
\chapter{Beaver Dam Eye Study}\label{bdes}   % level 1

\section{Introduction of BDES}

The Beaver Dam Eye Study (BDES) is an ongoing population-based study of age-related ocular disorders. 
It aims at collecting information related to the prevalence, incidence and severity of age-related 
cataract, macular degeneration and diabetic retinopathy. Between 1987 and 1988, $5925$ eligible people 
(age 43-84) were identified in Beaver Dam, WI. and of those, $4926(83.1\%)$ participated in the baseline 
exam. Five and ten year followup data have been collected and results are being reported. Many variables 
of various kinds are collected, including mortality between baseline and the followups. A detailed 
description of the study is given by Klein, Klein, Linton \& DeMets (1991). 
\nocite{klein:klein:linton:demets:1991}. Recent reports include Klein, Klein, Lee \& Cruickshanks (2001). 
\nocite{klein:klein:lee:cruickshanks:2001}. 

We are interested in the relation between five-year mortality for the non-diabetic study participants 
and possible risk factors at baseline. We focus on the non-diabetic participants since the pattern of 
risk factors for people with diabetes differs from that of the rest of the population. We consider 
$10$ continuous and $8$ categorical covariates, whose detailed information is given in the following 
tables. $Y$ is assigned $1$ if a patient participated in the baseline examination and died prior to 
the start of the first 5-year follow-up; $Y$ is assigned $0$ otherwise. 
\begin{itemize}
\item Continuous Covariates:
%\begin{table}
\begin{center}
\begin{tabular}{l|l|l}\hline
Variable & Name & Description \\
\hline
$X_1$: & {\em pky} &  pack years smoked, packs per day$/20)*$years \\
& & smoked \\
$X_2$: & {\em sch} &  highest year of school/college completed, years \\
$X_3$: & {\em inc} &  total household personal income, \\
& & thousands/month\\ 
$X_4$: & {\em bmi} &  body mass index, $kg/m^2$\\
$X_5$: & {\em glu} &  glucose (serum), $mg/dL$\\
$X_6$: & {\em cal} &  calcium (serum),$mg/dL$\\
$X_7$: & {\em chl} &  cholesterol (serum), $mg/dL$\\   
$X_8$: & {\em hgb} &  hemoglobin (blood), $g/dL$ \\
$X_9$: & {\em sys} &  systolic blood pressure, $mmHg$\\
$X_{10}$: & {\em age} &  age at examination, years\\
\hline
\end{tabular}
%\caption{Continuous covariates for BDES}
  \end{center}
%\end{table}

\newpage
\item Categorical Covariates: 
%\begin{table}
\begin{center}
\begin{tabular}{l|l|l|l}\hline
Variable & Name & Description & Definition\\
\hline
$Z_1$: & {\em cv} &  cardiovascular & $0 =$ no, $1 =$ yes\\
& & disease history & \\
$Z_2$: & {\em sex} &  gender & $0 =$ female, $1 =$ male\\ 
$Z_3$: & {\em hair} & hair color & $0 =$ blond/red, \\
& & &$1 =$ brown/black \\
$Z_4$: & {\em hist} & heavy drinking & $0 =$ never, $1 =$ past/currently \\
$Z_5$: & {\em nout} & winter leisure time & $0 =$ indoors,$1 =$ outdoors \\ 
$Z_6$: & {\em mar} &  marital status & $0 =$ no, $1=$ yes/ever\\
$Z_7$: & {\em sum} & day spent outdoors & $0 = <1/4$ day, $1 = >1/4$ day \\
& & in summer & \\
$Z_8$: & {\em vtm} & vitamin use & $0 =$ no, $1 = $yes \\
\hline
\end{tabular}
\end{center}
%\caption{Continuous covariates for BDES}
%\end{table}
\end{itemize}
There are $4422$ non-diabetic study participant in the baseline examination, and $395$ of them have 
missing data in the covariates. For the purpose of this study we assume the missing data are missing 
at random, thus these $395$ subjects are not included in our analysis. This assumption is not 
necessarily valid, age, blood pressure, body mass index, cholesterol, sex, smoking, hemoglobin may 
well affect the missingness, but a further examination of the missingness is beyond the scope of 
the present study. In addition, we exclude another $10$ participants who have either outlier values 
$pky > 158$ or very abnormal records $ bmi > 58$ or $ hgb < 6$. Thus we report an analysis of the 
remaining $4017$ non-diabetic participants from the baseline population. 

\section{Analysis of Five-Year Risk of Mortality}

The goal of this analysis is to select important risk factors for the five-year mortality event. A full 
and correct discovery of risk factors can increase our knowledge of the existing outcome pattern and 
help to reduce the mortality rate of the participants in the future study. Thus the variable selection is 
significantly meaningful for medical research. 

