provides internal analysis of two-way data of either a set of paired comparisons matrices or a rectangular, row-conditional matrix by means of a vector model, using a linear transformation of the data.
DATA:
2-Way, 2 mode dis/similarities, preferences (or 3-way, 2-mode dominance data
for pair-comparisons option)
TRANSFORM: Linear
MODEL:
Scalar Products (Vector)
INPUT
DATA
MDPREF accepts input data in either of two main forms:
(i) as a set of dominance (0,1) pair-comparisons matrices, or
(ii) as a set of rankings or ratings forming a rectangular, so-called
"first-score" matrix.
Options within the program differ with different data input. The type of input
is indicated by the DATA TYPE parameter in the PARAMETERS
command.
1.
The first-score matrix - DATA TYPEs 1-4
Suppose a set of N subjects is asked to rank in order of preference, or give a
rating to a set of p stimuli. The resultant data form a rectangular
'row-conditional' matrix with N rows (subjects) and p columns (stimuli), called
the "first score matrix" in MDPREF. Each row of the
matrix represents the preference rank or score assigned by that subject to the
stimuli.
Such
a matrix can also be obtained by taking the pair comparison matrix for a given
subject and summing each row. The resultant column of scores gives that
subject's (possibly weak) rank order of preference for the stimuli and these
may be collected to form the "first-score matrix".
Ranks
vs. Scores
Preference judgments may be represented for MDPREF (as in MINIRSA and other procedures) in four distinct ways.
The major distinction is that between a rank and a score. Ranked data may be
input to MDPREF in this form by specifying DATA TYPE(1) if
the order is most-preferred to least-preferred, or DATA TYPE(2) if it is
the reverse.
If the data represent 'scores', so that the lowest number is used to denote
the least preferred stimulus and the highest to represent the most preferred, the
option is indicated by DATA TYPE(3). Alternatively, the highest
number might have been used to represent the least preferred stimulus and if this is
so, DATA TYPE(4) should be specified.
The
pair-comparisons matrices DATA TYPE(0)
Suppose a subject is asked to consider all possible pairs of p stimuli and for
each pair to indicate which stimulus is preferred (or which stimulus
possesses more of a given attribute), they are asked to make p (p-1)/2
judgments of preference. The data obtained may be collected into a square,
asymmetric matrix whose rows and columns each represent the p stimulus points,
and entries a(i,j) take the value 1 if the subject prefers stimulus i to
stimulus j and 0 if the opposite is the case. The subject may be allowed to
express indifference between the stimuli, or express no opinion
on a particular pair comparison.
The
READ CODES command instructs the program to read in four code values,
the first of which represents preference, the second its opposite
("anti-preference"), the third indifference, and the fourth is a
missing data value. When using the input Wizard, you
will be prompted to enter these values, before proceeding to the spreadsheets to enter the subjects' preference
matrices using the codes specified.
If
there are N subjects performing this test of preference, then there will be N
such matrices. These are input to MDPREF by specifying DATA TYPE(0) in
the PARAMETERS command.
The
N pair comparisons matrices will be read by the READ MATRIX command.
They may be in free format (or entered according to an optional INPUT FORMAT
specification, to read one row of the input matrices), and the individual
matrices should follow each other without separation.
If
there are missing data, then MISSING(1) should be specified in the PARAMETERS
command.
The
MDPREF model represents the preferences of a subject for a group
of stimuli as a vector through the configuration of stimulus points. This
vector indicates the direction in which his (her) preference increases over the
space. Substantively this makes strong assumption about the nature of
preference, in that the model implies an "ideal" point - i.e. a point
of maximum preference - at infinity (which is similar to the classic
econometric assumption of insatiability). In MDPREF, where the point
of maximum preference is at infinity, the contours become perpendicular to the
vector).
MDPREF is a
linear (or metric) procedure and the measure of goodness-of-fit of the model to
the data is a product-moment correlation. Consider one subject vector passing
through a configuration of stimulus points with the perpendicular lines drawn
from the points onto the vector. It is the values given to the points at which
these perpendicular lines meet the vector which are maximally correlated with
that subject's data. (This is guaranteed by the Eckart Young decomposition).
The subject vectors are normalised (for convenience only) to the same length,
i.e. so that their ends lie at a common distance from the origin of the space,
forming a circle, sphere or hypersphere as the case may be. Thus when a
solution of more than 3 dimensions is represented (as it must be) as a set of
2-dimensional plots, some of the vectors will not, in fact, lie on the boundary
circle since they will have been projected down from the higher dimensions. The
length of the vector in the sub-space is related to the amount of variation in
that subject's data explained by those two dimensions of the solution space. In
the graphic displays of these results, an
additional menu item Vectors enables you to plot
or suppress the subject vectors if these are becoming too cluttered.
Dimensionality
The program lists the latent roots of the matrices. The number of positive
roots will be equal to the number of stimuli or the number of subjects,
whichever is the smaller. The magnitude of the roots gives an indication of the
amount of variation in the data accounted for by that dimension. The largest
root will always be first and the others will follow in decreasing order. Some
may be zero. An appropriate dimensionality may be chosen by means of the
scree-test.
Normalising
and Centering
With the data in the form of a first score matrix the user may choose how the
matrix is to be centred and normalised using the parameters CENTRE and NORMALISE.
The default for these parameters is 0 and means no action.
Other
options allow various courses:
CENT(1) instructs the program simply to subtract the row means. This
will, in a rating exercise, remove any effect due to differences in the actual
values used by particular subjects.
NORM(1) allows the
program not only to subtract the row means but also to take out any effect due
to differences in the range or spread of scores involved by normalising each
row by dividing it by its standard deviation.
