The PREFMAP program relates a stimulus space to preference data (the stimulus space must be produced by another PC-MDS program such as KYST or INDSCAL). A hierarchy of vector and ideal point models are performed. PREFMAP starts with coordinate input describing a stimulus configuration in a specified number of dimensions. Also input is a set of preference rankings, in which each subject ranks the stimuli. In context of this stimulus input, a set of preference scales is evaluated.
PREFMAP finds for each subject, an ideal point that is positioned within the stimulus space. For each of these ideal points, the squared Euclidean distances from each stimulus to the ideal point are linearly (for metric model) or monotonically (for non-metric model) related to the preferences expressed by the subject. The ideal point is defined differently in each of the four models.
Each subject's data is evaluated using any of the four models (called phases). When a model is selected, it is tested for its ability to describe the distance between the subject's ideal points (or preference vector in model 4) and the stimuli.
The PREFMAP output consists of an analysis for each phase requested and a Summary Table showing correlations within phases, F ratios for each phase and F ratios analyzing differences between phases. It is the analysis of correlations that shows how much of the variance in a subject's preferences are explained by a given phase or model. The F ratios show the improvement attributable to movement to a more sophisticated model.
While the Summary Table is developed for, and is technically correct for evaluating only metric data, the table may be used as a directional indicator of relative improvement attributable to model shifts when non-metric data is analyzed.
PREFMAP relates preference data to a multidimensional solution via a hierarchy of models, ranging from a linear 'vector model' through the Coombs, Bennett and Hays 'multidimensional unfolding model', and finally including generalizations of the multidimensional unfolding model. The method is described in Carroll's paper (1972).
Given a stimulus configuration of N points in K dimensional space and a set of preference scales in the form of a subject-by- stimulus matrix, the program finds for each individual an ideal point in the given stimulus space such that the squared Euclidean distances (defined differently in each of the four models) from each stimulus to the ideal point are linearly (in the metric case) or monotonically (in the nonmetric case) related to the preferences expressed by the subject.
PREFMAP consists of four phases corresponding to the four models. Phase 1 is the most general model in which each individual is allowed their own orientation and weighting of dimensions. Phase 2 allows each individual differential weighting of dimensions only. In Phase 3 all subjects are assumed to have the same orientation and weighting of dimensions, while Phase 4 corresponds to the vector model. The algorithm for the linear vector model is basically the same as that described in Chang and Carroll (1968), for the case in which linear correlation between the preference scales (data) and the projections of the stimulus points on the fitted vectors is maximized.
PREFMAP allows either metric or nonmetric analysis (or both) in all four phases.
The input data is in the form of a subjects-by-stimuli (or objects) matrix. Each row of the matrix represents a subject and the entries in the row are the preference judgments made by that subject on each of the stimuli (or objects). The maximum number of subjects that can be analyzed in PREFMAP is 99 and the maximum number of stimulus points allowed is 150. The maximum number of dimensions allowed is five. The coordinates of N stimulus points in K dimensional space must be provided by the user in PREFMAP.
Printed Output
The results of the analysis are printed for each subject within each phase consecutively. They will be listed separately for each phase.
A. Phase 1--Each subject is allowed their own orientation and pattern of weights of the rotated dimensions. For each subject, the output consists of:
B. Phase 2--Each subject is allowed differential weighting of dimensions only.
C. Phase 3--
D. Phase 4--
E.
Plotted Output
The program produces four kinds of plots, namely: 1) the individual ideal point plot which is only appropriate for Phase 1 where the individual orientation of the axes is allowed. The plot shows the ideal point in its rotated stimulus space; 2) the composite ideal point plot is drawn respectively at the end of Phases 1, 2 and 3. The plot shows all the ideal points in the group stimulus space; 3) the function plot of obtained squared distances versus the scale values which shows how linearly or monotonically the solution is related to the preference scales; 4) the vector plot which is appropriate only for Phase 4 and shows each subject as a vector in the group stimulus space.
Input File Arrangement
Each of the numbered lines listed below may refer to a set of images, e.g., line 3 is a set of images including the format and the matrix of the scale values.
Line 1. Parameter Line (Must Contain ALL 15 Parameters)
Line 2, 3. Stimulus Space Coordinates.
Line 2. A format line (use floating point format).
Line 3. X--a matrix of the coordinates of N stimuli in K dimensional space. X must be entered as indicated by IRX.
Lines 4, 5. These lines are present only when IRWT = 1.
Line 4. A format line (using floating point format).
Line 5. Weights--one value for each dimension. There should be K values.
Lines 6, 7. Scale values.
Line 6. Format line for reading (use floating point format).
