3(1) Example of Computing Descriptive Statistics

Research Question

:

How are the different measures of research output distributed? What is the average productivity of Indian scientists?

Methodology

:

Descriptive Statistics using TABLES programme

Dataset

:

ICSOPRU

SYNTAX
$RUN TABLES
   $FILES
   PRINT = G_TAB.LST
   DICTIN = R2R3CM.DIC
   DATAIN =R3CM.DAT
   $SETUP
   INCLUDE V1=360
   EXAMPLE OF TABLES PROGRAM
   PRINT=DICT
   TABLES
   TITLE='Univariate statistics for quantitative variables' -
      ROWVARS=(V363, V365,V367,V369,V371) -
        USTATS=(MEANSD,MEDMOD) PRINT=NOTABLES
Extract from Computer Output

The data matrix is 5 variables and 962 cases

 
0
Tabno
Wtnum
Wtsum
Mode
Median
Mean
Std.dev.
Skewness
Kurtosis
Minimum
Maximum
0V363S2A:#CMARTICLESNATL
 
1.00
0
828.00
.00
4.08
5.62
5.6540
2.1217
7.1281
.00000
40.000
0V365S2B:#CMARTICLESABROD
 
2.00
0
829.00
.00
.41
1.34
2.3825
2.9109
10.3761
.00000
15.000
0V367S3:#CMREVIEWS,BIBL
 
3.00
0
828.00
.00
.19
.80
1.7003
2.9542
11.1945
.00000
14.000
0V369S4:#CMINTLORIGREPOR
 
4.00
0
833.00
3.00
3.06
3.99
4.3469
3.2708
16.9291
.00000
40.000
0V371S5:#CMROUTINEREPORTS
 
5.00
0
824.00
3.00
3.91
6.79
8.3509
2.5838
7.5682
.00000
60.000
INTERPRETATION

IDAMS reports analysis specifications:
Number of cases =962
Number of variables = 5
V363     # Articles in national journals
V365     # Articles in foreign journals
V367     # Reviews and Bibliographies
V369     # Internal original reports
V371     # Routine reports

 

Third column Wtsum shows the number of valid cases used for computing univariate statistics of different variables.

Standard deviation is greater than the mean value for
Articles abroad
Reviews and bibliographies
Internal original reports
Routine reports

which indicates large dispersion in the data

Skewness is positive for all the variables, indicating a long tail on the right of the mean.

Peakedness of the distribution is high: highest for Internal Original Reports. The frequency distribution is highly Leptokurtic.

3(2) Example of Quantile Programme

Research Question

:

What is the extent of inequality in the research output of scientists in India? (Note: Research output is measured by two variables: V363: # of Articles published in the Indian journals; V365: # of Articles published in foreign journals).

Methodology

:

Gini Coefficient

Dataset

:

ICSOPRU

SYNTAX
$RUN QUANTILE
   $FILES
   PRINT = G_QUANT.LST
   DICTIN = R2R3CM.DIC
   DATAIN = R3CM.DAT
   $SETUP
   INCLUDE V1=360
   EXAMPLE OF QUANTILE PROGRAM
   BADDATA=MD1 -
     PRINT=DICT
   QUANTILE
   VAR=V363 N=25 PRINT=(FLOR,CLOR)
   VAR=V365 N=25 PRINT=(FLOR,CLOR) 
Extract from Computer Output

After filtering 962 cases read from the input data file
Analysis no. = 1     EXAMPLE OF QUANTILE PROGRAM
This analysis has missing data (code1) eliminated, adjusted N = 830
This analysis has missing data (code2) eliminated, adjusted N = 828

 

Distribution and Lorenz function for no. of sub-intervals = 25 Variable no. = 363

S2A:# CM ARTICLES NATL

***Distributionfunction***
***Lorenzfunction***
         
Minimum=
  .000
Minimum=
.000
(1)
  .000   .000
(2)
  .082   .001
(3)
  .376   .004
(4)
  .794   .004
(5)
  1.305   .011
(6)
  1.776   .023
(7)
  2.225   .037
(8)
  2.619   .054
(9)
  2.929   .076
(10)
  3.239   .097
(11)
  3.555   .120
(12)
  3.904   .148
(13)
  4.253   .177
(14)
  4.620   .208
(15)
  5.030   .243
(16)
  5.439   .279
(17)
  6.004   .321
(18)
  6.707   .365
(19)
  7.982   .417
(20)
  9.598   .480
(21)
  10.289   .551
(22)
  12.067   .631
(23)
  14.711   .726
(24)
  16.907   .834
Maximum=
  40.000
Maximum=
1.000
Ginicoefficient=
.495  
 

Lorenz plot variable no. = 363

S2A:# CM ARTICLES NATL

 

Fraction of ordered sample
Analysis no. = 2 EXAMPLE OF QUANTILE PROGRAM
This analysis has missing data (code1) eliminated, adjusted N = 831
This analysis has missing data (code2) eliminated, adjusted N = 829

 

Gini coefficient = .739

INTERPRETATION

IDAMS reports that 962 cases were read out of which 831 cases were taken for further analysis the remaining cases were deleted due to missing data.

