9(1) Example of Discriminant Analysis

Research Question

:

Research units spent time on different activities:

We assume that activity patterns may differ between two countries. Can we derive a rule for classifying research units into two groups [Country Code: 360 and on the basis 630] of their activity patterns?

 

Methodology

:

Discriminant analysis, using IDAMS module DISCRAN

Dataset

:

ICSOPRU2.DAT
SYNTAX
$RUN DISCRAN
$FILES
PRINT = DISCRAN.LST
DICTIN = ICSOPRU2.DIC
DATAIN = ICSOPRU2.DAT
$SETUP
INCLUDE V1=360, 638
prototype for DISCRAN program
BADDATA=MD1 -
 VARS=(V22 -V30) -
 MDHANDLING=(SAMPVAR,GROUPVAR,ANALVAR) -
 IDVAR=V2 -
 STEP=10-
 PRINT=(CDICT,GROUP) -
 GRVAR=V1 GR01=360 GR02=638
EXTRACT FROM COMPUTER OUTPUT

List of variables

V22 V23 V24 V25 V26 V27 V28 V29 V30

After filtering      464 cases read from the input data file
Valid cases processed: 464
     Number of cases in samples
Basic: 464    Test: 0    Anonymous: 0

Revised number of cases in samples
Basic: 464    Test: 0    Anonymous: 0

 

Means and standard deviations

(In the last column the global mean is printed)

         Group 1              Group 2
Variable  Mean     S.D.     Mean    S.D.    Mean
   22    55.3972 17.0743  28.0533 12.0499  42.1377
   23    5.2059   5.1370   6.8226  7.3723   5.9899
   24    6.8472   4.4405  12.7442  6.4885   9.7067
   25    2.1979   2.5754  13.4935 12.2004   7.6753
   26    6.0667   5.1025   2.6900  2.7588   4.4293
   27    2.8118   2.6504   2.7582  3.7750   2.7858
   28    12.5308 11.1494  25.9432 17.8515  19.0347
   29    5.7081   9.7895   2.8740  6.0661   4.3338
   30    3.2399   6.4383   4.6189  8.9578   3.9086
 

Step number 1

Variables entered : 22

Linear discriminant function

Variable number :
Constant        22
-2.829         .067

At each step, an observation is allocated :

Group 1 if its value for the linear discriminant function is strictly positive
Group 2 if is strictly negative
No allocation if it is zero

Classification table for basic sample

Allocated group
1    2

Original group
       1  191  48
       2   32   193

% correctly classified : 82.76

Step number 2

Variables entered :    22    26

  Linear discriminant function
****************************

                       Variable number :
Constant            22       26
-3.425               .065   .153

Classification table for basic sample

Allocated group
  1     2

Original group

1    205   34
2    23     202

% correctly classified : 87.72

Step number 3

Variables entered :   22   26   29

  Linear discriminant function
****************************

                    Variable number :
Constant         22      26     29
-3.705           .068   .137  .051

  Classification table for basic sample
*************************************

Allocated group
 1    2

Original group

1   212   27
2   23    202

% correctly classified : 89.22

Step number 4

Variables entered :   22   26   29   25

  Linear discriminant function
****************************

                 Variable number :
Constant    22   26    29    25
-2.944     .060 .122 .042 -.038

  Classification table for basic sample
*************************************

Allocated group
  1    2

Original group

1   218   21
2    24   201

% correctly classified : 90.30

Step number 5

Variables entered :   22   26   29   25   24

  Linear discriminant function
****************************

                 Variable number :

Constant      22     26      29      25      24
-2.456        .055  .118   .039  -.037  -.030

  Classification table for basic sample
*************************************

Allocated group
  1   2

Original group

1   222   17
2   23    202

% correctly classified : 91.38

The percentage of the basic sample decreases at the next step

 

Allocation and value of the linear discriminant function

for the observation in the basic sample

Group 1 Allocation Function


 101      1        1.145
 108      1         .023
 109      1        1.726
 201      2        -.856
 204      1        1.333
 419      2        -.035
 425      1         .034
 427      2        -.910
 430      1         .846
 501      2        -.101
 502      1         .667
 503      1        1.616
 504      1        1.770
 505      1        2.877
 506      1         .857
 507      2        -.230
*****
Group 2 Allocation Function


 101      2        -.919
 102      2        -.710
 202      1         .665
 301      2        -.195
 302      1        1.599
 303      2       -1.054
 901      1         .263
 902      1        1.254
1001      1        1.381
1601      1         .791
1602      1        2.289
1603      1         .233
***** 

Allocation and value of the linear discriminant function for the group means

Group 1 : 1.267
Group 2 : -1.346

INTERPRETATION

IDAMS reports analysis specifications.
Variables: V22-V28
No. of Cases:

      Basic sample:        464
      Test sample :          0
      Anonymous sample:      0

No. of groups:       2

 

IDAMS reports descriptive statistics (mean and standard deviation) of discriminant variable for each group and means of these variable for the entire sample.

