\section*{Background}
The LASSO-Patternsearch (LPS) algorithm
\cite{shi:wahba:wright:2008,LPSurl,Wri10a} is an effective approach
for identifying multiple dichotomous risk factors for outcomes of
interest in demographic and genomic studies. It uses an
$\ell_1$-regularized logistic regression formulation, targeting the
case in which only a small fraction of the large number of possible
candidate patterns are significant. The approach can be used to
consider simultaneously all possible patterns up to a specified
order. It can identify complicated correlation structures among the
predictor variables, on a scale that can cause serious difficulties
for algorithms that target problems of more modest size.

When applied to very large models with higher-order interactions
between the predictor variables, however, LPS quickly runs into
computational limitations. For example, a problem with two thousand
predictor variables yields a logistic-regression formulation with
about two million variables if both first- and second-order patterns
are included in the model. Problems of this size are at the limit of
LPS capabilities, yet current problems of interest in genetic
epidemiology consider ten thousand markers or more
\cite{valdar:solberg:2006}. For these kinds of data sets, a screening
stage can be added before applying LPS \cite{shi:wahba:lee:2007}.

In this article, we propose a Partitioned LASSO-Patternsearch
Algorithm (pLPS) scheme to tackle gigantic data sets in which we wish
to consider second- and possibly third-order interactions among the
predictors, in addition to the first-order effects. As in LPS, we
assume that all predictor variables are binary (or that they have been
dichotomized before the analysis). The model thus contains a huge
number of possible patterns, but the solution is believed to be
sparse, with only a few effects being significant risk factors for the
given outcome. In the first (screening) stage of pLPS, the predictors
are divided into partitions of approximately equal size, and LPS is
used to solve smaller subproblems in which just the predictors and
higher-order effects within a single partition, or the interactions
between variables in small groups of partitions, are considered as
variables in the optimization model. These reduced problems can be
solved independently, in parallel. By the end of the screening stage,
each predictor and each higher-order effect (up to the specified
order) has been considered in at least one of the subproblems. The
second stage of pLPS is an aggregation process, in which all
predictors identified in the first stage are considered, together with
all their interactions up to the specified order. An LPS process is
used to identify the final set of significant predictors and
interactions.


Tuning parameters in the first stage of pLPS are chosen by BGACV
criterion (see \cite{shi:wahba:wright:2008}). In the second stage, two tuning parameters
are used, one for main effects and one for interactions. These are
chosen by BGACV2, a variation of BGACV to be described below. We
examine the effectiveness of the pLPS strategy on simulated data and
on two large-scale genetic data sets.



\section*{Methods}
\label{sec.glps}

We now give further details of the pLPS scheme and its
implementation. For simplicity, most of our discussion focuses on the
case in which first-order effects and second-order interactions
between all predictors are considered. Extension of the approach to
include third-order effects as well is described briefly at the end of
the section.

Considering $n$ subjects with $p$ binary predictor variables, the
total number of interactions up to order $q$ is given by $N_B =
\sum_{\nu=0}^q{{p}\choose{\nu}}$. For $q=2$, we thus have $1+p(p+1)/2$
patterns. To apply pLPS, we first divide the $p$ variables into $k$
partitions so that each partition has $g=p/k$ variables. (For
simplicity of description, we assume that $p$ is divisible by $k$.)
% If the last group is not full some dummy variables that are not
% related to the response, such as zeros, may be added to this group.
The data set is $\{y,x_j, j = 1,2,\cdots,p\}$, where $y =
(y_1,y_2,\cdots,y_n) \in \{0,1\}$ is the response, $x_j =
(x_j(1),x_j(2),\cdots,x_j(n))$ is the $j$th covariate, and
$x_j(i)\in\{0,1\}$ for all $j=1,2,\cdots,p$ and $i=1,2,\cdots,n$.  By
relabelling the $p$ predictors as $x_{st}$, where $s = 1,2,\cdots,k$
denotes the partition number and $t=1,2,\cdots,g$ denotes the index
within the partition, we relabel the full data set as $\{y, x_{st}, \;
s = 1,2,\cdots,k, \, t=1,2,\cdots,g\}$.


% Let $q$ be the highest order of patterns we consider. Then there
% will be $N_B = \sum_{\nu=0}^q{{p}\choose{\nu}}$ patterns for LPS. We
% use $q=2$ to illustrate the pLPS algorithm. The case with larger $q$
% can easily be generalized.