Line 7. S--scale values. S is a matrix of (NSUB) subjects by stimulus. Each row represents a subject.
Line 8. Line 8. Labels for each of the n stimuli, followed by labels for each of the nsub subjects.
Line 1. Parameters (Free Format). There are 15 parameters. The first six parameters are defined as the 'data parameters' which are mandatory. The remaining 9 are 'option parameters' which will be discussed in detail in the "Parameter Options" section. A summary of these parameters and their default values are given in Table 1. N, K, NSUB, ISV, NORS, IRX, IPS, IPE, IRWT, LFITSW, IAV, MAXIT, ISHAT, IPLOT, CRIT.
N number of stimuli. (Usually # Rows in 1st matrix)
K number of dimensions. (Usually # Columns in 1st matrix)
NSUB number of subjects. (# Rows in 2nd matrix)
ISV 0, small scale value represents greater preference.
1, large scale value represents greater preference.
NORS 1, normalize scale values, for each subject make length = 1.
0, do not normalize scale values.
IRX 0, the coordinates of the stimulus points are entered as N by K.
(Stimuli x Dimensions)
1, the coordinates are entered as K by N. (Dim. x Stim.)
OPTIONAL PARAMETERS (See Next Section:):
IPS, IPE, IRWT
LFITSW, IAV, MAXIT, ISHAT
IPLOT
CRIT
OPTIONAL PARAMETERS:
Phase Parameter Options--IPS, IPE, IRWT
The program computes solutions for Phase 1 to Phase 4 sequentially. However, the starting and the ending phases can be changed simply by setting IPS equal to the phase where computation starts and IPE to the phase it stops, for example, if IPS = 3 and IPE = 4, the program will compute solutions for Phases 3 and 4 only. In case the computation starts on Phase 3, one has the option of applying different weights to the dimensions of the group stimulus space. This can be accomplished by setting IRWT = 1 and the weights are read, as described in section IV.
Nonmetric Parameter Options--LFITSW, IAV, MAXIT, ISHAT
The nonmetric analysis employs an iterative procedure for monotone regression. On each iteration the program finds for each subject the best monotone fit of d2 (the squared Euclidean distances between the stimuli and the ideal point) to the data matrix S (the preference scale values). (See Kruskal 1964 a,b)
LFITSW: option of how d2 is assumed to be related to S.
--0 linearly (or metrically).
--1 monotonically (plan monotone regression, appropriate when there are no ties in the data).
--2 monotonically (block monotone regression using the primary approach of treating ties among values for a group of equal data values, no restriction is placed on ordering within blocks for the fitted values).
--3 monotonically (block monotone regression using the secondary approach of treating ties among data values for a group of equal data values, the fitted values are required to be equal).
IAV--option on how to define the preference scale values of the average subject in each phase. In the metric analysis this option is irrelevant because once the average subject's scale values are computed in the starting phase, they remain the same. However, in the nonmetric case each subject's scale values are replaced in each phase by the best monotone fit values. Therefore, the average subject's scale values are different in each phase.
--0 average subject's scale values are computed in the starting phase and remain the same.
--1 average subject's scale values are computed for each new phase.
MAXIT--number of iterations on monotone fit. The default value for MAXIT is 15.
ISHAT--parameter deciding which set of scale values are used in each phase.
--1 at the beginning of a new phase, use the last monotone fit of the scale values from the previous phase.
--0 at the beginning of a new phase, use the original scale values.
Plotting Parameters
With respect to the four kinds of plot earlier described, option parameter IPLOT designates the kinds of plot to be produced. As default the program generates the average subject's ideal point plot and the composite ideal point plot for Phases 1, 2 and 3,. for Phase 4, the vector plot.
Symbols on Plot
The stimulus points are labeled by sequential numbers generated in the program. The ideal points may be labeled by either numbers or letters.
Parameters
IPLOT--option on kinds of plot for Phases 1 and 2.
--0 draw ideal point plot for the average subject only.
--1 draw ideal point plot for the average subject only and function plots for each subject.
--2 draw ideal point plot for the average subject and both function plots and ideal point plots for each subject.
Iterative Procedure Stop
CRIT is the criterion for stopping iterative procedure on monotone fit. If (SI-SI-1)2 <_ CRIT, iteration is terminated. (Where I denotes Ith iteration.) (CRIT=.001 seems to be a reasonable value)
Carroll, J.D. "Individual Differences and Multidimensional Scaling." In R.N. Shepard, A.K. Romney, and S.B. Nerlove (Eds.), Multidimensional Scaling: Theory and Applications in the Behavioral Sciences. Vol. 1, 1972.