These statistics give on idea of inter-group differences in the time devoted to different activities.

 

IDAMS reports the results of stepwise discriminant analysis. At each stage the most important discriminant variable is selected and a linear discriminant function (LDF) is set up. For each object, LDF score is computed and the object is allocated to:

Group – 1, if the LDF score is strictly positive
Group-2, if the LDF score is strictly negative

Not allocated to any group, if LDF score = 0.

Step 1:

At this stage, the most discriminant variable is identified, which is V22 and linear discriminant function is set up with this variable.

LDF = -2.829+.067´ V22

Objects are allocated to the groups according to the classification rule mentioned above .

Classification table
    Alloted Actual
1.   205     34
2.    23     202

No. of objects correctly classified = 82.76%.

Step 2:

At this stage, the second most discriminant variable is identified, viz. V26. A linear discriminant function is computed with variables V22 (identified at Step 1) and V26.

LDF = -3.425+.065V22+.153V26.

Object are allocated to each group according to the LDF score using the classification rule at .

No. of objects correctly classified = 87.72%

Step 3:

At this stage, the third most discriminant variable is identified, viz. V29. A linear discriminant function is computed with variables: V22, V26, and V29.

LDF = -3.075+.068V22+.137V26+.051V29

No. of objects correctly classified = 89.29%

Step 4:

At this stage, variable V25 is identified as the most discriminant variable after variables V22, V26 and V29.

A linear discriminant factor is computed with variables V22, V26, V29 and V25.

LDF = -2.944+.060V22+.122V26+.042V29-.038V25

No. of objects correctly classified = 90.30%.

Step 5:

At this stage, variable V24 is identified as the most discriminant variable after V22, V26, V29 and V25. The following linear discriminant function is computed.

LDF = -2.456+.055V22+.118V26+.039V29-.037V25-.030V24

No. of objects correctly classified = 91.38%.

The algorithm stops at this stage, since the entry of additional variable does not improve the classification accuracy. Hence the part of the print out for subsequent stages is omitted.

 

For each object, the value of the discriminant function is computed and the object is allocated to a group according to the allocation procedure mention at .

Table of Allocation of objects belong to Group-1.

To summarize the most important discriminant variables are: V22, V26, V29 and V25.

and the resulting discriminant function

LDF = -2.456+.055V22+.118V26+.039V29-.037V25-.030V24

achieves a classification accuracy of 91.38%.

    9(2) Example of Discriminant Analysis

Research Question

:

Research units spent time on different activities:

We assume that activity patterns may differ between two countries. Can we derive a rule for classifying research units into two groups [Country Code: 360 and on the basis 630] of their activity patterns?

Methodology

:

Discriminant analysis, using IDAMS module DISCRAN

Dataset

:

ANJU.DAT
SYNTAX
$RUN DISCRAN
$FILES
PRINT = SABINA1.LST
DICTIN = ANJU.DIC
DATAIN = ANJU.DAT
$SETUP
EXCLUDE V2=0
Pototype for DISCRAN program
BADDATA=MD1 -
 VARS=(V2-V8)-
 MDHANDLING=(SAMPVAR,GROUPVAR,ANALVAR) -
 IDVAR=V1 -
 STEP=7 -
 PRINT=(CDICT,DATA,GROUP) -
 SAVAR=V9 BASA=(1,2) ANSA=(3) -
 GRVAR=V15 GR01=1 GR02=2
EXTRACT FROM COMPUTER OUTPUT

Maximum of steps to be performed: 7
List of variables
     V2  V3  V4  V5  V6  V7  V8

After filtering 1063 cases read from the input data file
       3 cases contained illegal characters on filter variables and were skipped
Valid cases processed: 657

Number of cases in samples
Basic: 358 Test: 0 Anonymous: 299

Revised number of cases in samples
Basic: 358 Test: 0 Anonymous: 299

 

Basic sample

  a    
         *        2         3         4         5         6         7         8
*******************************************************************************
    1002 *     40.00     20.00     15.00      5.00     10.00       .00      10.00
    1015 *     60.00     30.00      5.00      4.00      1.00       .00        .00
    1016 *     50.00     10.00     16.00      5.00     12.00      2.00       5.00
    1018 *     20.00     20.00     20.00       .00     40.00       .00        .00

