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The DISTANCE Procedure

Example 32.1 Divorce Grounds – the Jaccard Coefficient

A wide variety of distance and similarity measures are used in cluster analysis (Anderberg; 1973; Sneath and Sokal; 1973). If your data are in coordinate form and you want to use a non-Euclidean distance for clustering, you can compute a distance matrix by using the DISTANCE procedure.

Similarity measures must be converted to dissimilarities before being used in PROC CLUSTER. Such conversion can be done in a variety of ways, such as taking reciprocals or subtracting from a large value. The choice of conversion method depends on the application and the similarity measure. If applicable, PROC DISTANCE provides a corresponding dissimilarity measure for each similarity measure.

In the following example, the observations are states. Binary-valued variables correspond to various grounds for divorce and indicate whether the grounds for divorce apply in each of the U.S. states. A value of "1" indicates that the ground for divorce applies, and a value of "0" indicates the opposite. The 0-0 matches are treated as totally irrelevant; therefore, each variable has an asymmetric nominal level of measurement. The absence value is 0.

The DISTANCE procedure is used to compute the Jaccard coefficient (Anderberg; 1973, pp. 89, 115, and 117) between each pair of states. The Jaccard coefficient is defined as the number of variables that are coded as 1 for both states divided by the number of variables that are coded as 1 for either or both states. Since dissimilarity measures are required by the CLUSTER procedure, the DJACCARD coefficient is selected. Output 32.1.1 displays the distance matrix between the first 10 states.

The CENTROID method is used to perform the cluster analysis, and the resulting tree diagram from PROC CLUSTER is saved into the tree output data set. Output 32.1.2 displays the cluster history.

The TREE procedure generates nine clusters in the output data set out. After being sorted by the state, the out data set is then merged with the input data set divorce. After being sorted by the state, the merged data set is printed to display the cluster membership as shown in Output 32.1.3.

The following statements produce Output 32.1.1 through Output 32.1.3: 

data divorce;
   input State $15.
         (Incompatibility Cruelty Desertion Non_Support Alcohol 
          Felony Impotence Insanity Separation) (1.) @@;
   if mod(_n_,2) then input +4 @@; else input;
   datalines;
Alabama        111111111    Alaska         111011110
Arizona        100000000    Arkansas       011111111
California     100000010    Colorado       100000000
Connecticut    111111011    Delaware       100000001
Florida        100000010    Georgia        111011110
Hawaii         100000001    Idaho          111111011
Illinois       011011100    Indiana        100001110
Iowa           100000000    Kansas         111011110
Kentucky       100000000    Louisiana      000001001
Maine          111110110    Maryland       011001111
Massachusetts  111111101    Michigan       100000000
Minnesota      100000000    Mississippi    111011110
Missouri       100000000    Montana        100000000
Nebraska       100000000    Nevada         100000011
New Hampshire  111111100    New Jersey     011011011
New Mexico     111000000    New York       011001001
North Carolina 000000111    North Dakota   111111110
Ohio           111011101    Oklahoma       111111110
Oregon         100000000    Pennsylvania   011001110
Rhode Island   111111101    South Carolina 011010001
South Dakota   011111000    Tennessee      111111100
Texas          111001011    Utah           011111110
Vermont        011101011    Virginia       010001001
Washington     100000001    West Virginia  111011011
Wisconsin      100000001    Wyoming        100000011
;

title 'Grounds for Divorce';
proc distance data=divorce method=djaccard absent=0 out=distjacc;
   var anominal(Incompatibility--Separation);
   id state;
run;

proc print data=distjacc(obs=10);
   id state; var alabama--georgia;
   title2 'First 10 States';
run;
title2;

proc cluster data=distjacc method=centroid 
             pseudo outtree=tree;
   id state;
   var alabama--wyoming;
run;

proc tree data=tree noprint n=9 out=out;
   id state;
run;

proc sort;
   by state;
run;

data clus;
   merge divorce out;
   by state;
run;

proc sort;
   by cluster;
run;

proc print;
   id state;
   var Incompatibility--Separation;
   by cluster;
run;

Output 32.1.1 Distance Matrix Based on the Jaccard Coefficient
Grounds for Divorce
First 10 States

State Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware Florida Georgia
Alabama 0.00000 . . . . . . . . .
Alaska 0.22222 0.00000 . . . . . . . .
Arizona 0.88889 0.85714 0.00000 . . . . . . .
Arkansas 0.11111 0.33333 1.00000 0.00000 . . . . . .
California 0.77778 0.71429 0.50000 0.88889 0.00000 . . . . .
Colorado 0.88889 0.85714 0.00000 1.00000 0.50000 0.00000 . . . .
Connecticut 0.11111 0.33333 0.87500 0.22222 0.75000 0.87500 0.00000 . . .
Delaware 0.77778 0.87500 0.50000 0.88889 0.66667 0.50000 0.75000 0.00000 . .
Florida 0.77778 0.71429 0.50000 0.88889 0.00000 0.50000 0.75000 0.66667 0.00000 .
Georgia 0.22222 0.00000 0.85714 0.33333 0.71429 0.85714 0.33333 0.87500 0.71429 0

Output 32.1.2 Clustering History
Grounds for Divorce

The CLUSTER Procedure
Centroid Hierarchical Cluster Analysis

Root-Mean-Square Distance Between Observations 0.694873

Cluster History
NCL Clusters Joined FREQ PSF PST2 Norm
Cent
Dist		 T
i
e
49 Arizona Colorado 2 . . 0 T
48 California Florida 2 . . 0 T
47 Alaska Georgia 2 . . 0 T
46 Delaware Hawaii 2 . . 0 T
45 Connecticut Idaho 2 . . 0 T
44 CL49 Iowa 3 . . 0 T
43 CL47 Kansas 3 . . 0 T
42 CL44 Kentucky 4 . . 0 T
41 CL42 Michigan 5 . . 0 T
40 CL41 Minnesota 6 . . 0 T
39 CL43 Mississippi 4 . . 0 T
38 CL40 Missouri 7 . . 0 T
37 CL38 Montana 8 . . 0 T
36 CL37 Nebraska 9 . . 0 T
35 North Dakota Oklahoma 2 . . 0 T
34 CL36 Oregon 10 . . 0 T
33 Massachusetts Rhode Island 2 . . 0 T
32 New Hampshire Tennessee 2 . . 0 T
31 CL46 Washington 3 . . 0 T
30 CL31 Wisconsin 4 . . 0 T
29 Nevada Wyoming 2 . . 0  
28 Alabama Arkansas 2 1561 . 0.1599 T
27 CL33 CL32 4 479 . 0.1799 T
26 CL39 CL35 6 265 . 0.1799 T
25 CL45 West Virginia 3 231 . 0.1799  
24 Maryland Pennsylvania 2 199 . 0.2399  
23 CL28 Utah 3 167 3.2 0.2468  
22 CL27 Ohio 5 136 5.4 0.2698  
21 CL26 Maine 7 111 8.9 0.2998  
20 CL23 CL21 10 75.2 8.7 0.3004  
19 CL25 New Jersey 4 71.8 6.5 0.3053 T
18 CL19 Texas 5 69.1 2.5 0.3077  
17 CL20 CL22 15 48.7 9.9 0.3219  
16 New York Virginia 2 50.1 . 0.3598  
15 CL18 Vermont 6 49.4 2.9 0.3797  
14 CL17 Illinois 16 47.0 3.2 0.4425  
13 CL14 CL15 22 29.2 15.3 0.4722  
12 CL48 CL29 4 29.5 . 0.4797 T
11 CL13 CL24 24 27.6 4.5 0.5042  
10 CL11 South Dakota 25 28.4 2.4 0.5449  
9 Louisiana CL16 3 30.3 3.5 0.5844  
8 CL34 CL30 14 23.3 . 0.7196  
7 CL8 CL12 18 19.3 15.0 0.7175  
6 CL10 South Carolina 26 21.4 4.2 0.7384  
5 CL6 New Mexico 27 24.0 4.7 0.8303  
4 CL5 Indiana 28 28.9 4.1 0.8343  
3 CL4 CL9 31 31.7 10.9 0.8472  
2 CL3 North Carolina 32 55.1 4.1 1.0017  
1 CL2 CL7 50 . 55.1 1.0663  

