location*row & 2 & 12 & 7.15 & 0.0090\\
      location*row*variety & 3 & 18 & 11.98 & 0.0002\\
      \hline
      \hline
    \end{tabular}
    \caption{Type III tests of fixed results for yield (1996)}
    \label{tab:11}
  \end{center}
\end{table}
}}
\end{slide}
%25 ===================================================================
\begin{slide}
\begin{figure}[tbhp]
  \begin{center}
    \includegraphics[height=2.8in, width=4.6in]{Y96yield.inter.ps}
    \caption{Interaction plot for yield in 1996 (digit 1 and 2 denote varieties)}
    \label{fig:6}
  \end{center}
\end{figure}
\end{slide}
% 26 =======================================================
\begin{slide}
{\small{
\begin{table}[tbph]
  \begin{center}
    \begin{tabular}{c||c|c|c|c|c|c}
      \hline
      \hline
      $H_0$ & Estimate & Std Error & DF & t Value & Pr $>$ t & Tukey P\\
      \hline
      \hline
      $\alpha_1 - \alpha_2 = 0$ & -202.50 & 552.01 & 12 & -0.37 & 0.7201 &1.000\\
      $\alpha_1 - \alpha_3 = 0$& 1759.19 & 552.01 & 12 & 3.19 & 0.0078 &0.0198\\
      $\alpha_2 - \alpha_3 = 0$ & 1961.69 & 552.01 & 12 & 3.55 & 0.0040 &0.0103\\ 
      \hline
      \hline
    \end{tabular}
    \caption{Row spacing effects testing for yield (1996)}
    \label{tab:12}
  \end{center}
\end{table}
}}
\end{slide}
%27 ====================================================
\begin{slide}
\Ti{Lodging fitting in 1996 (combined locations)}
{\small{
\begin{table}[tbph]
  \begin{center}
    \begin{tabular}{l|c|c|c|c}
      \hline
      \hline
      Effect & Num Df & Den Df & F Value & Pr $>$  F \\
      \hline
      location & 1 & 18 & 75.00 & $<$.0001\\
      row & 2 & 18 & 4.75 & 0.0221\\
      variety & 1 & 18 & 135.38 & $<$.0001\\
      row*variety & 2 & 18 & 2.91 & 0.0806\\
      location*row & 2 & 18 & 5.25 & 0.0160\\
      location*row*variety & 3 & 18 & 23.44 & $<$.0001\\
      \hline
      \hline
    \end{tabular}
    \caption{Type III tests of fixed results for lodging (1996)}
    \label{tab:13}
  \end{center}
\end{table}
}}
\end{slide}
% 28 ====================================================================
\begin{slide}
\begin{figure}[thbp]
  \begin{center}
    \includegraphics[height=2.8in, width=4.6in]{Y96lodge.inter.ps}
    \caption{Interaction plot for lodging in 1996 (digit 1 and 2 denote varieties
s)}
    \label{fig:7}
  \end{center}
\end{figure}
\end{slide}
% 29 =====================================================================
\begin{slide}
{\small{
\begin{table}[tbph]
  \begin{center}
    \begin{tabular}{c||c|c|c|c|c}
      \hline
      \hline 
      $H_0$ & Estimate & Std Error & DF & t Value & Pr $>$ t\\
      \hline
      \hline
      $\tau_1 - \tau_2 = 0$ & 1.2500 & 0.1443 & 18 & 8.66 &  $<$.0001\\
      $\beta_1 - \beta_2 = 0$ & 1.5833 & 0.1361 & 18 & 11.64 &  $<$.0001\\
      $\alpha_1 - \alpha_2 = 0$ & 0.0625 & 0.1768 & 18 & 0.35 & 0.7278\\
      $\alpha_1 - \alpha_3 = 0$& 0.5000 & 0.1768 & 18 & 2.83 & 0.0111\\
      $\alpha_2 - \alpha_3 = 0$ & 0.4375 & 0.1768 & 18 & 2.47 & 0.0235\\ 
      \hline
      \hline
    \end{tabular}
