Examples 8

8 Example of Multidimensional Scaling

Research Question	:	Find the structure of relationships between fields of research corporation between Indian and 35 major countries.
Methodology	:	Multidimensional Scaling
Dataset	:	COOP.DAT

SYNTAX

$RUN PEARSON
$FILES
PRINT   = PEARSON.LST
DICTIN  = COOP.DIC
DATAIN  = COOP.DAT
FT02    = COOP1.MAT
$SETUP
PEARSON CORRELATION MATRIX
BADDATA=MD1   -
   MDHANDLING=CASE  -
   ROWVARS=(V2-V12)  -
   WRITE=CORR

$RUN MDSCAL
$FILES
PRINT   = MDSCAL.LST
FT02    = MDSCAL.MAT
FT08    = COOP1.MAT
$SETUP
MULTIDIMENSIONAL SCALING
FILE=DATA  -
   COEFF=SIMI -
   WRITE=CONFIG  -
   PRINT=(SORT,MATR)

$RUN CONFIG
$FILES
PRINT   =CONFIG.LST
FT02    = CONFIG.MAT
FT09    = MDSCAL.MAT
$SETUP
CONFIGURATION ANALYSIS
WRITE=(CONFIG,DISTANCES)  -   
   PRINT=(PLOT)  
PRINT=(CONFIG,PLOT)  DIMEN=(1,2)

Multidimensional Scaling is carried out in a sequence of three programs:

PEARSON
MDSCAL
CONFIG

The output of PEARSON is a correlation matrix which becomes an input to MDSCAL. The output of this program is input to CONFIG for plotting the configuration yielded by MDSCAL.

EXTRACT FROM COMPUTER OUTPUT – MDSCAL.LST

Summary of options selected:

     DMAX=    2     DMIN=    2     DDIF=    1
     ITER=   50     R=    2.00     CUTO=  .00     STRM=  .01
     SFGR=  .00     SRAT= 1.00     COSA=  .66     ACSA=  .66
     INPUT=(STANDARD)                                                   
     STRESS=SQDIST
     COEFF=SIMILARITIES   
     TIES=DIFFER

Size of input matrix= 11
Input matrix dictionary:

Variable Number  Name
        2       v2:MAT                 
        3       V3:PHY                  
        4       V4;CHM                  
        5       V5:BIO                  
        6       V6:EAS                  
        7       V7:AGR                  
        8       V8:CLI                  
        9       V9:BIM                  
       10       V10:ENT                 
       11       V11:MTL                 
       12       V12:COM

Input matrix:

x	2	3	4	5	6	7	8	9	10	11	12
2
3	.857
4	.884	.952
5	.851	.920	.959
6	.889	.948	.975	.967
7	.810	.876	.896	.940	.914
8	.837	.920	.937	.954	.949	.934
9	.900	.947	.986	.959	.976	.911	.956
10	.954	.935	.961	.938	.957	.902	.938	.978
11	.962	.859	.919	.855	.906	.806	.853	.937	.956
12	.882	.954	.963	.941	.965	.901	.958	.972	.952	.915

History of computation. N= 11, Dimension= 2

Iteration Stress SRAT SRATAV CAGRGL COSAV ACSAV SFGR STEP

         0   .461   .800   .800   .000   .000   .000   .0236   .5437
         1   .372   .806   .802   .010   .006   .006   .0200   .5177
         2   .358   .962   .852  -.521  -.342   .346   .0277   .3040
         .     .      .      .      .      .      .      .       .  
        48   .078   .999   .998   .896   .817   .846   .0012   .0081
        49   .078   .999   .999   .266   .453   .463   .0020   .0106
        50   .078  1.002  1.000  -.739  -.334   .645   .0063   .0036

Final configuration of 11 points in 2 dimensions has stress .078

Final configuration

           1       2
    2   1.590   -.572
    3   -.458   -.922
    4    .033   -.004
    5   -.420    .682
    6   -.173    .195
    7  -1.311   1.087
    8   -.985   -.003
    9    .054    .036
   10    .467   -.041
   11   1.453   -.201
   12   -.251   -.257

Sorted configurations:

Dimension: 1

       7       8       3       5      12       6       4       9      10      11       2
   -1.311   -.985   -.458   -.420   -.251   -.173    .033    .054    .467   1.453   1.590

Dimension: 2

       3       2      12      11      10       4       8       9       6       5       7
    -.922   -.572   -.257   -.201   -.041   -.004   -.003    .036    .195    .682   1.087

Summary

 IPOINT      9     10      9      6     12      6     12     12     11     10      9      5     12     10      9     11
 JPOINT      4      9      6      4      9      5      6      4      2      4      5      4      8      6      8     10
 DATA     .986   .978   .976   .975   .972   .967   .965   .963   .962   .961   .959   .959   .958   .957   .956   .956
 DIST     .045   .420   .277   .287   .422   .546   .458   .380   .395   .435   .802   .823   .777   .682  1.040   .999
 DHAT     .045   .328   .328   .328   .422   .443   .443   .443   .443   .443   .771   .771   .771   .771   .936   .936


 IPOINT      8     10     12      4     12      8      6      9     12      7     10     10     11      8     10      8
 JPOINT      5      2      3      3     10      6      3      3      5      5      8      5      9      4      3      7
 DATA     .954   .954   .954   .952   .952   .949   .948   .947   .941   .940   .938   .938   .937   .937   .935   .934

 DIST     .887  1.241   .697  1.041   .749   .836  1.153  1.086   .954   .979  1.452  1.145  1.419  1.018  1.277  1.138
 DHAT     .936   .936   .936   .936   .936   .936  1.043  1.043  1.043  1.043  1.216  1.216  1.216  1.216  1.216  1.216


