Example 30.2 Crude Birth and Death Rates

This example uses the SAS data set Poverty created in the section Getting Started: CLUSTER Procedure. The data, from Rouncefield (1995), are birth rates, death rates, and infant death rates for 97 countries. Six cluster analyses are performed with eight methods. Scatter plots showing cluster membership at selected levels are produced instead of tree diagrams.

Each cluster analysis is performed by a macro called ANALYZE. The macro takes two arguments. The first, &METHOD, specifies the value of the METHOD= option to be used in the PROC CLUSTER statement. The second, &NCL, must be specified as a list of integers, separated by blanks, indicating the number of clusters desired in each scatter plot. For example, the first invocation of ANALYZE specifies the AVERAGE method and requests plots of three and eight clusters. When two-stage density linkage is used, the K= and R= options are specified as part of the first argument.

The ANALYZE macro first invokes the CLUSTER procedure with METHOD=&METHOD, where &METHOD represents the value of the first argument to ANALYZE. This part of the macro produces the PROC CLUSTER output shown.

The %DO loop processes &NCL, the list of numbers of clusters to plot. The macro variable &K is a counter that indexes the numbers within &NCL. The %SCAN function picks out the kth number in &NCL, which is then assigned to the macro variable &N. When &K exceeds the number of numbers in &NCL, %SCAN returns a null string. Thus, the %DO loop executes while &N is not equal to a null string. In the %WHILE condition, a null string is indicated by the absence of any nonblank characters between the comparison operator (NE) and the right parenthesis that terminates the condition.

Within the %DO loop, the TREE procedure creates an output data set containing &N clusters. The SGPLOT procedure then produces a scatter plot in which each observation is identified by the number of the cluster to which it belongs. The TITLE2 statement uses double quotes so that &N and &METHOD can be used within the title. At the end of the loop, &K is incremented by 1, and the next number is extracted from &NCL by %SCAN.

title 'Cluster Analysis of Birth and Death Rates';
ods graphics on;

%macro analyze(method,ncl);
   proc cluster data=poverty outtree=tree method=&method print=15 ccc pseudo;
      var birth death;
      title2;
   run;

   %let k=1;
   %let n=%scan(&ncl,&k);
   %do %while(&n NE);

      proc tree data=tree noprint out=out ncl=&n;
         copy birth death;
      run;

      proc sgplot;
         scatter y=death x=birth / group=cluster;
         title2 "Plot of &n Clusters from METHOD=&METHOD";
      run;

      %let k=%eval(&k+1);
      %let n=%scan(&ncl,&k);
   %end;
%mend;

The following statement produces Output 30.2.1, Output 30.2.3, and Output 30.2.4:

%analyze(average, 3 8)

For average linkage, the CCC has peaks at three, eight, ten, and twelve clusters, but the three-cluster peak is lower than the eight-cluster peak. The pseudo F statistic has peaks at three, eight, and twelve clusters. The pseudo statistic drops sharply at three clusters, continues to fall at four clusters, and has a particularly low value at twelve clusters. However, there are not enough data to seriously consider as many as twelve clusters. Scatter plots are given for three and eight clusters. The results are shown in Output 30.2.1 through Output 30.2.4. In Output 30.2.4, the eighth cluster consists of the two outlying observations, Mexico and Korea.

Output 30.2.1 Cluster Analysis for Birth and Death Rates: METHOD=AVERAGE
Cluster Analysis of Birth and Death Rates

The CLUSTER Procedure
Average Linkage Cluster Analysis

Eigenvalues of the Covariance Matrix
  Eigenvalue Difference Proportion Cumulative
1 189.106588 173.101020 0.9220 0.9220
2 16.005568   0.0780 1.0000

