%XXX why mann whitnet
%XXX why did others not report this earlier
%XXX change conclusion
%XXX walk the reviews looking for comments
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\begin{document}

\title{
Identifying the ``Best'' Software Prediction Models 
Requires a Combination of Methods}

\author{Tim~Menzies,~\IEEEmembership{Member,~IEEE,}
        Omid Jalali,
		Jairus Hihn,
	    Dan Baker,
		and Karen Lum
\thanks{ 
Tim Menzies, Omid Jalali, and Dan Baker
are with the Lane Department of
Computer Science and Electrical Engineering, West
Virginia University, USA: 
\protect\url{tim@menzies.us},
\protect\url{ojalali@mix.wvu.edu}, \protect\url{danielryanbaker@gmail.com}.}
\thanks{Jairus Hihn and Karen Lum at with NASA's Jet Propulsion Laboratory:
\protect\url{jhihn@mail3.jpl.nasa.gov},
\protect\url{karen.t.lum@jpl.nasa.gov}.}
\thanks{
The research described in this paper was carried out at West Virginia University
and the Jet 
Propulsion Laboratory, California Institute of Technology, under a contract with the US National Aeronautics and 
Space Administration. Reference herein to any specific 
commercial product, process, or service by trade name, 
trademark, manufacturer, or otherwise does not constitute 
or imply its endorsement by the US Government.}
\thanks{Download: \protect\url{http://menzies.us/pdf/07stability.pdf}.}
\thanks{
Manuscript received July 31, 2007; revised XXX, XXXX.}}

\markboth{Journal of ???,~Vol.~6, No.~1, January~2007}%
{Menzies \MakeLowercase{\textit{et.al.}}: Estimation and Conclusion Stability}

\IEEEaftertitletext{\vspace{-1\baselineskip}
\noindent\begin{abstract}
BACKGROUND: 
There exists a large and growing
number of estimation
methods but little conclusive evidence ranking one method over another.
Hence
industrial practitioners may be using
possibly  inferior methods.\newline
OBJECTIVE: 
There are many methods for effort estimation- 
so many that is impractical
to explore them all.
The research community should agree on
acceptable boundaries on experimentation. If studying 
all methods is 
too many, how many methods are enough?
\newline
METHOD:
20 times, ten rows were selected at random for
testing from
19 tables of COCOMO data.
158 estimation methods from the COSEEKMO toolkit~\cite{me06d}
were applied to the remaining data, then tested on the
ten rows.
Test performance was scored using three different  evaluation 
criteria and 
compared using ``loss'' statistics
generated from  a Mann-Whitney
analysis (95\% confidence).
\newline
RESULTS: 
Four methods always performed better than the other 154.
This result was stable
despite random variations in the training and test set,
and across the different evaluation criteria.
To the best of our knowledge, this is the first report
of stable conclusions over such a large space of data, methods,
evaluation criteria.
\newline
CONCLUSION: 
While  there exists no single universal ``best'' 
effort estimation method,
within the space of known estimation methods
there may exist a very small number of most useful methods.
This result both {\em complicates} and {\em simplifies} effort estimation research.
The {\em complication} is that 
any future effort estimation analysis should be preceded by a
``selection study'' that finds the best local estimator.
However,  
the {\em simplification} is that such a study need not be 
labor intensive: for COCOMO style datasets
this paper concludes that there are currently only four known candidates
for ``best local estimator''.
\end{abstract}\noindent
\begin{keywords}COCOMO, effort estimation, data mining, 
evaluation
\end{keywords}\vspace{1\baselineskip}}\maketitle

%------------------------------------------------------------------------- 

\section{Introduction}
Software effort estimates are often wrong by a factor 
of four~\cite{boehm81} or even more~\cite{kemerer87}. 
As a result,
the allocated funds may be inadequate to develop the required project.
In the worst case, over-running 
projects are canceled, wasting the entire development effort. 
For example, in 2003, NASA canceled
the CLCS system
after spending 
hundreds of millions of dollars on software development.
The project was canceled after the initial estimate
of
\$206 million was increased to between
\$488 million and \$533 million~\cite{clcs03}.
On cancellation, approximately 400 developers
lost their jobs\cite{clcs03}.


While the need for better estimates is clear, there exists
a very large number of effort estimation methods~\cite{jorg04,jorgensen05}
and few studies empirically
compare all these techniques. What is more usual are narrowly focused studies (e.g.
\cite{kemerer87,briand00,lum02,ferens98}) that test, say, linear regression models in different environments.


Unless we can {\em rank} methods and {\em prune}
inferior methods,
we will soon be overwhelmed by a growing number
of (possibly useless) effort estimation methods.
New open source data mining toolkits are
appearing with increasing frequency such as 
the R project\footnote{\url{http://www.r-project.org/}},
Orange\footnote{\url{http://www.ailab.si/orange/}}, and the
WEKA~\cite{witten05}.
All the learners in all these toolkits can be {\em stacked}
by 
{\em meta-learning} schemes 
where the conclusions of one 
data miner influences the next. 
There exists
one stacking for every ordering of $N$ learners; so
, five learners can be stacked $5!=120$ ways and ten learners
can be stacked in  millions of different ways.

%The abudence of such  tools tempt researchers to over-elaborate their effort estimation tools.
%For example, our COSEEKMO tool~\cite{me06d} takes nearly a 
%day to run its 158 methods.
%In this paper, we show that
%much of that execution is wasted since,
%$\frac{154}{158}$ of those methods are superfluous.
%In the experiments reported below, we show that despite:\bi
%\item two changes of the source of the data;
%\item three changes to the evaluation criteria;
%\item hundreds of random  to the training/test data;
%\ei
%that four methods consistently out-performed the rest.
%To the best of our knowledge, this is the first report of
%stable conclusions in such a comparison of a very large number of
%effort estimation mehtods.
%\subsection{Contributions}
% Can the new generation of
%data miners offer better estimates than traditional methods?
%Previously, we argued that this is the case 
%(\cite{me04h,me05a,me05c,me05d,me06d,me06e,me06f}). However,
%the experiments reported in this paper  have forced a revision of that view.
%
%Potentially,
%data mining methods can handle modeling tasks not suitable for 
%standard linear regression.
%For example:
%\bi
%\item {\em Model trees}~\cite{quinlan92b} applies regression
%to parts of the data, then builds a decision tree to decide which
%part is most relevant.
%This
%is useful if 
%estimations are  different for (say)
%very large
%or very small software systems. 
%\item 
%{\em Column pruning} 
%reduces estimate variance~\cite{miller02}
%by culling  noisy data resulting from
%poor collection practices.
%\item {\em Row pruning}
%methods
%build estimates from only relevant cases taken from some historical log.
%Like column pruning, row pruning can improve estimation by discarding extraneous
%influences that conflate the estimation and, potentially, increase the
%variance on the estimate.
%\ei
%Various research results \cite{kirsopp02,me06d}
%advocate 
%combing these methods.
%Do those results invalidate
%other papers that do not use multiple methods
%that combine, say,  row and/or column pruning?
%For example,
%Lokan \& Mendes~\cite{lokan06} do not apply column pruning
%or row pruning, even though they report
%large deviations in effort estimates\footnote{
%In Tables 11 \& 12 in \cite{lokan06} the reported effort estimates include
%two results with
%  efforts of  $\{\mu=4,704,\sigma=6,717\}$  
%and $\{\mu=4,310,\sigma=10,730\}$. Note that in both cases,
%the deviation is {\em larger} than the mean. This is a clear indicator
%of ``noisy'' data that could be enhanced via column or row pruning.}.
%
%To be fair to Lokan \& Mendes,
%we hasten to say that 
%they were following accepted practice
%in the field of effort estimation.
%Many 
%publications
%in this field
%only use  one estimation method- often,
%some variant on linear regression (e.g. see
%\cite{kemerer87,briand00,lum02,ferens98} and many of the 
%papers from the annual USC COCOMO forum).
%Also, 
%subsequently, Mendes did use multiple methods for effort 
%estimation~\cite{mendes07}. 
%
%Also,
%it is impractical to 
%try {\em all} methods. 
%Data mining toolkits are
%appearing with increasing frequency;
%e.g. R project\footnote{\url{http://www.r-project.org/}},
%Orange\footnote{\url{http://www.ailab.si/orange/}}, and the
%WEKA~\cite{witten05}.
%The learners in these toolkits can be
%{\em stacked} via 
%{\em meta-learning} schemes
%where
%the conclusions of one 
%data miner influences the next. 
%$N$ learners can be stacked in $N!$ ways so
%five learners can be stacked $5!=120$ ways and ten learners
%can be stacked millions of ways.
%
%
%Nevertheless, the research community should agree on
%acceptable boundaries on experimentation. If one method is too few and
%all methods are 
%too many, how many methods are enough?

Prior attempts to rank and prune different methods 
have been inconclusive.
Shepperd and Kadoda~\cite{sheppherd01} 
compared the effort models learned 
from a variant of regression, rule induction, case-based
reasoning (CBR), and neural networks. 
Their results  exhibited much {\em conclusion instability}
where the performance results:
\bi
\item
Differed markedly across different data sets;
\item
Differed markedly when they repeated their runs
using 
different random seeds.
\ei
Overall, while
no single best method was ``best''
they found weak evidence that two methods were generally
inferior (rule induction and neural nets)~\cite[p1020]{sheppherd01}.