 

Cumulative distribution of articles in 25 classes.
Minimum # of Articles = 0 and Maximum # of Articles = 40

 

Lorenz Curve: The area between the equidistribution line and the Lorenz curve indicates the inequality in the output of articles published in Indian journals.

 

The value of Gini coefficient = 0.495. It may be noted that the maximum value of Gini coefficient is 1.000.

  Similar analysis was performed for articles published abroad. However the computer output is not shown. The value of Gini coefficient for articles published in foreign journals is 0.739. This value is much higher than that for articles published in Indian journals. These results imply that there is greater inequality in the distribution of articles published abroad, compared with that of articles published in the country.

3(3) Example of Rank Programmes

Research Question

:

What are the factors that inhibit the dissemination of research results? Here is a list of seven factors:

  1. Inadequacy of supplies (paper, films, computer paper, etc.)
  2. Insufficient facilities for reproduction
  3. Complexity of administrative procedures
  4. Insufficient motivation in the unit
  5. Secrecy rules or practices
  6. Deficient publication policy within this country
  7. Other (please specify) ………………….

Rank these factors in order of importance based on the collective opinion of respondents.

Methodology

:

Ranking of alternatives

Dataset

:

ICSOPRU

SYNTAX
$RUN RANK
   $FILES
   PRINT = RANK.LST
   DICTIN = R4CM.DIC
   DATAIN = R4CM.DAT
   $SETUP
   INCLUDE V1=496
   FACTORS INHIBITIG DISSEMINATION OF RESEARCH RESULTS
   BADDATA=MD1 -
     METHOD=(CLASSICAL,NONDOMINATED,RANKS) -
     NALT=7 -
     VARS=(V351-V353) -
     NORMALIZE=YES -
     PRINT=CDICT
   PCON=60 DDIS=1 PDIS=5
   PCON=50 DDIS=1 PDIS=10
   PCON=40 DDIS=1 PDIS=15
Extract from Computer Output
***W* RNK003 --- No valid answer in case no 760
***W* RNK003 --- No valid answer in case no 778
***W* RNK001 --- Illegal code: 0 in case no 784
................................................
................................................
................................................

After filtering 980 cases read from the input data file
Valid cases processed: 686

 

Methods of fuzzy ranking

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

Input relation matrix:

.0000 .4038 .5204 .5000 .5160 .5510 .5437
.4052 .0000 .4942 .5058 .4825 .5496 .5510
.2507 .2362 .0000 .3120 .2668 .3382 .3280
.2157 .1837 .2595 .0000 .2493 .2668 .2638
.2551 .2668 .2857 .3411 .0000 .3630 .3571
.0962 .0758 .1224 .1122 .1239 .0000 .1254
.0933 .1122 .1108 .1137 .1137 .1195 .0000

The input relation is
  Fuzzy
  Asymmetric
  Antireflexive
  Non-trichotome
Absolute coherence: .3892 intensity: .5896 absolute dominance: .2295

 

Fuzzy method-1 N o n - d o m i n a t e d l a y e r s

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

Membership function of non-dominated subset:

Code/var no
: 6 3 1 4 2 5 7
Label
:              
Memb.val.
: .99854 1.00000 .73032 .67784 .73907 .52624 .54956
Core no 1
-- certainty level:1.00000
*************
--------------------------------

Credible alternatives (code or variable no):
3

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

Membership function of non-dominated subset:

Code/var no
: 6 3 1 4 2 5 7
Label
:              
Memb.val.
: 1.00000 ******* .73032 .71574 .73907 .54519 .54956
Core no 2
-- certainty level:1.00000
*************
--------------------------------

Credible alternatives (code or variable no):
6

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

Membership function of non-dominated subset:

Code/var no
: 6 3 1 4 2 5 7
Label
:              
Memb.val.
: ******* ******* 1.00000 .94752 ****** .78426 .78280
Core no 4
-- certainty level:1.00000
*************
--------------------------------

Credible alternatives (code or variable no):
1

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

Membership function of non-dominated subset:

Code/var no
: 6 3 1 4 2 5 7
Label
:              
Memb.val.
: ******* ******* ******* 1.00000 ******* .84548 .84985
Core no 5
-- certainty level:1.00000
*************
--------------------------------

Credible alternatives (code or variable no):
4

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

Membership function of non-dominated subset:

Code/var no
: 6 3 1 4 2 5 7
Label
:              
Memb.val.
: ******* ******* ******* ******* ******* 1.00000 .99417
Core no 6
-- certainty level:1.00000
*************
--------------------------------

Credible alternatives (code or variable no):

5

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

Membership function of non-dominated subset:

Code/var no
: 6 3 1 4 2 5 7
Label
:              
Memb.val.
: ******* ******* ******* ******* ******* ******* 1.00000
Core no 7
-- certainty level:1.00000
*************
--------------------------------

Credible alternatives (code or variable no):

7

Final coherence: .3892 intensity: .5896 final dominance: .2295

 

Normalized matrix

6
3
1
4
2
5
7
.0000
.4991
.6749
.6986
.6692
.8514
.8535
.5009
.0000
.6766
.7336
.6440
.8788
.8308
.3251
.3234
.0000
.5459
.4828
.7342
.7475
.3014
.2664
.4541
.0000
.4222
.7038
.6988
.3308
.3560
.5172
.5778
.0000
.7455
.7585
.1486
.1212
.2658
.2962
.2545
.0000
.5119
.1465
.1692
.2525
.3012
.2415
.4881
.0000
 

Fuzzy sets of ranks by alternatives
***********************************

Alternative (code or variable no) 6
******************************************
Rank no:
1
2
3
4
5
6
7
Memb.val:
.49910
.50090
.33081
.32514
.30143
.14865
.14645
Credible ranks:
2

Alternative (code or variable no) 3
******************************************

Rank no:
1
2
3
4
5
6
7
Memb.val:
.50090
.49910
.35603
.32335
.26638
.16923
.12121
Credible ranks:
1

Alternative (code or variable no) 1
******************************************

Rank no:
1
2
3
4
5
6
7
Memb.val:
.32335
.32514
.48285
.51715
.45408
.26582
.25249
Credible ranks:
4

Alternative (code or variable no) 4
******************************************

Rank no:
1
2
3
4
5
6
7
Memb.val: .26638 .30143 .42222 .45408 .54592 .30116 .29615
Credible ranks:
5

Alternative (code or variable no) 2
******************************************

Rank no:
1
2
3
4
5
6
7
Memb.val: .33081 .35603 .51715 .48285 .42222 .25449 .24149
Credible ranks:
3

Alternative (code or variable no) 5
******************************************

Rank no:
1
2
3
4
5
6
7
Memb.val: .12121 .14865 .25449 .26582 .29615 .51190 .48810
Credible ranks:
6

Alternative (code or variable no) 7
******************************************

Rank no:
1
2
3
4
5
6
7
Memb.val: .14645 .16923 .24149 .25249 .30116 .48810 .51190
Credible ranks:
7
 

1Fuzzy subsets of alternatives by ranks

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

Rank= 1
***********

Code/var no
: 6 3 1 4 2 5 7
Label
:              
Memb.val.
: .49910 .50090 .32335 .26638 .33081 .12121 .14645

Credible alternatives (code or variable no):

3

Rank= 2
***********

Code/var no
: 6 3 1 4 2 5 7
Label
:              
Memb.val.
: .50090 .49910 .32514 .30143 .35603 .14865 .16923

Credible alternatives (code or variable no):

6

Rank= 3
***********

Code/var no
: 6 3 1 4 2 5 7
Label
:              
Memb.val.
: .33081 .35603 .48285 .42222 .51715 .25449 .24149

Credible alternatives (code or variable no):

2

Rank= 4
***********

Code/var no
: 6 3 1 4 2 5 7
Label
:              
Memb.val.
: .32514 .32335 .51715 .45408 .48285 .26582 .25249

Credible alternatives (code or variable no):

1

Rank= 5
***********

Code/var no
: 6 3 1 4 2 5 7
Label
:              
Memb.val.
: .30143 .26638 .45408 .54592 .42222 .29615 .30116

0Credible alternatives (code or variable no):

4

Rank= 6
***********

Code/var no
: 6 3 1 4 2 5 7
Label
:              
Memb.val.
: .14865 .16923 .26582 .30116 .25449 .51190 .48810

Credible alternatives (code or variable no):

5

Rank= 7
***********

Code/var no
: 6 3 1 4 2 5 7
Label
:              
Memb.val.
: .14645 .12121 .25249 .29615 .24149 .48810 .51190

Credible alternatives (code or variable no):

7

1

 