Root-Mean-Square Total-Sample Standard Deviation 10.127

Root-Mean-Square Distance Between Observations 20.25399

Cluster History
Number
of
Clusters
Clusters Joined Freq Semipartial
R-Square
R-Square Approximate
Expected
R-Square
Cubic
Clustering
Criterion
Pseudo F
Statistic
Pseudo
t-Squared
Norm RMS
Distance
Tie
15 CL27 CL20 18 0.0035 .980 .975 2.61 292 18.6 0.2325  
14 CL23 CL17 28 0.0034 .977 .972 1.97 271 17.7 0.2358  
13 CL18 CL54 8 0.0015 .975 .969 2.35 279 7.1 0.2432  
12 CL21 CL26 8 0.0015 .974 .966 2.85 290 6.1 0.2493  
11 CL19 CL24 12 0.0033 .971 .962 2.78 285 14.8 0.2767  
10 CL22 CL16 12 0.0036 .967 .957 2.84 284 17.4 0.2858  
9 CL15 CL28 22 0.0061 .961 .951 2.45 271 17.5 0.3353  
8 OB23 OB61 2 0.0014 .960 .943 3.59 302 . 0.3703  
7 CL25 CL11 17 0.0098 .950 .933 3.01 284 23.3 0.4033  
6 CL7 CL12 25 0.0122 .938 .920 2.63 273 14.8 0.4132  
5 CL10 CL14 40 0.0303 .907 .902 0.59 225 82.7 0.4584  
4 CL13 CL6 33 0.0244 .883 .875 0.77 234 22.2 0.5194  
3 CL9 CL8 24 0.0182 .865 .827 2.13 300 27.7 0.735  
2 CL5 CL3 64 0.1836 .681 .697 -.55 203 148 0.8402  
1 CL2 CL4 97 0.6810 .000 .000 0.00 . 203 1.3348  

Output 30.2.2 Criteria for the Number of Clusters: METHOD=AVERAGE
Criteria for the Number of Clusters: METHOD=AVERAGE

Output 30.2.3 Plot of Three Clusters: METHOD=AVERAGE
Plot of Three Clusters: METHOD=AVERAGE

Output 30.2.4 Plot of Eight Clusters: METHOD=AVERAGE
Plot of Eight Clusters: METHOD=AVERAGE

The following statement produces Output 30.2.5 and Output 30.2.7:

%analyze(complete, 3)

Complete linkage shows CCC peaks at three, eight and twelve clusters. The pseudo F statistic peaks at three and twelve clusters. The pseudo statistic indicates three clusters.

The scatter plot for three clusters is shown.

Output 30.2.5 Cluster History for Birth and Death Rates: METHOD=COMPLETE
Cluster Analysis of Birth and Death Rates

The CLUSTER Procedure
Complete Linkage Cluster Analysis

Eigenvalues of the Covariance Matrix
  Eigenvalue Difference Proportion Cumulative
1 189.106588 173.101020 0.9220 0.9220
2 16.005568   0.0780 1.0000

Root-Mean-Square Total-Sample Standard Deviation 10.127

Mean Distance Between Observations 17.13099

Cluster History
Number
of
Clusters
Clusters Joined Freq Semipartial
R-Square
R-Square Approximate
Expected
R-Square
Cubic
Clustering
Criterion
Pseudo F
Statistic
Pseudo
t-Squared
Norm
Maximum
Distance
Tie
15 CL22 CL33 8 0.0015 .983 .975 3.80 329 6.1 0.4092  
14 CL56 CL18 8 0.0014 .981 .972 3.97 331 6.6 0.4255  
13 CL30 CL44 8 0.0019 .979 .969 4.04 330 19.0 0.4332  
12 OB23 OB61 2 0.0014 .978 .966 4.45 340 . 0.4378  
11 CL19 CL24 24 0.0034 .974 .962 4.17 327 24.1 0.4962  
10 CL17 CL28 12 0.0033 .971 .957 4.18 325 14.8 0.5204  
9 CL20 CL13 16 0.0067 .964 .951 3.38 297 25.2 0.5236  
8 CL11 CL21 32 0.0054 .959 .943 3.44 297 19.7 0.6001  
7 CL26 CL15 13 0.0096 .949 .933 2.93 282 28.9 0.7233  
6 CL14 CL10 20 0.0128 .937 .920 2.46 269 27.7 0.8033  
5 CL9 CL16 30 0.0237 .913 .902 1.29 241 47.1 0.8993  
4 CL6 CL7 33 0.0240 .889 .875 1.38 248 21.7 1.2165  
3 CL5 CL12 32 0.0178 .871 .827 2.56 317 13.6 1.2326  
2 CL3 CL8 64 0.1900 .681 .697 -.55 203 167 1.5412  
1 CL2 CL4 97 0.6810 .000 .000 0.00 . 203 2.5233  

Output 30.2.6 Criteria for the Number of Clusters: METHOD=COMPLETE
Criteria for the Number of Clusters: METHOD=COMPLETE

Output 30.2.7 Plot of Clusters for METHOD=COMPLETE
Plot of Clusters for METHOD=COMPLETE

The following statement produces Output 30.2.8 and Output 30.2.10:

%analyze(single, 7 10)

The CCC and pseudo F statistics are not appropriate for use with single linkage because of the method’s tendency to chop off tails of distributions. The pseudo statistic can be used by looking for large values and taking the number of clusters to be one greater than the level at which the large pseudo value is displayed. For these data, there are large values at levels 6 and 9, suggesting seven or ten clusters.

The scatter plots for seven and ten clusters are shown.