In the first stage of pLPS (the ``screening stage''), we solve two
types of reduced LPS subproblems. The first type is based on a pair of
partitions, denoted by $s_1$ and $s_2$, and defines the LPS variables
in the subproblems to be the first-order effects within each group
(for which there are $2g$ basis functions $\{B_{t_1} = x_{s_1 t}, \; t
= 1,2,\cdots,g\}$ and $\{B_{t_2} = x_{s_2 t}, \; t = 1,2,\cdots,g\}$) and
all the second-order interactions between a predictor in group $s_1$
and a predictor in group $s_2$. There are $g^2$ basis functions for
the latter effects, namely, $\{B_{t_1t_2} = x_{s_1t_1}\times
x_{s_2t_2}, \; t_1,t_2 = 1,2,\cdots,g\}$. Hence the total number of
patterns in the LPS model for each subproblem is $g^2+2g+1$, when we
include the constant basis function $B \equiv 1$.

The second type of reduced LPS problem is obtained from the first- and
second-order effects within a single partition.  Here, the basis
functions for group $s$ are $\{B_{t_1t_2} = x_{st_1}\times x_{st_2},
\; t_1,t_2 = 1,2,\cdots,g\}$ and $\{B_{t} = x_{st}, \; t =
1,2,\cdots,g\}$, making a total of $1+g(g+1)/2$ patterns, when we
include the constant basis function. Since each subproblem of the
second type has about half as many variables as each subproblem of the
first type, we define computational tasks of roughly equivalent
complexity by grouping two of the type-two problems together.
%
Figure 1 is a graphical presentation of the two types of
groups considered in the first stage of pLPS.

We now briefly describe the LPS methodology, which is applied to each
of these subproblems. By relabelling, we define the basis functions to
be $B_{\ell}(x)$, $\ell=1,2,\cdots,N_B$. Defining $p(x) :=
\mbox{Prob}[y= 1|x]$ and the logit (log odds ratio) $f(x) := \log
[p(x)/(1-p(x))]$, we estimate $f$ by minimizing
\begin{equation}\label{penlik}
I_{\lambda}(y,f) =  \cL (y,f) +\lambda J(f),
\end{equation}
where $\cL (y,f)$ is the negative log likelihood divided by $n$:
\begin{equation}\label{defL}
\cL(y,f) =\frac{1}{n}  \sum_{i=1}^n [ -y_if(x(i)) + \log (1+e^{f(x(i))})],
\end{equation}
with $f$ being expressed as a linear combination of the basis functions
\begin{equation}\label{f.def}
f(x) = \mu +  \sum_{\ell=1}^{N_B-1} c_{\ell}B_{\ell}(x),
\end{equation}
and the penalty function being defined by
\begin{equation}\label{J}
J(f) =  \sum_{\ell = 1}^{N_B-1} |c_{\ell}|.
\end{equation}
(We assume that the last basis function is the constant function $1$,
whose coefficient $\mu$ does not appear in $J$ and is therefore not
penalized.)  The penalty parameter $\lambda$ in (\ref{penlik}) is
chosen by BGACV.  We then build a parametric logistic regression model
on the remaining basis functions by minimizing (\ref{defL}) and
selecting the best model via backward elimination with the BGACV
criteria.  More details are given in
\cite[Section~3]{shi:wahba:wright:2008}.


If the outcomes can be predicted well using a small number of
patterns, the number of patterns surviving the first stage of pLPS
should be small. Suppose there are a total of $p^*$ unique predictor
variables in all these patterns. The second stage of pLPS --- the
``aggregation stage'' --- is an LPS problem in which just these
predictors and all their second-order effects are the patterns. There
will be $N_{B_1}= p^*$ basis function (denoted by $B_{1 \ell}$) for
the main effects and $N_{B_2} ~(={{p^*}\choose{2}})$ basis functions
(denoted by $B_{2 \ell}$) for the second-order interactions, plus one
constant basis function. In the aggregation stage, we use different
penalty parameters for the first- and second-order patterns, so the
objective function is
\begin{equation}
\label{penlik2}
I_{\lambda_1,\lambda_2}(y,f) = {\cal L}(y,f) + \lambda_1J_1(f) + \lambda_2J_2(f),
\end{equation}
where the link function $f$ is
\begin{equation}
\label{f.def2}
f(x) = \mu + \sum_{\ell=1}^{N_{B_1}}c_{1\ell} B_{1\ell}(x) + \sum_{\ell=1}^{N_{B_2}}c_{2\ell} B_{2\ell}(x),
\end{equation}
and the penalties are
\begin{equation}
\label{J2}
J_1(f) = \sum_{\ell = 1}^{N_{B_1}}|c_{1\ell}|,~~~ J_2(f) = \sum_{\ell = 1}^{N_{B_2}}|c_{2\ell}|.
\end{equation}

