MDPREF is a program for analysis of preferences. The analysis is usually conducted on a matrix of averaged preference evaluations that have been derived by aggregating preference evaluations for respondents. This data would most often show the average preference for the total sample, or subsample of subjects that we were interested in.
MDPREF analyzes a subject (vector) x stimuli (points) matrix containing the preference data.
In this situation, the term subject does not necessarily refer to people. Most often we are analyzing a matrix of average preferences for a set of objects that are evaluated on a set of attributes. Again, this attribute by object matrix contains the average preference evaluations.
Using the nomenclature of MDPREF, the input data matrix is defined as having subjects (rows that produce vectors on the map). These rows are often defined as attributes that describe the stimuli. The stimuli are the object defined by the columns of the matrix. In an application that does not involve the attribute-object matrix, the subject vectors could take on any number of forms, but are usually attributes descriptive of the stimuli (groups, entities, items, points) defined by the columns.
MDPREF is what is known as a "VECTOR MODEL". This means that the objective of the MDPREF analysis is to identify a perceptual map displaying subject (attribute) vectors. The vector model assumes a linear model such that preference is greatest at the end of the subject vector, and infinitely better as one moves an infinite distance along the vector. To form the subject vectors visually, lines are drawn from the origin of the plot to each subject point. Next, the stimuli (object) points are plotted by MDPREF. Each stimuli point projects (at a 90 degree angle to the vector) onto each subject vectors. This projection shows the average subjects metric preference of the stimuli with respect to the subject vectors.
Operationally, preferences may be measured as a simple ranking (1-8 if 8 items are ranked on attribute 1), or on a value scale.
MDPREF is designed to do multidimensional scaling of preference or evaluation data. MDPREF is a metric model based on a principal components analysis (Eckart-Young decomposition). In this analysis, a data matrix of dimension i subjects by j stimuli is decomposed into two smaller matrices, each of which approximates the original data matrix in a least squares sense.
The first of these resulting matrices is called a principal component score (or factor score) matrix of size (i x r), where r is the number of principal components. This matrix depicts the i subjects in the r principal component dimensions and is designated as [PCS].
The second matrix is called the principal component loading matrix (or factor loading matrix), and is of size (r x j). This matrix depicts the j stimuli in the r principal component dimensions and is designated as [PCL].
The original MDPREF program recognized two forms of input: paired comparisons data, and stimuli evaluation data. The PC-MDS version of MDPREF has deleted the paired comparisons data option because of the infrequent collection and use of such data. Originally it was from the paired-comparison matrices, that MDPREF derived a single matrix called the 'first score matrix' of dimension i rows and j columns. In the PC version of MDPREF, the first score matrix is the data matrix input by the user, and is designated S*. Each cell of the S* matrix contains a numerical entry (i,j), which represents the ith subject's rating of the jth stimuli, as measured by the researcher's survey instrument.
The 'first score matrix' which again, is a subject by stimuli matrix of evaluation scores, is decomposed into r dimensions or principal components. The first score matrix is additionally used to produce the [PCS] and [PCL] matrices discussed above. Subsequent to this analysis, a second score matrix is produced, having dimensions (i x j). The second score matrix contains derived projections of stimuli onto subject vectors. The values of the second score matrix agree as near as possible, in a least squares sense, with the first scores matrix.
MDPREF is valued as an analytical procedure because the resulting values in the [PCS] and [PCL] matrices project the stimuli onto subject vectors within the multidimensional stimuli attribute space. This multidimensional space allows for visual evaluation of the j stimuli an r dimensional space, where r<j.
MDPREF is simple to run. If operating from a your hard drive, the MDPREF program can be
accessed through either a DOS window or through Microsoft's EXPLORER. If using EXPLORER, simply access the
directory where MDPREF is located and click on the program icon. A DOS window will open and MDPREF will run.
If you choose to open your own DOS window, move to the directory where MDPREF is located and type:
MDPREF
The MDPREF program logo will appear, along with the three program options: CONTINUE, HELP, AND QUIT. The help file provides a listing of acceptable input parameter values and data file arrangement. A sample data set is provided for inspection in the help file. A test data set containing the same data is found on the program diskette under the name of MDPREF.DAT. At the completion of the help file, MDPREF begins.
The user must specify (1) a title, (2) the drive and name of the data file, and (3) the drive and name of the output file. For convenience, it is recommended that the data files use the '.DAT' extension to indicate 'data', and the output file maintain a '.PRN' extension for 'print'.
The final input required to run MDPREF is the specification of where the parameters are to found. If the first line of the data file contains the parameters, 0 is appropriate, otherwise 1 may be entered to enter parameters from the keyboard. The selection of option 1 brings to the screen a series of six questions requesting values for the six input variables identified as NP, NS, NF, NPF, IREAD, and NORP in the help file.
