### 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 50th 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.