Output 32.1.3 Cluster Membership
Grounds for Divorce

CLUSTER=1

State Incompatibility Cruelty Desertion Non_Support Alcohol Felony Impotence Insanity Separation
Arizona 1 0 0 0 0 0 0 0 0
Colorado 1 0 0 0 0 0 0 0 0
Iowa 1 0 0 0 0 0 0 0 0
Kentucky 1 0 0 0 0 0 0 0 0
Michigan 1 0 0 0 0 0 0 0 0
Minnesota 1 0 0 0 0 0 0 0 0
Missouri 1 0 0 0 0 0 0 0 0
Montana 1 0 0 0 0 0 0 0 0
Nebraska 1 0 0 0 0 0 0 0 0
Oregon 1 0 0 0 0 0 0 0 0

CLUSTER=2

State Incompatibility Cruelty Desertion Non_Support Alcohol Felony Impotence Insanity Separation
California 1 0 0 0 0 0 0 1 0
Florida 1 0 0 0 0 0 0 1 0
Nevada 1 0 0 0 0 0 0 1 1
Wyoming 1 0 0 0 0 0 0 1 1

CLUSTER=3

State Incompatibility Cruelty Desertion Non_Support Alcohol Felony Impotence Insanity Separation
Alabama 1 1 1 1 1 1 1 1 1
Alaska 1 1 1 0 1 1 1 1 0
Arkansas 0 1 1 1 1 1 1 1 1
Connecticut 1 1 1 1 1 1 0 1 1
Georgia 1 1 1 0 1 1 1 1 0
Idaho 1 1 1 1 1 1 0 1 1
Illinois 0 1 1 0 1 1 1 0 0
Kansas 1 1 1 0 1 1 1 1 0
Maine 1 1 1 1 1 0 1 1 0
Maryland 0 1 1 0 0 1 1 1 1
Massachusetts 1 1 1 1 1 1 1 0 1
Mississippi 1 1 1 0 1 1 1 1 0
New Hampshire 1 1 1 1 1 1 1 0 0
New Jersey 0 1 1 0 1 1 0 1 1
North Dakota 1 1 1 1 1 1 1 1 0
Ohio 1 1 1 0 1 1 1 0 1
Oklahoma 1 1 1 1 1 1 1 1 0
Pennsylvania 0 1 1 0 0 1 1 1 0
Rhode Island 1 1 1 1 1 1 1 0 1
South Dakota 0 1 1 1 1 1 0 0 0
Tennessee 1 1 1 1 1 1 1 0 0
Texas 1 1 1 0 0 1 0 1 1
Utah 0 1 1 1 1 1 1 1 0
Vermont 0 1 1 1 0 1 0 1 1
West Virginia 1 1 1 0 1 1 0 1 1

CLUSTER=4

State Incompatibility Cruelty Desertion Non_Support Alcohol Felony Impotence Insanity Separation
Delaware 1 0 0 0 0 0 0 0 1
Hawaii 1 0 0 0 0 0 0 0 1
Washington 1 0 0 0 0 0 0 0 1
Wisconsin 1 0 0 0 0 0 0 0 1

CLUSTER=5

State Incompatibility Cruelty Desertion Non_Support Alcohol Felony Impotence Insanity Separation
Louisiana 0 0 0 0 0 1 0 0 1
New York 0 1 1 0 0 1 0 0 1
Virginia 0 1 0 0 0 1 0 0 1

CLUSTER=6

State Incompatibility Cruelty Desertion Non_Support Alcohol Felony Impotence Insanity Separation
South Carolina 0 1 1 0 1 0 0 0 1

CLUSTER=7

State Incompatibility Cruelty Desertion Non_Support Alcohol Felony Impotence Insanity Separation
New Mexico 1 1 1 0 0 0 0 0 0

CLUSTER=8

State Incompatibility Cruelty Desertion Non_Support Alcohol Felony Impotence Insanity Separation
Indiana 1 0 0 0 0 1 1 1 0

CLUSTER=9

State Incompatibility Cruelty Desertion Non_Support Alcohol Felony Impotence Insanity Separation
North Carolina 0 0 0 0 0 0 1 1 1

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