The choice of penalty parameters $(\lambda_1, \lambda_2)$ in
(\ref{penlik2}) is critical to the performance of this formulation.
BGACV does not work well in this setting. Often, it tends to select
only second-order patterns, combining main effects with spurious
partners. Occasionally, it selects only main effects, breaking true
size-two patterns into separate main effects.  The large difference
between the number of basis functions makes the solutions sensitive to
the two penalty parameters.  Searching over a grid of values for
$\lambda_1$ and $\lambda_2$ is expensive and often does not give
satisfactory results.  As an alternative approach, we introduce the
following penalty function, known as BGACV2:
\begin{equation}
\label{BGACV2}
BGACV2(\lambda_1,\lambda_2) = BGACV(\lambda_1,\lambda_2)\times \left( 1+0.5\frac{|n_{b_1} - n_{b_2}|}{n_{b_1} + n_{b_2}} \right),
\end{equation}
where $n_{b_1}$ is the number of nonzero coefficients of main effects
and $n_{b_2}$ is the number of nonzero coefficients of size-two
patterns.  The additional penalty factor forces these two numbers to
be similar, reducing the possibility of the two extreme cases
discussed above. If the true model only contains main effects, the
BGACV2 penalty will tend to select fewer main effects than the BGACV
model.  However, BGACV is conservative (see discussion in
\cite{shi:wahba:wright:2008}), while BGACV2 is less so. We expect that
BGACV2 will not miss any important main effects, though it may also
produce some spurious second-order effects.  These spurious effects
will be further eliminated by the parametric logistic regression step
as noted above, followed by solving (\ref{penlik2}).

Minor extensions to the pLPS approach are needed when size-three
patterns ($q = 3$) are introduced.  In the screening phase of pLPS,
there are four types of subproblems (rather than two). These types are
distinguished by considering the labels $s_1,s_2,s_3$ of the three
partitions chosen to define the subproblem (with $s_1 \leq s_2 \leq
s_3$). The four types correspond to the cases $s_1<s_2<s_3$, $s_1=s_2<
s_3$, $s_1<s_2=s_3$, and $s_1=s_2=s_3$, respectively. In the
aggregation phase of pLPS, we will still be using two penalty
parameters, one for main effects and one for interactions; size-two
and size-three patterns share the same penalty parameter.  The
criterion function for choosing the appropriate values for penalty
parameters $\lambda_1$ and $\lambda_2$ is
\begin{equation}
\label{BGACV3}
BGACV3(\lambda_1,\lambda_2) = BGACV(\lambda_1,\lambda_2)\times \left(1+0.5\frac{|n_{b_1} - n_a|+|n_{b_2} - n_a|+|n_{b_3} - n_a|}{n_{b_1} + n_{b_2} + n_{b_3}}\right),
\end{equation}
where $n_{b_1}$ is the number of nonzero main effects, $n_{b_2}$ is
the number of nonzero size two patterns, $n_{b_3}$ is the number of
nonzero size three patterns and $n_a$ is the average of the three.

% In practice only $q = 2$ and $q = 3$ have been implemented. Higher
% order patterns can easily be generalized and will not be addressed
% in this paper.

In the remainder of the paper, we use pLPS to denote the $q = 2$ case
and pLPS3 for the $q = 3$ case.
% Sometimes, pLPS could also mean general pLPS with different $q$,
% depending on the setting.