Carroll, J.D., and Chang, J.J. "An Alternate Solution To The 'Metric Unfolding Problem.'" Paper presented at the Psychometric Society meeting, April 1971, St. Louis, Missouri.
Chang, J.J., and Carroll, J.D. "How To Use PROFIT, A Computer Program For Property Fitting By Optimizing Nonlinear And Linear Correlation." Unpublished paper, Bell Laboratories, 1968.
Kruskal, J.B. "Multidimensional Scaling By Optimizing Goodness-of-fit To a Nonmetric Hypothesis." Psychometrika, 1964, 29, 1-27. (a)
Kruskal, J.B. "Nonmetric Multidimensional Scaling: A Numerical
Method." Psychometrika, 1964, 29, 115-129. (b)
11 2 5 0 1 0 2 4 0 1 1 15 0 0 1 (2X,10F7.3) 1 -.504 -.054 MUSTANG 2 .332 -.156 CADILLAC SEVILLE 3 .174 -.018 LINCOLN CONTINENTAL 4 -1.429 .083 FORD ESCORT 5 .323 .506 CORVETTE 6 -1.402 -.512 CHEV. CHEVETTE 7 .493 -.565 NISSAN 300 ZX 8 -1.142 .498 RENAULT ALLIANCE 9 .712 .381 PORSCHE 944 10 1.154 -.367 JAGUAR XJ6 11 1.290 .205 MERCEDES 500 SEL (5X,11F2.0) 001 0210071103120408010506 002 0103040502070911100809 003 0609081007110405030201 004 0301020805100611040709 005 0805070904110210060301 MUSTANG CADILLAC LINCOLN ESCORT CORVETTE CHEVETTE 300ZS ALLINACE 944 XJ6 500SEL SUBJ1 SUBJ2 SUBJ3 SUBJ4 SUBJ5
P R E F M A P | 11 2 5 0 1 0 2 4 0 1 1 15 0 0
MDSCALING VIA A GENERALIZATION OF COOMBS UNFOLDING MODEL | (2X,10F7.3)
BY DR. J. D. CARROLL AND JIH JIE CHANG | 1 -.504 -.054 MUSTANG
PC - MDS VERSION | 2 .332 -.156 CADILLAC SEVILLE
ANALYSIS TITLE: | 3 .174 -.018 LINCOLN CONTINENTAL
DATA IS READ FROM FILE: NEWCARPR.DAT | 4 -1.429 .083 FORD ESCORT
OUTPUT FILE IS: NEWCARPR.PRN | 5 .323 .506 CORVETTE
| 6 -1.402 -.512 CHEV. CHEVETTE
****************************************************************-*******| 7 .493 -.565 NISSAN 300 ZX
N NO. OF STIMULI 11 | 8 -1.142 .498 RENAULT ALLIANCE
K NO. OF DIMENSIONS 2 | 9 .712 .381 PORSCHE 944
NSUB NO. OF SUBJECTS 5 |10 1.154 -.367 JAGUAR XJ6
ISV 0=SMALL SCALE VALUE REPRESENTS GREATER PREF. 0 |11 1.290 .205 MERCEDES 500 SEL
NORS 1=NORMALIZE SCALE VALUES 1 |(5X,11F2.0)
IRX 0=STIMULUS COORDINATES N BY K, OR 1 = K BY N 0 |001 0210071103120408010506 BRAND RANKS
IPS STARTING PHASE 2 |002 0103040502070911100809 FOR FIVE
IPE ENDING PHASE 4 |003 0609081007110405030201 SUBJECTS
IRWT 1=READ IN WEIGHTS, 0=NO WEIGHTS READ IN 0 |004 0301020805100611040709
LFITSW HOW D**2 IS RELATED TO SCALE VALUES 1 |005 0805070904110210060301
0=LINEARLY, |
1=MONOTONE WITH NO TIES, |
2=BLOCK MONOTONE WITH ORDERING IN BLOCKS |INPUT PARAMETERS TO PREFMAP
3=BLOCK MONOTONE WITH EQUALITY IN BLOCKS |
IAV 0=AVERAGE SUBJECTS COMPUTED ONCE FOR ALL PHASES, 1 |
1=CALCULATE EACH PHASE |
MAXIT MAXIMUM ITERATIONS, WHEN 0 IT IS SET TO 15 15 |
ISHAT 0=USE SCALE VALUES FROM PREVIOUS PHASE, 0 |
1=USE ORIG VALUES |
IPLOT 0=AVERAGE SUBJECTS, 0 |
1=AVERAGE SUBJECTS & SUBJECT FUNCTIONS, |
2=ALL PLOTS |
CRIT CRITERIA FOR STOPPING MONOTONE FIT .0010 |
***********************************************************************|
*****IDENTIFICATION KEY FOR PLOTS WITH IDENTIFIED POINTS***** |PLOT IDENTIFICATIONS FOR FIRST
PT # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 |50 STIMULI
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 |
|
POINTS 1 TO 11 ARE STIMULI AND POINTS 12 TO 16 ARE IDEAL POINTS |
|
VARIABLE FORMAT (STIMULUS COORDINATES) = (2X,10F7.3) |
|
ORIGINAL CONFIGURATION (X MATRIX) |PRINTOUT OF ORIGINAL INPUT
|FOR THIS EXAMPLE, THE CONFIGURATION
1 -.50400 -.05400 |MATRIX WAS PRODUCED BY A KYST ANALYSIS
2 .33200 -.15600 |OF A LOWER HALF MATRIX.