    3199 *     30.00     30.00     20.00     10.00     10.00     .00       .00
 
Anonymous sample
**************** 
b
         *        2         3         4         5         6         7         8
*******************************************************************************
    1006 *     60.00     15.00       .00     15.00     10.00       .00      .00
    1014 *     40.00     40.00     10.00       .00      5.00      5.00      .00
    1020 *     50.00     40.00       .00       .00     10.00       .00      .00
    1021 *     65.00     20.00       .00       .00     15.00       .00      .00
    1024 *     50.00     20.00     30.00       .00       .00       .00      .00
    6010 *     60.00     20.00     10.00     10.00       .00       .00      .00
    9009 *     40.00     25.00     20.00      5.00      5.00       .00      .00
 

Means and standard deviations

=============================

(In the last column the global mean is printed)

                    Group  1                    Group  2
 Variable       Mean        S.D.            Mean        S.D.           Mean

      2        34.2300     14.2926         36.0760     17.4503       35.0447
      3        23.0700     12.7850         18.8481     10.0833       21.2067
      4        21.7450     11.5174         17.0063     10.5701       19.6536
      5         3.6850      4.6460          6.5380      6.7305        4.9441
      6        10.1500     11.4021         11.8861     11.7365       10.9162
      7         1.9900      5.0110          5.7658      7.9062        3.6564
      8         5.1300      4.4736          3.8797      4.6462        4.5782

 ******************************************************************************

Step number 1

Variables entered : 7

Linear discriminant function

               Variable number :
Constant       7
    .306       -.084

At each step, an observation is allocated :

to group 1 if its value for the linear
discriminant function is strictly positive
To group 2 if it is strictly negative
No allocation if it is zero

Classification table for basic sample
  Allocated group
       1    2

Original group

1   159   41
2   76   82

% correctly classified : 67.32

******************************************************************************

Step number 2

Variables entered : 7 5

Linear discriminant function

              Variable number :
Constant      7     5
   .723     -.084 -.084

Classification table for basic sample
   
Allocated group
       1       2

Original group

1    136    64
2      48    110

% correctly classified : 68.72

******************************************************************************

Step number 3

Variables entered : 7 5 4

Linear discriminant function
****************************

                Variable number :
Constant           7        5      4
  .125             -.079 -.079 .028

Classification table for basic sample
*************************************

Allocated group
      1 2

Original group

1   149   51
2    52   106

% correctly classified : 71.23

The percentage of the basic sample decreases at the next step

Allocation and value of the linear discriminant function for the observation in the basic sample (only a few cases re-listed here)

   Group  1    Allocation      Function

   1002             1             .154
   1015             2            -.049
   1016             1             .024
   1018             1             .689
   3205             2            -.240
   3231             2            -.495
   3233             1             .243
   3235             1             .407
   3245             1             .390
   Group  2    Allocation      Function

   1010             2            -.240
   1032             2            -.381
   1054             2            -.635
   1115             2            -.663
   1182             1             .120
   1191             1             .548
   1199             1             .407
   3162             2            -.099
   3163             2            -.752
   3188             2            -.275
   3195             1             .179
   3199             2            -.099

Allocation and value of the linear discriminant function for the group means

Group 1 : .290
Group 2 : -.367

***********************************************************

Allocation and value of the linear discriminant function for the anonymous observations (only a few cases re-listed here)

  Number       Allocation      Function
   1006              2          -1.058
   1014              1            .011
   1020              1            .125
   1021              1            .125
   1024              1            .971
   6007              1            .689
   6008              1            .013
   6009              2          -1.173
   6010              2           -.381
   9009              1            .295
INTERPRETATION

IDAMS reports analysis specifications
Discriminant Variables: V2 V3 V4 V5 V6 V7 V8
No. of cases taken for analysis:

Basic sample: 358
Anonymous sample: 299

Grouping variable: V15
      No. of groups: 2

 

IDAMS prints the values of discriminant variables for the basic and anonymous samples perceptively.
      a Basic Sample
      b Anonymous Sample
These tables give an idea as to which activities relatively more or less prominent compared to the global mean.