After parameter input, MDPREF completes the analysis.
MDPREF Input Parameters
The commands must be provided in sequence given below. Each step refers to one line image, or several related images. To read in a data matrix, each row must begin a new image, but may continue on several consecutive lines if necessary.
1. Parameter line - FREE format
NP = Number of rows (vectors) in the data matrix (max. NP=100)
NS = Number of columns (points) in the data matrix (max. NS=100)
NP*NS must be less than 10000)
NF = Number of factors (or dimensions)
NF must be less than the Minimum of NP or NS.
And (NP*NF) must be less than 10000)
NPF = Number of factors to be plotted
(NPF must be less than or equal to NF )
IREAD = 1 read in the first score matrix (S*),
which must have NP rows and NS columns.
Each row of S* will be normalized by
subtracting the row mean.
= 2 Same as IREAD = 1, except in addition
each row of S* will be divided by the
standard deviation of values in the row
NORP = 0 Normalize subject vectors
= 1 Do not normalize
2. Format for reading the first score matrix up to 80 characters long the second line in the setup must contain a format statement for reading in the first score matrix.
3. Data images are read for the first score matrix. The first score matrix must be entered as a np by ns matrix, where, each subject begins a new line.
4. Labels for 3-D plots. 9 Characters maximum is allowed for each label. The column labels are read first, followed by the row labels. All labels must be present. Use a blank line if a label should be blank.
8 10 3 2 1 0
(10F5.2)
5.79 6.49 5.80 2.91 4.29 4.03 5.73 1.38 5.22 2.86
3.42 3.89 4.87 5.66 4.93 4.36 3.14 5.18 5.24 3.89
4.68 5.57 3.36 3.47 3.63 5.40 4.61 4.84 3.80 4.50
3.32 4.24 5.01 6.08 6.22 4.47 2.71 3.73 5.35 3.52
4.56 4.19 5.56 5.08 5.52 4.77 4.15 2.77 5.24 2.78
3.35 2.21 4.05 5.86 6.31 5.10 2.24 5.63 5.35 3.98
3.95 3.70 5.28 5.21 5.61 4.89 3.71 4.03 5.17 2.98
3.07 2.71 4.73 6.33 6.31 4.24 3.08 5.07 5.12 4.15
Fruity
Carbonation
Calories
Tart
Thirst
Popularity
Aftertaste
Pick-up
Coke
Coke Cl.
Diet Pepsi
Diet Slice
Diet 7-up
Dr Pepper
Pepsi
Slice
Tab
7-up
Fruity
Carbonation
Calories
Tart
Thirst
Popularity
Aftertaste
Pick-up
Notes
1. The MDPREF program was written by J. D. Carroll and J. J. Chang of Bell Telephone Laboratories. This guide is a revision of the paper 'How to use MDPREF, a Computer Program for Multidimensional Analysis of Preference Data' by J. J. Chang and J. D. Carroll. Revisions made by Scott M. Smith for the PC version of MDPREF.
SAMPLE MDPREF OUTPUT
M D P R E F |INPUT DATA: 8 ATTRIBUTES X 10 BRANDS
MULTIDIMENSIONAL ANALYSIS OF PREFERENCE DATA | 8 10 2 2 1 0
PROGRAM WRITTEN BY DR. J. D. CARROLL AND JIH JIE CHANG |(10F5.2)
PC - MDS VERSION |5.79 6.49 5.80 2.91 4.29 4.03 5.73 1.38 5.22 2.86
ANALYSIS TITLE: POP DATA ATTRIBUTE BY OBJECTS IN 2 DIMENSIONS |3.42 3.89 4.87 5.66 4.93 4.36 3.14 5.18 5.24 3.89
DATA IS READ FROM FILE: ATTXBR.POP |4.68 5.57 3.36 3.47 3.63 5.40 4.61 4.84 3.80 4.50
OUTPUT FILE IS: ATTXBR.PRN |3.32 4.24 5.01 6.08 6.22 4.47 2.71 3.73 5.35 3.52
|4.56 4.19 5.56 5.08 5.52 4.77 4.15 2.77 5.24 2.78
NP (NO. OF SUBJECTS) 8 |3.35 2.21 4.05 5.86 6.31 5.10 2.24 5.63 5.35 3.98
NS (NO. OF STIMULI) 10 |3.95 3.70 5.28 5.21 5.61 4.89 3.71 4.03 5.17 2.98
NF (NO. OF DIMENSIONS) 2 |3.07 2.71 4.73 6.33 6.31 4.24 3.08 5.07 5.12 4.15
NFP (NO. OF DIMENSIONS PLOTTED) 2 |
|INPUT SPECIFICATIONS:
IREAD 1=NP X NS SCORE MATRIX WITH ROW MEAN SUBTRACTED 1 | 8 = NUMBER OF SUBJECTS (ATTRIBUTE VECTORS)
2=SAME AS 1 WITH SCORES DIVIDED BY ROW S. D. |10 = NUMBER OF STIMULI (Objects)
|The parameters call for a 2 dimensional principal
NORP 0=NORMALIZE SUBJ. VECTORS 0 |components solution. Two dimensions will be
1=DO NOT |plotted.
