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\title{PREFMAP-3 User's Guide\footnotetext{The PREFMAP-3 project was started when the two
first authors were staying at Bell Labs in 1982. The authors would
like to thank Joseph Kruskal for his comments and advice regarding
the output of the computer program. Thanks are also due to Suzanne
Winsberg and Sandra Pruzansky for their comments on an earlier version
of the manuscript.}}
\author{Jacqueline Meulman$^*$ \and Willem J. Heiser\thanks{Department of
Data Theory, University of Leiden, The Netherlands} \and J. Douglas
Carroll\thanks{Bell Telephone Laboratories, Murray Hill, New Jersey
07974, U.S.A.}}
\date{1986}
\maketitle
\pagenumbering{roman}
\tableofcontents
\pagenumbering{arabic}
\chapter*{Introduction}
%
Preference mapping (PREFMAP) is a basic statistical technique for the
behavioral and social sciences, particularly psychology and
marketing, and has wide potential for application in other areas as
well (e.g., speech research, linguistics or psychoacoustics). Briefly
speaking, its purpose is to relate preference information on a number
of objects to a pre-existing spatial configuration of points, under
the assumption of a simple class of models. Here ``preference'' is
used as a generic name of any type of observations that indicate a
conditional dominance relation among the objects; thus
``properties'', ``attributes'', ``Q-sorts'', or -- in still another
language -- ``variables'' defined on ``cases'' can also be modeled
with preference mapping. What makes PREFMAP clearly distinct from a
straightforward correlation approach is the possibility to go beyond
monotonically increasing preference (or response-) functions, and to
study various forms of {\it single-peakedness}. This concept has a long
and complicated history in the behavioral sciences, some of which is
touched upon in Coombs and Avrunin (1977), and Heiser (1981). The
first regular description of the PREFMAP methodology as a hierarchy
of models and techniques was given by Carroll (1972). Further
useful references are Carroll (1980), Heiser and De Leeuw (1981),
and Coxon (1982).
This manual provides a comprehensive account of the PREFMAP models
and techniques in connection with the PREFMAP-3 program. PREFMAP-3 is
the successor to PREFMAP and PREFMAP-2 (Chang and Carroll, 1972),
preserving most of their features, but completely redesigned and
reprogrammed in order to obtain a greater flexibility, portability
and capacity. An outline of the differences is included in Appendix
\ref{chap:appendixA}, which also gives technical details and
some instructions for getting the program running properly. Nowhere,
is familiarity with the previous versions assumed.
The manual is organized into two parts. Part \ref{part_one} contains a
general introduction to the PREFMAP hierarchy of models and the
associated hierarchy of regression equations for fitting the models
to data. It also contains a section on preliminary transformations of
the spatial configuration to improve fit, and a short introduction to
the way in which nonmetric analyses are performed. Part
\ref{part_two} provides practical guidance in the use of the PREFMAP-3
program. The so-called standard input stream is described, and the
various mechanisms available to control the behavior of the program.
There is also an explanation of the layout of the output, and of the
trouble reports and program-specific warnings that may be issued
during execution. Part \ref{part_two} concludes with some
applications, merely giving a glimpse of the possibilities.
Finally, Appendix \ref {chap:appendixB} is a detailed statement of the
input records and their organization, attached there for ease of
references.
We welcome feedback from the users of PREFMAP-3, such as applications
of special interest, and novel but useful ways of using the program.
We also welcome information about any errors which may remains in the
program, especially if accompanied by sufficient information to
permit tracking them down.
\part{PREFMAP models and techniques}
\label{part_one}
\chapter{Basic data and objectives}
\label{chap:chap1}
%
The basic data for a PREFMAP analysis consists of two parts. The first
art is called the {\it external data matrix}, and can be depicted as
follows.
\begin{figure}[ht]
\centerline{$
\begin{array}{cccccc}
& 1 & \ldots & j & \ldots & m \\ \cline{2-6}
\multicolumn{1}{c|}{1} & & & & &\multicolumn{1}{c|}{}\\
\multicolumn{1}{c|}{\vdots}& & & & &\multicolumn{1}{c|}{}\\
\multicolumn{1}{c|}{i} & &&\delta_{ij}& &\multicolumn{1}{c|}{}\\
\multicolumn{1}{c|}{\vdots}& & & & &\multicolumn{1}{c|}{}\\
\multicolumn{1}{c|}{n} & & & & &\multicolumn{1}{c|}{}\\
\cline{2-6}
\end{array}$}
\caption{The external data matrix}
\label{fig:data}
\end{figure}
\noindent
The external data matrix contains entries $\delta_{ij}$ that are
interpreted as measures of the {\it dissimilarity} between row-object
$i$ and column-object $j$. In a psychological context the row-objects
are often called subjects, and the column objects stimuli. The
dissimilarities could be derived from preference judgments, in which
case they could be interpreted as measures of relative distance from
the subjects' ``ideal stimulus points'' in the space (a different
``ideal point'' being assumed for each subject, which may, in the case
of a model called the ``vector model'', be infinitely distant from the
actual stimuli, so that only a direction -- indicated by a subject
``vector'' -- is defined). The analysis of this type of data,
representing both the row- and the column-objects in a $p$-dimensional
space such that the Euclidean distances $d_{ij}$ between row-point $i$
and column-point $j$ resemble as closely as possible the
dissimilarities $\delta_{ij}$, has become known as {\it
Multidimensional Unfolding}. The type of analysis that is performed
by the PREFMAP-3 program is sometimes called {\it External Unfolding}.
Contrary to Internal Unfolding, where both the row-points and the
column-points must be solved for, External Unfolding finds a
representation for each row-object conditionally upon {\it given}
points for the column- objects.
This situation defines the second part of the basic input as the
target configuration (see Figure \ref{fig:target}).
\begin{figure}[ht]
\centerline{$
\begin{array}{lcccccc}
& & 1 & \ldots & s & \ldots & p \\ \cline{3-7}
& \multicolumn{1}{c|}{1} & & & & &\multicolumn{1}{c|}{}\\
& \multicolumn{1}{c|}{\vdots}& & & & &\multicolumn{1}{c|}{}\\
\mbox{object-}
& \multicolumn{1}{c|}{j} & & &y_{js}& &\multicolumn{1}{c|}{}\\
\mbox{points}
& \multicolumn{1}{c|}{\vdots}& & & & &\multicolumn{1}{c|}{}\\
& \multicolumn{1}{c|}{m} & & & & &\multicolumn{1}{c|}{}\\
\cline{3-7}
& & & \multicolumn{3}{c}{\mbox{dimensions}} & \\
\end{array}$}
\caption{The external data matrix}
\label{fig:target}
\end{figure}
The rows of the target configuration must correspond to the columns of
the data matrix. The entries $y_{js}$ are the coordinate values of
the $m$ points in $p$-dimensional space. These values may be obtained
from any multidimensional data analysis technique, or be derived from
some predefined structure. For historical reasons, the technique that
relates the external data to the given configuration is called
PREFerence Mapping. One should realize, however, that the entries of
the data matrix can consist of any score, rating or ranking by the
rows (subjects, scales, variables, properties) to the columns
(stimuli, objects, cases). From now on, we shall adhere to the
convention to denote the row-entities as {\it individuals} and the
column-entities as {\it objects}, and to assume that the scoring
direction is: ``small value $\leftrightarrow$ high preference" and
``large values $\leftrightarrow$ low preference" (hence that
$\delta_{ij}$ are sometimes mnemonically called dispreferences). If
the original data are preferences (or other {\it similarity} measures)
the values should be reversed in order to make them dispreferences (or
{\it dissimilarities}) say by subtracting them all from some large
number, or simply reversing their signs. (At present, this step must
be taken {\it before} input to PREFMAP-3.)
The individuals can be represented in the target configuration for the
objects in two different ways: either as a {\it vector} (which can be
thought of as indicating the direction of an infinitely distant ideal
point, as discussed earlier), or as an {\it ideal point}. In the
former representation, preference increases monotonically along one
single direction in space (indicated by the vector). In the simplest
case of the latter type of representation there is one single point of
maximum preference (called the ideal point), and preference decreases
along all directions in space as a function of the distance from the
ideal point. Thus in both cases the preference function is
single-peaked, provided we imagine the vector representation as a
"ridge of peaks" at infinity. As we shall see shortly there are
variations on this basic scheme that allow us to study single-dipped
functions, when there exists a point of minimum preference, or
mixtures of single-peakedness single-dippedness as well. The whole
range of models forms a nested sequence, or a hierarchy, and will be
discussed in chapter \ref{chap:chap2} in order of complexity.
For each member of the hierarchy, a specific set of regression
equations can be used to obtain least squares estimates of the model
parameters. A detailed derivation is given in chapter
\ref{chap:chap3}. For one member of the hierarchy - the simple
Unfolding or ideal point model - it can be useful to transform the
target configuration before the mapping process starts; these
preliminary transformations are fully explained in chapter
\ref{chap:chap4}. Regardless of the chosen model, the analysis can be
done metrically or nonmetrically (chapter \ref{chap:chap5}). In the
former case the relation between the values predicted by the
preference function and the data is assumed to be linear, in the
latter case merely the order of the data values is taken into account.
This option again enlarges the class of possible preference functions
a great deal (in anticipation we already freely used expressions like
``monotonically increasing preference", it being understood that in a
metric analysis monotonicity is constrained to be linear).
Due to the conditional nature of the preference mapping (i.e., a fixed
target configuration), each individual can be modeled independently
from the others. There are definite advantages to considering groups
of individuals simultaneously (therefore Figure \ref{fig:data} is a
table, not a single row of dissimilarities). But this circumstance
does not necessarily urge us to choose the same model for each
individual.
\chapter{The PREFMAP hierarchy of models}
\label{chap:chap2}
\section{The vector model}
The vector model is the simplest in the hierarchy. The individual is
represented in the $p$-dimensional configuration by a vector. The
direction of the vector is indicative of increasing preference. More
in particular, the dissimilarity values are approximated by the
perpendicular projection of the object points onto the individual
vector (where, in PREFMAP-3, the signs are adjusted to let a high
projection value correspond with a low dissimilarity value, i.e., by
convention, a high preference).
Although the representation of a row of the data matrix is very
similar to the results for, say a variable in Principal Components
analysis, we cannot conclude anything from a correlation point of view
without restraint. Only when the target configuration is orthonormal,
i.e. the coordinate values have sum of squares one and are
uncorrelated, the angles between vectors and the axes of the
configuration can be interpreted as correlations. When this condition
is not satisfied the vector must simply be interpreted as the best
{\it direction} of overall increasing preference.
\section{The Unfolding model}
%
This model is also called the ideal point model, or the simple
Euclidean model. The individual is represented as a point, located at
a position of an imaginary object point that would receive maximum
preference value (called the ideal point). The dissimilarities are
approximated by the squared Euclidean distances between the ideal
point and the object points. Here it is quite natural that a small
dissimilarity is represented by a small distance: the ideal point will
be close to the objects the individual prefers most. This explains
the dissimilarity/dispreference scoring convention (the Unfolding
model is, in many ways, most central in preference mapping). The name
Unfolding originates from the following metaphor. If we imagine - in
one or two dimensions - the object points as spots on a napkin, next
pick it up at the ideal point position and {\it fold} it, then the spots
will appear on the the folded napkin in order of preference. The data
analysis takes us in the reverse direction, hence the name Unfolding.
\section{The weighted Unfolding model}
%
The weighted Unfolding or weighted ideal point model also depicts the
individual as a point, but now the model allows any individual to
(re)weight the dimensions in his/her own way. The weights can be
thought of as importances of dimensions for a specific individual.
Accordingly, the dissimilarities are approximated by weighted squared
distances, with a distinct pattern of weights for every individual.
In terms of a preference function, more in particular a preference
surface, the weighted Unfolding model still implies a single peak at
the location of the ideal point, but now preference decreases more
rapidly in the direction of a more heavily weighted axis (and less
rapidly along a less heavily weighted one). The contours of equal
preference are ellipse (or ellipsoid) centered at the ideal point,
with axes parallel to the coordinate axes, and lengths of axes
inversely related to the weights. The simple Unfolding model is of
course the special case of equal weights, so that the ellipses
(ellipsoids) become circles (spheres) centered at the ideal point.
\section{The general Unfolding model}
%
Distances between an ideal point and the object points are invariant
under orthogonal rotation of the complete configuration. They do
become different, however, when we first allow an optimal rotation and
then apply weights to this new set of reference axes (and, if desired,
rotate back). This idea constitutes the final ideal point model,
sometimes called the general Euclidean model. An individual is
allowed to rotate the target configuration before weighting the new
(rotated) axes. Thus a group of individuals all modeled this way can
show differences in three respects: ideal point location, axis
orientation, and importance of the reoriented axes.
It will be clear that this generalization implies that the contours of
equal preference still are ellipses (in more than two dimensions:
ellipsoids) centered at the ideal point. For the general Unfolding
model is equivalent to the weighted Unfolding model {\it after} the
individual reorientation has been effected. Then, keeping the
position of the target configurations constant, ellipses of different
individuals will all have different orientations.
\section{Weights and their signs}
%
The parameters of the vector model are $p$ numbers fixing the
direction of the individual vector with respect to the coordinate axes
of the target configuration. If normalized (sum of squares equal to
1), these $p$ numbers are called direction cosines. In another
context, they are sometimes called ``weights'', but we have to be
careful in PREFMAP not to confuse them with the weights of the
weighted Unfolding model. It is not true, for instance, that the
vector model is a special case of the weighted Unfolding model in
which the ideal point is located in the origin. The unfolding model
``weights" apply to coordinate-wise squared differences (between the
ideal point and an object point); the vector model ``weights" weight
contributions to a linear sum of coordinate values (for each object
point). If it is not in terms of ``weights", in what sense, then, is
the vector model a special case of Unfolding model? When we move the
ideal point outwards along a fixed direction, say $v$, then the
contours of equal preference become circles (or spheres in higher
dimensions) with larger and larger radii. If moved far enough, the
circle (sphere) segments in the neighborhood of the object points are
approximately straight lines (planes or hyperplanes) perpendicular to
$v$, and this makes them indistinguishable from lines (hyperplanes) of
equal preference for a vector model with vector $v$.
Actually, the three ideal point models all accommodate weights. While
in the weighted and the general Unfolding model a {\it weight pattern}
is obtained, in the simple Unfolding model the {\it magnitude} of the
weights is equal to {\it one} for each dimension. Now, there is
nothing in the method that fits the models to the external data that
ensures the weights to be positive (see chapter \ref{chap:chap3}). In
the case of the simple ideal point model the weights for each
dimension can become minus one. We cannot interpret the individual
point any longer as an ideal point, but have to appreciate it as an
{\it anti-ideal point}: the (squared) distance between an individual
point and an object point will be small when the dissimilarity is
large. Preferences are represented in a way that shows how much an
individual {\it dislikes} an object. For the more complicated ideal
point models, the weight pattern can also turn out to be more
complicated. The same anti-ideal point interpretation holds when the
weights for each dimension are negative. When they are positive for
some dimensions and negative for others, the individual point is
called a {\it saddle point}. Along the dimensions with positive
weights distance decreases when preference increases, and for the
dimensions with negative weights it is just the other way around.
Finally a situation must be anticipated that complicates the
interpretation of the individual point for the simple ideal point.
This might come out when a preliminary transformation of the target
configuration is performed (see chapter \ref{chap:chap4}, for a
description and a mathematical account). When a preliminary
transformation is allowed for, we obtain equal weights with a {\it
sign pattern} for the ideal point model instead of a single weight
with an equal sign for each dimension. In that case we have to
interpret the individual point in the same way as we would do for the
weighted and the general model; i.e., we have to decide by inspecting
the signs of the weights whether the individual point is a saddle
point instead of simply an ideal or an anti-ideal point. In all cases
with mixed sign combinations the preference surface is no longer
single-peaked or single-dipped; this may occasionally give problems in
interpretation, but there seem to be no compelling general reasons for
excluding these possibilities from the hierarchy altogether. Another
target configuration might be tried, which is often a good idea
anyhow.
\chapter{The PREFMAP hierarchy of regression equations}
\label{chap:chap3}
%
Both the vector model and the Unfolding models can be expressed as a
set of equations that are linear in the unknowns. These equations can
be approximately solved by ordinary least squares regression if we
perform appropriate changes of variables and reparametrizations.
Carroll (1972) is the major reference here, see also Carroll (1980)
and Heiser and De Leeuw (1981). It will be shown in the next
paragraphs what operations have to be performed, starting again with
the vector model. In all models the data are assumed to be at the
interval level, so that the obtained model values must remain
invariant under linear transformations of the data (two parameters,
$a_i$ and $b_i$, are introduced to take care of this). The symbol
$\simeq$ is used to indicate least squares approximation, and $R_i$
denotes the multiple correlation coefficient. In all formulas the
index $i$ is retained to refer to individual $i$, although strictly
speaking it is superfluous (different individuals can be modeled
differently and independently). In particular, the introduction of
some model parameter, e.g. $w_{is}$, does not imply that it has to be
there for all $i=1, \ldots, n$. Whenever two parameters are
confounded, the identification conditions selected in PREFMAP-3 are
explicitly stated.
\section{Equations for the vector model}
%
In the vector model dissimilarities $\delta_{ij}$ are assumed to be
approximated as
\begin{eqnarray}
\delta_{ij} \simeq a_i \sum_{s=1}^p x_{is}y_{js} + b_i.
\label{for:delta_approx}
\end{eqnarray}
Here $a_i$ is the slope parameter, $b_i$ is the intercept term,
$x_{is}$ is the vector coordinate in dimension $s$, and $y_{js}$ is
the target point coordinate; $\delta_{ij}$ and $y_{js}$ are known;
$a_i, b_i$ and $x_{is}$ are unknown. Now we re-express the system
using the following convention: the index $q$ of the system, and $s$ or
other indices of the old system are coupled in order of enumeration.
\begin{eqnarray*}
{\tabcolsep2pt \begin{array}{r@{=}lr@{=}lr@{=}l}
\multicolumn{2}{c}{\mbox{\underline{Change of variables}}} &
\multicolumn{2}{c}{\mbox{\underline{Reparametrization}}} &
\multicolumn{2}{c}{\mbox{\underline{Index ranges and definition}}} \\
z_{j0}&1 & g_{i0}&b_i & q&0 \\
z_{jq}&y_{js} & g_{iq}&a_i x_{is} & q&s \ (s,q=1, \ldots,p) \\
\end{array}}
\end{eqnarray*}
Substitution into equation (\ref{for:delta_approx}) gives us the
transformed model
\begin{eqnarray}
\delta_{ij} \simeq g_{i0}z_{j0} + \sum_{q=1}^p g_{iq}z_{jq}
= \sum_{q=0}^p g_{iq}z_{jq}.
\label{for:delta_approx2}
\end{eqnarray}
Thus we end up with a set of $m$ nonhomogeneous linear equations in
$p+1$ unknowns. These can be approximately solved (in a least squares
sense) by multiple regression techniques. The predictor set contains
a vector of ones plus $p$ vectors of target coordinates. The
dissimilarities $\delta_{ij}$ will function as the criterion values.
Once the regression weights in equation (\ref{for:delta_approx2}) are
determined, we find values for the parameters of the original model by
applying
\begin{eqnarray*}
a_{i} &=&\left({\sum_{q=1}^p g_{iq}^2/R_i^2 \max_j \sum_{s=1}^p
y_{js}^2}\right)^{1/2}, \\
x_{is}&=& g_{iq}/a_i, \\
b_i &=& g_{i0}.
\end{eqnarray*}
In fact the normalization of the vector coordinates is free to be
chosen. In PREFPAM-3 it has been decided to normalize them in such a
way that the length of the vector is proportional to the fit $R_i$. The
overall size is determined by the target point that has largest
distance from the origin of the configuration. In this way the length
of the vectors will harmonize with the size of the target
configuration, while still displaying the relative fit.