3 .17400 -.01800 |
4 -1.42900 .08300 |
5 .32300 .50600 |
6 -1.40200 -.51200 |
7 .49300 -.56500 |
8 -1.14200 .49800 |
9 .71200 .38100 |
10 1.15400 -.36700 |
11 1.29000 .20500 |
|
VARIABLE FORMAT (SCALE VALUES) = (5X,11F2.0) |
PHASE 2 |PHASE 2 MODEL: ELLIPTICAL IDEAL POINT
|EACH SUBJECT REFLECTS DIFFERENTIAL
X MATRIX, (INPUT CONFIGURATION AFTER NORMALIZATION) |WEIGHTING OF THE DIMENSIONS ONLY.
1 -.5040 .3320 .1740 -1.4290 .3230 -1.|
.4930 -1.1420 .7120 1.1540 1.2900 |X MATRIX=ORIGINAL CONFIGURATION MATRIX
2 -.0540 -.1560 -.0180 .0830 .5060 -.| AFTER NORMALIZATION
-.5650 .4980 .3810 -.3670 .2050 |ITERATION PROCESS BEGINS, SOLVING FOR
|IDEAL POINT COORD. FOR EACH SUBJECT
PHASE 2 |
SUBJECT 1 |
SCALE VALUES BEFORE NORMALIZATION FOR SUBJECT 1 |
2.00000 10.00000 7.00000 11.00000 3.00000 |
4.00000 8.00000 1.00000 5.00000 6.00000 |
|S = NORMALIZED SUBJECT PREFERENCE DATA:
S (VECTOR OF SCALE VALUES, E.G. PREFERENCES) |EXAMPLE FOR SUBJECT 1:
-.36614 .31940 .06232 .40509 -.28045 |NORMALIZED ORIGINAL RANKS
-.19475 .14801 -.45183 -.10906 -.02337 |-.45183 = 1
BEGIN ITERATION ON MONOTONE FIT |-.36614 = 2
| . = .
ITERATION 1 | . = .
BETA VALUES (IN THE MOST GENERAL CASE THERE ARE (2K + K(K-1)/2 + 1) TE|+.31940 = 10
QUADRATIC, LINEAR, THEN A CONSTANT TERM) |+.40509 = 11
-.06728 -.14567 -.22842 .14894 -.47362 |
|ITERATION TO DETERMINE SUBJECT 1'S
(CORRELATION)= .75768 | DISTANCE FROM STIMULI
|
SIGNED DSQ, (SIGNED DISTANCE SQUARED FROM STIMULI TO IDEAL) |BETA VALUES ARE IMPORTANCES (WEIGHTS)
.13028 .00024 -.00880 .49816 -.26028 |ON EACH NEW OR ROTATED DIMENSION.
-.04967 .13746 -.17592 .05835 .00128 |NEGATIVE WEIGHTS INDICATE AN ANTI-
|IDEAL OR LEAST PREFERRED ITEM IN TERMS
ITERATION 2 |OF THAT PARTICULAR DIMENSION.
BETA VALUES (IN THE MOST GENERAL CASE THERE ARE (2K + K(K-1)/2 + 1) TE|
QUADRATIC, LINEAR, THEN A CONSTANT TERM) |BETA VALUES ARE THE UNSTANDARDIZED
-.12995 -.17341 -.23621 .20145 -.34357 |REGRESSION COEFFICIENTS FOR THE SCALE
|VALUES ON THE STIMULUS COORDINATES.
|THE FIRST BETA VALUE (-.13541) IS THE
|INTERCEPT . THE SIGN OF THE LAST BETA
|VALUE (-.27486) SHOWS WHETHER THE IDEAL
|POINT IS POSITIVE OR NEGATIVE. THE
|INTERMEDIATE AND LAST BETA VALUES SHOW
|THE IDEAL POINT COORDINATES.