\subsection{Including All Variables}
There are $18$ covariates listed above. Though some prior knowledge or experience of the doctors 
can be used to narrow down the number of variables for consideration, we deliberately take into 
account some ``noisy'' variables in the analysis. The variables {\em hair, nout} and {\em sum} are 
not directly related to mortality in general, and their inclusion is to show the performance of 
the proposed approach.  These variables are not expected to be picked out eventually by the model. 
The main effects model incorporating categorical variables in $(\ref{eq:obj3})$ is fitted. The 
sequential Monte Carlo bootstrap tests are shown as follows. 
\begin{figure}[H]
 \begin{center}
  \includegraphics[height=6.5in,width=5.7in]{bdes_ndb/boot0-6.eps}
 \caption{Monte Carlo bootstrap tests for the BDES (all variables included)}
   \label{fig:c52}
\end{center}
\end{figure}
In Figure \ref{fig:c52} Along the x-axis, the covariates are coded as 
{\em 0=age, 8=hgb, 1=pky, b=sex, 9=sys, a=cv, 5=glu, 3=inc, 7=chl, 4=bmi, g=sum, f=mar, d=hist, h=vtm, 
e=nout, c=hair, 2=sch, 6=cal}, 
and are listed in decreasing order of their $L_1$ norm scores. The tests for the first six covariates: 
{\em age, hgb, pky, sex, sys, cv} all have Monte Carlo $p$-values $1/51 \doteq 0.02$; while the 
test for {\em glu} is not significant with $p$-value $9/51 = 0.18$. The threshold is chosen as 
$q = L_{(6)} = 0.25$.

\begin{figure}[H]
 \begin{center}
  \includegraphics[height=2.8in,width=5.5in]{bdes_ndb/L1.eps}
   \caption{$L_1$ norm scores for the BDES (all variables included).}
   \label{fig:c51}
\end{center}
\end{figure}

Figure {\ref{fig:c51}} plots the $L_1$ norm scores for all the potential risk factors. Using the 
threshold (dashed line) $q = 0.25$ chosen by the bootstrap test procedure, the LBP model identifies 
six important variables: {\em age, hgb, pky, sex, sys, cv} for the five-year mortality. It is noticed 
that the $L_1$ curve decreases very quickly for the first few variables, then levels off at a slower speed.

\begin{figure}[H]
 \begin{center}
  \includegraphics[height=4.7in,width=5.5in]{bdes_ndb/uni_logit.ps}
  \caption{Estimated univariate logit component for important variables.}
   \label{fig:c53}
\end{center}
\end{figure}

Compared with the LBP model, the linear logistic model with stepwise selection using
{\em AIC} criterion, implemented with the function \texttt{glm} in {\em R} package,
misses the variable {\em sys} but selects three more variables: {\em inc, bmi} and 
{\em sum}. Figure \ref{fig:c52} depicts the estimated univariate logit components for 
the important continuous variables selected by the LBP model. All the curves can be 
approximated reasonably well by linear models except {\em sys}, whose functional form 
exhibits a quadratic shape. This explains why {\em sys} is not selected by the logistic 
model. When we refit the logistic regression model by including {\em sys}$^2$ in the model, 
the stepwise selection picked out both {\em sys} and {\em sys}$^2$.


\subsection{Excluding Noisy Variables}

Sometimes with help of the prior knowledge or experience of the experts, we know some 
variables are either irrelevant or unimportant to the event of interest before doing
our analysis. For example, in this data set, doctors might doubt that {\em hair, nout, sum} 
are directly related to mortality and question whether to take into account all the variables. 
Thus we refitted the LBP model by excluding these three variables. The bootstrap tests are 
shown in Figure \ref{fig:c54}. 

The null hypotheses of the first five tests are all rejected at level $\alpha = 0.05$ based 
on their Monte Carlo $p$-value $1/51 \doteq 0.02$. The sixth test has the $p$-value 
$3/51 \doteq 0.06$. If using the significance level $0.1$, the LBP model selects six 
important variables: {\em age, hgb, pky, sys, sex, cv}. If using the level $0.05$, $cv$ is
significant by the margin. This list of important risk factors is the same as the analysis
above by including all the variables. 

\begin{figure}[H]
 \begin{center}
  \includegraphics[height=5.9in,width=5.5in]{bdes_ndb2/boot0_6.eps}
 \caption{Monte Carlo bootstrap for the BDES (excluding ``noisy'' variables).}
   \label{fig:c54}
\end{center}
\end{figure}

Figure {\ref{fig:c55}} plots the $L_1$ norm scores for the model fitted without including 
{\em hair, nout, sum}. The threshold $q = 0.25$ chosen by the bootstrap test procedure is 
denoted by the dashed line. We note the rank of variables is a little bit different from 
Figure {\ref{fig:c51}}. Two pairs of variables switch their orders: {\em sex} and {\em sys}, 
{\em inc} and {\em chol}. Since the $L_1$ scores for the two variables in each pair are 
kind of lose, the order switch could be due to randomness in the data. 
\begin{figure}[H]
 \begin{center}
  \includegraphics[height=2.5in,width=5.5in]{bdes_ndb2/L1.eps}
  \caption{$L_1$ norm scores for the BDES (excluding ``noisy'' variables).}
   \label{fig:c55}
\end{center}
\end{figure}