  Descriptive statistics of discriminant variables for each group of the anonymous sample and global mean
 

Stepwise discriminant analysis

Step 1

At this stage, the most discriminant variable is identified, which is variable V7. The linear discriminant function computed with this variable.
= .306-.084V7

Classification table for basic sample

Allocated group
    1     2

Original group

1   159   41
2   76   82

% correctly classified : 67.32

Step 2

At this stage, the algorithm identifies V5 as the most important discriminant variable after V7. Variables V7 and V5 are used to construct the linear discriminant function

LDF = .723-.084V7-.084V5
No. of cases correctly classified = 68.73%

Step 3

At this stage, variable V4 is added to the linear discriminant function

LDF = -.125-.079V7-.079V5+.028V4
No. of cases correctly classified = 71.23%

The algorithm stops at this stage, since no others variable is able to improve the classification accuracy of the linear discriminant function.

 

Computation of LDF scores and classification of objects for each group:

Group: 1

Consider for example object Idcode: 1015. It has LDF score = -.049.

Since the score is negative, this object is assigned to Group 2.

Similar, interpretation for objects of Group – 2.

 

Computation of LDF scores of objects of the anonymous sample and assignment of objects to the groups.

Classification accuracy for the anonymous sample cannot be computed since their actual group membership is not known.

9.3 Example of Discriminant Analysis

Research Question

:

Drive a rule for classifying a set of 29 countries as Western, Asian and East European on the basis of priorities given in 10 fields: subfields of Physics.

Methodology

:

Multiple Discriminant analysis, using IDAMS module DISCRAN

Dataset

:

PHYSICS.DAT
SYNTAX
$RUN DISCRAN
$FILES
PRINT = PHYSICS.LST
DICTIN = PHYSICS.DIC
DATAIN = PHYSICS.DAT
$SETUP
EXCLUDE V12=4 AND V12=5
MULTIPLE DISCRIMINANT ANALYSIS OF PHYSICS DATA
BADDATA=MD1 -
 VARS=(V2-V11) -
 MDHANDLING=(SAMPVAR,GROUPVAR,ANALVAR) -
 IDVAR=V1 -
 STEP=8 -
 PRINT=(CDICT,DATA,GROUP) -
 GRVAR=V12 GR01=1 GR02=2 GR03=3
EXTRACT FROM COMPUTER OUTPUT

After filtering 36 cases read from the input data file
Valid cases processed: 36

     Number of cases in samples
Basic: 36 Test: 0 Anonymous: 0

Revised number of cases in samples
Basic: 29 Test: 0 Anonymous: 0
1Basic sample
************

 

Table of Means

 Variable      GR01        GR02        GR03        TOT.
      2       78.4706     52.5714    111.8000     77.9655
      3       93.1765    169.8571     86.8000    110.5862
      4      118.7059     90.8571    109.6000    110.4138
      5       73.5294     85.2857     87.6000     78.7931
      6      118.8235    109.5714    115.0000    115.9310
      7      116.7059     64.1429     56.4000     93.6207
      8      155.6471     71.7143     95.4000    125.0000
      9      100.8824     78.2857    104.6000     96.0690
     10      104.2353     69.8571    142.8000    102.5862
     11       87.7647     35.7143    135.2000     83.3793

 Table of Standard Deviations
 Variable      GR01        GR02        GR03
      2       40.1865     37.7921     50.1733
      3       21.5631     44.8344     34.9365
      4       34.1015     35.2032     26.1121
      5       20.9119     33.7965     59.9920
      6       48.5280     47.3493     64.1685
      7       57.2690     33.7107     34.0975
      8      103.1184     32.7445     34.9548
      9       41.2209     36.8112     53.2826
     10       73.9400     32.7476     63.5591
     11       45.7712     14.7039     53.7900
 

Step number 1

Variables entered : 3

  Classification table for basic sample
*************************************
         Allocated group
      GR01 GR02 GR03

Original group

GR01      10   0   7
GR02       1    6   0
GR03       1    1   3

% correctly classified : 65.52
******************************************************************************

 

Step number 1

Variables entered : 3 7

  Classification table for basic sample
*************************************
         Allocated group
      GR01 GR02 GR03

Original group

GR01      10   0   7
GR02       0    6   1
GR03       1    1   3

% correctly classified : 65.52
******************************************************************************

 

Step number 3

Variables entered : 3  7   11

  Classification table for basic sample
*************************************
         Allocated group
      GR01 GR02 GR03

Original group

GR01      10   0   6
GR02       0    7   0
GR03       2    0   3

% correctly classified : 72.41
******************************************************************************

 

Step number 4

Variables entered : 3   7   11   8

  Classification table for basic sample
*************************************
         Allocated group
      GR01 GR02 GR03

Original group

GR01      10   0   7
GR02       0    7   0
GR03       1    0   4

% correctly classified : 72.41
******************************************************************************