SIGNED DSQ, (SIGNED DISTANCE SQUARED FROM STIMULI TO IDEAL) |
.02038 -.13682 -.14988 .50725 -.37243 |
-.14269 .12797 -.29115 -.02137 -.08127 |SIGNED DSQ = SQUARES OF DISTANCE FROM
|THE IDEAL POINT TO EACH STIMULUS
ITERATION 3 |
BETA VALUES (IN THE MOST GENERAL CASE THERE ARE (2K + K(K-1)/2 + 1) TE|
QUADRATIC, LINEAR, THEN A CONSTANT TERM) |
-.13541 -.17683 -.24129 .19738 -.27486 |
|
SIGNED DSQ, (SIGNED DISTANCE SQUARED FROM STIMULI TO IDEAL) |
.01608 -.14141 -.15595 .49841 -.36440 |
-.12603 .13561 -.29309 -.02510 -.09609 |
END OF ITERATION, REACHED CRITERION |
|
(CORRELATION)= .92762 |
|
SIGNED DSQ, (SIGNED DISTANCE SQUARED FROM STIMULI TO IDEAL) |
.13814 -.01935 -.03389 .62048 -.24234 |
-.00397 .25768 -.17103 .09697 .02597 |
|
***********************************************************************|
|COORDINATES OF THE IDEAL POINT IN
SUBJECT 1 |THE OLD GROUP STIMULUS SPACE.
|
COORDINATES OF IDEAL POINT WITH RESPECT TO OLD AXES |COORDINATES OF IDEAL POINT IN THE NEW
.44796 -.43893 |INDIVIDUAL SPACE. THESE ARE THE SAME
|AS THE LAST BETA.
IMPORTANCES OF NEW AXES |IN PHASE 2 ANALYSIS, ONLY THE SIGN
.19738 -.27486 |IS IMPORTANT. IF +, THE IDEAL IS
|POSITIVE, IF -, THE IDEAL IS NEGATIVE.
***********************************************************************|
SUBJECT 2 |THE SIZE OF THE NUMBER DOES NOT
. . |INFLUENCE THE DIMENSIONAL WEIGHTS IN
. . |THE PHASE 2 ANALYSIS.
. . |
SUBJECT 5 |
***********************************************************************|
AVERAGE SUBJECT |S = NORMALIZED PREFERENCES FOR THE
|AVERAGE SUBJECT. FOR THIS SAMPLE,
S (VECTOR OF SCALE VALUES, E.G. PREFERENCES) | 11 9 10 2 7 1 4 3 5 6 8
-.17416 -.14835 -.15716 .26763 -.10830 |IS THE AVERAGE SUBJECT'S PREFERENCE
-.01852 .25334 -.09050 -.10212 -.12932 |ORDER.
BETA VALUES (IN THE MOST GENERAL CASE THERE ARE (2K + K(K-1)/2 + 1) TE|
QUADRATIC, LINEAR, THEN A CONSTANT TERM) |
-.18166 -.12423 -.02234 .12500 .55753 |
|
(CORRELATION)= .98120 |
|
SIGNED DSQ, (SIGNED DISTANCE SQUARED FROM STIMULI TO IDEAL) |
.12829 .02068 .01384 .46587 .13545 |
.19083 .46313 .07842 .13749 .09770 |
|
************************************************************************|
SUBJECT 6 |
COORDINATES OF IDEAL POINT WITH RESPECT TO OLD AXES |
.49692 .02004 |
|
IMPORTANCES OF NEW AXES |
.12500 .55753 |
***********************************************************************|
|
PHASE 3 |
|
X MATRIX, (INPUT CONFIGURATION AFTER NORMALIZATION) |
1 -.1782 .1174 .0615 -.5052 .1142 -.|
.1743 -.4038 .2517 .4080 .4561 |
2 -.0403 -.1165 -.0134 .0620 .3778 -.|
-.4219 .3718 .2845 -.2740 .1531 |
|
PHASE 3 |
SUBJECT 1 |
|
S (VECTOR OF SCALE VALUES, E.G. PREFERENCES) |
-.20230 -.00337 -.10730 .57895 -.20230 |
-.14639 -.00337 -.34045 -.10730 -.10730 |
BEGIN ITERATION ON MONOTONE FIT |
|
ITERATION 1 |
BETA VALUES (IN THE MOST GENERAL CASE THERE ARE (2K + K(K-1)/2 + 1) TE|
QUADRATIC, LINEAR, THEN A CONSTANT TERM) |