The choice of $g$ (the number of variables in each partition) is
determined by the computing power and the available memory. On our
super server (an AMD Dual-Core 2.8 GHz machine with 64 GB memory), we
usually set $g = 2,000$ for $q = 2$. This yields subproblems with $N_B
= 2,001,001$ basis functions, which can be handled comfortably by the
LPS code. On a more standard computer (Intel(R) Pentium(R) 4 2.80GHz
with 2 GB memory), we usually set $g = 200$ for $q = 2$ and $g = 35$
for $q = 3$. As we noted earlier, the subproblems in the first stage
of pLPS can be solved independently in parallel, on different
computers.  The grid-computing system Condor ({\tt
 http://www.cs.wisc.edu/condor/}) provides an ideal platform for
these parallel jobs.  In our Condor implementation, we request
machines from the pool with at least 2 GB of memory, and define our
group sizes to be $g = 200$ (for $q = 2$) and $g = 35$ (for $q = 3$).
Generally, for faster execution of pLPS, it is advantageous to set $g$
to the highest value that can be accommodated by the memory of the
computer. The final results of the computation do not depend strongly
on the choice of $g$.

\section*{Results and Discussion}
\subsection*{Simulation Studies}
\label{sec.glpssim}

In this section we study the empirical performance of pLPS through
three simulated examples. The first example is a relatively small data
set with independent predictor variables: One main effect and two
second order interactions are included in the link function. The
second example is a very large data set with strong correlations among
neighboring variables, in which two main effects and two second order
interactions are assumed to be important. The third example studies
the performance of pLPS3, which includes third-order interactions in
the model. Two main effects, one second order interaction and one
third order interaction are included.

We compare pLPS with three other methods:
\begin{itemize}
\item Logic Regression \cite{ruczinski:kooperberg:leblanc:2003}, as
 implemented in the R package   {\tt LogicReg},
\item Stepwise Penalized Logistic Regression
 (SPLR) \cite{park:hastie:2008}, as implemented in
the R package {\tt stepPlr},  and
\item Random Forest (RF) \cite{breiman:2001}, as implemented in  the R package
{\tt randomForest}.
\end{itemize}
%
The number of trees and number of leaves in Logic Regression are
selected by five-fold cross validation. The smoothing parameter in
SPLR is also selected by five-fold cross validation, and the model
size selected by BIC.

\subsubsection*{Simulation Example 1}
\label{sec.glps.sim1}

In our first example, 400 iid
Bernoulli(0.5) random variables were simulated. The sample size is 700
and the logit function is
\[
f(x) = -2+1.5X_{50} + 1.5X_{150}X_{250} + 1.5X_{251}X_{252}.
\]
One hundred data sets were generated according to this model and
analyzed by the four methods described above.

Table 1 presents the results of this simulation. Each
entry in the table shows the number of appearances of the pattern and the variables
in the 100 simulations. The main number (outside the parentheses) is
the {\em pattern count} showing how many times the given pattern is selected in the model. The numbers inside the parentheses are the {\em variable
 counts} showing how many times each variable in a given pattern appears
in the model, either as a main effect or in some
other interaction.  
%
% Two numbers are used to show the appearance frequency of a main
% effect. The first number counts how many times the pattern appears
% main effect. We call it the pattern count.  The second number
% (inside the parenthesis) counts how many times the variable appears
% in the model, either as a main effect or in some interactions. We
% call it the variable count. Similarly three numbers are needed to
% show the appearance frequency of a size two pattern, one pattern
% count and two variable counts. The pattern count is the number of
% times the pattern is selected and the variable counts are the number
% of times the variables appear in the model.
%
Random Forest does not generate an explicit model, but rather produces
an importance score for all variables. It is not possible to calculate
a pattern count, but we calculate the variable count according to
whether the variables in question appeared among the top 10 variables
identified by Random Forest.  For pLPS, Logic regression and SPLR, the last column labelled ``noise'' counts the
total number of appearances in the 100 trials by terms that are not
patterns in the model.  In this simulation, any pattern other than $X_{50}$,
$X_{150}X_{250}$, or $X_{251}X_{252}$ is taken to be noise.  For random forest, "noise" counts the total number of noisy variables selected in the 100 trials.  Any variable other than the five in the logit function is noise.  

On this example, pLPS selects all three patterns almost perfectly and
generates the least amount of noise in the form of spuriously selected
patterns. Logic Regression does not do well on the size-two patterns
and selects slightly more noise. Random Forest does well in selecting
the important variables but also selects many noisy variables. (If we
change the criterion for declaring that Random Forest has selected a
variable to the ``top eight'' or ``top five,'' we reduce the number of
noisy variables but also reduce the variable counts.)  SPLR has similar
performance to pLPS in selecting the patterns, but selects many more
spurious patterns.