%XXX prune intro . start at can't try all methods.
%XXX bring back old text

%XXX p2, column1 half way- jumps to just CBR. why?

%XXX p3: two "availables" too close together
%XXX p4: col [rune befpre row prine. add see appendix for more details
%XXX check all references to conclusion instabilty
%XXX p5: move % from RE
%XXX we use mann whitnet cause we are com[aring pruned to non-pruned


The genesis of this paper were
three observations suggesting  that it might be worthwhile revisiting the
Shepperd \& Kadoda
results. Firstly, their study was based on six data sets and our
work has let us access 19 data sets, from two very different sources.
With this data in hand, it seemed prudent to test the generality of their
conclusions on a larger number of examples.

Secondly, our 19 
data sets are expressed in terms of the COCOMO features~\cite{boehm81}.
These features were selected by
by Boehm (a widely-cited and experienced researcher with much industrial experience; 
e.g. see \cite{boehm86}) and subsequently tested
by a large  
research and industrial community (since 1985, the annual COCOMO forum has meet
to debate and  review the value of those features).
Perhaps, we speculated, conclusion
instability might be tamed by the use of better features.

Thirdly, we noted an interesting quirk in 
their experimental results.
Estimation methods can
prune tables of training data:
\bi
\item
CBR prunes away irrelevant {\em rows};
\item
Stepwise regression prunes away {\em columns} that
add little information to the regression equation;
\item
Shepperd \& Kadoda's rule induction and neural net
have no row/column pruning.
\ei
Note that
the 
 two methods found to be
``worst'' by Shepperd \& Kadoda had no
row or column pruning methods.
Pruning data can be useful to  remove outliers or ``noisy'' information
(spurious signals unconnected to the target variable). One symptom of outliers and noise is 
conclusion instability across different data sets and different random samplings.
Hence, we wondered if conclusion instability could be tamed via the application of more elaborate row {\em
and} column pruners.

The above observations lead to the experiment reported in this paper. 
Previously~\cite{me06d}, we have built the COSEEKMO effort estimation workbench that supports 158 
estimation
methods\footnote{
To be precise, COSEEKMO currently supports 15 learners and 8 row/column pre-processors
which can be applied to two different sets of internal tuning parameters.
In one view, this combination of (15+8*8)*2=158 different
estimation generation methods are not really different ``methods'';
rather it might be more accurate to call them ``instances of methods''.
This paper does not adopt that terminology for the following reason.
  To any user of our tools, our menagerie of estimation software
contains 158 {\em oracles} that may yield different effort estimates.
Hence, in the view adopted by this paper, they are 158
competing methods that must be assessed and (hopefully)
pruned to a more management size.}.
The methods are defined over COCOMO features, and make extensive use of
combinations of different row and column pruners.
When we ran this toolkit over our 19 data sets, we did not find conclusion instability.
In fact, a very clear pattern in the results was apparent:
\be
\item Many estimation methods offer  very little added value;
\item A small number (four) of estimation methods consistently out-perform the rest;
\item Within the set of four demonstrably useful methods, there is no
consistently best estimator. 
\ee
Therefore, we now 
say that
COSEEKMO
was an over-elaboration.
We are planning to prune that toolkit and we
advise other researchers not to make the same mistake.
For example, 
in this paper,
very simple extensions to decades-old
techniques out-performed 97\% of all the methods studies here.
If those percentages carry over to other effort estimation paradigms then
studies of commercial estimation tools
such as SEER-SEM~\cite{jensen83} or PRICE-S~\cite{park88} or SLIM~\cite{putnam92} 
could find that many superfluous modeling features in those tools.
If so, then
commercial practitioners are wasting their
 management budget purchasing needlessly elaborate tools.

The rest of this paper describes our experiments and 
discusses
their implications on prior and future work.

\section{Related Work}
There are many methods for effort estimation.
The rest of this paper offers brief notes on some
of them (with supporting details in the appendix).
For a more extensive survey of methods, see~\cite{shepperd07,jorgensen05}.

In order to introduce the reader to a range of estimation,
in this related work section we will just focus on two methods:
regression-based COCOMO and case-based reasoning.
This will be followed by a brief tutorial on row and column pruning.

\subsection{Regression-Based COCOMO}
\noindent
Two factors make us prefer COCOMO-based methods:
\bi
\item
{\em Public domain:}
Unlike other effort estimators such as PRICE-S~\cite{park88}, 
SLIM~\cite{putnam92}, or SEER-SEM~\cite{jensen83}.
 COCOMO is a public domain 
with published data and baseline results~\cite{chulani99}. 
\item
{\em Data availability:}
All the information we could access was in the
\mbox{COCOMO-I} format~\cite{boehm81}.
\ei
COCOMO is a based on linear regression which
assumes that the data can be
approximated by one linear model that includes lines of code
(KLOC) and other features $f$ seen in a software development project:
\[
effort = \beta_0 + \sum_i\beta_i\cdot f_i
\]
Linear regression adjusts $\beta_i$ to minimize the {\em prediction error} 
(predicted minus actual values for the project).

\begin{figure}[!b]
\begin{center}
{\scriptsize
\begin{tabular}{l|r@{:~}l|}\cline{2-3}
upper:   &acap&analysts capability\\
increase &pcap&programmers capability\\
these to &aexp&application experience\\
decrease &modp&modern programming practices\\
effort   &tool&use of software tools\\
         &vexp&virtual machine experience\\
         &lexp&language experience\\\cline{2-3}
middle   &sced&schedule constraint\\\cline{2-3}
lower:   &data&data base size\\
decrease &turn&turnaround time\\
these to &virt&machine volatility\\
increase &stor&main memory constraint\\
effort   &time&time constraint for CPU\\
         &rely&required software reliability\\
         &cplx&process complexity\\\cline{2-3}
\end{tabular}}
\end{center}
\caption{{
The $f_i$ features used in this study. From~\cite{boehm81}. Most
range from 1 to 6 representing ``very low'' to ``extremely high''.
}}\label{fig:em}
\end{figure}

After much research, Boehm advocated
the COCOMO-I features shown in 
\fig{em}. He also
argued that effort is exponential on KLOC~\cite{boehm81}:
\begin{equation}\label{eq:coc1}
effort = a \cdot KLOC^b  \cdot \prod_i\beta_i
\end{equation}
where $a$ and $b$ are domain-specific constants
and $\beta_i$ comes from looking up $f_i$ values
in \fig{effortmults}. 
When $\beta_i$ is used in the above equation, they yield estimates in months 
where one month is 152 hours (and includes development and management hours).

Exponential
functions like \eq{coc1}
can be learned via linear regression after a conversion to
a linear form:
\begin{equation}\label{eq:linear}
log(effort)=log(a)+b{\cdot}log(KLOC)+\sum_ilog(\beta_i)
\end{equation}
All our methods transform the data in this way
so 
when collecting performance statistics, 
the estimates must be unlogged.


\begin{figure}
\begin{center}
{\scriptsize
\begin{tabular}{|l|c|r@{~}|r@{~}|r@{~}|r@{~}|r@{~}|r|}
    \hline
    &&1&2&3&4&5&6\\
    \hline
upper&ACAP   &1.46   &1.19   &1.00   &0.86   &0.71   &\\
(increase&PCAP   &1.42 &1.17   &1.00   &0.86   &0.70 &\\
these to&AEXP   &1.29 &1.13   &1.00   &0.91   &0.82   &\\
decrease&MODP   &1.2  &1.10 &1.00 &0.91 &0.82 &\\
effort)&TOOL   &1.24 &1.10 &1.00 &0.91 &0.83 &\\
&VEXP   &1.21 &1.10 &1.00 &0.90 &  &\\
&LEXP   &1.14 &1.07 &1.00 &0.95 &  &\\\hline
middle&SCED   &1.23 &1.08 &1.00 &1.04 &1.10 &  \\\hline
lower&DATA   &    & 0.94 &1.00 &1.08 &1.16&\\
(increase&TURN   &       &0.87   &1.00   &1.07   &1.15   &\\
these to&VIRT   &       &0.87   &1.00   &1.15   &1.30   &\\
increase&STOR   &       &       &1.00   &1.06   &1.21   &1.56\\
effort)&TIME   &  &    &1.00   &1.11   &1.30   &1.66\\
&RELY   &0.75& 0.88& 1.00 & 1.15 & 1.40&\\
&CPLX   &0.70 &0.85 &1.00 &1.15 &1.30 &1.65\\
    \hline
\end{tabular}}
\end{center}
\caption{The COCOMO-I $\beta_i$ table~\cite{boehm81}. 
For example, the bottom right cell is saying that if CPLX=6, then
the nominal effort is multiplied by 1.65.}\label{fig:effortmults}
\end{figure}
Local calibration (LC) is a specialized form
of linear regression developed by Boehm~\cite[p526-529]{boehm81}
that assumes effort is model via the linear form \eq{linear}.
Linear regression would try to adjust all the $\beta_i$ values.
This is not practical when training on a very small number of projects.
Hence, LC fixes the $\beta_i$ values while adjusting the $<a,b>$ values to minimize
the prediction error. We shall refer to LC as ``standard practice'' since, in the COCOMO community
at least, it is the preferred method for calibrating standard COCOMO data~\cite{boehm00b}.