-.15549 -.56457 -.32391 .85609 |
|
(CORRELATION)= .86617 |
|
SIGNED DSQ, (SIGNED DISTANCE SQUARED FROM STIMULI TO IDEAL) |
.26596 .11859 .09673 .61070 .07023 |
.34033 .48916 .01298 .18893 .01478 |
|
ITERATION 2 |
BETA VALUES (IN THE MOST GENERAL CASE THERE ARE (2K + K(K-1)/2 + 1) TE|
QUADRATIC, LINEAR, THEN A CONSTANT TERM) |
-.15368 -.59209 -.34717 .84614 |
|
SIGNED DSQ, (SIGNED DISTANCE SQUARED FROM STIMULI TO IDEAL) |
-.00593 -.15960 -.18208 .34319 -.22064 |
.06588 .21123 -.27939 -.09572 -.28103 |
END OF ITERATION, REACHED CRITERION |
|
(CORRELATION)= .89734 |
|
SIGNED DSQ, (SIGNED DISTANCE SQUARED FROM STIMULI TO IDEAL) |
.28694 .13327 .11079 .63606 .07222 |
.35875 .50410 .01348 .19715 .01184 |
|
***********************************************************************|
SUBJECT 1 |
COORDINATES OF IDEAL POINT WITH RESPECT TO OLD AXES |
.34988 .20515 |
|
IMPORTANCES OF NEW AXES |
.84614 .84614 |
***********************************************************************|
SUBJECT 2 |
. . |
. . |
. . |
SUBJECT 5 |
***********************************************************************|
AVERAGE SUBJECT |THE BETA VALUES ARE HERE DIMENSION
|WEIGHTS. THE LAST BETA WEIGHT IS
S (VECTOR OF SCALE VALUES, E.G. PREFERENCES) |REPEATED BELOW AS IMPORTANCES OF
-.17419 -.14698 -.17851 .28420 -.09283 |THE AXES.
-.07514 .25897 -.09655 -.11103 -.11926 |
BETA VALUES (IN THE MOST GENERAL CASE THERE ARE (2K + K(K-1)/2 + 1) TE|
QUADRATIC, LINEAR, THEN A CONSTANT TERM) |
-.19387 -.37799 -.00314 1.06721 |
|
(CORRELATION)= .98137 |
|
SIGNED DSQ, (SIGNED DISTANCE SQUARED FROM STIMULI TO IDEAL) |
.13657 .01865 .01449 .50077 .15538 |
.19127 .50646 .09142 .13791 .10760 |
|
***********************************************************************|
SUBJECT 6 |
|
COORDINATES OF IDEAL POINT WITH RESPECT TO OLD AXES |
.17709 .00147 |
|
IMPORTANCES OF NEW AXES |
1.06721 1.06721 |
***********************************************************************|
STIMULI COORDINATES |
DIMENSION 1 2 |
STIMULI |
1 -.17819 -.04032 |
2 .11738 -.11648 |
3 .06152 -.01344 |
4 -.50524 .06197 |
5 .11420 .37782 |
6 -.49569 -.38230 |
7 .17430 -.42187 |
8 -.40376 .37185 |
9 .25173 .28448 |
10 .40801 -.27403 |
11 .45609 .15307 |
COORDINATES OF IDEAL POINTS |
DIMENSION 1 2 |
SUBJECTS |
1 .34988 .20515 |
2 -.08728 .00457 |
3 -.49957 -.01849 |
4 -.00849 -.04644 |
5 1.31268 -.21980 |
6 .17709 .00147 |
SUBJECT 6 IS THE AVERAGE SUBJECT |WEIGHTS OF AXES SHOW THE IMPORTANCE
WEIGHTS OF AXES |OF NEW AXES AND IF THE IDEAL POINT
DIMENSION 1 2 |IS + OR - . THIS IS A REPEAT OF THE
1 .84614 .84614 |FINAL BETA LISTED FOR EACH SUBJECT.
2 2.42537 2.42537 |THE SIZE OF THE BETA SHOWS IMPORTANCE
3 -.92105 -.92105 |OF THE DIMENSION AS A COMPONENT OF
4 2.62741 2.62741 |PREFERENCE. A SMALL WEIGHT MEANS
5 .32343 .32343 |THAT LARGER CHANGES IN THE DIMENSION
6 1.06721 1.06721 |CAN BE MADE WITHOUT INFLUENCING
SUBJECT 6 IS THE AVERAGE SUBJECT |PREFERENCE.