|The data form and normalization options are
*****IDENTIFICATION KEY FOR PLOTS WITH IDENTIFIED POINTS***** |specified
|
PT # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 |Codes for reading the two dimensional plots.
CHAR 1 2 3 4 5 6 7 8 9 A B C D E F |
PT # 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 |
CHAR G H I J K L M N O P Q R S T U |
PT # 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 |
CHAR V W X Y Z + / = * & $ @ % ? < |
PT # 46 47 48 49 50 |
CHAR ( ) " ; @ |
POINT NUMBERS ABOVE 50 IDENTIFIED AS >, MULTIPLE POINTS IDENTIFIED AS # |
|
IN JOINT SPACE PLOTS, THE FIRST 10 POINTS ARE STIMULI |
AND THE NEXT 8 POINTS ARE SUBJECTS |
|
INPUT FORMAT = (3X,10F7.4) |
|
MEAN OF THE RAW SCORES (BY SUBJECT) |Mean of the subject variables.
4.4543 4.4624 4.3913 4.4703 4.4666 4.4107|These are the row variables. In the example
4.4561 4.4860 |the means of each of the 8 attributes are given.
|
FIRST SCORE MATRIX (SUBJECT BY STIMULUS) |The original data matrix values minus the subject
1 1.3347 2.0367 1.3527 -1.5423 -.1563 -.4193|(row) mean. (i.e., 1.3347 = 5.789-4.4543 or in
1.2827 -3.0683 .7737 -1.5943 |other words, fruity/coke - average fruitiness
. |
. |The first score matrix is a preference score matrix,
. |where each entry is the preference rating made by
. |the ith subject on the jth stimulus.
8 -1.4160 -1.7670 .2510 1.8470 1.8300 -.2400|The first score matrix is decomposed by MDPREF into
-1.3980 .5840 .6370 -.3280 |NF dimensions.
|
CROSS PRODUCT MATRIX OF SUBJECTS |The cross products matrix is an intermediate matrix
1 24.5385 -6.2786 .7973 -1.8374 7.8784 -14.3155|used in the computation of the subject x subject
.6241 -10.7526 |(attribute x attribute) correlation matrix.
. |
. |
. |
8 -10.7526 8.6405 -6.5557 11.2627 4.1261 15.7736|
7.3943 14.8171 |
|
CORRELATION MATRIX OF SUBJECTS |The subject x subject (attribute x attribute)
1 1.0000 -.4985 .0680 -.1042 .5223 -.6536|correlation matrix is the basis for computing
.0475 -.5639 |the underlying dimensionality of the data matrix.
. |
. |
. |
8 -.5639 .8829 -.7200 .8223 .3520 .9268|
.7249 1.0000 |
|
CROSS PRODUCT MATRIX OF STIMULI |
1 7.6801 9.1070 .1489 -9.9256 -8.0465 -.7597|
9.7955 -5.9361 -3.1618 1.0983 |
. |
. |
. |The eigenvalues or characteristic roots of the
10 1.0983 .5196 -6.0104 -3.2425 -6.6664 -.5405|principal components factor analysis. For principal
3.3761 7.9758 -5.5578 9.0480 |components analysis, the eigenvalues equal the sum
|of the squared correlations (squared loadings)
ROOTS OF THE FIRST SCORE MATRIX |of the subjects (attributes) on stimuli (objects).
62.5203 30.2225 3.3362 2.0768 .9772 .5864|In other words, this is the sum of the r squares
.1766 .0212 |and shows the amount of variance accounted for by
|each component or dimension underlying the principal
PROPORTION OF VARIANCE ACCOUNTED FOR BY EACH FACTOR |components factor analysis.
1 2 |
.6257 .3025 |The proportion of variance accounted for by dimensions
|one and two shows that 62.57% of all variance is
CUMULATIVE PROPORTION OF VARIANCE ACCOUNTED FOR |accounted for by dimension 1 and 32.25% of variance
1 2 |is accounted for by dimension 2.