\section{Equations for the Unfolding model}
\label{sect:eqnunfolding}
%
Here the dissimilarities are assumed to be approximated as
\begin{eqnarray}
\delta_{ij} \simeq a_id_{ij}^2 + b_i,
\label{for:unfol}
\end{eqnarray}
with $d_{ij}^2$ the squared Euclidean distance between ideal point $i$
and target point $j$. Rewriting (\ref{for:unfol}) with coordinates
values $x_{is}$ and $y_{js}$ gives us
\begin{eqnarray}
\delta_{ij}
\simeq a_i \left\{{\sum_{s=1}^p(x_{is}-y_{js})^2}\right\}+b_i
= a_i \left\{{\sum_{s=1}^px_{is}^2
-2\sum_{s=1}^px_{is}y_{js}
+\sum_{s=1}^p y_{js}^2}\right\} + b_i.
\label{for:unfol_exp}
\end{eqnarray}
\begin{eqnarray*}
{\tabcolsep2pt \begin{array}{r@{=}lr@{=}lr@{=}l}
\multicolumn{2}{c}{\mbox{\underline{Change of variables}}} &
\multicolumn{2}{c}{\mbox{\underline{Reparametrization}}} &
\multicolumn{2}{c}{\mbox{\underline{Index ranges and definition}}} \\
z_{j0}&1 & g_{i0}&a_i \sum_{s=1}^px_{is}^2 + b_i & q&0 \\
z_{jq}&y_{js} & g_{iq}&-2a_i x_{is} & q&s \ (s,q=1, \ldots,p) \\
z_{j(p+1)}&\sum_{s=1}^p y_{js}^2 & g_{i(p+1)}&a_i & q&p+1 \\
\end{array}}
\end{eqnarray*}
\noindent
{\it Transformed model}
\begin{eqnarray}
\delta_{ij}
\simeq g_{i0}z_{j0}
+ \sum_{q=1}^pg_{iq}z_{jq} + g_{i(p+1)}z_{j(p+1)}
= \sum_{q=0}^{p+1}g_{iq}z_{jq}.
\end{eqnarray}
Now the transformed model is a set of $m$ linear equations in $p+2$
unknowns. The predictor set contains, in addition to the vector of
ones and the target coordinates, the sum of squares of the target
coordinates ($z_{j(p+1)}$). The parameters of the original model are
obtained as
\begin{eqnarray*}
a_i &=&|g_{i(p+1)}|, \\
x_{is}&=& -\frac{1}{2}g_{iq}/g_{i(p+1)}, \\
b_i &=& g_{i0}-a_i\sum_{s=1}^px_{is}^2.
\end{eqnarray*}
The slope is always identified as a nonnegative quantity; the sign of
the regression weight $g_{i(p+1)}$ determines whether we deal with an
ideal point or an anti-ideal point. When this regression weight
approaches zero, the ideal point will move to infinity.
\section{Equations for the weighted Unfolding model}
%
This model is equivalent to the Unfolding model in equation
(\ref{for:unfol}), except that the distances are defined by
\begin{eqnarray}
d_{ij}^2 = \sum_{s=1}^p w_{is}(x_{is}-y_{js})^2
\end{eqnarray}
and thus (\ref{for:unfol_exp}) can be written for the weighted model as
\begin{eqnarray}
\delta_{ij}
\simeq a_i \left\{{\sum_{s=1}^p w_{is}x_{is}^2
-2\sum_{s=1}^p w_{is}x_{is}y_{js}
+\sum_{s=1}^p w_{is} y_{js}^2}\right\} + b_i
\end{eqnarray}
giving for the new system
\begin{eqnarray*}
{\tabcolsep2pt \begin{array}{r@{=}lr@{=}lr@{=}l}
\multicolumn{2}{c}{\mbox{\underline{Change of variables}}} &
\multicolumn{2}{c}{\mbox{\underline{Reparametrization}}} &
\multicolumn{2}{c}{\mbox{\underline{Index ranges and definition}}} \\
z_{j0}&1 & g_{i0}&a_i \sum_{s=1}^pw_{is}x_{is}^2 + b_i & q&0 \\
z_{jq}&y_{js} & g_{iq}&-2a_i w_{is} x_{is} & q&s \ (s,q=1, \ldots,p) \\
z_{jq}&y_{js}^2 & g_{iq}&a_i w_{is} & q&p+s \ (s=1,\ldots,p; \\
\multicolumn{5}{c}{} &q=p+1,\ldots,2p) \\
\end{array}}
\end{eqnarray*}
{\it Transformed model}
\begin{eqnarray}
\delta_{ij} \simeq g_{i0}z_{j0} + \sum_{q=1}^{2p}g_{iq}z_{jq}
= \sum_{q=0}^{2p}g_{iq}z_{jq}.
\end{eqnarray}
Under the weighted Unfolding model we thus have to determine $2p+1$
regression weights. For that purpose the predictor matrix must
contain, apart from the vector of ones and $p$ vectors with the
target coordinates, another $p$ vectors with the squares of the
target coordinates. The solution for the original parameters is:
\begin{eqnarray*}
\begin{array}{rcll}
a_i &=&\left({\frac{1}{p}\sum_{q=p+1}^{2p}g_{iq}^2}\right)^{1/2},& \\
w_{is}&=& g_{iq}/a_i, & q=(p+1),\ldots,2p, \\
x_{is}&=& -\frac{1}{2}g_{iq}/g_{i(q+p)}, & q=1,\ldots,p,\\
b_i &=& g_{i0}-a_i\sum_{s=1}^pw_{is}x_{is}^2.&
\end{array}
\end{eqnarray*}
By choosing this identification for the $a_i$, the weights $w_{is}$
are normalized so that their sum of squares equals the number of
dimensions $p$. The sign pattern of the $w_{is}$'s indicates whether
we deal with an ideal point (all signs positive), an anti-ideal point
(all signs negative), or a saddle point (some signs positive, the
others negative).
\section{Equations for the general Unfolding model}
%
The general Unfolding model is essentially the same as the weighted
ideal point model, apart from the fact that both the ideal point and
the target points are jointly reoriented by an orthogonal rotation
matrix $\ma{T}_i$. When we define $\ma{X}^*=\ma{XT}_i$ and
$\ma{Y}^*=\ma{YT}_i$ the distances are given by
\begin{eqnarray}
d_{ij}^2 = \sum_{s=1}^p w_{is}(x_{is}^*-y_{js}^*)^2.
\end{eqnarray}
Defining the transformation matrix
$\ma{R}_i=\ma{T}_i\ma{W}_i\ma{T}_i'$, with $\ma{W}_i$ a diagonal
matrix containing the individual dimension weights, the regression
equations for the general Unfolding model are
\begin{eqnarray}
\lefteqn{\delta_{ij} \simeq
a_i \left\{{\sum_{s=1}^p \sum_{u=1}^p x_{is}r_{su}^i x_{iu}}
\right.} \nonumber \\
& & \left.{-2\sum_{s=1}^p \sum_{u=1}^p x_{is}r_{su}^i y_{ju}
+\sum_{s=1}^p \sum_{u=1}^p y_{js}r_{su}^i y_{ju}}\right\} + b_i.
\end{eqnarray}
(The notational convention used here is that the general entry in a
matrix denoted by a bold capital letter with {\it sub}script $i$ will
be indicated by the same letter, doubly subscripted and in small case
with a {\it super}script $i$; e.g. $r_{su}^i$ is the ($s,u$) element
of $\ma{R}_i$.)
\begin{eqnarray*}
{\tabcolsep2pt \begin{array}{r@{=}lr@{=}lr@{=}l}
\multicolumn{2}{l}{\mbox{Change of}} & \multicolumn{2}{c}{} &
\multicolumn{2}{c}{} \\
\multicolumn{2}{l}{\mbox{\underline{variables}}} &
\multicolumn{2}{l}{\mbox{\underline{Reparametrization}}} &
\multicolumn{2}{l}{\mbox{\underline{Index ranges and definition}}} \\
z_{j0}&1 &
g_{i0}&{\displaystyle
a_i \sum_{s=1}^p\sum_{u=1}^p x_{is}r_{su}^i x_{iu} + b_i} &
q&0 \\
z_{jq}&y_{js} &
g_{iq}&{\displaystyle -2a_i \sum_{u=1}^p x_{iu}r_{su}^i} &
q&s \ (s,q=1, \ldots,p) \\
z_{jq}&y_{js}^2 &
g_{iq}&a_i r_{ss}^i & q&p+s \ (s=1,\ldots,p; \\
\multicolumn{5}{c}{}&q=p+1,\ldots,2p) \\
z_{jq}&y_{js}y_{ju} &
g_{iq}&2a_ir_{su}^i(=2a_ir_{us}^i) &
q&2p+s(s-1)/2+u-1 \\
\multicolumn{5}{c}{}&(s__ 0$, under the assumption of
normally distributed errors, the following F-ratio and accompanying
degrees of freedom apply:
\begin{eqnarray*}
\frac{R_k^2/(k-1)}{(1-R_k^2)/(m-k)},
\mbox{ with }k-1\mbox{ and } m-k\ df.
\end{eqnarray*}
Here $R_k$ is the empirical estimate of $\rho_k$, the multiple
correlation with $k$ predictors. When $m = k$ we will obtain perfect
fit and the ratio is not defined. For testing the hypothesis
$\rho_a=\rho_b$, where model $a$ is the more complex model, the
following ratio applies:
\begin{eqnarray*}
\frac{(R_a^2-R_b^2/(k_a-k_b)}{(1-R_a^2)/(m-k_a)},
\mbox{ with }k_a-k_b\mbox{ and } m-k_a\ df,
\end{eqnarray*}
which can also be compared with the tabulated values for the
F-distribution.
\chapter{Preliminary transformations of the target configuration}
\label{chap:chap4}
\section{Room for improvement under the simple Unfolding model}
In some cases it is suitable to allow the target configuration to be
transformed before a group of individuals is fitted into it. There
are two possible linear transformations, which in spirit resemble the
weighted and the general Unfolding model: the original axes of the
target configuration can be differentially ``stretched", or a new set
of reference axes can be obtained by weighting after orthogonal
rotation. Either transformation will apply to all individuals, and is
optimal in the sense that the {\it proportion of total variance
accounted for} by the simple Unfolding model will be maximal. The
proportion of total variance accounted for equals the average squared
fit across individuals, for the {\it metric} case.
There are a number of situations where this type of preliminary
transformation seems suitable. When, e.g., the target configuration
is the group stimulus space from an INDSCAL individual differences
scaling analysis (cf. Carroll and Chang, 1970), the object point
coordinates are normalized for each axis. By performing a preliminary
weighting of the axes the configuration might become more meaningful.
Note that the full transformation, i.e. rotation and weighting, would
not be wise in this case since the orientation of axes obtained by an
INDSCAL analysis is uniquely related to the individual weights from
that analysis. Another application might be the case where we want to
use a target configuration that has been constructed from a set of
hypothetical variables. Since we would not be sure of their proper
orientation and their relative importance, it might be enlightening to
allow a preliminary rotation and stretching (reweighting) of the axes.
It is important to bear in mind that both options for the
transformation are especially designed for the simple ideal point
model. The weighted and the general ideal point model already provide
optimal weights by themselves, while the vector model will absorb
weights in the vector coordinates. Therefore we will obtain the same
fit when applying, e.g., the weighted Unfolding model directly
compared to performing optimal weighting beforehand and next fitting
the weighted model. The full transformation will affect both the
weighted and the simple ideal point model, but it is optimal only for
the latter. The vector model and the general model are not affected
by it, at least as far as the fit is concerned.
When a preliminary transformation is called for, the optimal
configuration is always determined across all rows, no matter what
models are fitted in the external analysis. This also implies that a
special feature of external analyses is lost when applying a
preliminary transformation. When we do not ask for a transformation,
the results across individuals are invariant under different
selections of subgroups of individuals in the analysis (since
individual results are obtained by separate multiple regressions).
When we do ask for a transformation, this is no longer true, because
the optimal configuration is solved for {\it given the selection of
individual data} in the analysis. Thus although the actual preference
mapping consists of separate regressions, the preliminary
transformation is determined ``jointly" across individuals.
In the next two sections we show how the transformed target
configuration is obtained. The solution for the full transformation
can also be found in Carroll (1980), the explicit solution for the
weighted case is new.
\section{The preliminary full transformation}
%
The problem that has to be solved can be written as
\begin{eqnarray}
\delta_{ij} \simeq a_i d_{ij}^2+b_i^*,
\label{for:fullproblem}
\end{eqnarray}
with
\begin{eqnarray}
d_{ij}^2 = (\ma{x}_i - \ma{Ty}_j)'(\ma{x}_i - \ma{Ty}_j),
\label{for:distgenlin}
\end{eqnarray}
where \ma{T} is a general linear transformation matrix defined as
$\ma{T} = \ma{WR}$, with \ma{R} an orthogonal rotation matrix and
\ma{W} a diagonal weights matrix. We use the notational convention
here to denote the $i$'th row of \ma{X} and the $j$'th row of \ma{Y}
by column vectors $\ma{x}_i$ and $\ma{y}_j$, resp. Substituting
(\ref{for:distgenlin}) into (\ref{for:fullproblem}) gives us:
\begin{eqnarray}
\delta_{ij} \simeq a_i \ma{x}_i'\ma{x}_i - 2a_i\ma{x}_i'\ma{Ty}_j
+a_i\ma{y}_j'\ma{T}'\ma{Ty}_j+b_i^*.
\label{for:fullproblem2}
\end{eqnarray}
When we define $\ma{C}=\ma{T}'\ma{T}$, and in addition the matrices
\ma{V}, \ma{B}, \ma{U}, and the vector \ma{w} (using the same coupling
of indices $q$ and $s$ as before):
\begin{eqnarray*}
\begin{array}{r@{=}lr@{=}ll}
v_{qj}&y_{js} &
b_{iq}&-2a_i \ma{x}_i'\ma{T} &
q=s \ (s=1,\ldots,p) \\
v_{(p+1)j}&1 &
b_{i(p+1)}&a_i \ma{x}_i'\ma{x}_i +b_i^* &
(q=p+1) \\
u_{qj}&y_{js}^2 &
w_{q}&c_{ss} &
q=s \ (s=1,\ldots,p) \\
u_{qj}&2y_{js}y_{jt} &
w_{q}&c_{st} &
\left\{{\begin{array}{l}
q=p-1+s(s-1)/2+t \\
s \gamma_l
\end{array}}\right..
\end{eqnarray}
Thus ties in the data may become untied by the monotone regression, in
either direction; the primary approach puts no additional constraints
on the modeling process. In contrast, the {\it secondary approach}
does constrain equal values to remain equal:
\begin{eqnarray}
\mbox{if }\delta_{ij}=\delta_{il} \mbox{ then }\gamma_j = \gamma_l .
\end{eqnarray}
It primarily depends on the precision and reliability of the data
which option is to be preferred; the secondary approach assumes more
precise and reliable data, and consequently will always give a worse
(or, at best, an equally bad) fit for a given model choice from the
PREFMAP hierarchy.
\part{Use of the PREFMAP-3 program}
\label{part_two}
\chapter{Description of the input}
%
As explained in chapter \ref{chap:chap1}, there are two basic pieces
of data for PREFMAP-3 to work with: an external data matrix, and a
target configuration. In addition, of course, the program has to be
told what exactly to do in one single run. In section
\ref{sect:generalsetup} a general description of the input
organization is given, while some parts of it are further explained in
sections \ref{sect:options}--\ref{sect:unitnumbers}. Finally, section
\ref{sect:sample} provides a small working example, which will also
serve in the discussion of the PREFMAP-3 output (chapter
\ref{chap:chap7}).
Since there any many ways to communicate with a computer, and many
control languages in current use, it is useful to state explicitly
what is meant by some of the words and phrases used throughout this
part of the PREFMAP-3 User's Guide. The program has been written in
ANSI-FORTRAN, and it obeys all rules and conventions from this
standard language.
First, the input must be coded on a special type of record, called
{\it card}, which is a record of fixed length and 80 positions long.
Of course, it does not have to be an actual ``card", but in some file
systems care must be taken to ensure that the input has this fixed
form. Secondly, cards are read by the program from a {\it unit},
which is an input device as defined by certain control phrases in the
operating system. In what will be called the {\it standard input
stream} it is assumed that all cards come from the same unit
(PREFMAP-3 is able to read parts of the input from distinct units, cf.
\ref{sect:generalsetup}, \ref{sect:unitnumbers}). Always make sure that a
card contains {\it blanks} on whatever positions where nothing else is
intentionally specified (although often PREFMAP-3 will not react
strangely to unexpected symbols). If the program expects a
specification on a certain position, but encounters a blank, it will
mostly perform a prechosen action called the {\it default}. If in the
sequel it is not indicated what the default action is, this can either
mean that PREFMAP-3 will not be able to respond, lacking essential
information, or that it will perform the action called `0' (zero).
Similarly, on output PREFMAP-3 generates a special type of records,
called {\it lines}, which have 132 positions if routed to a line
printer, and sometimes 133 positions if displayed on a screen (the
first extra character controls the line printer behavior). It is
possible for PREFMAP-3 to write different pieces of output to
different units, in the form of cards (again, not necessarily actual
cards, depending on the type of output device). However, this is not
assumed to be the case in the {\it standard output stream} as
described in chapter \ref{chap:chap7}. Cards and lines are discussed
in groups called {\it blocks}, independently from the ``blocks" that
might be present in a file system; a number of cards or blocks
together form a {\it deck}. The organization and interpretation of
the symbols on a card is ruled by a {\it format}, positions are also
called {\it columns}, and a group of columns a {\it field}. Constants
given to the program are called {\it parameters}, whereas the
statistical parameter estimates calculated by the program will always
be called by their specific name, as defined in Part \ref{part_one} of
this User's Guide.
\section{General set-up}
\label{sect:generalsetup}
%
The standard input stream consists of five blocks, which have a fixed,
predetermined order (the order of cards within blocks is fixed as
well). Schematically, we must have:
\begin{center}
\begin{tabular}{ll} \\ \\
\cline{1-1}
\multicolumn{1}{|l|}{Title card} & TITLE \\
\cline{1-1} & \\ \\
\cline{1-1}
\multicolumn{1}{|l|}{Data specification card} & CONTROL BLOCK \\
\multicolumn{1}{|l|}{Analysis specification card} & (
\ref{sect:DataSpecifications},
\ref{sect:AnalysisSpecifications},
\ref{sect:PrintPlot},
\ref{sect:unitnumbers}) \\
\multicolumn{1}{|l|}{Print/plot options card} & \\
\multicolumn{1}{|l|}{Unit number card} & \\
\cline{1-1} & \\ \\
\cline{1-1}
\multicolumn{1}{|l|}{Format card} & DATA BLOCK 1 \\
\multicolumn{1}{|l|}{\{Target configuration\} cards} & \\
\cline{1-1} & \\ \\
\cline{1-1}
\multicolumn{1}{|l|}{Format card} & DATA BLOCK 2 \\
\multicolumn{1}{|l|}{\{External data matrix\} cards} & \\
\cline{1-1} & \\ \\
\cline{1-1}
\multicolumn{1}{|l|}{\{Option table\} cards} & MODEL OPTION BLOCK
(\ref{sect:options}) \\
\cline{1-1} & \\ \\
\end{tabular}
\end{center}
\noindent
The {\it Title card} forms the first block. It simply identifies the
job, and may contain any alphanumeric symbols (up to 80 characters).
The PREFMAP-3 output will be labeled with this information.