STIMULI AND IDEAL POINTS: |
.*....*....*....*....*....*....*....*....*....*....*....*....*|THE PREFERENCE CONTOURS AROUND
1.50** | *|THE POINTS ARE HYPERBOLIC AND NOT
1.38** | *|ELLIPTICAL. THEREFORE, THE MIXED
1.27** | *|+ - IDEAL POSITIONS ARE DIFFICULT
1.15** | *|TO INTERPRET.
1.04** | *|
.92** | *|
.81** | *|
.69** | *|
.58** | *|
.46** | *|
.35** 8 | 5 *|
.23** | 9C *|
.12** 4 | B *| NOTE THAT POINT E IS A NEGATIVE
.00**-----------------------E---1-DF3-H--------------------------*| IDEAL POINT
-.12** | 2 *|
-.23** | A G *|
-.35** 6 | *|
-.46** | 7 *|
-.58** | *|
-.69** | *|
-.81** | *|
-.92** | *|
-1.04** | *|
-1.15** | *|
-1.27** | *|
-1.38** | *|
-1.50** | *|
.*....*....*....*....*....*....*....*....*....*....*....*....*|
. -1.6667. -1.0000. -.3333. .3333. 1.0000. 1.6667.|
-2.0000 -1.3333 -.6667 .0000 .6667 1.3333 2.|
PHASE 4 |PHASE 4 IS THE MOST GENERAL MODEL OF
|PREFMAP, PROVIDING NEGATIVE IDEAL
X MATRIX, (INPUT CONFIGURATION AFTER NORMALIZATION) |AND DIFFERENTIAL WEIGHTING OF
1 -.1782 .1174 .0615 -.5052 .1142 -.|DIMENSIONS. ELLIPTICAL PREFERENCE
.1743 -.4038 .2517 .4080 .4561 |CONTOURS AROUND THE IDEAL POINTS ARE
2 -.0403 -.1165 -.0134 .0620 .3778 -.|CREATED. ROTATION IS PRESENT
-.4219 .3718 .2845 -.2740 .1531 |
|
PHASE 4 |
SUBJECT 1 |
S (VECTOR OF SCALE VALUES, E.G. PREFERENCES) |
.14893 -.03206 .14893 -.42620 .14893 |
.14893 -.03206 .34697 .14893 .14893 |
BEGIN ITERATION ON MONOTONE FIT |
|
ITERATION 1 |
BETA VALUES (IN THE MOST GENERAL CASE THERE ARE (2K + K(K-1)/2 + 1) TE|
QUADRATIC, LINEAR, THEN A CONSTANT TERM) |
-.00005 .70151 .39606 |
|
(CORRELATION)= .84496 |
|
PROJECTIONS ON THE FITTED VECTOR |
-.17499 .04495 .04696 -.40949 .28519 |
-.05562 -.16879 .35907 .22057 .47242 |
|
ITERATION 2 |
BETA VALUES (IN THE MOST GENERAL CASE THERE ARE (2K + K(K-1)/2 + 1) TE|
QUADRATIC, LINEAR, THEN A CONSTANT TERM) |
-.00005 .72036 .37021 |
|
PROJECTIONS ON THE FITTED VECTOR |
-.14334 .04139 .03929 -.34106 .22209 |
-.03067 -.15324 .28661 .19242 .38517 |
END OF ITERATION, REACHED CRITERION |
|
(CORRELATION)= .85443 |
|
PROJECTIONS ON THE FITTED VECTOR |
-.17692 .05116 .04857 -.42104 .27427 |
-.03780 -.18915 .35393 .23763 .47562 |
|
SUBJECT 2 |
. . |
. . |
. . |
SUBJECT 5 |
|
END OF ITERATION, REACHED CRITERION |
|
(CORRELATION)= .98461 |
|
PROJECTIONS ON THE FITTED VECTOR |
-.17078 .13255 .06280 -.50893 .06012 |
.23172 -.45190 .20937 .44239 .43013 |
AVERAGE SUBJECT |
|
S (VECTOR OF SCALE VALUES, E.G. PREFERENCES) |
.10573 .09787 .11067 -.22826 .14746 |
.06973 -.29308 .13753 .13116 .13495 |
BETA VALUES (IN THE MOST GENERAL CASE THERE ARE (2K + K(K-1)/2 + 1) TE|
QUADRATIC, LINEAR, THEN A CONSTANT TERM) |
-.00002 .51639 .08465 |
|
(CORRELATION)= .87549 |
|
PROJECTIONS ON THE FITTED VECTOR |
-.18237 .09699 .05853 -.48856 .17382 |