    \caption{Main effect hypothesis testing for lodging (1996)}
    \label{tab:14}
  \end{center}
\end{table}
}}
\end{slide}
% 30 ==============================================================================
\begin{slide}
\Ti{Analysis for white mold over years (all data combined)}
\begin{center}
    $y_{ijklm} = \mu + \tau_l + \nu_m + \alpha_i + {b(\tau)}_{lk}
 + {e(\tau)}_{lik} + 
    \beta_{j} + {(\alpha*\beta)}_{ij} + {(\tau*\alpha)}_{li} + 
    {(\tau*\alpha*\beta)}_{lij} +  \epsilon_{ijklm}$
\end{center}
{\small{
\begin{table}[htbp]
  \begin{center}
    \begin{tabular}{l|c|c|c|c}
      \hline
      \hline
      Effect & Num Df & Den Df & F Value & Pr $>$  F \\
      \hline
      year & 3 & 177 & 11.89 & $<$.0001\\
      location & 1 & 177 & 2.68 & 0.1036\\
      variety & 1 & 177 & 35.88 & $<$.0001\\
      location*row & 2 & 177 & 0.06 & 0.9461\\
      row & 2 & 177 & 3.03 & 0.0508 \\
      row*variety & 2 & 177 & 1.26 & 0.2868\\
      location*row*variety & 3 & 177 & 0.01 & 0.9988\\
      \hline
      \hline
    \end{tabular}
    \caption{Type III tests of fixed results for white mold}
    \label{tab:15}
  \end{center}
\end{table}
}}
\end{slide}
% 30 ==============================================================================
\begin{slide}
{\small{
\begin{table}[tbph]
  \begin{center}
    \begin{tabular}{c||c|c|c|c|c}
      \hline
      \hline
      $H_0$ & Estimate & Std Error & DF & t Value & Pr $>$ t\\
      \hline
      \hline
      $\nu_1 - \nu_2 = 0$ & -12.2854 & 2.3435 & 177 & -5.24 &  $<$.0001\\
      $\nu_1 - \nu_3 = 0$ & -7.9104 & 2.3435 & 177 & -3.38 &  0.0009\\
      $\nu_1 - \nu_4 = 0$ & -11.9521 & 2.3435 & 177 & -5.10 & $<$.0001\\
      $\beta_1 - \beta_2 = 0$ & 9.9260   & 1.6571 & 177  & 5.99 & $<$.0001\\
      $\alpha_1 - \alpha_3 = 0$& 4.0422 & 2.0296 & 177 & 1.99 & 0.0479\\
      $\alpha_2 - \alpha_3 = 0$ & 4.5641 & 2.0296 & 177 & 2.25 & 0.0258\\ 
      \hline
      \hline
    \end{tabular}
    \caption{Main effect hypothesis testing for white mold over years}
    \label{tab:16}
  \end{center}
\end{table}
}}
\end{slide}
% 31 ==============================================================================
\begin{slide}
\Ti{Summary}
\begin{itemize}
\item [{\bf White mold}] Variety 2 has a significantly lower white 
mold incidence than variety 1; thus it is more resistant to
the disease. Wide row spacing is lower than narrow or medium spacing.
The best combination to prevent white mold is variety 2 and wide row spacing.

\item [{\bf Yield}] There is no significant difference in yield 
between variety 1 and 2. The yield is ranked as: medium, narrow 
and wide row spacing. There did not exist significant difference 
between narrow and medium row spacing in yield.

\item [{\bf Lodge}] Variety 2 has a lower lodging than variety 1. 
Wide row spacing is much lower than either narrow or medium spacing.
Thus the combination treatment of variety 2 and wide row spacing 
will make soybeans stand most erect. This also suggests white mold tends to 
flatten the plants. 
\end{itemize}
\end{slide}

\end{document}


