 

Step number 5

Variables entered : 3   7   11   8   4  

  Classification table for basic sample
*************************************
         Allocated group
      GR01 GR02 GR03

Original group

GR01      12   0   5
GR02       0    7   0
GR03       1    0   4

% correctly classified : 79.31

The percentage of the basic sample decreases at the next step

Allocation and distances of observations in the basic sample
      (In parentheses group number under consideration)

     Group   1   Allocation     Distances to each group 
      GR01
         1            3         2.356 ( 1)    7.120 ( 2)       .520 ( 3) 
         3            1         1.599 ( 1)    1.734 ( 2)      3.177 ( 3) 
         5            3         1.883 ( 1)    4.397 ( 2)      1.529 ( 3) 
         6            1         1.134 ( 1)    1.829 ( 2)      2.607 ( 3) 
         7            1         1.596 ( 1)    5.196 ( 2)      2.107 ( 3) 
         9            1         3.963 ( 1)   10.770 ( 2)     10.126 ( 3) 
        10            1         9.893 ( 1)   16.696 ( 2)     17.657 ( 3) 
        11            3         1.572 ( 1)    3.733 ( 2)      1.422 ( 3) 
        16            3         1.591 ( 1)    3.323 ( 2)      1.135 ( 3) 
        17            1         2.954 ( 1)    6.511 ( 2)      6.572 ( 3) 
        18            1         4.003 ( 1)    5.912 ( 2)     10.414 ( 3) 
        22            1         1.958 ( 1)    3.844 ( 2)      5.674 ( 3) 
        23            3         9.470 ( 1)   12.955 ( 2)      7.041 ( 3) 
        24            1         1.040 ( 1)    3.641 ( 2)      1.059 ( 3) 
        28            1         5.561 ( 1)   10.721 ( 2)      7.229 ( 3) 
        30            1         9.491 ( 1)   15.857 ( 2)     10.312 ( 3) 
        35            1       15.993 ( 1)    23.778 ( 2)     20.558 ( 3) 

     Group   2   Allocation     Distances to each group 
       GR02
         2            2         8.176 ( 1)    5.961 ( 2)      8.253 ( 3) 
        12            2         3.207 ( 1)     .279 ( 2)      4.337 ( 3) 
        15            2        16.687 ( 1)    6.353 ( 2)     16.651 ( 3) 
        21            2         4.564 ( 1)    1.059 ( 2)      4.217 ( 3) 
        25            2         3.994 ( 1)    2.398 ( 2)      6.420 ( 3) 
        26            2         5.229 ( 1)    3.755 ( 2)      6.529 ( 3) 
        29            2         5.882 ( 1)    2.421 ( 2)      7.177 ( 3) 
     Group   3   Allocation     Distances to each group 
       GR03
         4            3        10.175 ( 1)   11.813 ( 2)      7.909 ( 3) 
         8            3         4.062 ( 1)    3.880 ( 2)      3.175 ( 3) 
        14            3         4.049 ( 1)    8.732 ( 2)      1.331 ( 3) 
        19            1         1.216 ( 1)    3.622 ( 2)      2.337 ( 3) 
        33            3        10.878 ( 1)   15.002 ( 2)      5.898 ( 3) 

Discriminant factor analysis at step 5
************************************

Sum of eigenvalues = .89884

Discriminant power of first factor : .64040

Discriminant power of second factor : .25843

Discriminant power of third factor : .00000

First eigenvector

-.01455 .00635 .00644 .00251 -.00375

Second eigenvector

-.00208 -.00980 .01130 -.00275 -.00943

Values of discriminant factors for all observations and group means

Group GR01
             Fact 1   Fact 2
Mean -.10237 -1.89252

Group GR02
             Fact 1   Fact 2
Mean -1.99358 -1.63144

Group GR03
             Fact 1   Fact 2
Mean -.20409 -.50127

Codes used for presentation in the graph

The group near is indicated by *

The overlapping of 2 observations from different groups by $

One observation of the group GR01 is presented by 1
           The overlapping of 2 observations by A

One observation of the group GR02 is presented by 2
          The overlapping of 2 observations by B

One observation of the group GR03 is presented by 3
          The overlapping of 2 observations by C