In 2000, 
Boehm et.al. updated the COCOMO-I model~\cite{boehm00b}. After the update, numerous features 
remained the same:
\bi
\item
Effort is assumed to be exponential on model size.
\item
Boehm et.al. still recommends
local calibration
for tuning generic COCOMO to a local situation.
\item
Boehm et.al. advises that effort estimates can be improved via {\em manual stratification} (described later);
i.e. use domain knowledge to select relevant past data.
\ei

At the 2005 COCOMO forum, there were  
discussions about relaxing the access restrictions on the
\mbox{COCOMO-II} data.
To date, those discussions have not progressed.
Since other researchers do not have 
access to COCOMO-II, this paper will only report
results from COCOMO-I.

\subsection{Case-Based-Reasoning}
COCOMO's features are both the strongest and weakest part
of
that method. One the one hand, they have been selected and tested
by a large community of academic and industrial researchers lead
by Boehm. This group meets annually at the COCOMO forums (this
are large meetings that have been held annually since 1985).
One the other hand, these features may not be available in
the databases of a local site. 
Hence, regardless of the potential value added of using a well-researched
feature set, those features may not be available.

An alternative 
to COCOMO 
is the case-based reasoning (CBR) approach used by
Shepperd~\cite{shepperd07} and others~\cite{li07}. CBR accepts
data with any set of features.
Often, CBR uses a
{\em nearest neighbor}
method to makes predictions using past data that is
similar to a new situation. Some distance measure is used to find the $k$ nearest
older projects to each project in the $Test$ set. An effort estimate
can be generated from the mean effort of the $k$ nearest neighbors
(for details on finding $k$, see  below).

The benefit of nearest neighbor algorithms is that they
make the fewest domain assumptions. That is, they can process a 
broader range of the data available within projects:
\bi
\item Local calibration cannot be applied unless projects are described using the COCOMO ontology (\fig{em}). 
\item Linear regression is 
best applied to data where most of the values for the numeric factors are known.
\ei

The drawback of nearest neighbor is that, sometimes, they can
ignore important
domain assumptions. For
example, if effort is really exponential on KLOC, a standard nearest
neighbor algorithm has no way to exploit that.

\section{A Brief Tutorial on Row and Column Pruning}

Pruning can be useful since
project data collected in one context may not be relevant to another.
Kitchenham et.al.~\cite{kitch07} take great care to document this effect.
In a systematic review comparing estimates generated using 
historical data {\em within} the same company or {\em imported} from another,
Kitchenham et.al. found no
case where it was better to use data from other sites. Indeed, sometimes,
importing such data yielded significantly worse estimates.
Similar projects have less variation and so can be easier to calibrate: 
Chulani et.al.~\cite{chulani99} \& Shepperd and Schofield~\cite{shepperd97} report that row pruning
improves estimation accuracy.

Given a table $P*F$ containing one row for $P$ projects described using
$F$ features, row and column pruning 
prune {\em irrelevant} projects and features.
After pruning, the learner executes on a new table $P'*F'$
where $P' \subseteq P$ and $F' \subseteq F$.

Row pruning can be {\em manual} or {\em automatic}.
In {\em manual row pruning} (also called ``stratification'' in the COCOMO
literature~\cite{boehm00b}), an analyst applies their domain knowledge
to select project data that is similar to the new project to be
estimated.

Unlike other methods, the manual stratification used here uses
different subsets to create $Train$ sets.
Our 19 data sets
comes from two $sources$: Boehm's  COCOMO text~\cite{boehm81} and 
some more recent NASA data. Those sources divide into {\em subsets}
representing data from different companies, or different project types
(see the appendix for details).
\bi
\item In every case except for the manual stratification, $Train$ and $Test$ sets are created from the
subsets.
\item In manual stratification, the $Test$ set is created in the same manner from the subsets. 
However, the $Train$ set is created from the entire 
$source$ and not their subsets.
\ei

{\em Automatic row pruning} uses algorithmic techniques to select a subset
of the projects (rows). 
NEAREST and LOCOMO~\cite{jalali07} are automatic and use nearest neighbor methods on 
the $Train$ set to find the $k$ most relevant projects to generate predictions for the projects in the $Test$ set. 
The core of both automatic algorithms is a distance measure that must compare all pairs of projects.
Hence, these automatics methods take time $O(P^2)$. 
Both NEAREST and LOCOMO learn an appropriate $k$ from the $Train$ set and the $k$ with the lowest
error is used when processing the $Test$ set. 
NEAREST averages the effort associated with the $k$ nearest neighbors 
while LOCOMO passes the $k$ nearest neighbors to Boehm's local calibration (LC) method.

Column pruners fall into two groups:
\bi
\item WRAPPER and LOCALW are very thorough search algorithms that explore
subsets of the features, in no set order. This search takes time $O(2^F)$. 
\item COCOMIN~\cite{baker07} is far less thorough.
COCOMIN is a near linear-time pre-processor that selects the features
on some heuristic criteria and does not explore all subsets of the features.
It runs in $O(F{\cdot}log(F))$ for the sort and $O(F)$ time for the exploration of selected features.
\ei

\section{Experiments}

\subsection{Data}
%XXX different subsetsets, two sources
% previously we hare report that these data sets have very different properties

This paper is based on 19 subsets from two sources.
$COC81$\footnote{ 
\url{http://promisedata.org/repository/data/coc81/}.}
comes from Boehm's 1981 text on effort estimation. 
$NASA93$\footnote{
\url{http://promisedata.org/repository/data/nasa93/}.}
comes from
a study funded by the Space
Station Freedom Program.
$NASA93$ contains data from six different NASA centers including the Jet Propulsion
Laboratory.
For details on this data, see the appendix. 

\subsection{Experimental Procedure}

\noindent
Each of the 19 subsets of $COC81$ and $NASA93$ were expressed
as a table of data $P*F$.
The table stored {\em project} information in $P$ rows and
each row included the {\em actual} development effort. 
In the 19 subsets of $COC81$ and $NASA93$ used in the study,
$20 \le P \le 93$. The upper bound of this range ($P=93$)
is the largest data set's size.
The lower bound of this range ($P=20$) was selected based on
a prior study~\cite{me06d}.
For details on these data sets, see the appendix.

The table also has $F$ columns containing the project {\em features} $\{f_1,f_2,...\}$.
The features used in this study come from Boehm's COCOMO-I work (described in the appendix)
and include items such as
lines of code (KLOC), schedule pressure (sced), analyst capability (acap), etc. 

To build an effort model, 
table rows were divided at random into a $Train$ and $Test$ set
(and $|Train|+|Test|=P$). COSEEKMO's 
methods are then applied to the $Train$ set to generate a model which
was then used on the
$Test$ set.
In order to compare this study with our work~\cite{me06d},
we use the same $Test$ set size as the COSEEKMO
study; i.e. $|Test|=10$.

Effort models were assessed via three evaluation criteria:
\bi
\item $AR$: absolute residual; $abs(actual - predicted)$;
\item $MRE$: magnitude of relative error;  $\frac{abs(predicted - actual)}{actual}$;
\item $MER$: magnitude of error relative to estimate; $\frac{abs(actual - predicted)}{predicted}$;
\ei
\newcommand{\nope}{\ding{56}}
\newcommand{\yupe}{\ding{52}}
\begin{figure*}
\begin{center}
\scriptsize
\begin{tabular}{r@{~=~}l|l|l|p{1.5in}}
 method& name                  & row pruning & column pruning         & learner \\\hline
a    & LC           & \nope          & \nope                      & LC = Boehm's local calibration \\\hline
b    & COCOMIN +  LC     &  \yupe automatic $O(P^2)$        & \nope                      & local calibration \\\hline
c    & COCOMIN + LOCOMO + LC   &  \yupe automatic  $O(P^2)$       &  \yupe automatic $O(F{\cdot}log(F) + F)$ &  local calibration \\\hline
d    &  LOCOMO + LC      &  \nope         & \yupe automatic $O(F{\cdot}log(F) + F)$ &  local calibration\\\hline
e    &  Manual Stratification + LC &  \yupe manual  & \nope             &  local calibration \\\hline
f    & M5pW  + M5p       &  \nope        &  \yupe Kohavi's WRAPPER~\cite{kohavi97} calling M5p~\cite{quinlan92b}, $O(2^F)$ & model trees\\\hline
g    & LOCALW  + LC      &  \nope        &  \yupe Chen's WRAPPER~\cite{me06d}  calling LC, $O(2^F)$ & local calibration\\\hline
h    & LsrW  + LSR       &  \nope        &  \yupe Kohavi's WRAPPER~\cite{kohavi97} calling LSR, $O(2^F)$ & linear regression\\\hline
i    & NEAREST       &  \yupe automatic $O(P^2)$& \nope &      mean  effort of nearest neighbors\\\hline
\end{tabular}
\end{center}
\caption{Nine effort estimation methods explored
in this paper. F is the number of features (columns) and P is the number of projects (rows).}\label{fig:abcd}
\end{figure*}


For the sake of statistical validity, the above procedure was repeated 20 times
for each of the 19 subsets of $COC81$ and $NASA93$.
Each time, a different seed was used to generate the $Train$ and $Test$ sets.