Output 30.2.8 Cluster History for Birth and Death Rates: METHOD=SINGLE
Cluster Analysis of Birth and Death Rates

The CLUSTER Procedure
Single Linkage Cluster Analysis

Eigenvalues of the Covariance Matrix
  Eigenvalue Difference Proportion Cumulative
1 189.106588 173.101020 0.9220 0.9220
2 16.005568   0.0780 1.0000

Root-Mean-Square Total-Sample Standard Deviation 10.127

Mean Distance Between Observations 17.13099

Cluster History
Number
of
Clusters
Clusters Joined Freq Semipartial
R-Square
R-Square Approximate
Expected
R-Square
Cubic
Clustering
Criterion
Pseudo F
Statistic
Pseudo
t-Squared
Norm
Minimum
Distance
Tie
15 CL37 CL19 8 0.0014 .968 .975 -2.3 178 6.6 0.1331  
14 CL20 CL23 15 0.0059 .962 .972 -3.1 162 18.7 0.1412  
13 CL14 CL16 19 0.0054 .957 .969 -3.4 155 8.8 0.1442  
12 CL26 OB58 31 0.0014 .955 .966 -2.7 165 4.0 0.1486  
11 OB86 CL18 4 0.0003 .955 .962 -1.6 183 3.8 0.1495  
10 CL13 CL11 23 0.0088 .946 .957 -2.3 170 11.3 0.1518  
9 CL22 CL17 30 0.0235 .923 .951 -4.7 131 45.7 0.1593 T
8 CL15 CL10 31 0.0210 .902 .943 -5.8 117 21.8 0.1593  
7 CL9 OB75 31 0.0052 .897 .933 -4.7 130 4.0 0.1628  
6 CL7 CL12 62 0.2023 .694 .920 -15 41.3 223 0.1725  
5 CL6 CL8 93 0.6681 .026 .902 -26 0.6 199 0.1756  
4 CL5 OB48 94 0.0056 .021 .875 -24 0.7 0.5 0.1811 T
3 CL4 OB67 95 0.0083 .012 .827 -15 0.6 0.8 0.1811  
2 OB23 OB61 2 0.0014 .011 .697 -13 1.0 . 0.4378  
1 CL3 CL2 97 0.0109 .000 .000 0.00 . 1.0 0.5815  

Output 30.2.9 Criteria for the Number of Clusters: METHOD=SINGLE
Criteria for the Number of Clusters: METHOD=SINGLE

Output 30.2.10 Plot of Clusters for METHOD=SINGLE
Plot of Clusters for METHOD=SINGLEPlot of Clusters for METHOD=SINGLE, continued

The following statements produce Output 30.2.11 through Output 30.2.14:

%analyze(two k=10, 3)

%analyze(two k=18, 2)

For kth-nearest-neighbor density linkage, the number of modes as a function of k is as follows (not all of these analyses are shown):

k

 

modes

3

 

13

4

 

6

5-7

 

4

8-15

 

3

16-21

 

2

22+

 

1

Thus, there is strong evidence of three modes and an indication of the possibility of two modes. Uniform-kernel density linkage gives similar results. For K=10 (10th-nearest-neighbor density linkage), the scatter plot for three clusters is shown; and for K=18, the scatter plot for two clusters is shown.


Output 30.2.11 Cluster History for Birth and Death Rates: METHOD=TWOSTAGE K=10
Cluster Analysis of Birth and Death Rates

The CLUSTER Procedure
Two-Stage Density Linkage Clustering

Eigenvalues of the Covariance Matrix
  Eigenvalue Difference Proportion Cumulative
1 189.106588 173.101020 0.9220 0.9220
2 16.005568   0.0780 1.0000