Next, there must be a {\it Control block} formed by four cards. In the
Control block all details of the job are specified, including
information on the size and type of the three remaining blocks: Data
block 1, Data block 2, and the Model option block. The {\it Model
option block} contains specific information on the models to be fitted
for each individual, or group of individuals. An understanding of its
organization is needed for a proper specification of parameters in the
Control block, and therefore the Model option block will be fully
discussed first (section \ref{sect:options}); after that, each card of
the Control block is explained in a separate section
(\ref{sect:DataSpecifications}--\ref{sect:unitnumbers}).
{\it Data block 1} must contain the {\it target configuration}. The first
card is a format card, in particular a FORTRAN F-format card. Any of
the standard FORTRAN specifications may be used freely, but in many
cases the card will look like, e.g.,
\begin{eqnarray*}
\mbox{(3F10.4)}
\end{eqnarray*}
indicating that each following target configuration card will contain
three numbers of the {\it F}loating type, occupying 10 positions each,
with a decimal point in the sixth position, so that there remain four
positions for the fractional portion. After the format card there
must follow $m$ (number of objects) cards, each with $p$ (number of
dimensions) coordinate values. Any configuration can be given as
input to the program. If the coordinates are not in column deviation
form (column means equal to zero), PREFMAP-3 will perform a centering
operation.
{\it Data block 2} must contain the {\it external data matrix}. Like
in Data block 1, the first card is a format card, and the earlier
remarks about its construction apply here too. Next there must be $n$
(number of individuals) cards, each with $m$ (number of objects)
dissimilarity values. Make sure that the scoring direction is:
``small values $\leftrightarrow$ high preferences" and ``large value
$\leftrightarrow$ low preference", regardless of the model to be
fitted. If the data happen to be coded in the other direction, a
recoding facility {\it outside} PREFMAP-3 must be used. Most common
is to subtract all values from the largest possible one, or to change
all signs in the matrix.
Through specifications on the Unit number card of the Control block it
is possible to change the standard input stream in the sense that
PREFMAP-3 will read parts of the input from different sources (units).
Either the target configuration, and/or the external data matrix,
and/or the option table may be dropped from the standard input stream.
Notice that the Title card, the Control block, and the two format
cards {\it always} remain on the same, predetermined unit (with number
5, although it will mostly not be necessary for the user to specify
this anywhere).
A concise description of the complete deck set-up is given in {\it
Appendix \ref{chap:appendixB}}, which provides sufficient guidance for
the advanced user. The next sections implicitly refer to Appendix
\ref{chap:appendixB}; it is important to note that the order of
description in consecutive sections {\it does not correspond} to the
order of the blocks in the standard input stream (the Model option
block is explained first; the Title and Data blocks have already been
fully explained here).
\section{The option table}
\label{sect:options}
%
To the large extent, the option table embodies the flexibility of
PREFMAP-3; it monitors the variety of analyses to be done in a single
run. The rows of the table are called {\it option sets}, its columns
{\it analyses}, and each entry contains an {\it option}. An option,
in this context, is a particular combination of a model from the
PREFMAP hierarchy and a regression type. Since there are four
possible models and three possible regression types, there are 12
options to choose from. They are characterized by two-letter acronyms,
as shown in Table \ref{tab:ModRegr}. Both an option set, and
\begin{table}
\caption{Acronyms used for model-regression combinations.}
\protect\label{tab:ModRegr}
\begin{center}
{\footnotesize
\begin{tabular}{l|ccc}
\hline
& \multicolumn{3}{c}{Type of regression} \\
\cline{2-4}
Model & & Nonmetric & Nonmetric \\
&{\it M}etric& {\it P}rimary & {\it S}econdary \\
& & approach & approach \\
\hline
{\it V}ector & VM & VP & VS \\
{\it U}nfolding & UM & UP & US \\
{\it W}eighted Unfolding & WM & WP & WS \\
{\it G}eneral Unfolding & GM & GP & GS \\
\hline
\end{tabular} }
\end{center}
\end{table}
an analysis is a series of chosen options; the former refers to the
things to be done for any individual, the latter refers to the
presentation of results for a group of individuals. Let's first
consider the situation in which there is only one individual.
In the case of a single individual the option table can have only one
row (one option set), and at most four columns (four analyses). There
is no restriction on the order of the options. Examples are given in
Table \ref{tab:4M}, \ref{tab:one} and \ref{tab:3U}. Table
\ref{tab:4M} will inform PREFMAP-3 to successively apply the vector,
the Unfolding, the weighted Unfolding, and the general Unfolding model
all metrically.
\begin{table}
\caption{Option table for four metric analyses.}
\protect\label{tab:4M}
\begin{center}{\footnotesize
\begin{tabular}{|c|cccc|}
\hline
& \multicolumn{4}{c|}{analysis} \\
\cline{2-5}
option set & 1 & 2 & 3 & 4 \\
\hline
1 & VM & UM & WM & GM \\
\hline
\end{tabular}}
\end{center}
\end{table}
\begin{table}
\caption{Option table for one single analysis.}
\protect\label{tab:one}
\begin{center}{\footnotesize
\begin{tabular}{|c|cccc|}
\hline
& \multicolumn{4}{c|}{analysis} \\
\cline{2-5}
option set & 1 & 2 & 3 & 4 \\
\hline
1 & UP & & & \\
\hline
\end{tabular}}
\end{center}
\end{table}
\begin{table}
\caption{Option table for three different Unfolding analyses.}
\protect\label{tab:3U}
\begin{center}{\footnotesize
\begin{tabular}{|c|cccc|}
\hline
& \multicolumn{4}{c|}{analysis} \\
\cline{2-5}
option set & 1 & 2 & 3 & 4 \\
\hline
1 & UM & UP & US & \\
\hline
\end{tabular}}
\end{center}
\end{table}
Table \ref{tab:one} indicates that only one analysis has to be done,
with the Unfolding model, nonmetrically with primary approach to ties
(i.e., ties may become untied). Table \ref{tab:3U} will cause
PREFMAP-3 to perform three different unfolding analyses, first
metrically, next nonmetrically with primary approach to ties, and
finally nonmetrically with secondary approach to ties (i.e., ties must
remain tied in the latter analysis). If more than four analyses on a
single individual are desired, the data could be repeated and treated
as a case of multiple individuals.
If there are $n$ individuals in the external data matrix (as is most
commonly the case), the following three possibilities can be
distinguished:
\begin{itemize}
\item[(a)] every individual gets a different option set;
\item[(b)] there are groups of individuals that share the same option set;
\item[(c)] every individual gets same option set.
\end{itemize}
The most general situation is (a); each individual becomes associated
with one row of the option table. An example is given in Table
\ref{tab:GenOpt}, in which for every individual an entirely different
series of options is specified. PREFMAP-3 executes the options in
rowwise order: first all options of option set 1, next all options of
option set 2, etc.
\begin{table}
\caption{Option table, general type.}
\protect\label{tab:GenOpt}
\begin{center}{\footnotesize
\begin{tabular}{|c|cccc|}
\hline
& \multicolumn{4}{c|}{analysis} \\
\cline{2-5}
option set & 1 & 2 & 3 & 4 \\
\hline
1 & VM & UM & GM & WM \\
2 & VM & UM & & \\
3 & UM & UP & & \\
4 & UP & & & \\
5 & UP & & GP & \\
6 & UP & US & & WS \\
7 & VP & US & & \\
\hline
\end{tabular} }
\end{center}
\end{table}
After all rows have been processed (and interim results have been
printed), the program considers all options in a column of the option
table as a separate group, as one analysis. Clearly, the number of
individuals in each analysis may be different. According to Table
\ref{tab:GenOpt}, all individuals are in analysis 1; PREFMAP-3 will
give, upon request, a joint plot of the target configuration with
vectors for individuals 1, 2, and 7, and ideal points for the others.
Analysis 2 gives the unfolding results of individuals 1, 2, 3, 6, and
7; analysis 3 gives the general Unfolding results of 1 and 5; finally,
analysis 4 gives the weighted unfolding results of 1 and 6. Notice
that it is no problem to leave an intermediate entry in a row
unspecified. For instance, when PREFMAP-3 reaches option set 5, it
simply executes UP and GP consecutively; in analysis 2, it will give
individual 5 coordinates zero (marked with ``N.A.", not applied); the
GP results of individual 5 are presented in analysis 3.
It is also not a problem to have {\it redundancies} in the option
table. An example is Table \ref{tab:Redun}: there is a horizontal
redundancy in the last three option sets (VP under analysis 3 and 4),
and a vertical one in rows (1, 2, 3), (4, 5, 6, 7) and (8, 9, 10).
Apparently, there are three groups of data (for instance: preferences
from male subjects, preferences from female subjects, and a number of
properties characterizing the objects). Analysis 1, 2, and 3 each
focus on a separate group while analysis 4 gives a joint representation
of all groups under the same model. The horizontal redundancy will
cause PREFMAP-3 to compute VP twice in the last group (somewhat
superfluously, indeed, but no harm is done).
\begin{table}
\caption{Option table showing redundancies.}
\protect\label{tab:Redun}
\begin{center} {\footnotesize
\begin{tabular}{|c|cccc|}
\hline
& \multicolumn{4}{c|}{analysis} \\
\cline{2-5}
\multicolumn{1}{|c|}{option set} & 1 & 2 & 3 & 4 \\
\hline
1 & UP & & & VP \\
2 & UP & & & VP \\
3 & UP & & & VP \\
4 & & UP & & VP \\
5 & & UP & & VP \\
6 & & UP & & VP \\
7 & & UP & & VP \\
8 & & & VP & VP \\
9 & & & VP & VP \\
10 & & & VP & VP \\
\hline
\end{tabular} }
\end{center}
\end{table}
\begin{table}
\caption{Reduced version of Table \protect\ref{tab:Redun}.}
\protect\label{tab:Red}
\begin{center}{\footnotesize
\begin{tabular}{|c|cccc|}
\hline
& \multicolumn{4}{c|}{analysis} \\
\cline{2-5}
option set & 1 & 2 & 3 & 4 \\
\hline
1 & UP & & & VP \\
2 & & UP & & VP \\
3 & & & VP & VP \\
\hline
\end{tabular} }
\end{center}
\end{table}
The vertical redundancy is in fact possibility (b) mentioned above,
and can be communicated to PREFMAP-3 in a more economical way: through
a proper specification on the first card of the Control block (see
section \ref{sect:DataSpecifications}), Table \ref{tab:Red} will be
sufficient information for PREFMAP-3 to be able to execute each option
set repeatedly for all individuals within a group. The reduced form
of the table explains the introduction of the term option set: a row
may correspond with one individual, or with a group. Obviously,
possibility (c) --- every individual gets the same option set --- is a
special case of (b), and single-row tables like Table \ref{tab:4M},
\ref{tab:one} and \ref{tab:3U} suffice. A very small number of
distinct option sets (compared to the number of individuals) is most
common in applications of PREFMAP-3. The possibility to have a short
cut specification has been limited to the range of one up to four
groups. If more than four groups of individuals share the same
options, the user will still have to specify a complete option table.
The option table must be coded in the Model option block with one card
for each option set. Each two-letter acronym must occupy the first
two positions of four-column fields. For example:
\begin{center}{\tt \footnotesize\setlength{\tabcolsep}{0pt}
\begin{tabular}{*{17}{l}l}
\cline{1-16}
& & & & & & & & &1&1&1&1&1&1&1& ~~~column number \\
1&2&3&4&5&6&7&8&9&0&1&2&3&4&5&6& \\
\cline{1-16}
U&M& & &W&M& & &G&M& & &V&M& & & ~~~option codes \\
\cline{1-16}
\end{tabular}}
\end{center}
Empty cells of the option table must be coded as blanks; there are no
default options. The fact that the option acronyms have to link up to
the left might be confusing for some users, as it is in contrast to
the usual FORTRAN I-format convention. The options may be coded in
either upper to lower case characters.
\section{Data specifications}
\label{sect:DataSpecifications}
%
The data specification card is the first card of the Control block,
and may contain from 3 up to 7 parameters, coded in five-column fields
(linked up to the right). The parameters will each be described in
turn.
The first parameter indicates the {\it number of rows} in the external
data matrix, i.e. the number of individuals $n$. It has no default
value.
The second parameter indicates the {\it number of columns} in the
external data matrix, i.e. the number of objects $m$. It corresponds
to the number of rows in the target configuration, and has no default
value.
The third parameter indicates the {\it option set selection}: it
describes the way in which option sets in the option table are to be
linked to the individuals. If it is given value 0 (zero), PREFMAP-3
will apply the options in the first (and possibly only) row of the
option table to each and every individual. The value 1 designates the
situation that all individuals are to be analyzed with a different
option set; thus the program will expect an option table with $n$
rows. The value 2 tells PREFMAP-3 that some individuals should have
the same option set applied to them, in a way to be specified through
the next parameters.
The last four parameters are only needed when the third one (option
set selection) has value 2 (subgroups with the same option set).
Otherwise, they are ignored. Their values designate the {\it starting
points} in the external data matrix for which a new option set
applies. Thus each parameter must equal the row number of the first
individual of each subgroup sharing the same option set. As an
example, consider Table \ref{tab:Redun} as the complete specification
for what has to be done with 10 individuals. Under option set
selection 2 we can work with the reduced option Table \ref{tab:Red},
and have to give the 4th parameter value 1, the 5th parameter value 4,
and the 6th parameter value 8. The maximum number of subgroups is
four (hence at most four parameters). PREFMAP-3 expects an increasing
sequence of values. Whenever the specified sequence is not increasing
(e.g., the numbers 1, 5, 3, and 2 are given, in that order), the
conflicting starting points (3 and 2) will be ignored, and the program
acts as if a smaller number of subgroups has been specified (two
subgroups, starting with 1 and 5). If the same rule by which
PREFMAP-3 is able to determine the number of subgroups in general;
e.g., `` 1 4 8 0 " can only mean that there are three subgroups
(therefore, no separate parameter for number of subgroups is needed).
The rows of the external data matrix must be in the right order
according to the subgroups intended. If they are not, say we would
like to apply the vector model and the Unfolding model alternatingly,
we should specify that all rows have different option sets, and
alternate a ``vector card" with an ``Unfolding card" in the Model
option block.
\section{Analysis specifications}
\label{sect:AnalysisSpecifications}
%
The analysis specification card is the second card of the Control
block, and contains 10 parameters. It controls the general
characteristics of the analyses, and enables PREFMAP-3 to set up the
right amount of working area. The first nine parameters must be coded
as integer numbers in five-column fields, the last one is a floating
point number occupying 10 positions.
The first parameter indicates the {\it number of dimensions} of the
object space ($p$). It can be any number between 1 and $m- 1$ (the
number of objects minus one). Note that PREFMAP-3 will refuse to
attempt specific analyses for which the number of free parameters (a
function of $p$, see Table \ref{tab:pred_vars} in section
\ref{sect:ModelTesting}) exceeds $m$. Also note that the program
expects to be able to read $p$ coordinate values from the target
configuration cards in Data block 1 (which may contain more, but never
less than that number of values).
The second parameter indicates the {\it maximum number of analyses in
any option set}, or the number of columns in the option table. This
need not be the same as the maximum number of options in any option
set. For example, when using Tables \ref{tab:one}, \ref{tab:3U},
\ref{tab:GenOpt} or \ref{tab:Redun}, the value of this parameter must
be 1, 3, 4, and 4, resp.
The third parameter controls the application of a {\it preliminary
transformation of the configuration}. The default value (zero) will
leave the target configuration unchanged. A value of 1 designates
preliminary weighting of axes, a value of 2 designates preliminary
rotation and rotation and weighting (the full transformation).
The fourth parameter controls the {\it standardization of the external
data}. Since standardization does not affect the fit, and the
individual data are more comparable when they have zero mean and unit
variance, the data are standardized row-wise by default. In addition,
it is possible to apply only centering (zero mean) by specifying 1, or
only normalizing (sum of squares equal to $m$) by specifying 2. The
value 3 designates no standardization at all.
The next four parameters indicate simply whether or not a model is
going to be applied in any option set. This is necessary for
efficient array allocation. When a model, say the Unfolding model, is
not referred to on this card (by a `1'), but later on the program
encounters an Unfolding model in some option set, the Unfolding model
will not be applied, and a warning message will be printed. These
parameters are ordered according to the complexity of the models:
\begin{itemize}
\item[--] the fifth parameter: {\it application of the vector model};
\item[--] the sixth parameter: {\it application of the Unfolding model};
\item[--] the seventh parameter: {\it application of the weighted Unfolding model};
\item[--] the eighth parameter: {\it application of the general Unfolding model}.
\end{itemize}
In all cases, a value of 1 designates the affirmative, and a 0 (zero)
the negative specification.
The ninth parameter sets a {\it limit to the number of nonmetric
iterations}. Obviously, it is needed only if there are nonmetric
options specified. Its value is the maximum number of ALS cycles
allowed in each nonmetric analysis (cf. chapter \ref{chap:chap5});
the default value is 50. Usually PREFMAP-3 will be ready long before
this number of iterations when the standard convergence criterion
applies. If not, the maximum number of iterations generally has to be
adjusted, because it is not a proper stopping rule but merely a
safeguard.
The tenth parameter is the {\it convergence criterion}. for nonmetric
iterations, needed to decide when to stop the ALS process. When the
rate of change in the function (\ref{for:nonmetric}) drops below the
value of this parameter, the program concludes that the process has
converged. The default value is .00001; with a more stringent
criterion the number of iterations might increase beyond the limit of
50, the default value of the previous parameter. The user is advised
against making the criterion more lenient, unless it is absolutely
imperative to economize on computation time. The convergence
criterion must be coded with format F10.8.
\section{Print/plot options}
\label{sect:PrintPlot}
%
The print/plot options card is the third card of the Control block,
and contains 9 parameters of the five-column integer type. It
controls the output of PREFMAP-3. A detailed description of the
output will be given in chapter \ref{chap:chap7}; described here is
merely the way of getting various parts of it.
The first parameter controls how much of the {\it input data} is
printed. By default the first 10 rows of the external data matrix (at
most) are printed, to enable checking up on correct transfer. When a
value of 2 is specified, the complete external data matrix is printed.
The target configuration can be obtained by specifying 1 (to get it in
addition to the first 10 rows of the external data), or 3 (to get it
in addition to the complete external data).
The second parameter indicates what is to be printed {\it for each
option}. Complete results (value 1)
include the following:
\begin{itemize}
\item[--] the metric fit and the variance accounted for;
\item[--] the nonmetric fit (when applicable);
\item[--] the coordinates of the vector or the ideal point;
\item[--] the normalized weights (when applicable);
\item[--] the orthogonal rotation matrix (when applicable);
\item[--] the criterion and predicted values;
\item[--] the slope and the intercept.
\end{itemize}
The terms criterion values and predicted values refer to the fact that
PREFMAP-3 solves a regression problem. The criterion values either
are the (possibly standardized) external data, or the external data
transformed by a monotone function (for the nonmetric case). The
predicted values are the best approximation of the criterion values
under the chosen model (the right hand sides of the regression
equations, including the slope and the intercept terms). The
(nonmetric) fit is the correlation between the criterion values and
the predicted values. The slope and intercept terms make it possible
to construct the predicted values from the squared Euclidean distances
between an ideal point and the object points. Part of the results can
be suppressed by specifying a 2 (only the fit will be printed), or a 3
(fit as well as criterion and predicted values will be printed).
The third parameter controls the {\it selected results for each
analysis}. An analysis is a series of models for different
individuals brought together (as specified in the option table coded
in the Model option block). A value of 1 will cause PREFMAP-3 to
print a table of coordinate values, next a table of weights (if
applicable), and finally a series of rotation matrices (if
applicable). This is done for each analysis (column of the option
table) in turn. Individuals are always identified with their original
row number and the model that has been applied.