Methods performance scores were assessed using performance ranks rather than exact values.
To illustrate the process of replacing exact values with ranks,
consider the following example.
 If treatment $A$ generates $N_1=5$ values \{5,7,2,0,4\} 
and treatment $B$ generates $N_2=6$ values \{4,8,2,3,6,7\}, 
then these sort as follows:
\begin{center}
{\small
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c}
Samples& A & A & B & B & A & B & A & B & A & B & B\\
Values & 0 & 2 & 2 & 3 & 4 & 4 & 5 & 6 & 7 & 7 & 8\\
\end{tabular}}\end{center}
On ranking, averages are used when values are the same:
\begin{center}
{\small
\begin{tabular}{c|c@{~}|c@{~}|c@{~}|c@{~}|c@{~}|c@{~}|c@{~}|c@{~}|c@{~}|c@{~}|c}
Samples& A&  A  &B&  B&  A & B  &A&  B&  A&  B & B\\
Values&  0  &2  &2&  3&  4 & 4  &5&  6&  7&  7 & 8\\
Ranks&   1 &2.5&2.5 &4& 5.5&5.5 &7&  8& 9.5&9.5& 11\\
\end{tabular}}\end{center}
Note that, when ranked in this manner,
the largest value (8 in this case) gets the same rank
even if it was
ten to a hundred times larger.
That is, such rank tests
are less susceptible to large outliers.
This is very important for experiments with
 effort estimation.
In our experiments, we can build thousands to tens of thousands of estimators that exhibit
infrequent, but very large outliers.
For example, the relative error of an estimate is $RE=\frac{predicted-actual}{actual}$.
In our work we have seen data sets generate RE\% below 100 then suddenly spike
in one instance to over 8000\%.

After ranking the performance scores, we applied
the non-paired Mann-Whitney  paired ranked test~\cite{mann47}:
\bi
\item
Non-paired tests compare the performance of 
populations of treatments while paired test compare performance deltas of two methods
running on exactly the same train/test data. Since we are using row/column pruning,
paired tests are inappropriate since the underlying data distributions in the train test
can vary widely when (e.g.) a method that does use row and/or column pruning is compared to
one that does not.
\item
Mann-Whitney supports very succinct summaries of the results without intricate post-processing.
This is a very important requirement for our work since we are comparing 158 methods. 
Mann-Whitney  does not require that the sample sizes are the same.
So, in a single U test, learner $L_1$ can be compared to all its rivals. 
\ei

%has been extensively studied in the data mining community. 
%T-tests that assume Gaussian distributions are strongly deprecated.
%After reviewing a wide range of
%comparisons methods\footnote{Paired t-tests with and without the use of geometric means of the relative ratios;
%binomial tests with the Bonferroni correction;
%paired t-tests; ANOVA; Wilcoxon; Friedman},
%Demsar~\cite{demsar06} advocates the use of the 1945 Wilcoxon~\cite{wilcoxon45} 
%signed-rank test that compares the ranks for the positive and negative differences (ignoring the signs). 
%Writing five years earlier,
%Kitchenham et.al.~\cite{kitc01} comment that the Wilcoxon test has its limitations.
%Demsar's report offers the same conclusion, 
%noting that the Wilcoxon test requires that the sample sizes are the same.
%
%One test not studied by Demsar is Mann and Whitney's 1947
%modification~\cite{mann47} to Wilcoxon rank-sum test (proposed along with his signed-rank test)~\cite{demsar06}. We prefer this test since
%Consider how to rank 158 methods with
%Wilcoxon:
%\bi
%\item Rank all methods by their median scores, $r_1, r_2,r_3,...r_{158}$.
%\item Using Wilcoxon, find all methods that were statistically insignificantly different to the top
%ranked method. Call those methods $r_1, r_2,... r_i$, the ``best''.
%Remove these methods from further analysis.
%\item Repeat for the remaining methods to return sets of methods labeled ``second best'', ``third best'' etc.
%\ei
% Note that it is unclear how to handle
%the situation where $r_1, ... r_i$ have been labeled ``best''
%but $r_j$ ($j > i$) is statistically insignificantly different from $r_k$ for $2 \le k \le i$.
%The U test, on the other hand, does not require any post-processing
%(such as the Friedman test or the above ``home brew'' method) to conclude 
%if the median rank of one population (say, the $L_1$ results)
%is greater than, equal to, or less than the median rank of another (say, the $L_2,L_3,..,L_x$ results).

Mann-Whitney can be used to report ``wins'', ``loss'', or ``tie'' for
all pairs of methods 
$L_i,L_j, i\not=j$:
\bi
\item ``Tie'' ; i.e. the ranks are statistically the same;
\item ``Win'' ; i.e. not a tie and the median rank of one method has a lower error than the other;
\item ``Loss'' ; i.e. not a tie and the opposite to a win.
\ei
Given $L$ learning methods, the sum of $tie+win+loss$ for any one method is $L-1$.
When discussing discarding a method, an insightful metric is the number of losses. If this is non-zero,
then there is a case for discarding that method.

\subsection{158 Methods}

COSEEKMO's  158 methods combine:
\bi
\item Some {\em learners} such as standard linear regression, local calibration, and model trees.
\item Various {\em pre-processors} that may prune rows or columns.
\item Various {\em nearest neighbor} algorithms that can be used either as learners or as pre-processors to other learners.
\ei

Note that only some of the learners use pre-processors.
In all, COSEEKMO's methods combine
15 learners without a pre-processor and 8 
learners with 8 pre-processors; i.e. 
\mbox{$15+8*8=79$} combinations in total.

COSEEKMO's methods input project features described using the 
symbolic range {\em very low} to 
{\em extra high}.
Some of the methods map the symbolic range to numerics 1..6.
Other methods map the symbolic range into a set of {\em effort multipliers} and
{\em scale factors} developed by Boehm and are shown in the appendix (\fig{effortmults}).
Previously, we have queried the utility of these 
effort multipliers and scale factors~\cite{me06d}. COSEEKMO hence executes its 79 methods
twice: once using Boehm's values, then once again using
perturbations of those values. Hence, in all, COSEEKMO contains $2*79=158$ methods.

There is insufficient space in this paper to describe the 158 methods
(for full details, see~\cite{jalali07}).
Such a complete
description would be pointless since, as shown below, most of them are beaten by a very
small number of 
preferred methods. For example, our previous concerns regarding the effort multipliers and
scale factors proved unfounded (and so at least half the runtime of COSEEKMO is wasted).

\subsection{Brief Notes on Nine Methods}
This paper focuses on the
nine methods $(a,b,c,d,ef,g,h,i)$ of \fig{abcd}. Four of these, $(a,b,c,d)$,
are our preferred methods while the other four 
comment on premises of some prior publications~\cite{me05c}.
Each method may use a column or row pruner or, as with $(a,i)$, no pruning at all.

One way to categorize \fig{abcd} is by their relationship to accepted practice
(as defined in the COCOMO texts~\cite{boehm81,boehm00b}).
Methods $(a,e)$ are endorsed as best practice in the COCOMO community.
The others are our attempts to do better than current established practice using
e.g. intricate learning schemes or intelligent data pre-processors.

Method $f$ is an example of a more intricate learning schemes. 
Standard linear regression assumes that the data can be fitted to a single model.
On the other hand, the model trees used in $f$~\cite{quinlan92b} permit the
generation of multiple models (as well as a decision tree for selecting the appropriate model).

In methods $(f,h)$, the notation
M5pW and LsrW denotes a WRAPPER that uses M5p or LSR as its target
learner (respectively).

For more details on these methods, see the appendix.

\section{Results}

\noindent
Figures 
~\ref{fig:resultsBoth-MRE-Run1}, ~\ref{fig:resultsBoth-MER-Run1} 
and \ref{fig:resultsBoth-AR-Run1}
show results from
20 repeats of:
\bi
\item Dividing some subset into $Train$ and $Test$ sets; 
\item
Learning an effort model from the $Train$ set using COSEEKMO's 158 methods; 
\item
Applying that model to the $Test$ set; 
\item
Collecting performance statistics
from the $Test$ set using AR, MER, or MRE; 
\item Reporting the number of times a method losses, where ``loss'' is determined by
a 
Mann-Whitney U test
(95\% confidence);
\ei
\noindent
In these results,
conclusion instability due to 
{\em changing evaluation criteria} can be detected by
comparing  results across 
Figures
~\ref{fig:resultsBoth-MRE-Run1}, ~\ref{fig:resultsBoth-MER-Run1} 
and \ref{fig:resultsBoth-AR-Run1}.
Also,
conclusion instability due to {\em changing subsets} can be detected by comparing
results across 
different subsets generated by changing the random seed controlling $Train$ and
$Test$ set generation (i.e. the three  
runs of \fig{resultsBoth-AR-Run1} that used different random seeds).

A single glance shows our main result:
the plots are very similar.
Specifically, the $(a,b,c,d)$ results 
fall very close to $y=0$ losses.
The significance of this result is discussed below.

Each mark on these plots shows the 
number of times a method loses in seven $COC81$ subsets (left plots) and
twelve $NASA93$ subsets (right plots). 
The x-axis shows results from the methods
$(a,b,c,d,e,f,g,h,i)$ described in \fig{abcd}. 