.10376 -.33829 .29444 .35830 .47485 |
|
STIMULI COORDINATES |2 DIMENSIONAL COORDINATES OF 11
DIMENSION 1 2 | STIMULI
STIMULI |
1 -.17819 -.04032 |
2 .11738 -.11648 |
3 .06152 -.01344 |
4 -.50524 .06197 |
5 .11420 .37782 |
6 -.49569 -.38230 |
7 .17430 -.42187 |
8 -.40376 .37185 |
9 .25173 .28448 |
10 .40801 -.27403 |
11 .45609 .15307 |
|
STIMULI AND IDEAL POINTS: |POINTS 1 - 9, A, B = STIMULI
.*....*....*....*....*....*....*....*....*....*....*....*....*|POINTS C - H = IDEAL POINTS
1.50** | *|
1.38** | *|
1.27** | *|
1.15** | *|
1.04** | *|
.92** | *|
.81** | *|
.69** | *|
.58** | *|
.46** | C *|
.35** 8 | 5 E *|
.23** | 9 *|
.12** 4 | B H *|
.00**---------------------------1--03----------------------------*|
-.12** | 2 ; *|
-.23** D | A *|
-.35** 6 | *|
-.46** | 7 *|
-.58** | *|
-.69** | *|
-.81** | *|
-.92** | *|
-1.04** | *|
-1.15** | *|
-1.27** | *|
-1.38** | *|
-1.50** | *|
.*....*....*....*....*....*....*....*....*....*....*....*....*|
. -1.6667. -1.0000. -.3333. .3333. 1.0000. 1.6667.|
-2.0000 -1.3333 -.6667 .0000 .6667 1.3333 2.|
|
DIRECTION COSINES OF FITTED SUBJECT VECTORS |DIRECTION COSINES:
DIMENSION |IF THE VALUE IS > .9, LITTLE ROTATION
SUBJECT 1 2 |WILL OCCUR BECAUSE POINTS ARE ALREADY
1 .8894 .4571 |HIGHLY CORRELATED WITH THE DIMENSIONS.
2 -.9818 -.1899 |IF THIS OCCURS, THE PHASE 3 ANALYSIS
3 .9270 .3750 |IS AN APPROPRIATE PLACE TO START
4 .9915 -.1302 |(COMPLETE ANALYSIS FOR PHASES 1-3)
5 .9901 -.1402 |
AVG R .9868 .1618 |
CORRELATION (PHASE) F RATIO (PHASE) |WITHIN PHASE ANALYSIS:
R1 R2 R3 R4 F1 F2 F3 |
DF 5 5 4 6 3 7 |
SUBJ |
1 .000 .928 .897 .854 .000 9.252 9.646 1|
2 .000 .888 .888 .498 .000 5.583 8.675 |
3 .000 .979 .936 .928 .000 35.316 16.362 2|MODELS 2,4,3 FIT DATA FOR SUBJECT 3
4 .000 .997 .998 .670 .000 249.496 653.733 |MODELS 3,2 FIT SUBJECT 4
5 .000 .991 .987 .985 .000 81.634 88.161 12|=> MODELS 4,3,2 FIT SUBJECT 5
AVG .000 .981 .981 .875 .000 38.781 60.866 1|=> AVERAGE SUBJECT FOR PHASE 3 MODEL IS SIGNIFICANT.
F RATIO (BETWEEN PHASE) |
F12 F13 F14 F23 F24 F34 |
DF 1 5 2 5 3 5 1 6 2 6 1 7 |BETWEEN PHASE ANALYSIS:
SUBJ |
1 .000 .000 .000 2.377 2.805 2.702 |=> NO DIFFERENCE FOR SUBJECT 1
2 .000 .000 .000 .006 7.652 17.832 |MODEL 3 IS BETTER THAN 4 FOR SUB 2
3 .000 .000 .000 12.380 7.204 .772 |MODEL 2 IS BETTER THAN 3 FOR SUB 3
4 .000 .000 .000 -2.429 273.937 1078.809 |MODEL 3 IS BETTER THAN 4 FOR SUB 4
5 .000 .000 .000 2.574 2.079 1.293 |NO MODEL PREDICTS WELL FOR SUB 5
AVG .000 .000 .000 -.051 15.813 37.276 |AVERAGE SUB MOVING FROM PH. 3 TO 4
|IS NEARLY SIGNIFICANT. 2 TO 4 IS
ROOT MEAN SQUARE |SIGNIFICANT ALSO.
PHASE |
1 .000 |
2 .957 |
3 .942 |
4 .807 |
|
|
AN F - VALUE OF 1000.0 IN THE ABOVE TABLE INDICATES |
A POSSIBLE DIVISION BY ZERO. I.E. R IS VERY CLOSE TO 1.00 |