Method of classical logic ranking
*********************************

Analysis specifications:

Rank difference for concordance: 1
Minimum proportion for concordance: 60%
Rank difference for discordance: 1
Maximum proportion for discordance: 5%
***E* RNK004 --- All the alternatives are non-dominated

 

Rerun with new parameters
1

Method of classical logic ranking
*********************************

Analysis specifications:

Rank difference for concordance: 1
Minimum proportion for concordance: 50%
Rank difference for discordance: 1
Maximum proportion for discordance: 10%

****************
Core no 1
****************
Alternative codes/variable nos:
  6   3   1   4   2

****************
Core no 2
****************
Alternative codes/variable nos:
   5   7
1

Method of classical logic ranking
*********************************

Analysis specifications:

Rank difference for concordance: 1
Minimum proportion for concordance: 40%
Rank difference for discordance: 1
Maximum proportion for discordance: 15%

****************
Core no 1
****************
Alternative codes/variable nos:
   6   3   1   4   2

****************
Core no 2
****************
Alternative codes/variable nos:
   5   7

INTERPRETATION

IDAMS lists the cases containing illegal code or no valid answers. 294 such cases are identified. Out of 980 cases, 686 cases (980-294) are processed for analysis.

 

Input relational matrix. This is a matrix of fuzzy relations among the set of alternatives. The cells of this matrix indicate the dominance of one alternative over another. For example, the cell (Column 1, Row 2) indicates that the dominance of Alternative 1 over Alternative 2 is 0.4052, whereas the dominance of Alternative 2 over Alternative 1 is 0.4038.

In this matrix the values of only seven cells exceed the threshold of simple majority, which means that only a few preferences are expressed in a very clear manner. This is confirmed by the index of cohesion, 0.3892, which seems to be low. However, this index is not poor, when we take into account the small number of alternatives cited (3 out of 7).

Intensity = 0.5896. This index can be interpreted as the average credibility of the proposition "Alternative I is preferred to Alternative J" or "Alternative J is preferred to Alternative I"

Absolute dominance = 0.2295. This index indicates the average difference between the credibilities of the proposition "Alternative I is preferred to Alternative j" and of the opposite proposition "Alternative J is preferred to Alternative I".

 

Fuzzy Method – 1 (Non-dominated layers)

The cases are presented sequentially from the highest rank and for each of them, the following information is given:

  • Its sequential number, with the certainty level,
  • The code of the alternatives or the variable’s number,
  • The membership function value of the alternative, indicating how strongly they are connected to the core. The membership values of alternatives, belonging to the previous cores are substituted by asterisks.
  • List of alternatives, belonging to the core with the highest membership value (most credible alternatives).

3 > 6 > 1 > 2 > 4 > 7

 

Normalized Relational Matrix

It can be easily seen that the sum of the cell (1, 2) and cell (2, 1)

0.5009 + 0.4991 = 1.00

Cell( 1, 2) Cell (2, 1)

 

Fuzzy sets of Alternatives by rank:

All alternatives are listed sequentially with the following information:

  • The code of the alternative, or the variable’s number,
  • The membership function value of the alternative indicating how strongly it is connected to each rank,
  • The list of most credible rank(s) for that alternative.

Degree of credibility of places of Alternative:
Rank 2 (Membership Value = .50090)
Degree of credibility of places of Alternative 3
Rank 1 (Membership Value = .50090)
and so on.

We get the following results:

Alternative
Rank
6
2
3
1
1
4
4
5
2
3
5
6
7
7

3 > 6 > 2 > 1 > 4 > 5 > 7

 

Fuzzy subsets of ranks by alternatives

  • The rank’s number,
  • The membership function value of the alternative indicating how strongly it is connected to each rank,
  • The list of most credible alternative(s) for that rank.

Rank 1: Alternative with the highest membership value (.50090) is: 3
Rank 2: Alternative with the highest membership value (.50090) is: 6
and so on.

We get the following results:

Rank
Alternative
1
3
2
6
3
2
4
1
5
4
6
5
7
7

Thus the rank order is:

3 > 6 > 2 > 1 > 4 > 5 > 7

 

Method of Classical Ranking (Electre)

Parameters

Threshold for concordance: 1

Threshold for discordance: 1

Limiting proportion for concordance: 60%

Limiting proportion for discordance: 5%

All the alternative are non-dominating

No preference order

 

Method of Classical Logic (Electre)

Rerun with new parameters:

Threshold for concordance: 1

Threshold for discordance: 1

Limiting proportion for concordance: 50%

Limiting proportion for discordance: 10%

The following rank order is computed:

6, 3 , 1, 4, 2 >5, 7