In these plots, methods that generate {\em lower} losses are {\em better}.
For example,
the top-left plot of 
\fig{resultsBoth-AR-Run1}
shows results for ranking methods applied to $COC81$ using AR. 
In that plot, all of methods $(a,d)$ results from the seven $COCO81$ subsets
can be seen at $y=losses\approx 0$. That is,
in that plot,
these two methods {\em never} lose against the other 158 methods.

\begin{figure}
\begin{center}
\begin{tabular}{|p{.95\linewidth}|}\hline
\includegraphics[width=3.4in]{LossesColumnsBoth-MRE-Run1.pdf}
\\\hline\end{tabular}
\end{center}
\caption{MRE results. 
Mann-Whitey (95\% confidence).
These plots show number of losses of methods ${a,b..i}$ against
158 methods as judged by Mann-Whitney  (95\% confidence).
Each vertical set of marks shows results from 7 subsets of
COC81 or 12 subsets of NASA93.
}
\label{fig:resultsBoth-MRE-Run1}
\end{figure}
\begin{figure}
\begin{center}
\begin{tabular}{|p{.95\linewidth}|}\hline
\includegraphics[width=3.4in]{LossesColumnsBoth-MER-Run1.pdf}
\\\hline\end{tabular}
\end{center}
\caption{MER results.
Mann-Whitey (95\% confidence).
Same rig as \fig{resultsBoth-MRE-Run1}.}
\label{fig:resultsBoth-MER-Run1}
\end{figure}
\begin{figure}[!t]
\begin{center}
\begin{tabular}{|p{.95\linewidth}|}\hline
Results using random $seed_1$:\newline
\includegraphics[width=3.4in]{LossesColumnsBoth-AR-Run2.pdf}\\
Results using random $seed_2$:\newline
\includegraphics[width=3.4in]{LossesColumnsBoth-AR-Run1.pdf}\\
Results using random $seed_3$:\newline
\includegraphics[width=3.4in]{LossesColumnsBoth-AR-Run3.pdf}\\\hline\end{tabular}
\end{center}
\caption{AR results, repeated three different times
with three different random seeds.
Same rig as \fig{resultsBoth-MRE-Run1}.
}
\label{fig:resultsBoth-AR-Run1}
\end{figure}

There
are some instabilities in our results. For example, the exemplary
performance of methods $(a,d)$
in the top-left plot of
\fig{resultsBoth-AR-Run1} does {\em not} repeat in other plots.
For example in the $NASA93$ MRE and MER
results shown in 
\fig{resultsBoth-MRE-Run1} and \fig{resultsBoth-MER-Run1}, method $b$ loses much less than methods $(a,d)$.

However, in terms of number of losses generated by methods $(a,b,c,d,e,f,g,h)$, the following
two results holds across all evaluation criteria and all subsets:
\be
\item
One member of method $(a,b,c,d)$ always performs better (loses least)
than all members of methods
$(e,f,g,h)$. Also, all members of methods $(e,f,g,h)$ perform better than $i$.
\item
Compared to 158 methods, one member of $(a,b,c,d)$
always loses at some rate very close to zero.
\ee

In our results, there is no
single universal $best$ method.
Nevertheless, out of 158 methods, there are 154 clearly inferior methods.
Hence, we recommend ranking methods $(a,b,c,d)$ on
all the available historical data, then applying the best ranked method
to estimate new projects.

\section{Discussion}
In the following discussion, note that all these conclusions should not be generalized
beyond COCOMO-style data sets.

The superiority of $(a,b,c,d)$ is a strong endorsement of Boehm's 1981 estimation research.
These four methods are based around Boehm's preferred
method for calibrating generic COCOMO models to local data. 
Method $a$ is Boehm's {\em local calibration}
(or LC) procedure (defined in the appendix).
Methods $b$ and $d$ augment LC with pre-processors performing
simple column or row pruning
(and method $c$ combines both $b$ and $d$).
Methods $(a,b,c,d)$ endorse three of Boehm's 1981 assumptions about effort estimation: 
\begin{description}
\item[{\em Boehm'81 assumption 1:}]~\newline
Effort can be modeled as a single function that is exponential on lines of code~\ldots 
\item[{\em Boehm'81 assumption 2:}]~\newline
\ldots and linearly proportional to the product of a set of effort multipliers;
\item[{\em Boehm'81 assumption 3:}]~\newline
The effort multipliers influence the effort by a set of pre-defined constants that can be taken from Boehm's textbook~\cite{boehm81}.
\end{description}

Our results endorse some of Boehm's estimation modeling work, but not all of it.
Method $e$ is manual stratification, a commonly recommended method
in the COCOMO literature. This method performs surprisingly well and often
out-performs many intricate automatic methods.
However, as shown above, method $e$ is always inferior to 
more than one of $(a,b,c,d)$. Hence, contrary to the COCOMO literature,
we recommend replacing manual stratification with automatic methods.

Our results argue that there is little added value in
methods $(f,g,h)$.
This is a useful result since these methods contain some of our slowest algorithms.
For example,
the WRAPPER column selection method used in $(f,g,h)$  
is an elaborate heuristic search through, potentially, many combinations.

The failure of model trees in method $f$ is also interesting. 
If the model trees of method $f$ had out-performed $(a,b,c,d)$, that would
have suggested that effort is a multi-parametric phenomenon where, e.g.
over some critical size of software, different effects emerge.
This proved not to be the case, endorsing Boehm's assumption
that effort can be modeled as a single parametric log-linear equation.

Of all the methods in \fig{abcd}, 
$(a,b,c,d)$ perform the best and $i$ performs the worst.
One distinguishing feature of method $i$ is the {\em assumptions}
it makes about the domain. 
The NEAREST neighbor method $i$ is {\em assumption-less}
since it makes none of the {\em Boehm'81} assumptions listed above.
But, while
assumption-less, NEAREST is not {\em assumption-free}.
NEAREST uses a simple n-dimensional Euclidean distance to find similar projects.  
Wilson \& Martinez caution that this measure is inappropriate for 
sparse data sets~\cite{wilson97a}. Such sparse data sets arise when many of the
values of project features are unavailable. 
Shepperd \& Schofield argue that their case-based reasoning methods, like NEAREST
procedure used in method $i$, are better suited to sparse data domains where precise numeric
values are {\em not} available on all factors~\cite{shepperd97}.
All our data sets are non-sparse. Hence, it is not surprising that method $i$ performs
poorly on our data.
We suggest that researchers with access to sparse effort estimation data
sets repeat this kind of study on their data.

%\section{Related Work}
%
%We report here a set of comparative rankings of different estimation methods
%that are stable across:
%\bi
%\item
% hundreds of different random samples of data 
%\item taken
%from 19 projects, 
%\item assessed using three different evaluation criteria. 
%\ei To the best of our knowledge,
%this is the largest inter-method comparison yet reported in the literature.
%It is insightful to ask the question: why have not such  stable results
%been previously
%reported? 
%
%The first answer to this question is that it is not a widespread practice
%to compare different methods. As mentioned in the introduction, there
%it is common to report single-method results.
%
%Another answer comes from the statistical problems associated from
%conducting such inter-method comparisons.
%Elsewhere~\cite{me06d} we have documented the large deviations from central tendencies  
%that can occur
%in effort estimation.
%For example, in 30 experiments discarding ten examples (selected at random)
%the observed mean MRE values were less than 100\% while the standard deviations were
%\[\{median,max\} = \{150\%,649\%\}\]
%On investigation, the source of these large deviations were found to be the rare presence of 
%alarming large errors (in one extreme case, up to 8000\%).
%Large outliers can make mean calculations
%highly misleading. A single large outlier can make the mean
%value far removed from the median\footnote{Median: 
%the value below which 50\% of the values fall.}. 
%\item
%For data sets with only a small number of outliers (e.g. \fig{re}), the conclusions reached from
%different subsets can be very different,
%depending on the absence or presence of the infrequent outliers.
%\Such 
%One troubling result from the \fig{parametric} study is that the number of training
%examples
%was {\em not} connected to the size of standard deviation. A pre-experimental intuition
%was that the smaller the training set, the worse the prediction instability.
%On the contrary, we found small and large instability (i.e. MRE standard deviation)
%for both small
%and large training sets~\cite{me06d}. That is, instability
%cannot be tamed by further data collection. Rather, the data must be processed
%and analyzed in some better fashion (e.g. U test described below).
%
%These large instabilities explain the contradictory results in the effort estimation
%literature.
%Jorgensen reviews fifteen studies that compare model-based to expert-based estimation. 
%Five of those studies found in
%favor of expert-based methods; five found no difference; 
%and five found in favor of model-based estimation~\cite{jorg04}.
%Such diverse conclusions are to be expected if models exhibit large instabilities in their performance.
%
%
% from our prior work on the the papers that have attempted 
%
%19 projects, 
%thstable conclusions 
\section{External Validity}
This study has 
several biases, listed below. 

{\em Biases in the paradigm}: The paper explores 
model-based methods (e.g. COCOMO ) and not expert-based methods. 
Model-based methods use some algorithm to summarize old data and make predictions 
about new projects.
Expert-based methods use human expertise (possibly augmented with process guidelines
or checklists) to generate predictions.  
Jorgensen~\cite{jorg04} argues that most industrial effort estimation is expert-based
and lists 12 {\em best practices} for such effort-based estimation.
The comparative evaluation of model-based
vs. expert-based methods must be left for future work. 
The evaluation of expert-based methods will be a human-intensive study
and human subjects may rebel at participating in the lengthy
experimental
procedure described below. Before
 comparing any effort estimation
methods (be they model-based or expert-based) we must prune
the space of model-based methods.
For more on expert-based methods, see~\cite{jorg04,jorgensen04,shepperd97,chulani99}.