1         -3.784    -3.314    -2.843    -2.373    -1.903    -1.432     -.962     -.491     -.021      .449      .920
            +.........+.........+.........+.........+.........+.........+.........+.........+.........+.........+.
      .719  .                                                                                                    .   .719
      .631  .                                                                                       3            .   .631
      .543  .                                                                                                    .   .543
      .455  .                                                                                                    .   .455
      .367  .                                                                                                    .   .367
      .280  .                                                                                                    .   .280
      .192  .                                                                                                    .   .192
      .104  .                                                                                                    .   .104
      .016  .                                                                                                    .   .016
     -.072  .                                                                                                    .  -.072
     -.160  .                                                                                        3           .  -.160
     -.248  .                                                                                                    .  -.248
     -.336  .                                                                                 1                  .  -.336
     -.424  .                                                                       3  MEAN3                          .  -.424
     -.512  .                                                                           *            1           .  -.512
     -.600  .                                                                                                    .  -.600
     -.688  .                                                                                                    .  -.688
     -.776  .                                                                                                    .  -.776
     -.864  .                                                                                                    .  -.864
     -.952  .                                                           3                                        .  -.952
    -1.040  .                                                                     1    1                         . -1.040
    -1.128  .                                                                       1                            . -1.128
    -1.216  .                                        2                                 1                         . -1.216
    -1.304  .                                               2                                                    . -1.304
    -1.392  .                                                                               1              1     . -1.392
    -1.480  . 2                                                                                                  . -1.480
    -1.568  .                                     MEAN2                                                               . -1.568
    -1.656  .                                      *                                3                            . -1.656
    -1.744  .                                         2       2           1                        1             . -1.744
    -1.832  .                                      2                   1                                         . -1.832
    -1.920  .                                                                             * MEAN1                     . -1.920
    -2.008  .                                                                                                    . -2.008
    -2.096  .                                                                                                    . -2.096
    -2.184  .                                                2                                                   . -2.184
    -2.272  .                                                                                                    . -2.272
    -2.360  .                                                                                                    . -2.360
    -2.448  .                                                                                                    . -2.448
    -2.536  .                                                                  1        1                        . -2.536
    -2.624  .                                                                                                    . -2.624
    -2.711  .                                                                                                    . -2.711
    -2.799  .                                                                                                  1 . -2.799
    -2.887  .                                                                                                    . -2.887
    -2.975  .                                                                                                    . -2.975
    -3.063  .                                                                                                    . -3.063
    -3.151  .                                                                                                    . -3.151
    -3.239  .                                                                                                    . -3.239
    -3.327  .                                                                                           1        . -3.327
    -3.415  .                                                                                                    . -3.415
    -3.503  .                                                                                                    . -3.503
    -3.591  .                                                               1                                    . -3.591
    -3.679  .                                                                                                    . -3.679
    -3.767  .                                                                                                    . -3.767
    -3.855  .                                                                                                    . -3.855
    -3.943  .                                                                                         1          . -3.943
    -4.031  .                                                                                                    . -4.031
            +.........+.........+.........+.........+.........+.........+.........+.........+.........+.........+.
          -3.784    -3.314    -2.843    -2.373    -1.903    -1.432     -.962     -.491     -.021      .449      .920

 

Step number 6

Variables entered : 3   7   11   8   4   9

  Classification table for basic sample
*************************************
         Allocated group
      GR01 GR02 GR03

Original group

GR01      11   0   6
GR02       0    7   0
GR03       1    0   4

% correctly classified : 75.86
******************************************************************************

 

Step number 7

Variables entered : 3   7   11   8   4   9   2

  Classification table for basic sample
*************************************
         Allocated group
      GR01 GR02 GR03

Original group

GR01      12   0   5
GR02       0    7   0
GR03       1    0   4

% correctly classified : 79.31
******************************************************************************

 

Step number 8

Variables entered : 3   7   11   8   4   9   2   6

  Classification table for basic sample
*************************************
         Allocated group
      GR01 GR02 GR03

Original group

GR01      12   0   5
GR02       0    7   0
GR03       1    0   4

% correctly classified : 79.31

Allocation and distances of observations in the basic sample
    (In parentheses group number under consideration)