The fourth parameter indicates that a {\it scatter and transformation
plot} must be made for each fitted model of the first $N$ individuals;
the parameter value $N$ can be any integer in the range from 0 up to
$n$, the number of individuals. In a single plot the criterion values
are plotted against the data (the transformation), along with the
predicted values (showing the vertical scatter around the
transformation). Note that the individuals for whom the plots are to
be obtained should come first in the data matrix. In metric analyses,
the transformation will always be a straight line; in nonmetric
analyses, it will be the optimal monotonically increasing step
function. The scatter not only visualizes the fit, it provides in
fact a pictorial breakdown of fit into its $m$ components, each
associated with one of the objects.
The fifth parameter calls for a {\it plot of ideal points} and/or {\it
vectors} in the target configuration, again for each analysis in turn.
The ideal points and vectors are labeled by integers
$(1,2,...,8,9,0,1,...)$, and the object points are labeled by
characters (A,B,...,H,I,J,A,...). When the weighted Unfolding model
or the general Unfolding model has been applied for more than one
individual, the ideal point plot is followed by a plot of the weights.
Further details are given in section \ref{sect:OverviewOutput}. The
value of this parameter, $K$, indicates that all pairs of the first
$K$ dimensions will be plotted. Thus if $K = 3$, then three plots are
obtained: dimension 1 versus 2, 1 versus 3, and 2 versus 3. When $K =
2$, simply one plot is obtained; when $K = 1$, a two- dimensional plot
of the first dimension is made (a straight line in a square box). Of
course, $K$ should not be larger than $p$, the dimensionality of the
analysis.
The sixth parameter indicates printing of a {\it history of
computation in the nonmetric regression} (1 = yes, 0 = no). When
requested, a history is given for each nonmetric option. It will
provide an impression of the course of the iterative process, which
can be worthwhile looking at, especially when convergence is suspected
to be slow or irregular.
The seventh parameter indicates computation and printing of the {\it
F-statistic} (1 = yes, 0 = no). The F-ratios for significance testing
will be printed for each model in an option set, while each model will
also be compared to the nearest simpler model from the hierarchy
available. This is done for each individual in turn. For example:
when the weighted Unfolding model and the vector model have been
applied, the F-statistics for testing the null hypotheses $\rho_W = 0,
\rho_V = 0$, and $\rho_U = \rho_V$ are given (cf. section
\ref{sect:ModelTesting}). In case the simple Unfolding model has been
applied too, the program gives -- in addition to the single model
F-statistics -- the F-ratios for the hypotheses $\rho_W = \rho_U$ and
$\rho_U = \rho_V$. These F- ratios are always computed on the basis of
the metric fit, and when nonmetric regressions were performed. In the
latter case the F-statistics approach is no longer meaningful. When a
model fits the data perfectly ($R_2 = 1$), the F-ratio is undefined
and thus will not be given.
The last two parameters arrange {\it storing of the individual
results} on an output device to be specified on the unit number card
(section \ref{sect:unitnumbers}). The eighth parameter indicates
storage of coordinates (code 1), of coordinates and weights (code 2),
or of coordinates, weights and rotation matrices (code 3). When in
operation, the results are always preceded by the target coordinates.
The ninth parameter indicates storage of the external data together
with the predicted values (code 1), or the external data together with
the predicted values and the criterion values (code 2).
The prime usage of the storage facility is to be able to produce
graphics on a plotting device outside PREFMAP-3. However, there can
be another reason for using it. This is because the third and fifth
parameter of the print/plot options card affect the efficiency of
array allocation by the program itself. When a very large number of
individuals must be dealt with, and the core memory area available to
the program is limited, the storage facilities can be utilized as a
substitute for printing. In general, then, it can be advised:
\begin{itemize}
\item[(a)] When the number of individuals is small, to ask for
\begin{itemize}
\item[--] complete results for each option (2nd parameter,
especially if one is interested in the predicted values and
the monotone transformation of the data);
\item[--] selected results for each analysis (3rd parameter);
\item[--] plotting of ideal points and/or vectors in the target
configuration (5th parameter).
\end{itemize}
\item[(b)] When the number of individuals is large, to ask for
\begin{itemize}
\item[--] the fit for each option only (2nd parameter);
\item[--] selected results for each analysis (3rd parameter);
\item[--] routing of individual results to other output units
(8th and 9th parameter);
\item[--] plotting of ideal points and/or vectors in the target
configuration (5th parameter).
\end{itemize}
\item[(c)] When a number of individuals is very large, to ask for
\begin{itemize}
\item[--] routing of individual results to other output
units (8th and 9th parameter).
\end{itemize}
\end{itemize}
If the default values for storage are used on the unit number card,
PREFMAP-3 will actually print, rather than store, the individual
results. Nevertheless, the program does not need additional array
area for doing that, and the above-mentioned recommendations remain
valid.
\section{unit numbers for I/O blocks}
\label{sect:unitnumbers}
The unit number card is the fourth card of the Control block, and
contains 9 parameters of the five-column integer type. The first {\it
three} parameters (with default value 5) all refer to {\it input
units}, and can be used to alter the standard input stream (cf.
section \ref{sect:generalsetup}). The next {\it two} parameters
(having no default value) refer to {\it scratch files}, which are
needed during execution of the program. The last {\it four}
parameters (with default value 6) refer to {\it output units} for
receiving part of the results outside the standard output stream. See
Appendix B for details. Whenever a unit number specification deviates
from the default value, the user has to define a file in the operating
system associated with the same number.
Note that the codes on the input files should satisfy the format
specification given on the format cards, which always reside in the
same unit as the parameter cards (section \ref{sect:generalsetup}).
If the target configuration, the external data and the option table
share the same unit number, they should always be there in that order.
It is also possible to have the target configuration and the option
table on one unit, say 22, and the external data on another, say 21.
The required space for the two scratch files depends upon the size of
the problem and the organizational features of the operating system.
Whether or not the output files are really used is controlled by the
last two parameters of the print/plot options cards (section
\ref{sect:PrintPlot}). Various partitions of the results are possible
by specifying different output unit numbers. Since the results are
written in the order of computation, we would otherwise obtain
coordinates, weights, and rotation matrices alternatingly on the same
output file, which would make them rather inaccessible for plotting
afterwards.
\section{Sample input stream}
\label{sect:sample}
A sample input is given here to illustrate the complete input to
PREFMAP-3 for a single run; the output of this small example will be
discussed in section \ref{sect:SampleOutput}. The column numbers are
given on top, for easy counting out.
\noindent
{\footnotesize
\begin{verbatim}
5 10 15 20 25 30 35 40 45 50 55
-------------------------------------------------------
*** TEST PREFMAP 3 ***
5 5 2 1 3
3 3 02 0 1 1 0 0 50 0
3 1 1 2 2 1 1 0 0
5 5 5 9 8 0 0 0 0
(8X,3F12.7)
0.2863523 0.1391261 -0.4
0.2459524 -0.0714838 -0.2
0.0495586 0.1090570 0.0
-0.1291228 -0.1841551 0.2
-0.4527405 0.0074558 0.4
(5F7.3)
1.500 3.500 1.500 1.500 3.000
6.500 6.000 4.895 5.273 1.000
-9.000 -7.677 -8.115 -7.625 -5.182
3.000 3.000 3.000 3.000 3.000
9.000 8.667 8.077 8.375 8.000
VM VP
VS UP WM
-------------------------------------------------------
\end{verbatim}}
\chapter{Description of output}
\label{chap:chap7}
\section{General output structure}
%
The standard output stream consists of three major blocks, with
contents highly dependent upon the specification of the print/plot
options card. Schematically, we get:
\noindent\centerline{\footnotesize
\begin{tabular}{ll} \\ \\
\cline{1-1}
\multicolumn{1}{|l|}{Heading} & \\
\multicolumn{1}{|l|}{Overview of chosen parameters} & JOB INFORMATION BLOCK \\
\multicolumn{1}{|l|}{Report of data \& options} & \\
\cline{1-1} \\ \\
\cline{1-1}
\multicolumn{1}{|l|}{Results for individual 1} & \\
\multicolumn{1}{|l|}{\hspace{1.5cm}$\vdots$} & \\
\multicolumn{1}{|l|}{Results for individual $i$}& INDIVIDUAL MODELS BLOCK \\
\multicolumn{1}{|l|}{\hspace{1.5cm}$\vdots$} & \\
\multicolumn{1}{|l|}{Results for individual $n$} & \\
\multicolumn{1}{|l|}{Summary of results} & \\
\cline{1-1} \\ \\
\cline{1-1}
\multicolumn{1}{|l|}{Summary tables for consecutive analyses} & ANALYSIS BLOCK \\
\multicolumn{1}{|l|}{Joint plots for consecutive analyses} & \\
\cline{1-1} \\ \\
\end{tabular}}
\noindent
The output routed to other units has a form comparable to the
Individual models block, and will actually be inserted there if the
default unit numbers are in operation.
\section{Overview of the output blocks}
\label{sect:OverviewOutput}
The {\it Job information block} is the least elastic part of the
output; it is only the report of the data that can be compressed
(first parameter of the print/plot options card). The overview of
chosen parameters is given in two forms; as an {\it echo} of the
parameters cards, enabling the user to verify the literal input
instructions, and as a {\it list of interpreted instructions}
providing the user with feedback on the actions actually selected.
The subheadings of the list are numbered with the corresponding card
numbers printed in form of the echo. When the target configuration
turns out not to be centered at the mean, this is remedied by the
program. The mean for each original dimension will be printed. When
a preliminary transformation has been requested (third parameter of
the analysis specification card) there will always be a print of the
transformation target configuration, as well as the orthogonal
rotation matrix and/or the normalized weights.
After the job information block PREFMAP-3 starts giving results for
each individual in turn, across options, except when {\it all}
parameters on the print/plot options card have default values. In the
latter case only the last part of the {\it Individual models block} is
given: a summary of results consisting of the {\it average fit} across
individuals sharing the same option (for each of the options present,
not necessarily in the same column of the option table). For the
metric options the total variance, the total variance accounted for,
and the proportion of total variance accounted for are printed as
well. The user is referred to section \ref{sect:PrintPlot} for an
explanation of the selections of individual results possible; here
merely a number of additional details are mentioned.
The vector coordinates are normalized such that their length is
proportional to the individual fit. Their overall size is determined
by the size of the target configuration. For each Unfolding model it
is indicated whether the fitted point is an ideal or an anti-ideal
point. Saddle points are recognized by inspection of the sign pattern
of the fitted weights. When a coordinate value ``9999.0'' is printed
for an ideal point this indicates that the weight for the squared
predictor term in the regression equation (cf. chapter
\ref{chap:chap3}) has become almost zero. In such a case the ideal
point is located at infinity (or is behaving like a vector, one could
say). Thus it might be wise to reanalyse the individual with the
vector model. For the weighted models, the weights are normalized
such that their sum squares equals the number of dimensions.
All parts of the {\it Analysis block} are optional. The summary
tables for consecutive analyses are controlled by the third parameter
of the print/plot options card, the joint plots by the fifth
parameters on that card. Firstly all coordinates for the first
analysis (e.g., the general Unfolding model for each individual) are
given, followed by individual normalized weights and rotation
matrices. These results are printed next to the row number and the
model that has been applied (especially useful for mixed models within
an analysis). If an individual did not participate in an analysis,
the coordinates will be given as ``0.0 0.0" with the label ``N.A."
(not applied) as the option description. When plotted, such an
individual is to be found in the origin, the natural point of
inconspicuousness. After the results for analysis 1 the program
continues with the results for subsequent analysis, if any.
In the joint plots for consecutive analyses the ideal points and
vectors are labeled by integers (1,2,...,8,9,0,1,...) and the object
points are labeled by characters (A,B ...,H,I,J,A,...). The
convention to plot non-participating individuals in the origin is also
maintained for fitted anti-ideal points or saddle points. These will
be displayed in a separate plot to prevent interpretational confusion.
In all cases the original row number is retained to label the points.
When two or more points coincide, the location is labeled with ``M"
for ``more points". As was mentioned before, an ideal point can be
far outside the configuration of objects points (maximally ``9999.0").
Including such a point in the joint plot would make it impossible to
adequately display the other points. To prevent such a ``degenerate"
plot, all points that are far outside the range of the object space
are omitted from the plot, and a warning is issued.
For the weighted and the general unfolding model the individual points
are plotted in the so-called {\it common space}, i.e. the original
target configuration. Individually reshaped plots could be made
outside PREFMAP-3 by applying the appropriate (rotation and) weights
to both the target configuration and the coordinates of the individual
point. Finally, the weights are displayed in a separate plot. Under
the general model we have to bear in mind that the weights pertain to
differently oriented reference axes. We can still compare, however,
the relative importance of preferred directions across individuals
(e.g., individual 1 has a ratio of 2:1, whereas individual 2 has a
ratio of 3:1 for the first axis against the second in an idiosyncratic
orientation).
\section{Sample output stream}
\label{sect:SampleOutput}
This section will show the output of the program in response to the
sample input stream from section \ref{sect:sample}. The example has
not been designed to demonstrate a serious analysis. On the contrary,
the example was created to show a number of possible peculiarities of
which the user should be aware when using the program. The complete
printout is presented at the end of this section.
In the first place the full preliminary transformation of the target
configuration is asked for. This transformation cannot be performed:
the number of target points should at least be 7 to fit the $p(p+1)/2
= 6$ parameters of the preliminary transformation in $p = 3$
dimensions. Next the input data is read, and the program detects that
one row of data matrix has zero variance; hence this row will be
omitted from the analysis. The other rows will maintain their
original row number for easy identification.
When the program reads the option sets, where the first two rows of
the data matrix share the same set, it finds the option WM in the
second option set. However, the weighted Unfolding model cannot be
applied: in the first place, there are too many parameters to be
estimated $(2p + 1 = 7)$, and, moreover, the weighted model has not
been referred to on the analysis specification card.
The results for individual 1 show the following peculiarities. The
first option, the vector model applied metrically, gives a very poor
fit (.306). This result is depicted in the accompanying scatter plot,
where the points for the criterion values (labeled by a star) and the
predicted values (labeled by a ``D") are far apart. When the vector
model is next applied nonmetrically, the fit proves considerably
(.985); but the criterion values, which are the external data
transformed by a monotone function, are hardly informative. There
remain only two distinct values; -2.0 (the star in the lower left
corner of the plot) and 0.5 (the ``M" in the upper left corner, in this
case indicating two stars on the right). The fact that the ``D" points
are relatively close to the stars is reflected in the relatively high
correlation. One should always mistrust such a dramatic difference in
fit between the metric and the nonmetric application of a model. The
analysis for individual 2 shows a rather high fit for the metric
vector model (.961). When next the nonmetric model is applied, the
fit is perfect, at the cost of obtaining a three-step transformation
shown in the corresponding scatter/transformation plot. The ``M"
labels indicate that the criterion values and the predicted values
coincide completely.
For individual 3 the metric Unfolding model fits perfectly. Hence
there is no improvement possible for the nonmetric model, and the
F-statistics are not defined. The same applies to the analysis for
individual 5, and these results could already have been anticipated
since the number of target points equals exactly the number of
parameters that has to be estimated $(p+2 = 5)$. When we inspect
the selected results across analyses, we see that for individual 4
zeros have been assigned to the coordinates and weights, while for
analysis 3 all individuals obtain zeros (since 1 and 2 were not
included in this analysis, and the option ``WM" for 3 and 5 could not
be applied).
The first joint plot of the target points and the vectors from
analysis 1 shows the coordinates for individual 4 plotted in the
origin (``M"). In the joint plot for analysis 2, which shows vectors
for individuals 1 and 2, and an ideal point for individual 3, we do
not find the ideal point for individual 5. Because individual 5
obtained negative weights under the Unfolding model, the point should
be depicted in a separate plot as an anti-ideal point. However, we do
not actually obtain this plot, because the coordinates of the
anti-ideal point are too much outside the range of the target
coordinates.
\newpage
{\scriptsize \def\baselinestretch{.75}
\begin{verbatim}
P R E F M A P - 3
*** EXTERNAL UNFOLDING *** JACQUELINE MEULMAN
VERSION - 1.0 *** & *** WILLEM HEISER
FEBRUARY 1985 *** PROPERTY FITTING *** J. DOUGLAS CARROLL
BELL LABORATORIES
MURRAY HILL, NJ
1 JOB TITLE: *** TEST PREFMAP 3 ***
ECHO OF PARAMETER CARDS:
2 5 5 2 1 3 0 0
3 3 3 2 0 1 1 0 0 500.00000000
4 3 1 1 2 2 1 1 0 0
5 5 5 5 9 8 0 0 0 0
2 DATA SPECIFICATIONS:
NUMBER OF ROW POINTS (THESE ARE FITTED) IS 5
NUMBER OF COLUMN POINTS (THESE ARE GIVEN) IS 5
OPTION SET SELECTION: 2
0 = ALL ROWS SAME OPTION SET
1 = ALL ROWS DIFFERENT OPTION SETS
2 = SPECIFIED ROWS SAME OPTION SET
OPTION SETS START WITH ROWS 1 3
3 ANALYSIS SPECIFICATIONS:
THE NUMBER OF DIMENSIONS IS 3
MAXIMUM NUMBER OF ANALYSES IN ANY OPTION SET 3
PRELIMINARY TRANSFORMATION OF CONFIGURATION 2
0 = REMAINS UNCHANGED
1 = WEIGHTED
2 = ROTATED AND WEIGHTED
STANDARDIZE EXTERNAL DATA 0
0 = YES (=1+2)
1 = CENTER ONLY
2 = NORMALIZE ONLY
3 = NONE OF THE ABOVE
MODELS: 0 = NOT APPLIED, 1 = APPLIED
VECTOR MODEL 1
UNFOLDING MODEL 1
WEIGHTED UNFOLDING MODEL 0
GENERAL UNFOLDING MODEL 0
THE NUMBER OF NON-METRIC ITERATIONS IS 50
THE CONVERGENCE CRITERION IS 0.10E-04
4 PRINT/PLOT OPTIONS: 0 = NONE
PRINT INPUT: 3
1 = TARGET CONFIGURATION
2 = EXTERNAL DATA
3 = 1 & 2
PRINT RESULTS ACROSS OPTION SETS 1
1 = COMPLETE RESULTS
2 = FIT
3 = 2 & CRITERION-PREDICTED VALUES
SELECTED RESULTS ACROSS ANALYSES 1
SCATTER&TRANSFORMATION PLOT FOR N ROWS, N = 2
PAIRWISE PLOTS IDEAL POINTS/VECTORS, DIM = 2
HISTORY OF NON-METRIC REGRESSION 1
STATISTICS FOR METRIC ANALYSES 1
STORE OUTPUT: 0
1 = COORDINATES
2 = LIKE 1 + WEIGHTS
3 = LIKE 2 + ROTATION MATRICES
STORE OUTPUT: 0
1 = PREDICTED VALUES
2 = LIKE 1 + CRITERION
\end{verbatim}
\newpage
\begin{verbatim}
5 UNIT NUMBERS FOR INPUT/OUTPUT
THE CONFIGURATION WILL BE READ FROM UNIT 5
THE EXTERNAL DATA WILL BE READ FROM UNIT 5
THE OPTIONS WILL BE READ FROM UNIT 5
UNIT NUMBER FOR SCRATCH FILE 1 9
UNIT NUMBER FOR SCRATCH FILE 2 8
OUTPUT UNIT FOR THE COORDINATES 0
OUTPUT UNIT FOR THE WEIGHTS 0
OUTPUT UNIT FOR THE ROTATION MATRICES 0
OUTPUT UNIT FOR PREDICTED&CRITERION VALUES 0
*** WARNING *** THE PRELIMINARY TRANSFORMATION WILL NOT BE PERFORMED, BECAUSE THE
NUMBER OF TARGET POINTS IS NOT SUFFICIENT.