 IPOINT      8      5     11     12      7      9     11     10     12      9      7      6      4     12      7     11
 JPOINT      3      3      4     11      6      7      6      7      7      2      4      2      2      2      3      3
 DATA     .920   .920   .919   .915   .914   .911   .906   .902   .901   .900   .896   .889   .884   .882   .876   .859
 DIST    1.060  1.605  1.433  1.705  1.446  1.723  1.673  2.106  1.712  1.651  1.732  1.922  1.656  1.867  2.183  2.043
 DHAT    1.216  1.519  1.519  1.576  1.576  1.698  1.698  1.797  1.797  1.797  1.797  1.797  1.797  1.867  2.093  2.093


 IPOINT      3     11     11      5      8      7     11
 JPOINT      2      5      8      2      2      2      7
 DATA     .857   .855   .853   .851   .837   .810   .806
 DIST    2.077  2.071  2.446  2.369  2.637  3.341  3.050
 DHAT    2.093  2.093  2.408  2.408  2.637  3.196  3.196

EXTRACT FROM COMPUTER OUTPUT –CONFIG.LST

1 Vector plots

 Vector    2 plotted against vector    1                     Vector
                                                                2
            -200 -180 -160 -140 -120 -100  -80  -60  -40  -20   *   20   40   60   80  100  120  140  160  180  200
              * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
        200  *                                                  *                                                  * 200
        192  *                                                  *                                                  * 192
        184  *                                                  *                                                  * 184
        176  *                                                  *                                                  * 176
        168  *                                                  *                                                  * 168
        160  *                                                  *                                                  * 160
        152  *                                                  *                                                  * 152
        144  *                                                  *                                                  * 144
        136  *                                                  *                                                  * 136
        128  *                                                  *                                                  * 128
        120  *                                                  *                                                  * 120
        112  *                 6                                *                                                  * 112
        104  *                                                  *                                                  * 104
         96  *                                                  *                                                  *  96
         88  *                                                  *                                                  *  88
         80  *                                                  *                                                  *  80
         72  *                                       4          *                                                  *  72
         64  *                                                  *                                                  *  64
         56  *                                                  *                                                  *  56
         48  *                                                  *                                                  *  48
         40  *                                                  *                                                  *  40
         32  *                                                  *                                                  *  32
         24  *                                                  *                                                  *  24
         16  *                                             5    *                                                  *  16
          8  *                                                  * 8                                                *   8
 Vector  1    * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
         -8  *                         7                        * 3         9                                      *  -8
        -16  *                                                  *                                                  * -16
        -24  *                                          11      *                                    10            * -24
        -32  *                                                  *                                                  * -32
        -40  *                                                  *                                                  * -40
        -48  *                                                  *                                                  * -48
        -56  *                                                  *                                       1          * -56
        -64  *                                                  *                                                  * -64
        -72  *                                                  *                                                  * -72
        -80  *                                                  *                                                  * -80
        -88  *                                                  *                                                  * -88
        -96  *                                       2          *                                                  * -96
       -104  *                                                  *                                                  *-104
       -112  *                                                  *                                                  *-112
       -120  *                                                  *                                                  *-120
       -128  *                                                  *                                                  *-128
       -136  *                                                  *                                                  *-136
       -144  *                                                  *                                                  *-144
       -152  *                                                  *                                                  *-152
       -160  *                                                  *                                                  *-160
       -168  *                                                  *                                                  *-168
       -176  *                                                  *                                                  *-176
       -184  *                                                  *                                                  *-184
       -192  *                                                  *                                                  *-192
       -200  *                                                  *                                                  *-200
              * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
            -200 -180 -160 -140 -120 -100  -80  -60  -40  -20   *   20   40   60   80  100  120  140  160  180  200

INTERPRETATION

IDAMS reports analysis specifications
DMAX = Max # of dimensions = 2
DMIN = Min # of dimensions = 2
ITER = # of iterations
R = 2 indicates that Minkowski r-metric used is Euclidean
Cut-off = 0 Data values = 0 are discarded
STRMIN = .01 Minimum stress for which the scaling process is stopped.
SFGR = 0 Minimum value of the scale factor of the gradient.
SRAT = 1 The stress ratio. Scaling computation stops if the stress ratio between successive steps reaches 1.
ACSA = .66 The weighting factor for the average absolute value of the cosine of the angle between successive gradients.

TIES = DIFFER Unequal distances corresponding to equal data values do not contribute to the stress coefficient and no attempt is made to equalize these differences
STRESS = SQDIS

Dictionary and input correlation matrix.

History of Computation
Initial stress at step 0 = 0.461
Find stress at 50^th iteration = 0.078

Final Configuration
Coordinates of points representing the variables on a two-dimensional vector space.

Sorted Configuration
For each dimension

Row 1: Labels of points representing the variables ordered from minimum value of coordinate to the maximum value.

Row 2: Coordinates of corresponding points

Sorted configuration shows which points are close to each other and which points are for apart.

For example, points (4) and (9) are close to each other on Dimension 1. Points (4) and (8) are close to each other on Dimension 2.

Summary

I POINT	Variable subscript i
J POINT	Variable subscript j
DATA	For each variable point is the input similarity
DIST	Distance between the points in the final configuration
DHAT	Number which minimize the stress, subject to the constraint that the d-hats have he same rank order as the input data; they are "appropriate" distances, estimated from the input data.

Consider for example
Column 8 and Row 4 in the input matrix. The value of similarity = 0.920
Distance computed between these two points = 0.045
DHAT = 0.045

Graphical representation of the final configuration without any transformation.

Please see the text for general principles of interpretation of the MDS configuration.