Property-fitting
is typically used as a form of interpretation of a configuration, using external
information about the stimuli, and mapping each into the configuration as
(increasing) directions or vectors . It takes as input both a configuration of
stimulus points and a set of rankings or ratings of the same set of stimuli.
These rankings and ratings are usually estimates of different properties of the
stimuli. The program locates each property as a vector through the configuration
of points, so that it indicates the direction over the space in which the
property is increasing. The fitting is accomplished by maximising the
correlation between the original property values and the projection of the
stimuli onto the vector. This correlation may be either linear or non-linear
(continuity). PROFIT using the linear option is formally identical to Phase 4
(vector model) of the preference mapping program PREFMAP, also using the linear option.
An
internal form of the point-vector model (i.e. where the input configuration is
not fixed but is generated from the data) is available in MDPREF.
An
option within PARAMAP allows a rectangular or
row-conditional (two-way, two mode) array of data to be input for internal
analysis using a continuity (kappa) transformation between the data and the
solution. But only the stimuli are represented in the
solution.
There
are two parts to the input data for PROFIT:
The
configuration
The configuration consists of the coordinates for a set of
objects (stimuli) on a number of dimensions. This may be an a priori
configuration, or one resulting from another multidimensional scaling analysis,
or, indeed, from a factor analysis. The configuration is input to the program by
means of the READ CONFIG command, in free format (or under an associated
INPUT FORMAT specification), and may be presented either stimuli (rows)
by dimensions (columns) or dimensions (rows) by stimuli (columns). In the latter
case the parameter MATFORM should be given the value 1. Since the
configuration is not substantially altered by the PROFIT algorithm, analysis can
only take place in a given dimensionality and attempts to specify more than one
value in the DIMENSIONS command will cause an error.
The
properties
Each of the "properties" which PROFIT will seek to represent as
vectors in the configuration, is a set of values which distinguish the stimuli
on a particular criterion. These may be physical values or subjective
evaluations of the stimuli on criteria other than those used to generate the
original configuration. For instance, a simple use of the program might be to
map into a MINISSA representation of the perceived
similarities between a set of stimuli, information about the subjects'
preferences of the same stimuli.
Input
of properties
Each property consists of a set of values, one for each
stimulus in the configuration. Values may be input in free format.
(Alternatively, ALL properties must be in the same format, given by an
associated INPUT FORMAT specification which precedes the READ
MATRIX command which reads the properties.) Each property, however, is
preceded by a line containing a label, which appears in the
output.
PROFIT
seeks to represent the properties as vectors over the configuration of points.
The analysis is external in as much as the
configuration is regarded as being fixed: the stimulus points cannot be moved to
make the fit of the vectors better.
The
fitted vector is regarded as indicating the direction in which the given
property is increasing. As a theory this implies that preference increases
continually, never reaching a maximum (corresponding to the economic concept of
insatiability).
The
linear procedure:
1. The columns of the configuration are normalised.
2.
The XMAT matrix is computed.
For each property in
turn:
3. The direction cosines of the vectors are computed.
4. The
projections of the points onto the vectors are computed.
5. The correlation
between the projections and the property
values is
computed.
6. The cosines corresponding to the angles between each
pair
of vectors are computed.
7. The configuration
and vector-ends are plotted using both
normalised and
original coordinates.
The
non-linear procedure:
1. The configuration is
normalised.
For each property:
2. KAPPA and ZSQ
measures of alienation and correlation
respectively
are computed.
3. The cosines of the angles between the vectors and the
original
axes are calculated.
4. The projections
of the points onto the vectors are calculated.
When
all properties have been treated in this way :
5. The cosine of the angle
between each pair of vectors is
calculated.
6. The
configuration of points and vectors is plotted
in
original and normalised
co-ordinates.
The use
of the WEIGHT parameter
The weighting function plays a
crucial role in the definition of KAPPA. This function can take on three
different values and each value defines a different "flavour" of KAPPA. The
choice of flavour depends crucially on the characteristics of the property
values:
WEIGHT (0)
This is the general definition of non-linear
correlation and no restrictions are placed on the data. Therefore, this index
can always be applied to examine the extent to which the property values (data)
and the projections of the stimulus points (solution) are related by a smooth or
continuous function.
WEIGHT (1)
In this case, it is assumed that the
property values are equally spaced. So the level of measurement of the
properties is in effect taken to be ordinal if the order is specified with equal
intervals. To do this any equally spaced values may be chosen, such as 1, 2,
3,...N or 5, 10, 15,...5N.
There is no restriction on the characteristics of
the stimulus configuration when using this option. This option limits the
calculation of KAPPA to adjacent points. In this case, K becomes equivalent to
Von Neumann's Eta (the ratio of the mean square successive difference). See
below for the use of BCO in conjunction with this
option.