THE CONFIGURATION WILL BE READ WITH FORMAT (8X,3F12.7)
THE TARGET CONFIGURATION WILL BE CENTERED: MEAN ORIGINAL DIMENSION 1 0.000
THE TARGET CONFIGURATION WILL BE CENTERED: MEAN ORIGINAL DIMENSION 2 0.000
THE TARGET CONFIGURATION WILL BE CENTERED: MEAN ORIGINAL DIMENSION 3 0.000
TARGET CONFIGURATION (CENTERED)
-------------------------------
1 2 3
1 0.286 0.139 -0.400
2 0.246 -0.071 -0.200
3 0.050 0.109 0.000
4 -0.129 -0.184 0.200
5 -0.453 0.007 0.400
THE EXTERNAL DATA WILL BE READ WITH FORMAT (5F7.3)
5 ROWS OF THE EXTERNAL DATA
-------------------------------
1 2 3 4 5
1 1.500 3.500 1.500 1.500 3.000
2 6.500 6.000 4.895 5.273 1.000
3 -9.000 -7.677 -8.115 -7.625 -5.182
4 3.000 3.000 3.000 3.000 3.000
ROW 4 WILL BE OMITTED FROM THE ANALYSIS SINCE IT HAS ZERO VARIANCE
5 9.000 8.667 8.077 8.375 8.000
MODEL: V = VECTOR
U = UNFOLDING
W = WEIGHTED UNFOLDING
G = GENERAL UNFOLDING
REGRESSION: M = METRIC
P = NON-METRIC, PRIMARY APPROACH TO TIES
S = NON-METRIC, SECONDARY APPROACH TO TIES
ANALYSIS
OPTION SET: 1 2 3
1 VM VP
2 VS UP WM
*** WARNING *** OPTION 3 WILL NOT BE APPLIED. EITHER THE MODEL HAS NOT BEEN REFERRED TO
ON CARD 3 OR THE NUMBER OF TARGET POINTS IS NOT SUFFICIENT TO FIT THIS MODEL.
\end{verbatim}
\newpage
\begin{verbatim}
ROW= 1 ANALYSIS= 1 VECTOR MODEL METRIC REGRESSION
=================================================================
METRIC FIT 0.306 VARIANCE 1.000 V.A.F. 0.094
COORDINATES
-----------
1 2 3
1 0.108 0.112 0.100
CRITERION VALUES (EXTERNAL DATA, POSSIBLY TRANSFORMED BY MONOTONE FUNCTION)
----------------
1 2 3 4 5
1 -0.803 1.491 -0.803 -0.803 0.918
PREDICTED VALUES
----------------
1 2 3 4 5
1 -0.176 0.040 -0.480 0.397 0.219
SLOPE 27.33366 INTERCEPT 0.00000
\end{verbatim}
\newpage
\begin{verbatim}
SCATTER PLOT. DATA (X-AXIS) VS CRITERION (*) AND PREDICTED VALUES (D) (M=*+D), ROW 1 VECTOR MODEL
.--+------+------+------+------+------+------+------+------+------+------+---.
1.491 I * I
1.450 I I
1.408 I I
1.367 I I
1.325 I I
1.283 I I
1.242 I I
1.200 I I
1.159 I I
1.117 I I
1.076 I I
1.034 I I
0.993 I I
0.951 I I
0.909 I * I
0.868 I I
0.826 I I
0.785 I I
0.743 I I
0.702 I I
0.660 I I
0.619 I I
0.577 I I
0.535 I I
0.494 I I
0.452 I I
0.411 I D I
0.369 I I
0.328 I I
0.286 I I
0.245 I I
0.203 I D I
0.162 I I
0.120 I I
0.078 I I
0.037 I D I
-0.005 I I
-0.046 I I
-0.088 I I
-0.129 I I
-0.171 I D I
-0.212 I I
-0.254 I I
-0.296 I I
-0.337 I I
-0.379 I I
-0.420 I I
-0.462 I D I
-0.503 I I
-0.545 I I
-0.586 I I
-0.628 I I
-0.670 I I
-0.711 I I
-0.753 I I
-0.794 I M I
.--+------+------+------+------+------+------+------+------+------+------+---.
-0.803 -0.574 -0.344 -0.115 0.115 0.344 0.574 0.803 1.032 1.262 1.491
\end{verbatim}
\newpage
\begin{verbatim}
ROW= 1 ANALYSIS= 2 VECTOR MODEL NON-METRIC REGRESSION PRIMARY APPROACH TO TIES
=========================================================================================
METRIC FIT 0.306 VARIANCE 1.000 V.A.F. 0.094
HISTORY OF NON-METRIC REGRESSION
--------------------------------
DIFFERENCE WITH
ITERATION FIT PRECEDING ITERATION
1 0.92946 0.62341
2 0.94096 0.01150
3 0.94956 0.00860
4 0.95847 0.00891
5 0.96705 0.00858
6 0.97473 0.00769
7 0.98120 0.00646
8 0.98526 0.00406
9 0.98549 0.00023
10 0.98549 0.00000
NON-METRIC FIT 0.985
COORDINATES
-----------
1 2 3
1 0.369 0.258 0.390
CRITERION VALUES (EXTERNAL DATA, POSSIBLY TRANSFORMED BY MONOTONE FUNCTION)
----------------
1 2 3 4 5
1 0.500 0.500 -2.000 0.500 0.500
PREDICTED VALUES
----------------
1 2 3 4 5
1 0.612 0.241 -1.942 0.714 0.375
SLOPE 41.88269 INTERCEPT 0.00000
\end{verbatim}
\newpage
\begin{verbatim}
SCATTER PLOT. DATA (X-AXIS) VS CRITERION (*) AND PREDICTED VALUES (D) (M=*+D), ROW 1 VECTOR MODEL
.--+------+------+------+------+------+------+------+------+------+------+---.
0.714 I D I
0.665 I I
0.616 I D I
0.566 I I
0.517 I M * * I
0.468 I I
0.419 I I
0.370 I D I
0.321 I I
0.272 I I
0.222 I D I
0.173 I I
0.124 I I
0.075 I I
0.026 I I
-0.023 I I
-0.073 I I
-0.122 I I
-0.171 I I
-0.220 I I
-0.269 I I
-0.318 I I
-0.367 I I
-0.417 I I
-0.466 I I
-0.515 I I
-0.564 I I
-0.613 I I
-0.662 I I
-0.712 I I
-0.761 I I
-0.810 I I
-0.859 I I
-0.908 I I
-0.957 I I
-1.006 I I
-1.056 I I
-1.105 I I
-1.154 I I
-1.203 I I
-1.252 I I
-1.301 I I
-1.351 I I
-1.400 I I
-1.449 I I
-1.498 I I
-1.547 I I
-1.596 I I
-1.646 I I
-1.695 I I
-1.744 I I
-1.793 I I
-1.842 I I
-1.891 I I
-1.940 I D I
-1.990 I * I
.--+------+------+------+------+------+------+------+------+------+------+---.
-1.013 -0.741 -0.470 -0.199 0.073 0.344 0.616 0.887 1.158 1.430 1.701
STATISTICS ACROSS OPTIONS FOR ROW 1
--------------------------------------
THE STATISTICS GIVEN BELOW DEPEND ONLY ON THE METRIC FIT, NOT ON THE NON-METRIC FIT.
VECTOR MODEL, P.VAF. = 0.094, F = 0.034 WITH 3 AND 1 DEGREES OF FREEDOM
\end{verbatim}
\newpage
\begin{verbatim}
ROW= 2 ANALYSIS= 1 VECTOR MODEL METRIC REGRESSION
=================================================================
METRIC FIT 0.961 VARIANCE 1.000 V.A.F. 0.923
COORDINATES
-----------
1 2 3
2 -0.520 0.250 -0.060
CRITERION VALUES (EXTERNAL DATA, POSSIBLY TRANSFORMED BY MONOTONE FUNCTION)
----------------
1 2 3 4 5
2 0.907 0.650 0.083 0.277 -1.916
PREDICTED VALUES
----------------
1 2 3 4 5
2 0.724 1.074 -0.012 -0.073 -1.712
SLOPE 8.02532 INTERCEPT 0.00000
\end{verbatim}
\newpage
\begin{verbatim}
SCATTER PLOT. DATA (X-AXIS) VS CRITERION (*) AND PREDICTED VALUES (D) (M=*+D), ROW 2 VECTOR MODEL
.--+------+------+------+------+------+------+------+------+------+------+---.
1.074 I D I
1.020 I I
0.965 I I
0.911 I * I
0.857 I I
0.803 I I
0.749 I D I
0.695 I I
0.640 I * I
0.586 I I
0.532 I I
0.478 I I
0.424 I I
0.370 I I
0.316 I I
0.261 I * I
0.207 I I
0.153 I I
0.099 I * I
0.045 I I
-0.009 I D I
-0.064 I D I
-0.118 I I
-0.172 I I
-0.226 I I
-0.280 I I
-0.334 I I
-0.388 I I
-0.443 I I
-0.497 I I
-0.551 I I
-0.605 I I
-0.659 I I
-0.713 I I
-0.768 I I
-0.822 I I
-0.876 I I
-0.930 I I
-0.984 I I
-1.038 I I
-1.093 I I
-1.147 I I
-1.201 I I
-1.255 I I
-1.309 I I
-1.363 I I
-1.417 I I
-1.472 I I
-1.526 I I
-1.580 I I
-1.634 I I
-1.688 I D I
-1.742 I I
-1.797 I I
-1.851 I I
-1.905 I * I
.--+------+------+------+------+------+------+------+------+------+------+---.
-2.000 -1.701 -1.402 -1.103 -0.804 -0.505 -0.206 0.093 0.392 0.691 0.990
\end{verbatim}
\newpage
\begin{verbatim}
ROW= 2 ANALYSIS= 2 VECTOR MODEL NON-METRIC REGRESSION PRIMARY APPROACH TO TIES
=========================================================================================
METRIC FIT 0.961 VARIANCE 1.000 V.A.F. 0.923
HISTORY OF NON-METRIC REGRESSION
--------------------------------
DIFFERENCE WITH
ITERATION FIT PRECEDING ITERATION
1 0.99313 0.03253
2 0.99790 0.00478
3 0.99934 0.00143
4 0.99979 0.00045
5 0.99993 0.00014
6 0.99998 0.00005
7 0.99999 0.00001
8 1.00000 0.00000
NON-METRIC FIT 1.000
COORDINATES
-----------
1 2 3
2 -0.247 0.405 0.374
CRITERION VALUES (EXTERNAL DATA, POSSIBLY TRANSFORMED BY MONOTONE FUNCTION)
----------------
1 2 3 4 5
2 1.034 1.034 -0.200 -0.200 -1.667
PREDICTED VALUES
----------------
1 2 3 4 5
2 1.033 1.037 -0.201 -0.202 -1.666
SLOPE 6.29833 INTERCEPT 0.00000
\end{verbatim}
\newpage
\begin{verbatim}
SCATTER PLOT. DATA (X-AXIS) VS CRITERION (*) AND PREDICTED VALUES (D) (M=*+D), ROW 2 VECTOR MODEL
.--+------+------+------+------+------+------+------+------+------+------+---.
1.096 I I
1.045 I M M I
0.994 I I
0.943 I I
0.891 I I
0.840 I I
0.789 I I
0.738 I I
0.687 I I
0.636 I I
0.585 I I
0.534 I I
0.482 I I
0.431 I I
0.380 I I
0.329 I I
0.278 I I
0.227 I I
0.176 I I
0.125 I I
0.073 I I
0.022 I I
-0.029 I I
-0.080 I I
-0.131 I I
-0.182 I M M I
-0.233 I I
-0.285 I I
-0.336 I I
-0.387 I I
-0.438 I I
-0.489 I I
-0.540 I I
-0.591 I I
-0.642 I I
-0.694 I I
-0.745 I I
-0.796 I I
-0.847 I I
-0.898 I I
-0.949 I I
-1.000 I I
-1.051 I I
-1.103 I I
-1.154 I I
-1.205 I I
-1.256 I I
-1.307 I I
-1.358 I I
-1.409 I I
-1.460 I I
-1.512 I I
-1.563 I I
-1.614 I I
-1.665 I M I
-1.716 I I
.--+------+------+------+------+------+------+------+------+------+------+---.
-1.916 -1.634 -1.352 -1.069 -0.787 -0.505 -0.223 0.060 0.342 0.624 0.907
STATISTICS ACROSS OPTIONS FOR ROW 2
--------------------------------------
THE STATISTICS GIVEN BELOW DEPEND ONLY ON THE METRIC FIT, NOT ON THE NON-METRIC FIT.
VECTOR MODEL, P.VAF. = 0.923, F = 3.981 WITH 3 AND 1 DEGREES OF FREEDOM
\end{verbatim}
\newpage
\begin{verbatim}
ROW= 3 ANALYSIS= 1 VECTOR MODEL NON-METRIC REGRESSION SECONDARY APPROACH TO TIES
=========================================================================================
METRIC FIT 0.898 VARIANCE 1.000 V.A.F. 0.807
HISTORY OF NON-METRIC REGRESSION
--------------------------------
DIFFERENCE WITH
ITERATION FIT PRECEDING ITERATION
1 0.98603 0.08762
2 0.99826 0.01223
3 0.99979 0.00153
4 0.99997 0.00019
5 1.00000 0.00002
6 1.00000 0.00000
NON-METRIC FIT 1.000
COORDINATES
-----------
1 2 3
3 0.488 0.224 0.277
CRITERION VALUES (EXTERNAL DATA, POSSIBLY TRANSFORMED BY MONOTONE FUNCTION)
----------------
1 2 3 4 5
3 -0.898 -0.724 -0.724 0.728 1.617
PREDICTED VALUES
----------------
1 2 3 4 5
3 -0.897 -0.725 -0.724 0.729 1.617
SLOPE 14.88134 INTERCEPT 0.00000
\end{verbatim}
\newpage
\begin{verbatim}
ROW= 3 ANALYSIS= 2 UNFOLDING MODEL NON-METRIC REGRESSION PRIMARY APPROACH TO TIES
=========================================================================================
METRIC FIT 1.000 VARIANCE 1.000 V.A.F. 1.000
HISTORY OF NON-METRIC REGRESSION
--------------------------------
DIFFERENCE WITH
ITERATION FIT PRECEDING ITERATION
1 1.00000 0.00000
NON-METRIC FIT 1.000
COORDINATES IDEAL POINT
-----------------------
1 2 3
3 -0.891 -0.162 -0.903
CRITERION VALUES (EXTERNAL DATA, POSSIBLY TRANSFORMED BY MONOTONE FUNCTION)
----------------
1 2 3 4 5
3 -1.167 -0.124 -0.469 -0.083 1.843
PREDICTED VALUES
----------------
1 2 3 4 5
3 -1.167 -0.124 -0.469 -0.083 1.843
SLOPE 15.93727 INTERCEPT -28.74530
STATISTICS ACROSS OPTIONS FOR ROW 3
--------------------------------------
THE STATISTICS GIVEN BELOW DEPEND ONLY ON THE METRIC FIT, NOT ON THE NON-METRIC FIT.
PERFECT FIT FOR OPTION 2. NO STATISTICS.
VECTOR MODEL, P.VAF. = 0.807, F = 1.395 WITH 3 AND 1 DEGREES OF FREEDOM
\end{verbatim}
\newpage
\begin{verbatim}
ROW= 5 ANALYSIS= 1 VECTOR MODEL NON-METRIC REGRESSION SECONDARY APPROACH TO TIES
=========================================================================================
METRIC FIT 0.997 VARIANCE 1.000 V.A.F. 0.994
HISTORY OF NON-METRIC REGRESSION
--------------------------------
DIFFERENCE WITH
ITERATION FIT PRECEDING ITERATION
1 1.00000 0.00285
2 1.00000 0.00000
NON-METRIC FIT 1.000
COORDINATES
-----------
1 2 3
5 0.311 0.232 0.463
CRITERION VALUES (EXTERNAL DATA, POSSIBLY TRANSFORMED BY MONOTONE FUNCTION)
----------------
1 2 3 4 5
5 1.502 0.770 -0.960 -0.227 -1.086
PREDICTED VALUES
----------------
1 2 3 4 5
5 1.502 0.770 -0.960 -0.227 -1.086
SLOPE 23.54395 INTERCEPT 0.00000
ROW= 5 ANALYSIS= 2 UNFOLDING MODEL NON-METRIC REGRESSION PRIMARY APPROACH TO TIES
=========================================================================================
METRIC FIT 1.000 VARIANCE 1.000 V.A.F. 1.000
HISTORY OF NON-METRIC REGRESSION
--------------------------------
DIFFERENCE WITH
ITERATION FIT PRECEDING ITERATION
1 1.00000 0.00000
NON-METRIC FIT 1.000
COORDINATES ANTI-IDEAL POINT
----------------------------
1 2 3
5 -2.369 -1.185 -2.932
CRITERION VALUES (EXTERNAL DATA, POSSIBLY TRANSFORMED BY MONOTONE FUNCTION)
----------------
1 2 3 4 5
5 1.547 0.653 -0.931 -0.131 -1.138
\end{verbatim}
\newpage
\begin{verbatim}
PREDICTED VALUES
----------------
1 2 3 4 5
5 1.547 0.653 -0.931 -0.131 -1.138
SLOPE 2.73786 INTERCEPT 43.19877
STATISTICS ACROSS OPTIONS FOR ROW 5
--------------------------------------
THE STATISTICS GIVEN BELOW DEPEND ONLY ON THE METRIC FIT, NOT ON THE NON-METRIC FIT.
PERFECT FIT FOR OPTION 2. NO STATISTICS.
VECTOR MODEL, P.VAF. = 0.994, F = 58.233 WITH 3 AND 1 DEGREES OF FREEDOM
SUMMARY OF RESULTS: (PROPORTION OF TOTAL VARIANCE ACCOUNTED FOR ONLY GIVEN FOR METRIC OPTIONS)
MODEL ' VM ' N = 2 AVERAGE FIT = 0.633
ROOT MEAN SQUARED FIT = 0.713
TOTAL VARIANCE = 2.000 TOTAL VARIANCE ACCOUNTED FOR = 1.016
PROPORTION OF TOTAL VARIANCE ACCOUNTED FOR = 0.508
MODEL ' UP ' N = 2 AVERAGE FIT = 1.000
ROOT MEAN SQUARED FIT = 1.000
MODEL ' VP ' N = 2 AVERAGE FIT = 0.993
ROOT MEAN SQUARED FIT = 0.993
MODEL ' VS ' N = 2 AVERAGE FIT = 1.000
ROOT MEAN SQUARED FIT = 1.000
ANALYSIS 1 COORDINATES
----------------------
1 VM 0.108 0.112 0.100
2 VM -0.520 0.250 -0.060
3 VS 0.488 0.224 0.277
4 VS 0.000 0.000 0.000
5 VS 0.311 0.232 0.463
ANALYSIS 2 COORDINATES
----------------------
1 VP 0.369 0.258 0.390
2 VP -0.247 0.405 0.374
3 UP -0.891 -0.162 -0.903
4 UP 0.000 0.000 0.000
5 UP -2.369 -1.185 -2.932
ANALYSIS 2 WEIGHTS
------------------
3 UP 1.000 1.000 1.000
4 UP 0.000 0.000 0.000
5 UP -1.000 -1.000 -1.000
ANALYSIS 3 COORDINATES
----------------------
1 N.A. 0.000 0.000 0.000
2 N.A. 0.000 0.000 0.000
3 N.A. 0.000 0.000 0.000
4 N.A. 0.000 0.000 0.000
5 N.A. 0.000 0.000 0.000
\end{verbatim}
\newpage
\begin{verbatim}
IDEAL POINTS AND/OR VECTORS (INTEGERS) IN TARGET CONFIGURATION. ANALYSIS 1 DIM 1 (X-AXIS) VS DIM 2 (Y-AXIS)
.--+------+------+------+------+------+------+------+------+------+------+---.