{\em Evaluation bias:}
We will show conclusion stability 
across three criteria; absolute residual; magnitude of error
relative to estimate; or magnitude of relative error 
(these criteria are defined below).
This does not
mean that we have shown stability across {\em all possible} evaluation biases. 
Other evaluation biases
may offer different
rankings to our estimation methods. 
For example, we do not explore PRED(30)\footnote{PRED(N) is
the \% of the magnitude of relative error estimates
$\le N\%$.} since Shepperd (personal communication)
depreciates it. Several other prominent publications eschew its
use as well (\cite{foss05,myrtveit05}).

{\em Sampling bias:}
Model-based estimation use data
and so are only useful in organizations that maintain historical
data. Such data collection is rare in organizations
with low process maturity. However, it is common elsewhere; e.g. amongst government
contractors whose contract descriptions include process auditing requirements.
For example, United States
government contracts often
require a model-based estimate at each
project milestone.  Such models are used to generate estimates or
to double-check an expert-based estimate.

Another source of sampling bias is that the 19 datasets used in this study
come from two sources:
(1) Boehm's 1981 text on Software Engineering textbook~\cite{boehm81}
and (2) data collected from NASA in the 1980s and 1990s from six different
NASA centers including the Jet Propulsion Laboratory (for details
on this data, see the appendix). 
Numerous prior research has argued that 
conclusions from NASA data are relevant to the general software
engineering industry.
Basili, Zelkowitz, et.al.~\cite{basili02}, for example, published
extensively for decades their conclusions taken from NASA data.
NASA makes extensive use of contractors.
Our NASA
data comes from different teams working at geographical locations spread
through-out the United States using a variety of programming languages.
While some of our data is from flight systems (a particular NASA specialty),
most are ground systems and share many of the properties of other terrestrial
software (same operating systems, development languages, development practices).
Much of NASA's software is written by 
contractors who service a wide range of clients (not just NASA).
These contractors are contractually
obliged (ISO-9001) to demonstrate their understanding and usage of
current industrial best practices. 

Yet another source of sampling bias is that our conclusions are based
only on the 19 data sets studied here. 
The data used in this study is
largest public domain set of COCOMO-style data available. Also, our 
data source is as large
as the proprietary    COCOMO data sets  used in prior 
TSE publications~\cite{chulani99}.


{\em Biases in the model:} This study adopts
the COCOMO model for all its work. This decision
was forced on us: the
COCOMO-style data sets described in the appendix are 
the only public domain data we could access.  Also, all
our previous work was based on COCOMO data since our funding body (NASA)
makes extensive use of COCOMO. The implications of our work on other estimation
frameworks is an open and (as mentioned in the introduction) pressing issue.
We strongly urge researchers with access to non-COCOMO data to repeat the
kind of row/column pruning analysis described here.


Note that we make no claim that this study explores the entire space
of possible
effort estimation methods. 
Indeed, when we review the space of known methods
(see Figure~1 in~\cite{myrtveit05}), it is clear
that COSEEKMO covers only a small part of that total space. 
The reader
may know of other effort estimation methods they believe we should try.
Alternatively, the reader may have a design or an implementation of a new kind
of effort estimator. In either case, before it can be shown that an existing or new method
is better than the four we advocate here,
we first need a demonstration that it is possible to make stable conclusions regarding
the relative merits of different estimation methods.
This paper offers such a demonstration.

\section{Conclusion}


This paper concludes   five years of research that began with the following question;
can the new generation of data miners offer better effort estimates that traditional methods?
%Curiously, as discussed 
%in {\em Related Work}, this paper
%offers almost the opposite conclusions to a prior study
%by Shepperd \& Kadoda~\cite{sheppherd01}.
%Based on artifically generated data sets
%based on
%non-COCOCOMO-style data, those authors reported
%little or no conclusion stability regarding the relative benefits
%of different estimation methods.
%We offer below several hypothesis for our differing results.
%Shepped 
%\& Kadoda  used whatever attributes were locally available
%while we map all our data into the features endorsed by the COCOMO research.
%Also, we make extensive use of {\em full pruning}
%of both rows and columns to cull extraneous
%or ``noisey'' portions of the training tables of data.
%Case-based reasoning, on the other hand, only uses row pruning
%when it select relevant cases for analysis.
%We conjecture  that our conclusion stability
%is a result of COCOMO features+full pruning.
%
%
%{\em Contributions of This Paper:}
%The contributions of this paper are based on the above conclusions.
%Conclusion \#1 lets us discard many estimation methods. It also demands
%that researchers proposing new estimation methods need to carefully
%demonstrate that their new, supposedly more sophisticated method, is actually
%an improvement over current practice.
%
%Conclusion \#2 simplifies deploying  effort estimation methods
%at an industrial site (the search for useful estimation methods can be constrained
%to a very small set).
%
%Conclusion \#3 cautions that our four ``best'' methods cannot be pruned
%to a smaller set. Rather, when processing a new domain, all four methods should be tried.
%
%More generally, we recommend the use of COCOMO-style features (with full
%pruning) to avoid the instability problems reported by Shepperd and Kadoda
%(but this recommendation needs more study-
%see  
% our {\em Future Work} section).
%

In other work~\cite{baker07}
we detected no improvement using   bagging~\cite{brieman96} and boosting~\cite{FreSch97} methods for
COCOMO-style data sets.
In this work, we have found
that one of four methods is always better than another 154 methods:
\bi
\item
A single linear model is adequate for the purposes of effort estimation.
All the methods that assume multiple linear models, such as model trees $(f)$,
or no parametric form at all, such as nearest neighbor $(i)$, perform
relatively poorly.
\item
Elaborate searches do not add value to effort estimation.
All the $O(2^F)$ column pruners do worse than near-linear-time
column pruning. 
\item
The more intricate methods such as model trees do no better than other methods. 
\ei

The implications for none-COCOMO-style effort estimation is an open question. At the very
least, we would argue that
the above results
lead to the following methodological principle. Before releasing some new, supposedly more sophisticated
effort estimation method, it should be a requirement that the author of that new method demonstrates
that the new method offers some advantage over current practice.
This principle may to be an obvious statement for any scientific discipline. However, it
is not a commonly applied principle in effort estimation.
Apart from Shepperd \& Kadoda~\cite{sheppherd01},
there are few examples of 
studies using multiple methods or multiple
data sets. For the record, those examples
include:
\be
\item
Chulani \& Boehm's comparison of the Bayesian methods of COCOMO-II against COCOMO-I~\cite{chulani99};
\item
 Mendes' use of two estimation methods on data collected within one company or from other companies~\cite{mendes07};
\item Our prior work on COSEEKMO~\cite{me06d}.
 \ee
However, our reading of the literature is that the ranking and pruning of many methods using many data sets
and multiple evaluation criteria is not common practice. 
Note that, of the above examples, only example \#1 offered a comparative assessment of alternate
estimation methods. Example \#2 and \#3 only {\em used} alternative methods; they
did not show that there existed some stable {\em ranking} between the methods (i.e. stable across
data subsets, stable across evaluation criteria).


The conclusion of this paper is that effort estimation research should be both 
{\em complicated} and 
{\em simplified}:
Paradoxically,
these results both
{\em complicates} and {\em simplifies}
effort estimation
research:
\bi
\item
{\em Complication \#1:}
One reason for {\em not} using COCOMO is that the available data may not
be expressed in terms of the COCOMO features. Nevertheless, our results
suggest that it may be worth the effort to use COCOMO-style data collection,
it only to reduce the instability in the conclusions.
\item {\em Complication \#2:}
Simply applying one or two methods in a new domain is not
enough. In the study reported in this paper,
one method out a set of four was always best but {\em that best method was
dataset-specific}.
Therefore, prior to researchers drawing conclusions about aspects
of effort estimation
properties in a particular context, there should be a
{\em selection study} to rank and prune the 
available estimators according to the details of a local domain.
\item
{\em Simplification:}
Fortunately, our results also suggest that
such {\em selection studies}
need not be very elaborate.
At least for COCOMO-style data,
we report that $\frac{154}{158}=97\%$ of the methods implemented
in our COSEEKMO toolkit~\cite{me06d}
added little or nothing
to Boehm's 1981 regression procedure~\cite{boehm81}.
\ei

Such a selection study could proceed as follows.
For COCOMO-style data sets, the following methods should be tried
and the one that does best on historic data (assessed using Mann-Whitney U test)
should be used to predict new projects:
\bi
\item Adopt the three Boehm'81 assumptions and use LC-based methods.
\item While some row and column pruning can be useful, elaborate column pruning (requiring
an $O(2^F)$ search) is not. 
Hence, try LC with zero or more of LOCOMO's row pruning or COCOMIN's column pruning.
\ei

For future work, we recommend an investigation
of an ambiguity in our results:
\bi
\item
Prior experiments reported conclusion
instability after limited application
of row and column pruning to non-COCOMO features.
\item
Here, we have report conclusion stability after
extensive row and column pruning to COCOMO-style
features.
\ei
It is hence unclear what removed the conclusion
instability. Was it pruning? Or the use of the
COCOMO features? To test this, we require a
new kind of data set. Given examples expressed
in whatever local features are  available,
those examples should be augmented with COCOMO
features. Then, this study should be repeated:
\bi
\item With and without the local features;
\item With and without the COCOMO features;
\item With and without pruning;
\ei
We would be interested in contacting any industrial
group with access to this new kind of data set.