CENT(2) and NORM(2)
perform the same operations on the column elements, i. e. subtracting column
means and column normalising respectively. This latter option has the effect of
taking out the unanimity effect in subjects judgements and leaving only the significant
differences in judgements.
CENT(3) instructs
the program to double centre the matrix by subtracting both row and column
means. NORM(3) does this and normalises the entire matrix.
Weighting
of pair comparison matrices
Since pairwise judgements are often difficult to make, the user may sometimes
wish to accord to each judgement a 'weight'. This might represent the degree of
confidence which the subject attaches to his/her judgement, or perhaps the
reliability which the researcher ascribes to each judgement.
If
weights are input (indicated by WEIGHTS(1) in the PARAMETERS
command) there must be one weights matrix per subject. The weights matrix
immediately follows its associated pair comparisons matrix. This may be input
in free format, or read according to an optional WEIGHTS FORMAT
specification, which should be suitable for real (F-type) numbers.
Recurring
patterns of input
If, as often happens, there is more than one identical weights matrix, then the
number of such matrices may be specified as the SAME PATTERN parameter.
In this case, the weights matrix follows the first pair comparisons matrix.
Those pair comparisons matrices having the same pattern of weights then follow
each other without separation.
Blocking
of pair-comparisons data
If the number of possible pair-comparisons judgements has been thought too
great then the researcher may resort to the use of incomplete data, i.e.
certain element-pairs may not be presented to the subjects. The resulting data
matrix will have 'blocks' missing. If one of these strategies is used and the
data are arranged in blocks, then BLOCK(1) must be specified in the PARAMETERS
command so that allowance can be made in the calculation of row- and
column-sums.
INPUT
PARAMETERS
Keyword
Default
Function
N OF SUBJECTS [number]
Number of subjects
N OF STIMULI [number]
Number of stimuli
DIMENSIONS [number]
Dimensions of the data
DATA TYPE 0 0:
Data are in a pair-comparisons
matrix
1: Data are ranks (I-scales) of column
indices in decreasing
order of preference
2: As 1
but in increasing order of
preference.
3: Data are scores in order of column
indices - high score means high preference.
4:
As 3 but high scores mean low preference.
OPTIONS
WITH THE FIRST-SCORE MATRIX
Keyword
Default
Function
MATFORM 0 0:
The matrix is entered subjects
(rows) by stimuli (columns).
1: The matrix is entered
stimuli
(rows) by subjects (columns).
GROUPS 0 The
number of groups present in an
analysis of variance should be specified.
CENTRE 0:
The data are not centred.
1:
Row-means only are subtracted.
2: Column-means
only are subtracted.
3: Matrix is double centred.
NORMALISE 0 0:
Matrix is not normalised.
1:
Rows are centred and normalised.
2: Columns are
centred and normalised.
3:
Both rows and columns are
centred and normalised.
OPTIONS
WITH PAIRED-COMPARISONS MATRICES
Keyword
Default
Function
SAME PATTERN 0 Sets the number of subjects
whose
pattern
of missing data or weights
are the same.
WEIGHTS 0
0: No weights are input.
1: Weights are input.
BLOCK 0 0:
The data are not arranged in blocks.
1: The non-empty cells are arranged
in blocks or are to be treated as such.
NOTE:
Weights cannot be used with this option.
MISSING 0
0: There are no missing data.
1: There are missing data in the matrix.
GROUPS 0 The
number of groups present in an
analysis of variance should be specified.
CENTRE 0 0:
The data are not centred.
1: Row-means only are subtracted.
2:
Column-means only are subtracted.
3:
The matrix is double centred.
NORMALISE 0 0:
Matrix is not normalised.
1:
Rows are centred and normalised.
2: Columns
are centred and normalised.
3:
Both rows and columns are
centred and normalised.
NOTES
1. READ CONFIG, is not valid with MDPREF.
2. Even if only two or three codes are used in the paired comparisons matrices READ
CODES must specify four (integer) codes which must be in the order
specified.
PRINT
options
(to main output file)
Option
Form Description
FINAL p
x r The stimulus matrix,
followed by subject
N x r matrix.
FIRST
N x p The
first-score matrix. (This is the
input matrix after
being centred/normalised.
Means and standard
deviations of
subjects are
printed.
CROSS-PRODUCTS Four
matrices are printed:
N x N
1: the cross-product matrix of subjects;
p x p
2: the cross-product matrix of stimuli;
N x N
3: the correlation matrix of subjects;
p x p
4: the correlation matrix of stimuli.
SECOND N
x p The second-score matrix.
ROOTS
The latent roots.
RESIDUALS N x
p The first-score matrix minus
the
second-score.
CORRELATIONS N
The correlation for each subject between
the data and the
stimulus projections
is printed.
By
default, only the final configuration is printed.
PLOT
options
(to main output file)
Option
Description
SUBJECTS
The n(n-1)/2 plots of the subject
vectors in the chosen
dimensionalities.
STIMULI
The p(p-1)/2 plots of the stimulus
points
in the chosen
dimensionalities.
JOINT
Both of the above.
SHEPARD A
quasi-Shepard-plot - in this case
simply
the first-score plotted
against the second-score.
ROOTS
A scree diagram of the latent roots.
By
default, the first two dimensions of the joint space only are plotted
PUNCH
options
Option
Description
SUBJECT SPACE The final configuration
of subjects is output.
STIMULUS SPACE The final configuration of stimuli
is output.
By
default no secondary output file is produced.
PROGRAM
LIMITS
Maximum
no. of subjects = 200
Maximum no. of stimuli = 200
Maximum dimensions = 8
See also