0.537 I I
0.519 I I
0.500 I I
0.482 I I
0.464 I I
0.446 I I
0.427 I I
0.409 I I
0.391 I I
0.373 I I
0.354 I I
0.336 I I
0.318 I I
0.300 I I
0.281 I I
0.263 I I
0.245 I 2 I
0.226 I 5 3 I
0.208 I I
0.190 I I
0.172 I I
0.153 I I
0.135 I A I
0.117 I C 1 I
0.099 I I
0.080 I I
0.062 I I
0.044 I I
0.026 I I
0.007 I E M I
-0.011 I I
-0.029 I I
-0.048 I I
-0.066 I B I
-0.084 I I
-0.102 I I
-0.121 I I
-0.139 I I
-0.157 I I
-0.175 I D I
-0.194 I I
-0.212 I I
-0.230 I I
-0.248 I I
-0.267 I I
-0.285 I I
-0.303 I I
-0.322 I I
-0.340 I I
-0.358 I I
-0.376 I I
-0.395 I I
-0.413 I I
-0.431 I I
-0.449 I I
-0.468 I I
.--+------+------+------+------+------+------+------+------+------+------+---.
-0.520 -0.420 -0.319 -0.218 -0.117 -0.016 0.085 0.186 0.286 0.387 0.488
\end{verbatim}
\newpage
\begin{verbatim}
IDEAL POINTS AND/OR VECTORS (INTEGERS) IN TARGET CONFIGURATION. ANALYSIS 2 DIM 1 (X-AXIS) VS DIM 2 (Y-AXIS)
.--+------+------+------+------+------+------+------+------+------+------+---.
0.740 I I
0.718 I I
0.695 I I
0.672 I I
0.649 I I
0.626 I I
0.603 I I
0.581 I I
0.558 I I
0.535 I I
0.512 I I
0.489 I I
0.467 I I
0.444 I I
0.421 I I
0.398 I 2 I
0.375 I I
0.352 I I
0.330 I I
0.307 I I
0.284 I I
0.261 I 1 I
0.238 I I
0.216 I I
0.193 I I
0.170 I I
0.147 I A I
0.124 I I
0.101 I C I
0.079 I I
0.056 I I
0.033 I I
0.010 I E M I
-0.013 I I
-0.035 I I
-0.058 I I
-0.081 I B I
-0.104 I I
-0.127 I I
-0.150 I I
-0.172 I 3 I
-0.195 I D I
-0.218 I I
-0.241 I I
-0.264 I I
-0.287 I I
-0.309 I I
-0.332 I I
-0.355 I I
-0.378 I I
-0.401 I I
-0.423 I I
-0.446 I I
-0.469 I I
-0.492 I I
-0.515 I I
.--+------+------+------+------+------+------+------+------+------+------+---.
-0.891 -0.765 -0.639 -0.513 -0.387 -0.261 -0.135 -0.009 0.117 0.243 0.369
ANALYSIS 2 *** NOT PLOTTED *** 5 UP -2.369 -1.185 *** SEVERELY OUT OF RANGE ***
\end{verbatim}
\def\baselinestretch{1} }
\chapter{Some applications of PREFMAP-3}
In this chapter three applications of PREFMAP-3 will be presented.
because it is not possible to cover the whole range of applications
here, the user might find it useful to consult one of the following
applications: Coxon (1974), Cermak and Cornillon (1976), Davison and
Jones (1976), Falbo (1977), Nygren and Jones (1977), Seligson (1977),
Coxon and Jones (1978), Davison {\it et al.} (1980), and Kuyper (1980).
Heiser and De Leeuw (1981) have made an attempt to classify social
science applications of preference mapping into prototypical groups.
They distinguish:
\begin{itemize}
\item[--] {\it general Thurstonian attitude scaling}: scale values for
the attitude items are determined by a separate experimental procedure
(cf. Torgerson (1958), chapter 4), and the problem of finding
attitude scores for the individuals, given their list of
``endorsements" of preference ratings, can be solved by PREFMAP (also
if we switch from one-dimensional to multidimensional representations
of the attitude items);
\item[--] {\it trade-off studies}: suppose we have a collection of objects
known to differ on two negatively correlated desirable traits, e.g. a
set of insurance policies varying in prize and in cover. We now may
want to characterize customers in terms of their {\it safety bias} on
the basis of reported preferences;
\item[--] {\it multidimensional psychophysics}: suppose we have a collection
of objects selected as to differ on two physical attributes, e.g. a
set of taste mixtures (say, alanine and glutamic acid combined in
various concentrations), which are to be judges as to their {\it
sweet-sourness}; or a set of odour mixtures (say, jasmine bergamot in
various concentrations), to be judges on their {\it hedonic tone};
\item[--] {\it multidimensional psychophysics}: here the objects are
varied systematically on psychological attributes. Typically, one
confronts subjects with hypothetical ``stimulus persons", differing on,
e.g., intelligence and dominance, and asks for a judgment of overall
{\it likeableness}. A large amount of research has been dedicated to
the discovery of the rule by which a subject combines different pieces
of information into one final impression.
\end{itemize}
It will be clear that in all cases the prime advantage of the PREFMAP
methodology is to be able to go beyond the assumption of monotonicity
of the dependent variable with respect to the independent (i.e.,
varied or selected by the experimenter) variables. For a discussion
of applying PREFMAP as a kind of quality control for Multidimensional
Scaling and Internal Unfolding (study of {\it discriminant-,
convergent-, or cross-validity}) the user is referred to Heiser and de
Leeuw's original (1981) paper, and to Bechtel (1976, 1981).
\section{Facial expressions}
Our first application is a reanalysis of a classical example
concerning facial expressions. The target configuration has been
obtained by performing a nonmetric Multidimensional Scaling analysis
of the dissimilarities collected by Abelson and Sermat (1962). The
objects are selected photographs from the Lightfoot-series, in which
an actress expresses 13 different emotions. They are listed in the
stub of Table \ref{tab:facial}. The external data are the median
ratings by 96 male undergraduates on three nine-point scales, obtained
for the same 13 emotions by Engen {\it et al.} (1958), and reflecting
judgments on the attributes ``pleasant-unpleasant" (P-U),
``attention-rejection" (A-R), and ``tension-sleep" (T-S).
\begin{table}
\caption{Median rating scale values of 13 facial expressions.}
\protect\label{tab:facial}
\begin{center}{\footnotesize
\begin{tabular}{rl|rrr} \hline
& \multicolumn{1}{c|}{Expression} & P-U & A-R & T-S \\ \hline
1. & Grief at death of mother & 3.8 & 4.2 & 4.1 \\
2. & Savoring a Coke & 5.9 & 5.4 & 4.8 \\
3. & Very pleasant surprise & 8.8 & 7.8 & 7.1 \\
4. & Maternal love -- baby in arms & 7.0 & 5.9 & 4.0 \\
5. & Physical exhaustion & 3.3 & 2.5 & 3.1 \\
6. & Anxiety -- something is wrong with her plane & 3.5 & 6.1 & 6.8 \\
7. & Anger at seeing dog beaten & 2.1 & 8.0 & 8.2 \\
8. & Physical strain -- pulling hard on seat of chair & 6.7 & 4.2 & 6.6 \\
9. & Unexpectedly meets old boy friend & 7.4 & 6.8 & 5.9 \\
10. & Revulsion & 2.9 & 3.0 & 5.1 \\
11. & Extreme pain & 2.2 & 2.2 & 6.4 \\
12. & Knows her plane will crash (disaster) & 1.1 & 8.6 & 8.9 \\
13. & Light sleep & 4.1 & 1.3 & 1.0 \\ \hline
\end{tabular} }
\end{center}
\end{table}
The data in Table \ref{tab:facial} are to be interpreted
dissimilarities with respect to the right pole of the bipolar
attribute scales; e.g., emotion 12, disaster, is judged to be very
unpleasant, and emotion 3, very pleasant surprise, is indeed judged to
be very pleasant, i.e. very dissimilar to unpleasant. On the other
hand, remember the convention in PREFMAP-3 to let a small
dissimilarity correspond to a {\it large} projection score under the
vector model (and to a {\it small} distance under the Unfolding
model). For these data we consider the vector model as the most
appropriate, because it most readily incorporates the bipolarity of
the scales (viz., as two opposing directions along the vector). Only
when the fit would be very poor we might consider to apply a more
complex model, in which one of the poles would have to be favoured as
the central one (to be mapped as the peak/dip of the preference
function).
The data were analyzed both metrically and nonmetrically, and the
resulting fit measures are collected in Table \ref{tab:facial_fit}. Notice that the
metric fit is already quite good for all attributes, and that it is
only slightly improved by allowing an optimal monotone transformation.
We may conclude that the attribute judgments are fairly coherent with
the dissimilarity judgments, a conclusion also reached by Heiser and
Meulman (1983), who analyzed the two sets of data simultaneously. The
results for the present analysis are presented in Figure \ref{fig:facial}, which
displays the points for the facial expressions and the vectors for the
scales in a joint plot.
\begin{table}
\caption{Fit measures for facial expressions, vector model.}
\protect\label{tab:facial_fit}
\begin{center}{\footnotesize
\begin{tabular}{c|ccc} \hline
Regression type & P-U & A-R & T-S \\ \hline
Metric & .961 & .861 & .947 \\
Nonmetric & .988 & .922 & .991 \\ \hline
\end{tabular} }
\end{center}
\end{table}
\begin{figure}
\centerline{\footnotesize
\setlength{\unitlength}{0.7cm}
\begin{picture}(14.8,14.8)(0,-1)
\linethickness{0.3pt}
\put(0,-1){\framebox(14.8,14.8){}}
\put( 2.3 ,11.7 ){\makebox(0,0){disaster}}
\put(13.5 ,10.45){\makebox(0,0){surprise}}
\put( 9.8 , 9.9 ){\makebox(0,0){strain}}
\put( .7 , 9.1 ){\makebox(0,0){anger}}
\put( 4.05, 8.0 ){\makebox(0,0){anxiety}}
\put(12.1 , 8.1 ){\makebox(0,0){meeting}}
\put(10.1 , 7.5 ){\makebox(0,0){savor}}
\put(12.25, 5.75){\makebox(0,0){maternal}}
\put( 4.2 , 5.1 ){\makebox(0,0){pain}}
\put( 5.35, 4.55){\makebox(0,0){grief}}
\put( 4.6 , 3.3 ){\makebox(0,0){revulsion}}
\put( 7.7 , 3.2 ){\makebox(0,0){exhaust}}
\put( 9.05, 1.5 ){\makebox(0,0){sleep}}
\put( 4.7 ,13.3 ){\makebox(0,0){T}}
\put(10.3 , .2 ){\makebox(0,0){S}}
\put( 6.5 ,13.3 ){\makebox(0,0){A}}
\put( 8.45, .2 ){\makebox(0,0){R}}
\put(13.8 , 9.55){\makebox(0,0){P}}
\put( 1.0 , 4.0 ){\makebox(0,0){U}}
\bezier{204}( 4.7 ,13.3 )(7.5 ,6.75 )(10.3 , .2 )
\bezier{ 6}( 9.85, .5 )(10.075,.35 )(10.3 , .2 )
\bezier{ 6}(10.4 , .7 )(10.35, .45 )(10.3 , .2 )
\bezier{200}( 6.5 ,13.3 )(7.475,6.75 )( 8.45, .2 )
\bezier{ 6}( 8.1 , .4 )(8.275, .3 )( 8.45, .2 )
\bezier{ 6}( 8.75, .5 )(8.6 , .35 )( 8.45, .2 )
\bezier{190}(13.8 , 9.55)(7.4 ,6.775)( 1.0 , 4.0 )
\bezier{ 6}( 1.3 , 4.4 )(1.15 ,4.2 )( 1.0 , 4.0 )
\bezier{ 6}( 1.5 , 3.9 )(1.25 ,3.95 )( 1.0 , 4.0 )
\end{picture} }
\caption{PREFMAP-3 solution for three attributes of facial
expression}
\protect\label{fig:facial}
\end{figure}
It is evident from Figure \ref{fig:facial} that ``attention-rejection"
and ``tension-sleep" partition the facial expressions in much the same
way, and that both are psychologically independent from the
``pleasant-unpleasant" contrast.
\section{Parliament 1972}
Parliament 1972\protect\footnote{The data of the Parliament Survey
were collected by a team of political scientists of the Department of
Political Science at Leiden University. The project was supported in
part by the Netherlands Organization for the Advancement of Pure
Research (ZWO), under grants 43-03 and 43-09.}
is a questionnaire study among the members of the
Second Chamber of the Dutch Parliament (comparative with the House of
Representatives in the U.S.). The members (MPs) expressed their
preferences for the political parties residing in Parliament, and also
their position with respect to 7 political issues on a nine-point
scale. The political parties are given in Table \ref{tab:party_abbr},
and the issues in Table \ref{tab:pol_issues}. The objects in the
PREFMAP-3 analysis are the Members, identified only by their party
allegiance; the issues serve as the PREFMAP-3 individuals, thus the
issue self-ratings are the external data of this application. To
obtain a target the preference data have been analyzed with a metric
{\it internal} Unfolding program (SMACOF3, Heiser and de Leeuw, 1979).
From the SMACOF3 results the {\it coordinates for the MPs} served as
the target configuration for the PREFMAP-3 analysis.
\begin{table}
\caption{Party allegiance of MPs in the Parliament 1972 study.}
\protect\label{tab:party_abbr}
\begin{center}{\footnotesize
\begin{tabular}{llcc} \hline
Party & Description & No. of MPs
& MP-label \\ \hline \\
PSP & pacifist-socialist & 2 & P \\
PPR & radical & 1 & R \\
PvdA & labor & 36 & L \\
D'66 & pragmatic liberal & 8 & 6 \\
ARP & protestant & 11 & A \\
KVP & catholic & 32 & K \\
CHU & protestant & 10 & U \\
VVD & conservative-liberal & 15 & V \\
GPV & calvinist & 2 & G \\
SGP & calvinist & 1 & S \\
DS'70 & conservative-socialist & 6 & 7 \\ \\ \hline
\end{tabular} }
\end{center}
\end{table}
\begin{table}
\caption{Political issues in the Parliament 1972 study.}
\protect\label{tab:pol_issues}
\begin{center}{\footnotesize
\begin{tabular}{lp{4cm}p{4cm}} \hline
Issue &
\multicolumn{2}{c}{lower end (1) \protect\dotfill \ (9) upper end}\\ \hline
\\
1. DEVELOPMENT {\it AID} &
Government should spend {\it less} money on aid to developing
countries &
Government should spend {\it more} money on aid to developing
countries \\ \\
2. {\it ABORTION} &
A {\it woman has the right} to decide for herself about abortion &
Government should {\it prohibit} abortion completely \\ \\
3. LAW \& {\it ORDER} &
Government should take {\it stronger} action against public
disturbances &
Government takes {\it too strong} action against public
disturbances \\ \\
4. {\it INCOME} DIFFERENCES &
Income differences should become {\it much less} &
Income differences should {\it remain} as they are \\ \\
5. {\it PARTICIP}ATION &
{\it Workers too} must have participation in decisions important
in industry &
{\it Only management} should decide important matters in industry
\\ \\
6. {\it TAXES} &
Taxes should be {\it decreased} so that people can decide for
themselves &
Taxes should be {\it increased} for general welfare \\ \\
7. {\it ARMIES} &
Government should insist on {\it maintaining} strong Western
armies &
Government should insist on {\it shrinking} the Western
armies \\ \\ \hline
\end{tabular} }
\end{center}
\end{table}
Since it can be argued that at least for some issues the vector model
adage ``the more the better" is not too suitable (more money to
developing countries, but not the whole budget; taxes could be
increased, but not to 100\%) both the vector model and the Unfolding
model have been tried. The fit measures are collected in Table
\ref{tab:parl_fit}.
\begin{table}
\caption{Fit measures for Parliament 1972.}
\protect\label{tab:parl_fit}
\begin{center}{\footnotesize
\begin{tabular}{l|cccc} \hline
Issue & VM & VP & UM & UP \\ \hline
AID & .692 & .845 & .714 & .870 \\
ABORTION & .609 & .836 & .665 & .839 \\
ORDER & .793 & .925 & .794 & .930 \\
INCOME & .703 & .867 & .710 & .876 \\
PARTICIP & .654 & .875 & .727 & .877 \\
TAXES & .791 & .902 & .813 & .908 \\
ARMIES & .772 & .872 & .772 & .880 \\ \hline
\end{tabular} }
\end{center}
\end{table}
On the basis of the fit measures we would decide that the vector
vector is yet the more appropriate model; the Unfolding model, either
metrically or nonmetrically, does not fit much better. When
inspecting the issue points in the Unfolding analysis it became clear
that all of them are located at the outskirts of, or outside the
target configuration. This implies that the regression weight for the
adequate term is small. Besides, ABORTION, ORDER, PARTICIP, and
ARMIES turned out to be mapped as ideal-points. These are easy to
interpret in this context, since we only have to reverse the
interpretation of a small value on the issue scales.
Nevertheless, the combination of the just slightly better fit and the
small (negative) regression weights for the quadratic term leads us to
prefer the vector model over the Unfolding model. We see the joint
plot in Figure \ref{fig:parl_joint}. Apart from the target points
(MP's, labeled with small capitals) the column points from the
SMACOF3 internal Unfolding analysis are plotted as well (the political
parties, labeled with large capitals). The latter have not been used
in the PREFMAP-3 analysis, but are given for completeness; they are in
fact located at the centroid of the MPs belonging to that party (a
restriction used in the internal Unfolding analysis). The vectors
should be interpreted according to the lower end descriptions of the
issue scales. The horizontal axis clearly reflects the left-right
distinction in Dutch politics (plotted in reversed order), coherent
with issues like income differences (``much less" versus ``remain"),
armies (``shrinking" versus ``maintaining"), participation (``workers
too" versus ``management only"), and law \& order (``too strong" versus
``stronger"). But the second dimension should not be disregarded.