     Group   1   Allocation     Distances to each group 
       GR01
        1             3          3.712 ( 1)       9.107 ( 2)     1.991 ( 3) 
        3             1          3.473 ( 1)       4.135 ( 2)     5.260 ( 3) 
        5             3          4.951 ( 1)       7.754 ( 2)     4.380 ( 3) 
        6             1          1.291 ( 1)       1.934 ( 2)     3.270 ( 3) 
        7             1          2.714 ( 1)       5.709 ( 2)     3.670 ( 3) 
        9             1          5.540 ( 1)      12.311 ( 2)    12.788 ( 3) 
       10             1         11.016 ( 1)      18.649 ( 2)    18.901 ( 3) 
       11             3          2.795 ( 1)       4.652 ( 2)     2.614 ( 3) 
       16             3          5.507 ( 1)       8.813 ( 2)     3.904 ( 3) 
       17             1          8.908 ( 1)      13.420 ( 2)    10.915 ( 3) 
       18             1          6.774 ( 1)       8.307 ( 2)    14.724 ( 3) 
       22             1          5.254 ( 1)       7.595 ( 2)     7.854 ( 3) 
       23             3         16.312 ( 1)      19.895 ( 2)    15.910 ( 3) 
       24             1          7.261 ( 1)       9.272 ( 2)     7.976 ( 3) 
       28             1          9.260 ( 1)      13.796 ( 2)    10.494 ( 3) 
       30             1         12.990 ( 1)      18.726 ( 2)    14.945 ( 3) 
       35             1         16.070 ( 1)      24.131 ( 2)    20.797 ( 3) 

     Group   2   Allocation     Distances to each group 
       GR02
         2            2         10.159 ( 1)       7.820 ( 2)     9.653 ( 3) 
        12            2          6.492 ( 1)       2.407 ( 2)     9.006 ( 3) 
        15            2         23.166 ( 1)      14.579 ( 2)    23.009 ( 3) 
        21            2          7.518 ( 1)       3.288 ( 2)     7.129 ( 3) 
        25            2          5.451 ( 1)       3.352 ( 2)     8.440 ( 3) 
        26            2          7.361 ( 1)       5.925 ( 2)     8.689 ( 3) 
        29            2          8.778 ( 1)       5.046 ( 2)    11.636 ( 3) 
 

     Group   3   Allocation     Distances to each group 
       GR03
        4            3          13.573 ( 1)      14.864 ( 2)    10.981 ( 3) 
        8            3           5.731 ( 1)       5.106 ( 2)     4.758 ( 3) 
       14            3           9.590 ( 1)      16.108 ( 2)     5.891 ( 3) 
       19            1           6.470 ( 1)       9.831 ( 2)     6.514 ( 3) 
       33            3          13.243 ( 1)      17.049 ( 2)     9.708 ( 3) 

Discriminant factor analysis at step 8
************************************

Sum of eigenvalues = .96221

Discriminant power of first factor : .67570

Discriminant power of second factor : .28651

Discriminant power of third factor : .00000

First eigenvector

-.01420 .00528 .00484 .00410 -.00687 -.00301 .00701 .00333

Second eigenvector

-.00264 -.01202 .00958 -.00053 -.01130 -.00914 .00523 .00141

Values of discriminant factors for all observations and group means

Group GR01
             Fact 1   Fact 2
Mean .18115 -2.57679

Group GR02
             Fact 1   Fact 2
Mean -1.73343 -2.22830

Group GR03
             Fact 1   Fact 2
Mean .20855 -1.11043

Codes used for presentation in the graph

The group near is indicated by *

The overlapping of 2 observations from different groups by $

One observation of the group GR01 is presented by 1
   The overlapping of 2 observations by A

One observation of the group GR02 is presented by 2
   The overlapping of 2 observations by B

One observation of the group GR03 is presented by 3
   The overlapping of 2 observations by C