\subsubsection*{Simulation Example 2}
\label{sec.sim.glps2}

Example 2 studies the behavior of pLPS on a large
data set ($n = 1000$, $p = 8000$) with correlations among the
covariates.  To generate the binary variables $X_i$, $i=1,2,\cdots,p$,
we start with normal distributions, choosing $X_i^*\sim N(0,1)$, $i =
1,2,\cdots,p$ so that corr$(X_i^*,X_{i+1}^*) = 2/3$ and
corr$(X_i^*,X_{i+2}^*) = 1/3$, $i = 1,2,\cdots,p-2$. ($X_i^*$ and
$X_j^*$ are independent if $|i-j|>2$.)  We then set $X_i = 1$ if
$X_i^*>0$ and $X_i=0$ otherwise, for each $i = 1,2,\cdots,p$. The logit
function is
\[
f(x) = -4 + 2X_{500} + 3X_{5000} + 2X_{1000}X_{3000} +
3X_{7000}X_{7002}.
\]
The simulation was repeated 50 times (each run is quite
time-consuming).  We could not run Logic Regression on this example,
as the dimensions exceed the limit of that code.

Table 2 shows the results, in the same format as
Table 1. pLPS misses the pattern $X_{1000}X_{3000}$ twice
but selects the rest perfectly, and generates a smaller number of
spurious noise patterns than the other methods. In Random Forest, we
declared a variable to be selected if it was ranked in the top 12.
% \noprint{SJW: You use top-12 here but top-10 in the first
%   example. Perhaps should say something about sensitivity of the
%   results to this parameter.}
It misses the variabls in the pattern $X_{1000}X_{3000}$ with some frequency.  SPLR
selects all four patterns perfectly, but at the cost of a large number
of spurious patterns. SPLR requires the user to set the maximum number
of parameters allowed in the model, and selects the actual number by
BIC.  We set this maximum to $20$, and it was reached on all $50$
runs. (The maximum is still reached on every run when we set this
parameter to $50$.)
% \noprint{SJW: It seems questionable to ask SPLR to pick 20 patterns
%   on each run and then complain that most of them are noise. What
%   happens to the pattern and variable counts when you ask for only 5
%   or 8 patterns? A referee might well ask.}

% $(50\times4+800)/50 = 20$.


\subsubsection*{Simulation Example 3}
\label{sec.sim.glps3}
Example 3 studies the behavior of pLPS3 on a large
data set, with sample size $n = 1000$ and $p = 500$ variables. The marginal distribution and correlation structure are the same as in
Example 2.

The logit function is
\[
f(x) = -4 + 2X_{100} + 3X_{200} + 2X_{300}X_{400} +
3X_{150}X_{450}X_{451}.
\]
Again this simulation was repeated 50 times. As we can see from Table
3, pLPS3 selects all patterns quite well with a
reasonable number of noisy patterns. Logic Regression selects fewer
noisy patterns but does not do well in identifying the two interaction
terms. Random Forest does well in the size-three pattern but misses
the size two pattern quite often. (We declared the top 12 variables
identified by Random Forest to be ``selected'').

As in the previous examples, SPLR does well at selecting the important
patterns but also selects many noise patterns.
% \noprint{SJW: What was the ``max pattern'' parameter set to in SPLR,
%   for this example?  Also, as asked in an earlier footnote, what
%   happens when you decrease this parameter to a level closer to 4?}


To summarize the results obtained from simulated data: Logic
Regression cannot handle very large data sets and does not reliably
identify the interaction terms. Random Forest does not provide an
explicit model of the interactions. It frequently scores well, but can
perform poorly if the signal is not strong enough. SPLR scores well at
selecting the right patterns, but selects too many noise patterns. By
contrast, pLPS usually selects the right patterns without adding too
many noise patterns.