WEIGHT (2)
If the property values tend to be highly clustered into
two or more groups of values, then PROFIT can be used to determine
whether this is also the case for the projections of the stimuli on the fitted
vector. To do this we must choose the property values in such a way that it
becomes possible to discriminate the clusters. Ordinal level of measurement is
sufficient, provided the property values are equally spaced. By defining the
maximum distance between two points which are to be taken as falling in the same
grouping, the program then selects the clusters. This maximum distance is set
using the BCO parameter.
The weight factor will now have the effect of
restricting attention to property distances which are close to each other (in
effect, in the same grouping) and ignoring values outside the BCO value.
In this case, K can be shown to be the equivalent of the "correlation ratio".
The use
of the BCO parameter In the general
case a value of 0 for BCO (the default) will make the weighting function
undefined for equal property values. If there are equal property values and
BCO the program will terminate. This option in effect assumes that there
are no ties between the property values. If ties do occur among your property
values then a small value of BCO (say .001) should be used. This will
allow calculation of the weight factor even when the property values are equal.
A large value for BCO has the effect of allowing Kappa to decrease
indefinitely and is not recommended. WEIGHT
(1) When Von
Neumann's Eta is approximated, then the value of the BCO parameter has a
more simple explanation than in the previous case. Now BCO simply gives
the size of the equal intervals. Note that woth WEIGHT (1), which is the default
value, then BCO (0) has no meaning and some other value must be
specified. WEIGHT
(2) In this case the
BCO parameter gives the maximum distance allowed between points in the
hypothetical clusters described above. Again, in this case, the default value
BCO (0) has no meaning, and must be over-ridden by some other
value. INPUT COMMANDS
Keyword
Function Keyword
Default
Function NOTES PRINT
options (to main output
file)
property (linear
regression). PLOT
options (to main output
file) By
default only the first two dimensions of the joint space are
plotted. PUNCH
options (to secondary
output
file) By
default, no secondary output file is produced. PROGRAM
LIMITS See also
This parameter has a different use and meaning
when used in conjunction with different WEIGHT options:
WEIGHT
(0)
N OF SUBJECTS/
[number]
Number of subjects or "properties"
PROPERTIES
N
OF STIMULI
[number]
Number of stimuli in the analysis
DIMENSIONS
[number]
Dimensionality for the analysis (one
only)
LABELS [followed
by a Optionally identify the stimuli,
followed
by
series
of labels the properties. All labels
required must
be
each
on
a entered,
without
omissions.
separate line]
PARAMETERS
REGRESSION 1 1:
Linear regression will be
performed.
2:
Non-linear regression will be
performed.
3:
Both regressions performed
(independently).
MATFORM 1 1: The
configuration is input
stimuli
(rows) by dimensions
(columns).
2:
The configuration is input
dimensions
(rows) by stimuli
(columns).
WEIGHT 0
(See above for relation to
BC0)
0:
Carrolls index of
continuity.
1:
Van Neumann's ration of the
mean.
2:
The "correlation
ratio".
BC0 0
(See above for relation to WEIGHT)
1. N
OF PROPERTIES may be used in PROFIT in place
of
N OF SUBJECTS.
2. READ CONFIG is
obligatory.
3. LABELS Allows you to add optional labels, on successive
lines
following the command, to identify the stimuli
and properties.
4. Since the non-linear option involves calculation of large
powers
of the data values, exponent overflow may
occur. In this case
the data values should be made
smaller. This might be done by
changing the format
statement so as to divide the values by,
say,
100.
Option
Form
Description
INITIAL p
x r The matrix of stimulus
points
as
normalised by the program. This
will
differ in linear and non-linear approaches.
CORRELATIONS
The
following are
output:
(Default) 1(a) the correlations for each
(b)
the eigenroots associated with
each
vector
(non-linear
regression).
PROPERTIES
The following
are
output:
N x r 1. the direction
cosines between
each
of
the fitted vectors and
dimensions
in
the normalised space.
N
x r 2. the direction cosines
between
each
of
the fitted vectors and
dimensions
of
the original
space.
N x N 3. the cosines of the
angles
between
the
vectors.
RESIDUALS A
table of residuals is
output
i.e.
obtained distances - original distances.
Option
Description
INITIAL The
stimulus configuration plotted in pairs
of
dimensions
with both original and
normalised
co-ordinates
marked (up to r(r-1)/2
plots).
FINAL Both
stimulus points and property
vectors
plotted
together; original and
normalised
co-ordinates
(up to r(r-1)/2
plots).
SHEPARD N plots
of original property values
against
projections
on fitted vectors giving
the
shape
of the linking
function.
RESIDUALS Histogram of residual
values.
Option Description
SPSS This
command produces a file containing
the
following
variables:
i
property
j
stimulus
DATA
original value on property i
of
stimulus
j
FITTED
projection on fitted
vector
RESID
difference between original and
fitted
values.
SOLUTION Two matrices
are
output:
i)
the matrix of stimulus points
as
normalised,
and
ii)
the matrix of direction cosines for
the
fitted
vectors.
Maximum no. of stimuli = 200
Maximum no. of subjects/properties =
200
Maximum no. of dimensions = 10