\section*{Acknowledgments}
Martin Shepperd was kind enough to make suggestions about
different evaluation biases and the design of the
NEAREST and LOCOMO methods.

\appendix
\subsection{Data Used in This Study} 

In this study, effort estimators were built using all or some {\em part} of data from two sources:
\bdd
\item[{\em $COC81$:}]~~63 records in the COCOMO-I format. Source: \cite[p496-497]{boehm81}. 
Download from \url{http://unbox.org/wisp/trunk/cocomo/data/coc81modeTypeLangType.csv}.
\item[{\em $NASA93$:}]~~~93 NASA records in the COCOMO-I format.  
Download from \url{http://unbox.org/wisp/trunk/cocomo/data/nasa93.csv}.
\edd

Taken together, these two sets are the largest
COCOMO-style data source in the public domain (for reasons
of corporate confidentiality, access to Boehm's COCOMO-II data set is
highly restricted).
$NASA93$ was originally collected to create a NASA-tuned version of
COCOMO, funded by the Space Station Freedom Program and 
contains data from six NASA centers including the Jet Propulsion
Laboratory. For more details on this dataset, see~\cite{me06d}.

Different subsets and number of subsets used (in parenthesis) are:
\bdd
\item[{\em All(2):}]~selects all records from a particular source.
\item[{\em Category(2):}]~~~~~~~$NASA93$ designation selecting the type of project; e.g. avionics.
\item[{\em Center(2):}]~~~~~$NASA93$ designation selecting records relating to where the software was built.
\item[{\em Fg(1):}]~$NASA93$ designation selecting either ``$f$'' (flight) or ``$g$'' (ground) software.
\item[{\em Kind(2):}]~~$COC81$ designation selecting records relating to the development platform; e.g. max is mainframe.
\item[{\em Lang(2):}]~~~$COC81$ designation selecting records about different development languages; e.g ftn is FORTRAN.
\item[{\em Mode(4):}]~~~designation selecting records relating to the COCOMO-I development mode: one of semi-detached, embedded, and organic.
\item[{\em Project(2):}]~~~~~$NASA93$ designation selecting records relating to the name of the project.
\item[{\em Year(2):}]~~is a $NASA93$ term that selects the development years, grouped into units of five; e.g. 1970, 1971, 1972, 1973, 1974 are labeled ``1970''.
\edd
There are more than 19 subsets overall. Some have fewer than 20 projects and 
hence were not used. The justification for using 20 projects or more is offered in~\cite{me06d}.

\subsection{Learners Used in This Study}

\subsubsection{Learning with Model Trees}

Model trees are a generalization of linear regression.
Instead of fitting the data to {\em one linear model}, 
model trees learn {\em multiple linear models}, and a
decision tree that decides which linear model to use.
Model trees are useful when the projects form regions and
different models are appropriate for different regions.
COSEEKMO includes the M5p model tree learner defined by Quinlan~\cite{quinlan92b}.
\subsubsection{Other Learning Methods}

See the {\em Related work} section
for notes on
learning
with linear regression;
local calibration; and nearest neighbor methods.




\subsection{Pre-Processors Used in This Study}

\subsubsection{Pre-processing with Row Pruning}

The LOCOMO tool~\cite{jalali07} in COSEEKMO
is a row pruner that combines a nearest neighbor method
with LC. LOCOMO prunes away all projects except those $k$ ``nearest'' to the $Test$ set data.

To learn an appropriate value for $k$, LOCOMO uses the $Train$ set as follows:
\bi
\item For each project $p_0\in Train$, LOCOMO sorts the remaining 
$Train - p_0$ examples by their Euclidean distance from $p_0$. 
\item LOCOMO then passes the $k_0$ examples closest to $p_0$
to LC. The returned $<a,b>$ values are used to estimate effort for $p_0$.
\item After trying all possible $k_0$ values, $2 \le k_0 \le |Train|$, 
$k$ is then set to the $k_0$ value that yielded the smallest mean MRE\footnote{A
justifications for using the mean measure within LOCOMO is offered at the end of the appendix.}.
\ei
This calculated value $k$ is used to estimate the effort for projects in the
$Test$ set. For all $p_1\in Test$, the $k$ nearest neighbors
from $Train$ are passed to LC. The returned $<a,b>$ values are then 
used to estimate the effort for $p_1$. 

\subsubsection{Pre-Processing with Column Pruning}

Kirsopp \& Schofeld~\cite{kirsopp02} 
and Chen \& Menzies \& Port \& Boehm~\cite{me05c}
report that column pruning improves effort estimation.
Miller's research~\cite{miller02} explains why.
Column pruning (a.k.a. feature subset selection~\cite{hall03} or variable
subset selection~\cite{miller02}) reduces the deviation of a
linear model learned by minimizing least squares error~\cite{miller02}.
To see this, consider a linear model with constants $\beta_i$ 
that inputs features $f_i$ to predict for $y$:
\[y = \beta_0 + \beta_1\cdot f_1 + \beta_2\cdot f_2 + \beta_3\cdot f_3 ...\]
The variance of $y$ is some function of the variances in $f_1, f_2$, etc.
If the set $F$ contains noise 
then random variations in $f_i$ can increase the uncertainty of $y$.
Column pruning methods decrease the number of features $f_i$, thus
increasing the stability of the $y$ predictions. 
That is, the fewer the features (columns), the more restrained are the model predictions.

Taken to an extreme, column pruning can reduce $y$'s variance 
to zero (e.g. by pruning the above equation back to $y=\beta_0$)
but increases model error (the equation $y=\beta_0$ will ignore all project data when generating estimates).
Hence, intelligent column pruners experiment with some proposed subsets $F' \subseteq F$ before
changing that set. COSEEKMO currently contains three intelligent column pruners: WRAPPER, LOCALW, and COCOMIN.

WRAPPER~\cite{kohavi97} is a standard best-first search through the space of possible features. 
At worst, the WRAPPER must search an space exponential on the number of features $F$; i.e. $2^F$.
However, a simple best-first heuristic makes WRAPPER practical for effort estimation.
At each step of the search, all the current subsets are scored by passing them to
a {\em target leaner}. If a set of features does not score better than a smaller subset,
then it gets one ``mark'' against it. If a set has more than $STALE=5$ number of marks, it is deleted.
Otherwise, a feature is added to each current set and the algorithm continues. 

In general, a WRAPPER can use any target learner. Chen's LOCALW is a WRAPPER specialized
for LC. Previously~\cite{me06d,me05c}, we have explored LOCALW for effort estimation.

Theoretically, WRAPPER (and LOCALW)'s exponential time search is more
thorough, hence more useful, than simpler methods that try fewer options. To 
test that theory, we will compare WRAPPER and LOCALW to a linear-time column pruner 
called COCOMIN~\cite{baker07}.

COCOMIN is defined by the following operators:
\[\{sorter, order, learner, scorer\}\]
The algorithm runs in linear time over a {\em sorted} set of features, $F$.
This search can be {\em order}ed in one of two ways:
\bi
\item A ``backward elimination'' process starts with all features $F$ and throws some away, one at a time.
\item A ``forward selection'' process starts with one feature and adds in the rest, one at a time.
\ei
Regardless of the search order, at some point the current set of features $F' \subseteq F$ 
is passed to a {\em learner} to generate a performance {\em score} by applying 
the model learned on the current features to the $Train$ set. 
COCOMIN returns the features associated with the highest score.

COCOMIN pre-sorts the features on some heuristic criteria. Some of
these criteria, such as standard deviation or entropy, are gathered
without evaluation of the target learner. Others are gathered by
evaluating the performance of the learner using only the feature in
question plus any required features, such as KLOC for COCOMO, to
calibrate the model. After the features are ordered, each feature is
considered for backward elimination, or forward selection if chosen,
in a single linear pass through the feature space, $F$. The decision to
keep or discard the feature is based on an evaluation measure
generated by calibrating and evaluating the model with the training data.

Based on~\cite{baker07}, the version of COCOMIN used in this study:
\bi
\item sorted the features by the highest median MRE;
\item used a backward elimination search strategy;
\item learned using LC;
\item scored using mean MRE.
\ei
Note that mean MRE is used 
internally to COCOMIN (and LOCOMO, see above) since it is fast and simple to compute. Once the search  
terminates, this paper strongly recommends the more thorough (and hence more intricate 
and slower) median non-parametric measures to assess the learned effort estimation model.

\bibliographystyle{IEEEtran}
\bibliography{refs}
\end{document}
\subsection{Symptoms of Instability}

KFM
caution that,
historically, ranking estimation methods has been done quite poorly. 
Based on an analysis of two (non-COCOMO)
data sets as well as simulations over artificially generated data set,
Foss et.al. and Myrtveit et.al.
concluded that numerous commonly used methods such as the mean
MRE\footnote{MRE = magnitude of relative error $=abs(actual-predicted)/actual$.} are
unreliable measures of estimation effectiveness.
Also, the conclusions reached from these standard measures can vary wildly
depending on which subset of the data is being used for testing~\cite{myrtveit05}.


\begin{figure}
\begin{center}
\includegraphics[width=3.5in]{Parametric.pdf}
\end{center}
\caption{Results of 2 different runs of COSEEKMO comparing
two methods using mean MRE values. Points on the
Y-axis show the difference in mean relative error (MRE) between method1 and method2.
Lower values endorse method2 since, when such values occur,
method2 has a lower error than method1.}
\label{fig:parametric}
\end{figure}

\fig{parametric} demonstrates conclusion instability.
It shows two experimental runs. In each run,
30 times, effort estimate models were built for our 19 subsets using
two methods. Each time, an effort model was built from a randomly selected 90\% of the data.
Results are expressed in terms of the difference in mean MRE between the
two subsets; e.g. in Run~\#1, method1 had a much larger mean MRE than method2.