\subsection*{The Gene Expression Barcode Data}
\label{sec.genebar}

With current microarray technology we are able to measure thousands of
RNA transcripts at one time. This capability allows for richer
characterization of cells and tissues. However, feature
characteristics such as probe sequence can cause the observed
intensity to be far away from the actual expression. Although the
``probe effect'' is large, it is  consistent across different
hybridizations, meaning that the effect is quite similar when
comparing the intensities of different hybridizations for the same
gene. Therefore, the majority of microarray data analysis uses
relative expression rather than absolute expression. To overcome this
limitation in measurement, a gene expression bar code (GEBC)
\cite{zilliox:irizarry:2007} was proposed recently. The goal is to
investigate what intensity measurement constitutes ``no expression''
for a given gene and microarray platform. GEBC starts by preprocessing
all genes using Robust Multi-array Analysis (RMA)
\cite{irizarry:hobbs:2003}. For each gene, an empirical density
smoother is used to estimate the density function of this gene across
tissues, and the smallest mode of the density function is taken to be
the expected intensity of an unexpressed gene. Gene expressions to the
left of this mode are used to estimate the standard deviation of
unexpressed genes.  If the log expression estimate of a gene is $K$
standard deviations larger than the unexpressed mean, then this gene
is considered to be expressed. The constant $K$ is chosen to be 6 by
cross-validation. For the purpose of our model, expressed genes are
coded as 1 and unexpressed genes as 0.

GEBC \cite{zilliox:irizarry:2007} downloaded publicly available raw data
from 40 different studies and created a database of 1094 human samples
representing 118 different tissues. Of these samples, 503 are normal,
500 are breast tumors, and 91 are other diseases. A total of 22,215
genes are available for each sample.

We apply pLPS on this data set, with genes dichotomized by GEBC, as
described above. Many genes have extremely unbalanced expression
levels, being expressed (or unexpressed) in a very small percentage of
the tissues. We removed these genes from our analysis, after which
7,654 genes remained. In our first analysis, we took all normal
tissues as ``controls'' and all non-breast tumor tissues as ``cases.''
In the second analysis we analyze the survival time of breast cancer
patients after dichotomization. We define subjects with survival time
less than 5 years as ``cases'' and those with survival time longer
than 10 years as ``controls.''

\subsubsection*{Cancer}

In this analysis, all normal and non-breast cancer tissues are
used. Breast tumors were excluded because no normal breast tissue was
available. The data set contains 503 normal tissues and 70 cancer
tissues, giving a malignancy rate of $12.2\%$.


The model fitted by pLPS is shown in (\ref{glps.cancer}).  Five size two interactions are selected.

\begin{eqnarray}
\label{glps.cancer}
f &=& - 8.15 + 3.58\times CALU \times ERBB3 + 1.93\times LAMC1\times CD24 \nonumber\\
&~& + 3.29\times LPCAT1\times ACY1 + 3.75\times FXYD3\times GNL3 \nonumber\\
&~& + 2.34\times NOTCH3\times CD24.
\end{eqnarray}


Most of these genes are known to be related to one or more types of
cancer. For example, ERBB3 is very important in the development of
breast cancer \cite{Perez-Nadales:2004} and prostate cancer \cite{
 koumakpayi2006expression}.  LPCAT1 is shown to be highly
overexpressed in colorectal adenocarcinomas, when compared to normal
mucosas \cite{Mansilla:2009}.  ACY1 is found to be underexpressed in
small-cell lung cancer (SCLC) cell lines and tumors
\cite{miller1989lack}. FXYD3 is overexpressed in pancreatic ductal
adenocarcinoma and influences pancreatic cancer cell growth
\cite{kayed2006fxyd3}. Notch3 overexpression is common in pancreatic
cancer \cite{dang2007role}. Finally, CD24, one of the most well-known
genes in this model, is related to breast cancer, ovarian cancer,
NSCLC, and colorectal cancer \cite{kristiansen2002cd24,kristiansen2003cd24:2,kristiansen2003cd24,weichert2005cytoplasmic}.


To compare the performance of pLPS with the alternative methods
discussed in the simulation section, the number of predictor genes
must be reduced further, because Logic Regression cannot handle more
than 1,000 variables. A screen step \cite{shi:wahba:lee:2007} was
implemented to perform the reduction. We fitted a simple logistic
regression on each gene and selected the most significant genes based
on the p-values from the regression models.  This step yields 636
genes.