.6257 .9282 |
|The cumulative sum of variance accounted for shows
SECOND SCORE MATRIX (SUBJECT BY STIMULUS) |that 92.82% of all preference variance is accounted
1 .2667 .3995 .2930 -.2647 -.0529 -.0627|for by the first two dimensions.
.2813 -.6385 .1074 -.3291 |
. |The second score matrix is derived projections of
. |stimuli (objects) onto subject (attribute) vectors.
. |This is as nearly proportional as possible to the
8 -.3323 -.4352 .0518 .4608 .4299 .0421|first score matrix.
-.4577 .1791 .1969 -.1354 |
POPULATION MATRIX |
FACTOR |The population matrix is the dimension 1 and 2 plot
1 .6091 .7931 |projections of subjects (attributes) vectors.
2 -.9974 .0726 |The coordinates of the subject vectors are on the
3 .8406 -.5416 |unit circle (Euclidean distance from origin = 1.0).
4 -.8559 .5172 |
5 -.3318 .9434 |
6 -.9953 -.0972 |
7 -.7584 .6518 |
8 -.9989 .0460 |
|
STIMULUS MATRIX (NORMALIZED) |
FACTOR |The projections of ten stimuli (brands) on to
1 .3362 .0781 |dimensions 1 and 2. These are the coordinates used
2 .4431 .1634 |in graphs 2 and 3.
3 -.0336 .3953 |
4 -.4604 .0198 |
5 -.4186 .2549 |
6 -.0442 -.0451 |
7 .4583 .0027 |
8 -.2090 -.6446 |
9 -.1843 .2769 |
10 .1125 -.5013 |
|
STIMULUS MATRIX (STRETCHED BY SQ. ROOT OF THE EIGENVALUES) |
FACTOR |By stretching stimuli (objects) relative to the
1 2.6584 .4291 |square root of the eigenvalues, the scales are
2 3.5039 .8983 |weighted for the amount of variance explained by
3 -.2659 2.1731 |each dimension (analogy: weighting by importance).
4 -3.6401 .1090 |This matrix is not included in the plots.
5 -3.3101 1.4012 |
6 -.3496 -.2480 |
7 3.6236 .0146 |
8 -1.6522 -3.5437 |
9 -1.4575 1.5225 |
10 .8893 -2.7561 |
|
|
PLOT OF SUBJECTS IN DIMENSIONS 1 AND 2 |
+....+....+....+....+....+....+....+....+....+....+....+....+ |
. | . |
. | . |
.97+ 5 | + |
. | 1 . | 1-8 = 8 Subjects (attributes)
. 7 | . |
.55+ 4 | + |
. | . |
. | . |
.14+ 2 | + |
.---------------8--------------0------------------------------. |
. 6 | . |
-.28+ | + |
. | . |
. | 3 . |
-.69+ | + |
. | . |
. | . |
-1.11+ | + |
+....+....+....+....+....+....+....+....+....+....+....+....+ |
-2.0 -1.7 -1.3 -1.0 -.7 -.3 .0 .3 .7 1.0 1.3 1.7 2.0 |
PLOT OF STIMULUS POINTS IN DIMENSIONS 1 AND 2 |
+....+....+....+....+....+....+....+....+....+....+....+....+ |
. | . |1-9 + A = 10 Stimuli (brands)
. | . |B - I = 8 subjects (attribute) vectors
.97+ | + |
. | . |The projection of stimuli (objects) on each subject
. | . |vector is as similar as possible to the order of
.55+ | + |preference expressed by the subject in the original
. 3| . |preference data.
. 5 9 | . |
.14+ | 1 2 + |
.----------------------4------60------7-----------------------. |
. | . |
-.28+ | + |
. | . |
. | A . |
-.69+ 8 | + |
. | . |
. | . |
-1.11+ | + |
+....+....+....+....+....+....+....+....+....+....+....+....+ |
-2.0 -1.7 -1.3 -1.0 -.7 -.3 .0 .3 .7 1.0 1.3 1.7 2.0 |
|
|
PLOT OF STIMULI AND SUBJECTS IN DIMENSIONS 1 AND 2 |
+....+....+....+....+....+....+....+....+....+....+....+....+ |
. | . |
. | . |
.97+ F | + |1-9 + A = 10 Stimuli (objects)
. | B . |
. H | . |
.55+ E | + |
. 3| . |
. 5 9 | . |
.14+ C | 1 2 + |
.---------------I-------4-----60------7-----------------------. |
. G | . |
-.28+ | + |
. | . |
. | A D . |
-.69+ 8 | + |
. | . |
. | . |
-1.11+ | + |
+....+....+....+....+....+....+....+....+....+....+....+....+ |
-2.0 -1.7 -1.3 -1.0 -.7 -.3 .0 .3 .7 1.0 1.3 1.7 2.0 |