1         -2.939    -2.516    -2.092    -1.669    -1.245     -.821     -.398      .026      .449      .873     1.296
            +.........+.........+.........+.........+.........+.........+.........+.........+.........+.........+.
     -.526  .                                                                                                    .  -.526
     -.602  .                                                                                                  3 .  -.602
     -.678  .                                                                                 3                  .  -.678
     -.754  .                                                                                                    .  -.754
     -.829  .                                                                                                    .  -.829
     -.905  .                                                                                                    .  -.905
     -.981  .                                                                 3                                  .  -.981
    -1.057  .                                                                                                    . -1.057
    -1.133  .                                                                          *                         . -1.133
    -1.208  .                                                                                                    . -1.208
    -1.284  .                                                                           1            1           . -1.284
    -1.360  .                                                                                                    . -1.360
    -1.436  .                                                                                                    . -1.436
    -1.512  .                                                   3                                                . -1.512
    -1.587  .                                                                                                    . -1.587
    -1.663  .                            2           2                              1                            . -1.663
    -1.739  . 2                                                               1         1                        . -1.739
    -1.815  .                                                                                                    . -1.815
    -1.891  .                                                                         3                          . -1.891
    -1.966  .                                                                                                    . -1.966
    -2.042  .                                                                1                                   . -2.042
    -2.118  .                                                                                                    . -2.118
    -2.194  .                            *                                         1                             . -2.194
    -2.270  .                                          2                                    1                    . -2.270
    -2.345  .                                                                                                    . -2.345
    -2.421  .                                                      1                                             . -2.421
    -2.497  .                                                      1                       1      1              . -2.497
    -2.573  .                                                                         *                          . -2.573
    -2.649  .                         2                                                                          . -2.649
    -2.725  .                                                               1                                    . -2.725
    -2.800  .                          2         2                                                               . -2.800
    -2.876  .                                                                                                    . -2.876
    -2.952  .                                                                                                    . -2.952
    -3.028  .                                                                                                    . -3.028
    -3.104  .                                                                                                    . -3.104
    -3.179  .                                                                                                    . -3.179
    -3.255  .                                                                                                    . -3.255
    -3.331  .                                                                                                    . -3.331
    -3.407  .                                                                                                    . -3.407
    -3.483  .                                                                                               1    . -3.483
    -3.558  .                                                                                                    . -3.558
    -3.634  .                                                                                                    . -3.634
    -3.710  .                                                                                                    . -3.710
    -3.786  .                                                                                                    . -3.786
    -3.862  .                                                                                                    . -3.862
    -3.937  .                                                                                                    . -3.937
    -4.013  .                                                                                                    . -4.013
    -4.089  .                                                                                                    . -4.089
    -4.165  .                                                                                                    . -4.165
    -4.241  .                                                                                                    . -4.241
    -4.316  .                                                                                    1               . -4.316
    -4.392  .                                                                                                    . -4.392
    -4.468  .                                                                                                    . -4.468
    -4.544  .                                                   1                                    1           . -4.544
    -4.620  .                                                                                                    . -4.620
            +.........+.........+.........+.........+.........+.........+.........+.........+.........+.........+.
          -2.939    -2.516    -2.092    -1.669    -1.245     -.821     -.398      .026      .449      .873     1.296

0Memory at disposal : 5000
Memory used : 932

INTERPRETATION

IDAMS reports analysis specifications
No. of cases read = 36
No. of cases used in the analysis = 29
(Countries of South America and Africa are omitted in this example)
No. of Discriminant variables = 10

 

Table of means and standard deviation of variables separately for each group and for the entire sample. This table gives an idea to which fields are more or less prominent in which group.

 

At this stage, the most important discriminant variable is identified and a linear discriminant function is computed and used for classification of countries according to their distance from the group centroids.

The most important discriminant variable is V3 (condensed matter physics)

Classification table shows that No. of cases correctly classified = 65.52%
Group 2 (East Europe is better classified) than the other regions.

Average value of V3 is higher in this group than in any other group.

 

Step 2

At this stage the most important variable after V3 is V7 (Partial Physics)

The discriminant function computed with these variables, achieves a classification accuracy is 65.52%.

It is interesting to note there is no change in the overall classification accuracy, but there is a change in the classification matrix.

One country from group 2 which was misclarified into Group 1 is now misclarified into Group 3

 

Step 3

At this stage the most important discriminant variable after V2 and V7 is V11 (Acoutics).

The discriminant function set up with these three variables achieve a clarification accuracy of 72.41%.

 

Step 4

At this stage, the most important discriminant variable is Mathematical Physics, but the overall clarification accuracy remains uncharged, but there is some adjustment is the classification matrix.

 

At this stage the most important discriminant variable after V3, V7, V11 and V8 is V4
(Chemical Physics)

The discriminant function computed with these variables achieves a classifcation accuracy of 79.35%.

Note that:

All East-European countries are correctly classified.
4 out of 5 Asian countries are correctly classified.
12 out of 17 Western Countries are correctly classified.
Classification frequency decreases after this step.

 

Allocation Table and distances from the centroids of each group
Name of the countries in this group are misclassified 9 East-European.
The following countries are

Allocation Table and Distances from the centroid of each group

This table provides information as to how different countries are situated from the centroids of different groups.

The countries are allocated to the nearest group.

The following countries of Group-1 which are allocated to Group 1 are still quite away from their centroid.

Idcode        Distance from the centroid of Grouping

35                    15.999
36                    9.993
37                    9.470
38                    9.491

These countries seen to be ‘outliers’.

5 countries of this region are misclassified into Group 3 (Asia). Their idcodes are printed.

Group 2.
No country is misclassified

Group 3
Only one country (Idcode 19) is misclassified into Group 1 (Western countries).

 

Discriminant Function Analysis

The first discriminant function is linear combination of the variables that best discriminant between the groups.

The second discriminant function is orthgond to the first and is the next best combination of the variable.