K = 10

Root-Mean-Square Total-Sample Standard Deviation 10.127

Cluster History
Number
of
Clusters
  Freq Semipartial
R-Square
R-Square Approximate
Expected
R-Square
Cubic
Clustering
Criterion
Pseudo F
Statistic
Pseudo
t-Squared
Normalized
Fusion Density
Maximum Density
in Each Cluster
Tie
Clusters Joined Lesser Greater
15 CL16 OB94 22 0.0015 .921 .975 -11 68.4 1.4 9.2234 6.7927 15.3069  
14 CL19 OB49 28 0.0021 .919 .972 -11 72.4 1.8 8.7369 5.9334 33.4385  
13 CL15 OB52 23 0.0024 .917 .969 -10 76.9 2.3 8.5847 5.9651 15.3069  
12 CL13 OB96 24 0.0018 .915 .966 -9.3 83.0 1.6 7.9252 5.4724 15.3069  
11 CL12 OB93 25 0.0025 .912 .962 -8.5 89.5 2.2 7.8913 5.4401 15.3069  
10 CL11 OB78 26 0.0031 .909 .957 -7.7 96.9 2.5 7.787 5.4082 15.3069  
9 CL10 OB76 27 0.0026 .907 .951 -6.7 107 2.1 7.7133 5.4401 15.3069  
8 CL9 OB77 28 0.0023 .904 .943 -5.5 120 1.7 7.4256 4.9017 15.3069  
7 CL8 OB43 29 0.0022 .902 .933 -4.1 138 1.6 6.927 4.4764 15.3069  
6 CL7 OB87 30 0.0043 .898 .920 -2.7 160 3.1 4.932 2.9977 15.3069  
5 CL6 OB82 31 0.0055 .892 .902 -1.1 191 3.7 3.7331 2.1560 15.3069  
4 CL22 OB61 37 0.0079 .884 .875 0.93 237 10.6 3.1713 1.6308 100.0  
3 CL14 OB23 29 0.0126 .872 .827 2.60 320 10.4 2.0654 1.0744 33.4385  
2 CL4 CL3 66 0.2129 .659 .697 -1.3 183 172 12.409 33.4385 100.0  
1 CL2 CL5 97 0.6588 .000 .000 0.00 . 183 10.071 15.3069 100.0  


3 modal clusters have been formed.


Output 30.2.12 Cluster History for Birth and Death Rates: METHOD=TWOSTAGE K=18
Cluster Analysis of Birth and Death Rates

The CLUSTER Procedure
Two-Stage Density Linkage Clustering

Eigenvalues of the Covariance Matrix
  Eigenvalue Difference Proportion Cumulative
1 189.106588 173.101020 0.9220 0.9220
2 16.005568   0.0780 1.0000


K = 18

Root-Mean-Square Total-Sample Standard Deviation 10.127

Cluster History
Number
of
Clusters
  Freq Semipartial
R-Square
R-Square Approximate
Expected
R-Square
Cubic
Clustering
Criterion
Pseudo F
Statistic
Pseudo
t-Squared
Normalized
Fusion Density
Maximum Density
in Each Cluster
Tie
Clusters Joined Lesser Greater
15 CL16 OB72 46 0.0107 .799 .975 -21 23.3 3.0 10.118 7.7445 23.4457  
14 CL15 OB94 47 0.0098 .789 .972 -21 23.9 2.7 9.676 7.1257 23.4457  
13 CL14 OB51 48 0.0037 .786 .969 -20 25.6 1.0 9.409 6.8398 23.4457 T
12 CL13 OB96 49 0.0099 .776 .966 -19 26.7 2.6 9.409 6.8398 23.4457  
11 CL12 OB76 50 0.0114 .764 .962 -19 27.9 2.9 8.8136 6.3138 23.4457  
10 CL11 OB77 51 0.0021 .762 .957 -18 31.0 0.5 8.6593 6.0751 23.4457  
9 CL10 OB78 52 0.0103 .752 .951 -17 33.3 2.5 8.6007 6.0976 23.4457  
8 CL9 OB43 53 0.0034 .748 .943 -16 37.8 0.8 8.4964 5.9160 23.4457  
7 CL8 OB93 54 0.0109 .737 .933 -15 42.1 2.6 8.367 5.7913 23.4457  
6 CL7 OB88 55 0.0110 .726 .920 -13 48.3 2.6 7.916 5.3679 23.4457  
5 CL6 OB87 56 0.0120 .714 .902 -12 57.5 2.7 6.6917 4.3415 23.4457  
4 CL20 OB61 39 0.0077 .707 .875 -9.8 74.7 8.3 6.2578 3.2882 100.0  
3 CL5 OB82 57 0.0138 .693 .827 -5.0 106 3.0 5.3605 3.2834 23.4457  
2 CL3 OB23 58 0.0117 .681 .697 -.54 203 2.5 3.2687 1.7568 23.4457  
1 CL2 CL4 97 0.6812 .000 .000 0.00 . 203 13.764 23.4457 100.0  


2 modal clusters have been formed.


Output 30.2.13 Plot of Clusters for METHOD=TWOSTAGE K=10
Plot of Clusters for METHOD=TWOSTAGE K=10

Output 30.2.14 Plot of Clusters for METHOD=TWOSTAGE K=18
Plot of Clusters for METHOD=TWOSTAGE K=18

In summary, most of the clustering methods indicate three or eight clusters. Most methods agree at the three-cluster level, but at the other levels, there is considerable disagreement about the composition of the clusters. The presence of numerous ties also complicates the analysis; see Example 30.4.