Here the abortion issue brings together the left-wing parties (PSP,
PVA, D66) and the (economically conservative) liberal parties D70 and
VVD. They are opposed to the denominational parties (KVP, ARP, CHU,
SGP, GPV) and, although to a lesser extent, to the left wing party
PPR. The presence of this latter party on the ``prohibit" side of the
abortion vector can be easily explained by the fact that this MP used
to belong to the catholic party KVP.
\begin{figure}
\centerline{\footnotesize
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\begin{picture}(15.3,14.8)(0,0)
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\put( 1.6 ,10.1 ){\makebox(0,0){TAXES}}
\put( 2.1 ,10.1 ){\makebox(0,0){AID}}
\put( 1.1 , 7.4 ){\makebox(0,0){ARMIES}}
\put( .7 , 6.85){\makebox(0,0){ORDER}}
\put(13.4 , 6.0 ){\makebox(0,0){PARTICIP}}
\put(13.25, 5.4 ){\makebox(0,0){INCOME}}
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\bezier{221}(7.35,7.35)( 4.475, 8.725)( 1.6 ,10.1 )
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\bezier{217}(7.35,7.35)( 4.225, 7.375)( 1.1 , 7.4 )
\bezier{228}(7.35,7.35)( 4.025, 7.1 )( .7 , 6.85)
\bezier{214}(7.35,7.35)(10.375, 6.675)(13.4 , 6.0 )
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\put( 4.4 , 9.45){\makebox(0,0){D70}}
\put( 4.15, 9.3 ){\makebox(0,0){VVD}}
\put( 9.55, 9.0 ){\makebox(0,0){D66}}
\put(10.3 , 7.2 ){\makebox(0,0){PVA}}
\put( 6.9 , 6.8 ){\makebox(0,0){KVP}}
\put(13.75, 6.8 ){\makebox(0,0){PSP}}
\put( 4.6 , 6.5 ){\makebox(0,0){CHU}}
\put( 6.75, 5.8 ){\makebox(0,0){ARP}}
\put(12.2 , 5.3 ){\makebox(0,0){PPR}}
\put( 2.5 , 2.05){\makebox(0,0){SGP}}
\put( 3.15, 2.15){\makebox(0,0){GPV}}
\put(13.6 , 7.0 ){\makebox(0,0){\tiny P}}
\put(13.85, 6.6 ){\makebox(0,0){\tiny P}}
\put(12.2 , 5.3 ){\makebox(0,0){\tiny R}}
\put( 8.2 , 8.5 ){\makebox(0,0){\tiny L}}
\put(10.65, 8.5 ){\makebox(0,0){\tiny L}}
\put(10.15, 8.2 ){\makebox(0,0){\tiny L}}
\put(10.95, 8.25){\makebox(0,0){\tiny L}}
\put(10.95, 8.15){\makebox(0,0){\tiny L}}
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\put( 9.45, 7.65){\makebox(0,0){\tiny L}}
\put( 9.5 , 7.5 ){\makebox(0,0){\tiny L}}
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\put(10.15, 7.65){\makebox(0,0){\tiny L}}
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\put(10.4 , 7.6 ){\makebox(0,0){\tiny L}}
\put( 9.75, 7.4 ){\makebox(0,0){\tiny L}}
\put(10.9 , 7.4 ){\makebox(0,0){\tiny L}}
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\put(10.45, 6.5 ){\makebox(0,0){\tiny L}}
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\put(10.2 , 6.4 ){\makebox(0,0){\tiny L}}
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\put(11.15, 5.8 ){\makebox(0,0){\tiny L}}
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\put( 8.55, 9.3 ){\makebox(0,0){\tiny 6}}
\put( 9.2 , 9.25){\makebox(0,0){\tiny 6}}
\put( 9.9 , 9.2 ){\makebox(0,0){\tiny 6}}
\put( 9.9 , 8.7 ){\makebox(0,0){\tiny 6}}
\put(10.1 , 9.1 ){\makebox(0,0){\tiny 6}}
\put(10.4 , 8.1 ){\makebox(0,0){\tiny 6}}
\put(10.85, 8.65){\makebox(0,0){\tiny 6}}
\put( 4.35, 4.4 ){\makebox(0,0){\tiny A}}
\put( 4.95, 6.1 ){\makebox(0,0){\tiny A}}
\put( 5.45, 6.4 ){\makebox(0,0){\tiny A}}
\put( 5.55, 6.3 ){\makebox(0,0){\tiny A}}
\put( 6.1 , 6.1 ){\makebox(0,0){\tiny A}}
\put( 7.8 , 5.95){\makebox(0,0){\tiny A}}
\put( 8.55, 6.2 ){\makebox(0,0){\tiny A}}
\put( 8.55, 5.8 ){\makebox(0,0){\tiny A}}
\put( 7.1 , 5.6 ){\makebox(0,0){\tiny A}}
\put( 7.65, 5.55){\makebox(0,0){\tiny A}}
\put( 7.95, 5.0 ){\makebox(0,0){\tiny A}}
\put( 9.05, 3.6 ){\makebox(0,0){\tiny K}}
\put( 9.2 , 4.55){\makebox(0,0){\tiny K}}
\put( 5.15, 5.1 ){\makebox(0,0){\tiny K}}
\put( 7.75, 9.6 ){\makebox(0,0){\tiny K}}
\put( 7.65, 8.7 ){\makebox(0,0){\tiny K}}
\put( 6.9 , 7.95){\makebox(0,0){\tiny K}}
\put( 8 , 8 ){\makebox(0,0){\tiny K}}
\put( 9.6 , 5.45){\makebox(0,0){\tiny K}}
\put( 8.8 , 6.05){\makebox(0,0){\tiny K}}
\put( 9.3 , 6.45){\makebox(0,0){\tiny K}}
\put( 7.6 , 5.75){\makebox(0,0){\tiny K}}
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\put( 5.25, 6 ){\makebox(0,0){\tiny K}}
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\put( 3.9 , 6.95){\makebox(0,0){\tiny K}}
\put( 5.55, 6.9 ){\makebox(0,0){\tiny K}}
\put( 5.85, 6.85){\makebox(0,0){\tiny K}}
\put( 6.2 , 6.8 ){\makebox(0,0){\tiny K}}
\put( 6.5 , 7.1 ){\makebox(0,0){\tiny K}}
\put( 6.7 , 7.05){\makebox(0,0){\tiny K}}
\put( 5.35, 7.1 ){\makebox(0,0){\tiny K}}
\put( 5.65, 7.25){\makebox(0,0){\tiny K}}
\put( 5.95, 7.25){\makebox(0,0){\tiny K}}
\put( 6.1 , 7.4 ){\makebox(0,0){\tiny K}}
\put( 7 , 7.35){\makebox(0,0){\tiny K}}
\put( 5.35, 7.85){\makebox(0,0){\tiny K}}
\put( 5.85, 7.65){\makebox(0,0){\tiny K}}
\put( 6.5 , 7.8 ){\makebox(0,0){\tiny K}}
\put( 7 , 7.5 ){\makebox(0,0){\tiny K}}
\put( 4.4 , 4.35){\makebox(0,0){\tiny U}}
\put( 3.75, 6.05){\makebox(0,0){\tiny U}}
\put( 3.4 , 6.3 ){\makebox(0,0){\tiny U}}
\put( 3.95, 6.35){\makebox(0,0){\tiny U}}
\put( 4.1 , 6.45){\makebox(0,0){\tiny U}}
\put( 4.35, 6.7 ){\makebox(0,0){\tiny U}}
\put( 4.95, 7.1 ){\makebox(0,0){\tiny U}}
\put( 5.7 , 7.0 ){\makebox(0,0){\tiny U}}
\put( 6.45, 7.4 ){\makebox(0,0){\tiny U}}
\put( 4.9 , 6.85){\makebox(0,0){\tiny U}}
\put( 2.55, 7.85){\makebox(0,0){\tiny V}}
\put( 3.35, 8.3 ){\makebox(0,0){\tiny V}}
\put( 2.85, 8.55){\makebox(0,0){\tiny V}}
\put( 3.25, 8.6 ){\makebox(0,0){\tiny V}}
\put( 3.0 , 8.7 ){\makebox(0,0){\tiny V}}
\put( 2.25, 8.8 ){\makebox(0,0){\tiny V}}
\put( 3.6 , 9.4 ){\makebox(0,0){\tiny V}}
\put( 3.7 , 9.4 ){\makebox(0,0){\tiny V}}
\put( 4.6 ,10.2 ){\makebox(0,0){\tiny V}}
\put( 5.15, 9.8 ){\makebox(0,0){\tiny V}}
\put( 5.2 , 9.85){\makebox(0,0){\tiny V}}
\put( 5.35,10 ){\makebox(0,0){\tiny V}}
\put( 5.5 ,10 ){\makebox(0,0){\tiny V}}
\put( 5.55,10 ){\makebox(0,0){\tiny V}}
\put( 6.3 ,10.65){\makebox(0,0){\tiny V}}
\put( 2.85, 2.35){\makebox(0,0){\tiny G}}
\put( 3.5 , 1.9 ){\makebox(0,0){\tiny G}}
\put( 2.5 , 2.05){\makebox(0,0){\tiny S}}
\put( 2.8 , 8.9 ){\makebox(0,0){\tiny 7}}
\put( 3.3 , 9.05){\makebox(0,0){\tiny 7}}
\put( 6.1 , 8.85){\makebox(0,0){\tiny 7}}
\put( 4.85,10.1 ){\makebox(0,0){\tiny 7}}
\put( 5.4 ,10.6 ){\makebox(0,0){\tiny 7}}
\put( 3.75, 9.3 ){\makebox(0,0){\tiny 7}}
\end{picture} }
\caption{Joint plot of the MPs, parties and issues for the
Parliament 1972 study.}
\protect\label{fig:parl_joint}
\end{figure}
On the whole, the representation of MPs, parties and issues is quite
convincing, because it accumulates evidence against the idea of a
one-dimensional polity on the one hand, and, on the other hand, it
demonstrates the apparent possibility of predicting the stands on
seven major political issues from a two-dimensional preference space.
It summarizes, with little loss of information, 2232 observations in
284 parameters. The close resemblance to Heiser's (1981) analysis is
also worth mentioning.
\section{Preference for family composition}
Our final application is a partial reanalysis of preference data
collected by Delbeke (1978)\footnote{We are indebted to Luc Delbeke for
kindly making these data available to us.}. The objects in this study
are 16 family types, the individuals 82 undergraduates at Leuven
University in Belgium, who ranked the family types in order of
preference. Family types are defined as all combinations of {\it
number of sons} and {\it number of daughters}, each ranging from 0 to
3. Thus (2,1) indicates two sons and one daughter, and (0,3) means no
sons and three daughters. The number of sons and the number of
daughters have been used to create two variables, defining the target
configuration (see Figure \ref{fig:family_target}). When we would
connect the points horizontally, we find family types with an equal
number of daughters. Connecting the points vertically gives families
with an equal number of boys.
\begin{figure}
\centerline{\footnotesize
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\put(3,0){\makebox(0,0){(3,0)}}
\put(3,1){\makebox(0,0){(3,1)}}
\put(3,2){\makebox(0,0){(3,2)}}
\put(3,3){\makebox(0,0){(3,3)}}
\end{picture} }
\caption{Target configuration for the analysis of family composition
preference.}
\protect\label{fig:family_target}
\end{figure}
In Coombs' ({\it et al.}, 1973) theory regarding family composition
preferences two new variables are defined in terms of the old ones.
These are the {\it size} and the {\it sex} of the families. The main
idea of the present analysis is to apply PREFMAP-3 with a {\it
preliminary full transformation} of the target configuration. This
might give us a clue on the question which of the two possibilities is
more appropriate for describing Delbeke's family composition
preferences.
\begin{figure}
\centerline{\footnotesize
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\put( 6.55, 7.55){\circle*{.07}}
\put( 9.05, 9.10){\circle*{.07}}
\put( 8.55, 8.50){\circle*{.07}}
\put( 7.55, 8.10){\circle*{.07}}
\put( 7.15, 8.05){\circle*{.07}}
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\put( 6.30, 8.55){\circle*{.07}}
\put( 7.70, 7.75){\circle*{.07}}
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\put( 8.85, 7.05){\circle*{.07}}
\end{picture} }
\caption{Transformed target configuration and ideal points from the analysis
of preferences for family composition.}
\protect\label{fig:family_trans}
\end{figure}
The average fit for the 82 subjects, analyzing their preferences with
the Unfolding model, metrically, with preliminary full transformation,
is .906; the proportion of the total variance accounted for is .823.
Without a preliminary full transformation, these figures are .899 and
.814, resp. The increase in fit is obviously not dramatic. For the
preliminary transformation weights of 1.060 and .936 were obtained,
for the first new axis and the second, resp. Thus the axes are only
slightly differentially weighted. But the new axes' orientation is
interesting (see Figure \ref{fig:family_trans}). Here the subjects
points are labeled with a ``x" (nine individuals are not included in
the plot because their coordinates were severely out of range). It is
clear that the transformation recovered the ``size/sex" set of
variables. Looking at the family points we now have a large bias
towards daughters at the top, moving to a large bias towards sons at
the bottom. The horizontal axis coincides with the total number of
children. Evidently, this sample of Belgium students has high
preference for large families, while sons are preferred over
daughters.
For further analyses on this kind of data, we refer to Coxon (1974),
Bechtel (1976), and Rodgers and Young (1981).
\vspace{3cm}
Jacqueline Meulman
Willem J. Heiser
J. Douglas Carroll
\chapter*{References}
\begin{description}
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facial expressions. {\it J. Exp. Psych.}, {\it 63}, 546-554.
\item
Bechtel, G.G. (1976). {\it Multidimensional Preference Scaling}. The
Hague: Mouton.
\item
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Ann Arbor, Mich.: Mathesis Press, 441-478.
\item
Carroll, J.D. (1972). Individual differences and multidimensional
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{\it Vol. I: Theory}. New York: Seminar Press, 105-155.
\item
Carroll, J.D. (1980). Models and methods for multidimensional
analysis of preferential choice (or other dominance) data. In E.D.
Lantermann \& H. Feger (Eds.), {\it Similarity and Choice}. Bern: Hans
Huber Publ., 234-289.
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Carroll, J.D. \& Chang, J.J. (1970). Analysis of individual
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Cermak, G.W. \& Cornillon, P.C. (1976). Multidimensional analyses of
judgments about traffic noise. {\it J. Acoust. Soc. Am.}, {\it 59}, 1412-1420.
\item
Chang, J.J. \& Carroll, J.D. (1972). {\it How to use PREFMAP and
PREFMAP-2} -- Programs which relate preference data to multidimensional
scaling solutions. Unpublished manuscript, Bell Telephone Labs,
Murray Hill, NJ.
\item
Coombs, C.H. \& Avrunin, G.S. (1977). Single-peaked functions and
the theory of preference. {\it Psych. Rev.}, {\it 84}, 216-230.
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Coombs, C.H., McClelland, G.H. \& Coombs, L. (1973). {\it The measurement
and analysis of family composition preferences}. Internal Report,
Mich. Math. Psych. Progr. no. 73-5, University of Michigan.
\item
Coxon, A.P.M. (1974). The mapping of family composition preferences:
a scaling analysis. {\it Social Science Research}, {\it 3}, 191-210.
\item
Coxon, A.P.M. (1982). {\it The User's Guide to Multidimensional
Scaling}. London: Heinemann Educ. Books.
\item
Coxon, A.P.M. \& Jones, C.L. (1978). {\it The images of Occupational
Prestige}. New York: MacMillan.
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Davison, M.L. \& Jones, L.E. (1976). A similarity-attraction model
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Davison, M.L., King, P.M., Kitchener, K.S. \& Parker, C.A. (1980).
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Developm. Psych.}, {\it 16}, 121-131.
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Delbeke, L. (1979). {\it Enkele analyses op voorkeuroordelen voor
gezinssamenstellingen}. Internal Report, Centrum voor Math. Psych.
en Psych. Meth., University of Leuven, Leuven, Belgium.
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Engen, T., Levy, N. \& Schlosberg, H. (1958). The dimensional analysis of a
new series of facial expressions. {\it J. Exp. Psych.}, {\it 55}, 454-458.
\item
Falbo, T. (1977). Multidimensional scaling of power strategies.
{\it J. Pers. Soc. Psych.}, {\it 35}, 537-547.
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Gifi, A. (1981). {\it Nonlinear Multivariate Analysis}. Leiden, The
Netherlands: Department of Data Theory, University of Leiden.
\item
Heiser, W.J. (1981). {\it Unfolding Analysis of Proximity Data}. Doctoral
Dissertation, University of Leiden, Leiden, The Netherlands.
\item
Heiser, W.J. \& De Leeuw, J. (1979). {\it How to use SMACOF-3}. Internal
Report, Department of Data Theory, University of Leiden, Leiden, The
Netherlands.
\item
Heiser, W.J. \& De Leeuw, J. (1981). Multidimensional mapping of
preference data. {\it Math. Sci. Hum.}, {\it 19}, 39-96.
\item
Heiser, W.J. \& Meulman, J. (1983). Constrained multidimensional
scaling, including confirmation. {\it Appl. Psych. Meas.}, {\it 7}, 381-404.
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Kruskal, J.B. (1964). Multidimensional scaling by optimizing
goodness of fit to a nonmetric hypothesis. {\it Psychometrika}, {\it
29}, 1-28.
\item
Kruskal, J.B. (1965). Analysis of factorial experiments by
estimating monotone transformation of the data. {\it J. Roy. Stat.
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Kruskal, J.B. \& Shepard, R.N. (1974). A nonmetric variety of
linear factor analysis. {\it Psychometrika}, {\it 39}, 123-157.
\item
Kuyper, H. (1980). {\it About the Saliency of Social Comparison
Dimensions}. Doctoral Dissertation, University of Groningen,
Groningen, The Netherlands.
\item
Nygren, T.E. \& Jones, L.E. (1977). Individual differences in
perceptions and preferences for political candidates. {\it J. Exp.
Soc. Psych.}, {\it 13}, 182-197.
\item
Rodgers, J.L. \& Young, F.W. (1981). Successive unfolding of family
preferences. {\it Appl. Psych. Meas.}, {\it 5}, 51-62.
\item
Seligson, M.A. (1977). Prestige among peasants: a multidimensional
analysis of preference data. {\it Am. J. Sociol.}, {\it 83}, 632-652.
\item
Shepard, R.N. (1962). The analysis of proximities, I \& II.
{\it Psychometrika}, {\it 27}, 125-140 and 219-246.
\item
Torgerson, W.S. (1958). {\it Theory and Methods of Scaling}. New York: Wiley.
\item
Young, F.W. (1981). Quantitative analysis of qualitative data. {\it
Psychometrika}, {\it 46}, 357-387.
\end{description}
\appendix
\chapter{Comparison with PREFMAP-2 and implementation}
\label{chap:appendixA}
%Comparison with PREFMAP-2 and miscellaneous technical information for
%implementation of PREFMAP-3.
\begin{sloppypar}
Only the most important differences between the PREFMAP-2 and the
PREFMAP-3 program will be described here; there are many disparities
in details, most notably in the input and the output, but these will
be evident to anyone switching from one program to the other.
\end{sloppypar}
From a practical point of view the PREFMAP-3 program has been designed
to be as {\it flexible} as possible. The array allocation is done
dynamically, so that -- given a large enough array to start with --
there are no severe limitations to the number of individuals, objects,
or dimensions to be analyzed (see Table \ref{tab:declar} for
examples). It is even possible to analyze an ``{\it infinite}" number
of individuals without any increase in space beyond what is required
to analyze one single individual. In that case the fact that every
row of the external data matrix is analyzed separately is maximally
exploited. Although we don't yet have experience with it, we expect
this feature will make it feasible to run PREFMAP-3 smoothly on a
Personal Computer (or other machines with modest core memory). Note,
however, that it will still require space to obtain print output
dealing with all individuals simultaneously (e.g., plotting the ideal
points in the target configuration); thus one should refrain from
those specifications when the data matrix is very large (and array
area is limited). However, the individual results will never be
completely lost, since they can be routed to other output units to
provide for the possibility of constructing composite plots
afterwards. Specific suggestions for dealing with various kinds of
situations are given in section \ref{sect:PrintPlot}.
A second major difference with the PREFMAP-2 program consists in the
choice of the {\it organization of the model fitting process}. In the
PREFMAP-2 program the same model must necessarily be fitted, at each
phase of the analysis, to each row of the data matrix. One has to
start at one stage in the hierarchy and to stop at another, solving
for all intermediate models. Moreover, the choice of the starting
point affects the results for the simpler models in the PREFMAP-2
program. Those organizational features have been changed quite
radically in the present program. Any ordered combination of models
can be chosen, for example the most complex (the general ideal point
model) and the simplest (the vector model), and they will be applied
to the same (transformed) target configuration. In addition, the
separate rows need not be fitted by the same models, although that is
still possible. The possibility exists to give each row its own set
of models.
The final major difference between PREFMAP-2 and PREFMAP-3 that we
want to mention consists in the fact that the present program performs
only {\it purely external analyses}. In the PREFMAP-2 program one out
of a number of possible object configurations could be generated.
This object configuration was obtained from the very same data matrix,
whereas the rows would be fitted in afterwards in the second stage of
the analysis. This two-stage process in fact renders an internal
Unfolding solution. Since there are many general MDS and PCA programs
available that perform internal analyses under various criteria, this
possibility does not exist anymore in PREFMAP-3. If the user desires
to employ, in PREFMAP-3, an object configuration derived from the
external data, this configuration must be calculated with some
auxiliary procedure, and must then be inserted in the input stream to
the PREFMAP-3 program.