We ran five-fold cross validation for all methods and summarized the
results in Table 4. (Performance measures in this
table are the average of the five-fold cross validation.)  We tabulate
the number of selected genes ($\#$ Gene), the number of non-zero
coefficients ($\#$ Para), the highest order of interactions ($q$) and
the summation of these three quantities (Total). The individual
parameters measure the complexity of the model from different points
of view, while the total provides an overall criterion.  For
prediction accuracy we present the area under the ROC curve in the
column labelled ``AUC''.  We can observe from these results that pLPS
and pLPS3 select fewer genes; pLPS, pLPS3, and Logic use fewer
parameters than SPLR; pLPS and pLPS3 do not go to high order
interactions because these are precluded by the model. In the total
complexity criterion, there is a tie for first between pLPS and pLPS3.
As for prediction, pLPS is the clear winner in AUC.



\subsubsection*{Breast Cancer Survival Time}

The survival of breast cancer patients depends on many factors, such
as grade, stage and oestrogen-receptor status. In this section we
study the possible genetic effects using the gene expression barcode
data. We denote patients who lived less than 5 years after diagnosis
as ``cases'' and patients who lived more than 10 years after diagnosis
as ``controls.'' Patients with a censored death time less than 10
years and patients that died between 5 and 10 years are excluded. The
remaining pool contains 243 patients, among which 80 are cases. The
five-year death rate is $80/243 = 32.9\%$. As in the previous
subsection, we used a screen step to reduce the size of the
model. This step yielded 592 genes.

We applied the same methods with five-fold cross validation on the
breast cancer survival data, summarizing the results in Table
5.  Among the five measures presented, pLPS does the
best in terms of the highest order of interactions and AUC measure,
winning by a large margin over the other methods in the latter
measure.  Logic Regression performs surprisingly well in model
complexity, selecting the smallest number of genes and
parameters. However its prediction, as measured by AUC, has been
sacrificed by the use of simple models.


(\ref{modelbreast}) shows one model fitted by pLPS. There are one main effect and four size two interactions.

\begin{eqnarray}\label{modelbreast}
f &=& 3.21 - 1.59\times PODXL - 2.00\times SYNE2\times AKAP11 + 2.05\times CD20\times CREB1\nonumber\\
&~&  - 1.88\times STAT5A\times MAPT -1.89 \times MAOB\times IFFO1.
\end{eqnarray}

Among these selected genes, CDC20, CREB1, STAT5A and MAPT are known to
be related to breast cancer.  It was noted in \cite{kidokoro:2008}
that CDC20 is overexpressed in a large subset of malignancies such as
colorectal, breast, lung and bladder cancers. The study
\cite{chhabra:2007} reports that CREB1 is much higher in breast tumor
tissues as compared to non-neoplastic mammary tissues.  Active STAT5
has been identified as a tumor marker of favorable prognosis in human
breast cancer, and STAT5 activation is lost during metastatic
progression \cite{sultan:2005}. It has been pointed out by
\cite{ikeda:2010} that MAPT inhibits the function of taxanes and high
expression of MAPT decreased the sensitivity to taxanes.

\section*{Conclusions}

We have described a partitioned version of the LASSO-Patternsearch
algorithm (named pLPS) that extends the range of this method to data
sets with a higher number of predictors, and allows parallel execution
of much of the computation.
%
% LASSO-Patternsearch Algorithm is a global method that works on
% binary data with possible interactions. It has a limitation on the
% number of predictor variables. Traditional methods for small $n$
% large $p$ problem handles each variable separately. Grouped
% LASSO-Patternsearch Algorithm proposed in this article is in between
% a global method and a univariate method. It breaks down big data
% sets into smaller problems that can be solved by current computing
% facility.
We show through simulations that pLPS is better than competing methods
in selecting the correct variables and patterns while controlling for
the amount of noise in the selected model. By testing on two gene
expression data sets, we also show that pLPS gives smaller models with
much better prediction accuracy than competing approaches.

Unlike LPS, two smoothing parameters with modified tuning criterion
are used in pLPS and pLPS3.  We impose a penalty on the difference
between the number of main effects and the number of interactions for
pLPS and a penalty on the difference among the numbers of main effects
(size-two interactions in pLPS and size-three interactions in
pLPS3). These penalties eliminate the extreme cases in which only main
effects or interactions come up in the LASSO step. (These extreme
cases appear too often with the original, unmodified criterion.)  On
the other hand, if an extreme case is the truth the LASSO step will
generate some noisy patterns, but the parametric step tends to
eliminate the noise and thus select the correct model.
\label{sec.glpsdisc}