After Run~\#1, the results endorse method2 since that method either (a)~did better (lower errors)
as method1 or (b)~had similar performance to method1.
However, that conclusion is not stable. Observe in Run~\#2 that:
\bi
\item
The improvements of method2 over method1 disappeared 
in subsets 1,2,3,7, and 11.
\item
Worse, in subsets 1,2, and 11 method1 performed
dramatically better than method2.
\ei
The deviations seen in 30 repeats of the above procedure
were quite large: within each data set, the standard deviation
on the MREs were $\{median,max\} = \{150\%,649\%\}$~\cite{me06d}.
Port (personal communication) has proposed a bootstrapping method to determine the true performance distributions
of COSEEKMO's methods. That method would require $10^2$ to $10^3$ re-samples and, given
COSEEKMO's current runtimes, it would take $10^2$ to $10^3$ days to terminate.

One troubling result from the \fig{parametric} study is that the number of training
examples
was {\em not} connected to the size of standard deviation. A pre-experimental intuition
was that the smaller the training set, the worse the prediction instability.
On the contrary, we found small and large instability (i.e. MRE standard deviation)
for both small
and large training sets~\cite{me06d}. That is, instability
cannot be tamed by further data collection. Rather, the data must be processed
and analyzed in some better fashion (e.g. U test described below).

These large instabilities explain the contradictory results in the effort estimation
literature.
Jorgensen reviews fifteen studies that compare model-based to expert-based estimation. 
Five of those studies found in
favor of expert-based methods; five found no difference; 
and five found in favor of model-based estimation~\cite{jorg04}.
Such diverse conclusions are to be expected if models exhibit large instabilities in their performance.

\begin{figure}[!t]
\begin{center}
\includegraphics[width=2.5in]{graphs/re_nasa93_fg_g.pdf}
\includegraphics[width=2.5in]{graphs/re_nasa93_year_1975.pdf}
\includegraphics[width=2.5in]{graphs/re_nasa93_mode_embedded.pdf}
\includegraphics[width=2.5in]{graphs/re_coc81_langftn.pdf}
\end{center}
\caption{Relative errors seen in COSEEKMO's
experiments on four data sets. Thin lines show the actual
values. Thick lines show a Gaussian distribution that uses
mean and standard deviation of the actual values.
From top to bottom, the plots are of:
{\em (top)} NASA ground systems;
NASA software written around 1975;
NASA embedded software (i.e. software developed within tight hardware, software, and operational constraints);  
{\em (bottom)} some FORTRAN-based software systems.
}\label{fig:re}
\end{figure}

\subsection{Diagnosing the Cause}

The thin line of \fig{re} is drawn by sorting the 
relative error\footnote{RE = $(predicted - actual)/actual$} (RE)
seen in four of the subsets studied in \fig{parametric}.
Observe that while most of the actual RE values are nearly zero, an {\em infrequent}
number
(on the right hand side)
are {\em extremely large} (up to 8000 in the second plot). 
Such large {\em spikes} in RE result when
the predicted values are 
much larger than the actual values and result from
(1)~noise in the data or (2)~a training set that learns an overly steep
exponential function for the effort model. 

The size of the spikes in \fig{re} are remarkable. Research papers typically report RE values in the range 
$0\le RE\le3$\cite{me06d}.   
Such values are completely 
dwarfed by errors in the range of 8000 such as those seen in the second plot of \fig{re} (hence, most
of the thin lines in \fig{re} are flat). 
These very large, but
infrequent, outliers explain conclusion instability:
\bi
\item
Large outliers can make mean calculations
highly misleading. A single large outlier can make the mean
value far removed from the median\footnote{Median: 
the value below which 50\% of the values fall.}. 
\item
For data sets with only a small number of outliers (e.g. \fig{re}), the conclusions reached from
different subsets can be very different,
depending on the absence or presence of the infrequent outliers.
\ei

\fig{re} also illustrates how poorly standard methods assess the performance of effort estimation data.
Demsar~\cite{demsar06} offers a definition of {\em standard methods} in data mining.
In his study of four years of proceedings from the {\em International Conference
on Machine Learning}, Demsar
found that the standard
method of comparative assessment 
were t-tests over some form of repeated sub-sampling
such as cross-validation, separate subsets, or randomized re-sampling. 
Such t-tests assume that the distributions being studied are Gaussian and, as shown by the
thick line of \fig{re}, effort estimation results can be highly
non-Gaussian.
These thick lines show a Gaussian cumulative 
distribution function computed from the means and standard deviations of 
the actual RE values (the thin lines).
For example, the Gaussian approximation to the {\em actual values} of
\[\{1.1, 1.3, 1.5, 1.7, 2.1, 2.3, 2.7, 800\}\]
has a mean of 101.6 and a standard deviation of 282.2.
Observe how poorly such Gaussian distributions represent the actual  RE values:
\bi
\item
There exists orders of magnitude differences 
between the {\em actual} plots (the thin lines) and the {\em Gaussian} approximations
(the thick lines). 
\item
The Gaussian goes negative while none of our effort estimation methods assume that
it takes less than no time to build software.
\ei
\subsection{Fixing Instability}

The problem of comparatively assessing $L$ learners run on multiple sub-samples of $D$ data sets
has been extensively studied in the data mining community. 
T-tests that assume Gaussian distributions are strongly deprecated.
For example, Demsar~\cite{demsar06} argues that non-Gaussian populations are 
common enough to require a methodological change in data mining. 

After reviewing a wide range of
comparisons methods\footnote{Paired t-tests with and without the use of geometric means of the relative ratios;
binomial tests with the Bonferroni correction;
paired t-tests; ANOVA; Wilcoxon; Friedman},
Demsar advocates the use of the 1945 Wilcoxon~\cite{wilcoxon45} 
signed-rank test that compares the ranks for the positive and negative differences (ignoring the signs). 
Writing five years earlier,
Kitchenham et.al.~\cite{kitc01} comment that the Wilcoxon test has its limitations.
Demsar's report offers the same conclusion, 
noting that the Wilcoxon test requires that the sample sizes are the same.
To fix this problem, Demsar augments Wilcoxon with the Friedman test.

One test not studied by Demsar is Mann and Whitney's 1947
modification~\cite{mann47} to Wilcoxon rank-sum test (proposed along with his signed-rank test).
We prefer this test since:
\bi
\item
The Mann-Whitney U test does not require that the sample sizes are the same.
So, in a single U test, learner $L_1$ can be compared to all its rivals. 
\item
The U test does not require any post-processing
(such as the Friedman test) to conclude if the median rank of one population (say, the $L_1$ results)
is greater than, equal to, or less than the median rank of another (say, the $L_2,L_3,..,L_x$ results).
\ei

We used Mann-Whitney to compare a set of methods, computing the
$wins$, $ties$, and $losses$ for each.
Since we seek methods that can be rejected, the value of interest to us is
how often methods {\em lose}.
\begin{figure}
\begin{tabular}{|p{.95\linewidth}|}\hline
\footnotesize
The sum and median of A's ranks is
\[\begin{array}{ccl}
sum_A &=& 1 + 2.5 + 5.5 + 7 + 9.5 = 25.5\\
median_A& =&5.5
\end{array}\]
and the sum and median of B's ranks is
\[\begin{array}{ccl}
sum_B&=& 2.5 + 4 + 5.5 + 8 + 9.5 + 11 = 40.5\\
median_B&=& 6.75
\end{array}\]
The $U$ statistic is calculated from 
\mbox{$U_x=sum_x-(N_1(N_2+1))/2$}:
\[\begin{array}{c}
U_A = 25.5 - 5*6/2 = 10.5\\
U_B = 40.5 - 6*7/2 = 19.5
\end{array}\]
These can be converted to a Z-curve using:
\[
\begin{array}{rclcl}
\mu&= &(N_1N_2)/2&=&516.4\\
\sigma & = & \sqrt{\frac{N_1N_2(N_1 + N_2 + 1)}{12}}&=&5.477\\
Z_A & = & (U_A - \mu)/\sigma&=&-0.82\\
Z_B & = & (U_B - \mu)/\sigma&=&0.82\\
\end{array}
\]
(Note that $Z_A$ and $Z_B$ have the same absolute value.
In all case, these will be the same, with opposite signs.)

If $abs(Z) < 1.96$ then the samples $A$ and $B$ have the same median rankings (at
the 95\% significance level). In this case, we add one to both $ties_A$ \& $ties_B$.
Otherwise, their median values can be
compared, using some domain knowledge. In this work, {\em lower} values
are better since we are comparing errors. Hence:
\bi
\item
If $median_A < median_B$ add 1 to both $wins_A$ \& $losses_B$.
\item
Else if $median_A > median_B$ add 1 to both $losses_A$ \& $wins_B$.
\item
Else, add 1 to both $ties_A$ \& $ties_B$. 
\ei
\\\hline
\end{tabular}
\caption{An example of the Mann-Whitney U test.}\label{fig:mw}
\end{figure}

end{document}