\subsection*{Implementation}
PREFMAP-3 is a portable ANSI FORTRAN-IV program. It has been
successfully tested on IBM, Honeywell, and VAX computers. The program
contains a FORTRAN array allocation routine, called DECLAR. This
routine allocates a superarray of a certain specified length.
\begin{table}
\caption{Examples of analyses that can be performed within a
specified superarray size of 25100.}
\protect\label{tab:declar}
\begin{center}{\footnotesize\setlength{\tabcolsep}{3pt}
\begin{tabular}{rr*{9}{c}|c} \hline
NROW& NCOL& NDIM& NANA& IPRE& VP& UP& WP& GP& IPRI& IPLO& NWORDS \\ \hline
350& 20 & 3 & 4 & 2 & 1 & 1 & 1 & 1 & 1 & 3 & 24033 \\
500& 65 & 2 & 4 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 24934 \\
1000& 75 & 2 & 3 & 2 & 1 & 1 & 1 & 0 & 1 & 2 & 24307 \\
1000& 115 & 2 & 1 & 2 & 0 & 1 & 0 & 0 & 1 & 2 & 24831 \\
2000& 85 & 2 & 1 & 2 & 0 & 1 & 0 & 0 & 1 & 2 & 24643 \\
3000& 40 & 2 & 1 & 2 & 0 & 1 & 0 & 0 & 1 & 2 & 24263 \\
3500& 10 & 2 & 1 & 2 & 0 & 1 & 0 & 0 & 1 & 2 & 25093 \\
99999& 130 & 3 & 1 & 2 & 0 & 1 & 0 & 0 & 0 & 0 & 24266 \\
99999& 730 & 4 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 24934 \\
99999& 890 & 2 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 24794 \\
\hline
\end{tabular}
\begin{tabular}{llll}
\\ \\
{\it Legend:} \\ \\
NROW & = number of rows &
UP & = Unfolding model, primary app. \\
NCOL & = number of columns &
WP & = weighted Unfolding model, ditto \\
NDIM & = number of dimensions &
GP & = general Unfolding model, ditto \\
NANA & = number of analyses &
IPRI & = print selected results \\
IPRE & = preliminary transformation &
IPLO & = plot pairwise dimensions \\
VP & = vector model, primary approach &
NWORDS & = total array area \\
\end{tabular}}
\end{center}
\end{table}
The required size of the superarray depends on a number of parameters,
some of which vary with the size of the problem (number of objects,
dimensions), while others depend on the user-selected analysis and
print specifications. If the superarray is not large enough, the
program will return from DECLAR with an error message and with the
correct size of the superarray. Table \ref{tab:declar} gives an overview of
different analyses that can be performed within the internally fixed
upper bound of the superarray (25100 words). This upper bound can, of
course, be enlarged when implementing the program. If a genuine
dynamic storage allocation facility is derived, subroutine DECLAR
should be replaced by a machine assembler routine.
On some installations the plots produced by the program might come out
rectangular instead of square. Several statements in the subroutine
PRPLOT should be adapted according to the comments in this subroutine.
The logical unit numbers to read the standard input stream and to
write the standard output stream are defined in the main routine by
the statements INPARA = 5 and IWRITE = 6, respectively.
\subsection*{Program structure}
In terms of its subroutines, the program is structured as indicated in
Figure \ref{fig:flow}.
% (unfortunately, the name PLOT appears in the
%Figure instead of the correct subroutine name PRPLOT).
Here, successive subroutine calls are given from top to bottom,
whereas further calls into a deeper level are depicted horizontally.
The subroutine MAIN3 controls the flow of the program. PREREC is used
to perform printing of the input data and configuration, PREPO
performs the preliminary transformation, PRED1 up to PRED4 construct
predictor matrices suitable for the selected models, and PSINV
computes their pseudo inverses.
\begin{figure}
\centerline{\footnotesize
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%
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%
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%
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%
\put( 0 ,15 ){\line(1,0){1}}
\put( 1 ,15 ){\makebox(0,0)[l]{~MONO}}
\put( 4 ,15 ){\line(1,0){2}}
\put( 5 ,13 ){\line(0,1){4}}
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\put(10 ,12 ){\makebox(0,0)[l]{~MRMNH}}
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\put(12 ,14 ){\line(1,0){1}}
\put(13 ,14 ){\makebox(0,0)[l]{~SHEL9}}
%
\put( 0 ,17 ){\line(1,0){1}}
\put( 1 ,17 ){\makebox(0,0)[l]{~ESTIMA}}
%
\put( 0 ,18 ){\line(1,0){1}}
\put( 1 ,18 ){\makebox(0,0)[l]{~ZEROI}}
%
\put( 0 ,19 ){\line(1,0){1}}
\put( 1 ,19 ){\makebox(0,0)[l]{~PSINV}}
\put( 4 ,19 ){\line(1,0){2}}
\put( 6 ,19 ){\makebox(0,0)[l]{~SIVAD}}
%
\put( 0 ,20 ){\line(1,0){1}}
\put( 1 ,20 ){\makebox(0,0)[l]{~PRED4}}
%
\put( 0 ,21 ){\line(1,0){1}}
\put( 1 ,21 ){\makebox(0,0)[l]{~PRED3}}
%
\put( 0 ,22 ){\line(1,0){1}}
\put( 1 ,22 ){\makebox(0,0)[l]{~PRED2}}
%
\put( 0 ,23 ){\line(1,0){1}}
\put( 1 ,23 ){\makebox(0,0)[l]{~PRED1}}
%
\put( 0 ,24 ){\line(1,0){1}}
\put( 1 ,24 ){\makebox(0,0)[l]{~PREPRO}}
\put( 4 ,24 ){\line(1,0){2}}
\put( 5 ,22 ){\line(0,1){4}}
\put( 6 ,24 ){\makebox(0,0)[l]{~TRED2}}
\put( 5 ,26 ){\line(1,0){1}}
\put( 6 ,26 ){\makebox(0,0)[l]{~PRED4}}
\put( 5 ,25 ){\line(1,0){1}}
\put( 6 ,25 ){\makebox(0,0)[l]{~PSINV}}
\put( 5 ,23 ){\line(1,0){1}}
\put( 6 ,23 ){\makebox(0,0)[l]{~IMTQL2}}
\put( 5 ,22 ){\line(1,0){1}}
\put( 6 ,22 ){\makebox(0,0)[l]{~PREREC}}
%
\put( 0 ,25 ){\line(1,0){1}}
\put( 1 ,25 ){\makebox(0,0)[l]{~ZERO}}
%
\put( 0 ,26 ){\line(1,0){1}}
\put( 1 ,26 ){\makebox(0,0)[l]{~PREREC}}
%
\put( 0 ,20.5){\line(-1,0){1}}
\put(-3 ,20.5){\makebox(2,0){MAIN3}}
\put(-3 ,20.5){\line(-1,0){1}}
\put(-6 ,20.5){\makebox(2,0){MAIN2}}
\put(-6 ,20.5){\line(-1,0){1}}
\put(-10 ,20.5){\makebox(0,0)[l]{~DECLAR}}
%
\put(-8.5,21 ){\line(0,1){1}}
\put(-8.5,22.5){\oval(3,1)}
\put(-10 ,22 ){\makebox(3,1){START}}
%
\put(-8.5,20 ){\line(0,-1){1}}
\put(-8.5,18.5){\oval(3,1)}
\put(-10 ,18 ){\makebox(3,1){STOP}}
%
\put(-1 ,17.5){\vector(1,0){1}}
\put(-1 , 2.5){\vector(1,0){1}}
\put(-1 , 2.5){\line(0,1){15}}
\put(-0.7 , 4 ){\shortstack{O\\P\\T\\I\\O\\N\\S}}
%
\put(-2 ,18.5){\vector(1,0){2}}
\put(-2 , 1 ){\vector(1,0){2}}
\put(-2 , 1 ){\line(0,1){17.5}}
\put(-2.7 ,10 ){\shortstack{L\\O\\O\\P\\ \\A\\C\\R\\O\\S\\S}}
\put(-1.7 , 8 ){\shortstack{R\\O\\W\\S}}
\end{picture} }
\caption{Flow of the PREFMAP-3 program.}
\protect\label{fig:flow}
\end{figure}
The program contains two major loops. The inner one is across
options, the outer one is across rows of the data matrix. The inner
loop computes for each option successively:
\begin{itemize}
\item[(1)] the predicted values for the metric case (ESTIMA);
\item[(2)] the optimal monotone function for the nonmetric case (by cycling
between PROJEC and MONOR, the Alternating Least Squares iterations);
\item[(3)] the predicted values for the nonmetric case (ESTIMA);
\item[(4)] the parameter estimates of the original model (UNRA1 up to UNRA4);
\item[(5)] RESUL1 and RESUL2 take care of the various output options.
\end{itemize}
When no composite plotting or printing is asked for, the individual
results will be lost from here on. After the loop across options is
finished, statistics across options are given (STATIS). Then the
program continues by finishing the outer loop across individuals.
Composite results are provided for by OUTP3. Of course, the program
skips subroutines when the activities performed by them are not called
for by the user.
\chapter{Deck set-up PREFMAP-3}
\label{chap:appendixB}
{\footnotesize
\noindent
\begin{tabular}{r@{--}lcp{8cm}}
\multicolumn{4}{c}{{\it CARD 1: TITLE CARD} (section \ref{sect:generalsetup})} \\
\multicolumn{4}{c}{} \\
\multicolumn{2}{c}{\underline{Col.}} & {\underline{Format}} &
\multicolumn{1}{l}{{\underline{Information}}} \\
1&80 & 20A4 & Any alphanumeric information to title the printout \\
\end{tabular}
\vspace{1cm}
\noindent
\begin{tabular}{r@{--}lcp{8cm}}
\multicolumn{4}{c}{{\it CARD 2: DATA SPECIFICATION} (section
\ref{sect:DataSpecifications})} \\
\multicolumn{4}{c}{} \\
\multicolumn{2}{c}{\underline{Col.}} & {\underline{Format}} &
\multicolumn{1}{l}{{\underline{Information}}} \\
1 & 5 & I5 & number of row points \\
6 & 10 & I5 & number of column points \\
11 & 15 & I5 & option set selection \\
\multicolumn{3}{c}{}& 0 = all rows same option set \\
\multicolumn{3}{c}{}& 1 = all rows different option set \\
\multicolumn{3}{c}{}& 2 = specified rows same option set \\
16 & 20 & I5 & row number to start first option set (only needed if
col. 11--15 equals 2; ditto for the next three parameters) \\
21 & 25 & I5 & row number to start second option set \\
26 & 30 & I5 & row number to start third option set \\
31 & 35 & I5 & row number to start fourth option set \\
\end{tabular}
\vspace{1cm}
\noindent
\begin{tabular}{r@{--}lcp{8cm}}
\multicolumn{4}{c}{{\it CARD 3: ANALYSIS SPECIFICATION} (section
\ref{sect:AnalysisSpecifications})} \\
\multicolumn{4}{c}{} \\
\multicolumn{2}{c}{\underline{Col.}} & {\underline{Format}} &
\multicolumn{1}{l}{{\underline{Information}}} \\
1 & 5 & I5 & number of dimensions \\
6 & 10 & I5 & maximum number of analyses in any option set \\
11 & 15 & I5 & preliminary transformation of configuration \\
\multicolumn{3}{c}{}& 0 = remains unchanged \\
\multicolumn{3}{c}{}& 1 = weighted \\
\multicolumn{3}{c}{}& 2 = rotated and weighted \\
16 & 20 & I5 & standardize external data \\
\multicolumn{3}{c}{}& 0 = yes (default, amounts to 1 + 2) \\
\multicolumn{3}{c}{}& 1 = center only \\
\multicolumn{3}{c}{}& 2 = normalize only \\
\multicolumn{3}{c}{}& 3 = none of the above \\
21 & 25 & I5 & application vector model (in any option set) \\
\multicolumn{3}{c}{}& 0 = no \\
\multicolumn{3}{c}{}& 1 = yes \\
26 & 30 & I5 & application Unfolding model (in any option set) \\
\multicolumn{3}{c}{}& 0 = no \\
\multicolumn{3}{c}{}& 1 = yes \\
31 & 35 & I5 & application weighted Unfolding model (in any option set) \\
\multicolumn{3}{c}{}& 0 = no \\
\multicolumn{3}{c}{}& 1 = yes \\
36 & 40 & I5 & application general Unfolding model (in any option set) \\
\multicolumn{3}{c}{}& 0 = no \\
\multicolumn{3}{c}{}& 1 = yes \\
41 & 45 & I5 & number of nonmetric iterations (default = 50) \\
46 & 55 & F10.8 & convergence criterion for nonmetric regression (default =
.10E-04) \\
\end{tabular}
\vspace{1cm}
\noindent
\begin{tabular}{r@{--}lcp{8cm}}
\multicolumn{4}{c}{{\it CARD 4: PRINT/PLOT OPTIONS} (section
\ref{sect:PrintPlot})} \\
\multicolumn{4}{c}{} \\
\multicolumn{2}{c}{\underline{Col.}} & {\underline{Format}} &
\multicolumn{1}{l}{{\underline{Information}}} \\
1 & 5 & I5 & print input \\
\multicolumn{3}{c}{}& 0 = 10 rows of the external data at most \\
\multicolumn{3}{c}{}& 1 = target configuration \\
\multicolumn{3}{c}{}& 2 = external data \\
\multicolumn{3}{c}{}& 3 = target configuration as well as external data \\
6 & 10 & I5 & print results for each option \\
\multicolumn{3}{c}{}& 0 = none \\
\multicolumn{3}{c}{}& 1 = complete results \\
\multicolumn{3}{c}{}& 2 = fit only \\
\multicolumn{3}{c}{}& 3 = fit + criterion and predicted values \\
11 & 15 & I5 & print selected results for each analysis \\
\multicolumn{3}{c}{}& 0 = none \\
\multicolumn{3}{c}{}& 1 = coordinates, weights, rotation matrices \\
16 & 20 & I5 & scatter and transformation plots \\
\multicolumn{3}{c}{}& 0 = no plots \\
\multicolumn{3}{c}{}& N = plot for first N rows \\
21 & 25 & I5 & plot ideal points in target configuration \\
\multicolumn{3}{c}{}& 0 = no plots \\
\multicolumn{3}{c}{}& 1 = plot first dimension only \\
\multicolumn{3}{c}{}& 2 = plot first two dimensions \\
\multicolumn{3}{c}{}& K = plot K dimensions (pairwise) \\
26 & 30 & I5 & print history of nonmetric regression \\
\multicolumn{3}{c}{}& 0 = no print \\
\multicolumn{3}{c}{}& 1 = print for each row \\
31 & 35 & I5 & print F-statistic for metric analyses \\
\multicolumn{3}{c}{}& 0 = no statistics \\
\multicolumn{3}{c}{}& 1 = F-statistic for each row \\
36 & 40 & I5 & store output 1 \\
\multicolumn{3}{c}{}& 0 = no storage \\
\multicolumn{3}{c}{}& 1 = store coordinates \\
\multicolumn{3}{c}{}& 2 = store coordinates and weights \\
\multicolumn{3}{c}{}& 3 = store coordinates, weights and rotation
matrices \\
41 & 45 & I5 & store output 2 \\
\multicolumn{3}{c}{}& 0 = no storage \\
\multicolumn{3}{c}{}& 1 = store predicted values \\
\multicolumn{3}{c}{}& 2 = store predicted values and criterion values \\
\end{tabular}
\vspace{1cm}
\noindent
\begin{tabular}{r@{--}lcp{8cm}}
\multicolumn{4}{c}{{\it CARD 5: UNIT NUMBERS FOR INPUT/OUTPUT} (section
\ref{sect:unitnumbers})} \\
\multicolumn{4}{c}{} \\
\multicolumn{2}{c}{\underline{Col.}} & {\underline{Format}} &
\multicolumn{1}{l}{{\underline{Information}}} \\
1 & 5 & I5 & unit number to read the target configuration from
(default = 5) \\
6 & 10 & I5 & unit number to read the external data from (default = 5) \\
11 & 15 & I5 & unit number to read the option table (default = 5) \\
16 & 20 & I5 & unit number for scratch file 1 \\
21 & 25 & I5 & unit number for scratch file 2 \\
26 & 30 & I5 & unit number to store the coordinates (default = 6) \\
31 & 35 & I5 & unit number to store the weights (default = 6) \\
36 & 40 & I5 & unit number to store the rotation matrices (default = 6) \\
41 & 45 & I5 & unit number to store the predicted and the criterion
values (default = 6) \\
\end{tabular}
\vspace{1cm}
\noindent
\begin{tabular}{r@{--}lcp{8cm}}
\multicolumn{4}{c}{{\it CARD 6: FORMAT CARD FOR THE CONFIGURATION} (section
\ref{sect:generalsetup})} \\
\multicolumn{4}{c}{} \\
\multicolumn{2}{c}{\underline{Col.}} & {\underline{Format}} &
\multicolumn{1}{l}{{\underline{Information}}} \\
1 & 80 & 20A4 & FORTRAN F-format to read the configuration \\
\multicolumn{4}{c}{}\\
\multicolumn{3}{c}{}& {\it TARGET CONFIGURATION CARDS} (section
\ref{sect:generalsetup}) \\
\multicolumn{3}{c}{}& The target configuration must appear here if
the input medium is unit 5. The configuration must have the form as
given in Figure \ref{fig:target}, i.e. as many (groups of) cards as
there are objects, and as many fields as there are dimensions \\
\end{tabular}
\vspace{1cm}
\noindent
\begin{tabular}{r@{--}lcp{8cm}}
\multicolumn{4}{c}{{\it FORMAT CARD FOR THE EXTERNAL DATA} (section
\ref{sect:generalsetup})} \\
\multicolumn{4}{c}{} \\
\multicolumn{2}{c}{\underline{Col.}} & {\underline{Format}} &
\multicolumn{1}{l}{{\underline{Information}}} \\
1 & 80 & 20A4 & FORTRAN F-format to read the external data \\
\multicolumn{4}{c}{}\\
\multicolumn{3}{c}{}& {\it EXTERNAL DATA MATRIX CARDS} (section
\ref{sect:generalsetup}) \\
\multicolumn{3}{c}{}& The external data must appear here if
the input medium is unit 5. The data matrix must have the form as
given in Figure \ref{fig:data}, i.e. as many (groups of) cards as
there are individuals, and as many fields as there are objects \\
\end{tabular}
\vspace{1cm}
\noindent
\begin{tabular}{r@{--}lcp{8cm}}
\multicolumn{4}{c}{{\it OPTION TABLE CARDS} (section
\ref{sect:options})} \\
\multicolumn{4}{c}{} \\
\multicolumn{2}{c}{\underline{Col.}} & {\underline{Format}} &
\multicolumn{1}{l}{{\underline{Information}}} \\
1 & 16 & 4A4 & The option table must appear here if
the input medium is unit 5. Each card contains one option set,
consisting of 1 up to 4 options. Each option is coded with two
characters; the first option code starts in column 1, the second
starts in column 5, etc.\\
\multicolumn{4}{c}{}\\
\multicolumn{3}{c}{}& First characters may be: \\
\multicolumn{4}{c}{}\\
\multicolumn{3}{c}{}&
{\begin{tabular}{r@{=}l}
V & Vector model \\
U & Unfolding model \\
W & Weighted Unfolding model \\
W & General Unfolding model \\
\end{tabular}} \\
\multicolumn{3}{c}{}&\\
\multicolumn{3}{c}{}& Second characters may be: \\
\multicolumn{4}{c}{}\\
\multicolumn{3}{c}{}&{
\begin{tabular}{r@{=}l}
M & Metric regression \\
P & Nonmetric regression, primary approach to ties \\
S & Nonmetric regression, secondary approach to ties \\
\end{tabular} }\\
\end{tabular}
}
\end{document}
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