The MODECLUS Procedure

Example 78.3 Cluster Analysis with Significance Tests

This example uses artificial data containing two clusters. One cluster is from a circular bivariate normal distribution. The other is a ring-shaped cluster that completely surrounds the first cluster. Without significance tests, the ring is divided into several sample clusters for any degree of smoothing that yields reasonable density estimates. The JOIN= option puts the ring back together. Output 78.3.1 displays a short summary generated from the first PROC MODECLUS statement. Output 78.3.2 contains a series of tables produced from the second PROC MODECLUS statement. The lack of p-value in the JOIN= option makes joining continue until only one cluster remains (see the description of the JOIN= option). The cluster memberships are then plotted as displayed in Output 78.3.1 through Output 78.3.8.

The following statements produce Output 78.3.1 through Output 78.3.8:

```title  'Modeclus Analysis with the JOIN= option';
title2 'A Normal Cluster Surrounded by a Ring Cluster';

data circle; keep x y;
c=1;
do n=1 to 30;
x=rannor(5);
y=rannor(5);
output;
end;

c=2;
do n=1 to 300;
x=rannor(5);
y=rannor(5);
z=rannor(5)+8;
l=z/sqrt(x**2+y**2);
x=x*l;
y=y*l;
output;
end;
run;
```
```proc modeclus data=circle m=1 r=1 to 3.5 by .25 join=20 short;
run;
```
```proc modeclus data=circle m=1 r=2.5 join out=out;
run;
```
```proc sgplot data=out noautolegend;
yaxis values=(-10 to 10 by 5);
xaxis values=(-15 to 15 by 5);
scatter y=y x=x / group=cluster Markerchar=cluster;
by _NJOIN_;
run;
```

Output 78.3.1: Significance Tests with the JOIN=20 and SHORT Options

 Modeclus Analysis with the JOIN= option A Normal Cluster Surrounded by a Ring Cluster

The MODECLUS Procedure

Cluster Summary
R Number of
Clusters
Joined
Maximum
P-value
Number of
Clusters
Frequency of
Unclassified
Objects
1 36 0.9339 1 301
1.25 20 0.7131 1 301
1.5 10 0.3296 1 300
1.75 5 0.1990 2 0
2 5 0.0683 2 0
2.25 3 0.0504 2 0
2.5 4 0.0301 2 0
2.75 3 0.0585 2 0
3 5 0.0003 1 0
3.25 4 0.1923 2 0
3.5 4 0.0000 1 0

Output 78.3.2: Significance Tests with the JOIN Option

 Modeclus Analysis with the JOIN= option A Normal Cluster Surrounded by a Ring Cluster

The MODECLUS Procedure
R=2.5 METHOD=1

Cluster Statistics -Saddle Test: Version 92.7-
Cluster Frequency Maximum
Estimated
Density
Boundary
Frequency
Estimated
Density
Mode
Count
Count
Overlap
Count
Z Approx
P-value
1 103 0.00617328 22 0.00308664 39 19 0 2.495 0.5055
2 71 0.00571029 20 0.0043213 36 27 9 1.193 0.999
3 53 0.00509296 18 0.00401263 32 25 10 0.986 0.9999
4 45 0.00478429 19 0.00354964 30 22 14 1.429 0.9924
5 30 0.00462996 0 . 29 0 . 3.611 0.0301
6 28 0.00370397 17 0.00354964 23 22 9 0.000 1

 Cluster 6 with P-value 1.0000 will be joined to cluster 4.

Cluster Statistics -Saddle Test: Version 92.7-
Cluster Frequency Maximum
Estimated
Density
Boundary
Frequency
Estimated
Density
Mode
Count
Count
Overlap
Count
Z Approx
P-value
1 103 0.00617328 22 0.00308664 39 19 0 2.495 0.5055
2 71 0.00571029 20 0.0043213 36 27 9 1.193 0.999
3 53 0.00509296 18 0.00401263 32 25 10 0.986 0.9999
4 73 0.00478429 13 0.00293231 30 18 0 1.588 0.9778
5 30 0.00462996 0 . 29 0 . 3.611 0.0301

 Cluster 3 with P-value 0.9999 will be joined to cluster 1.

Cluster Statistics -Saddle Test: Version 92.7-
Cluster Frequency Maximum
Estimated
Density
Boundary
Frequency
Estimated
Density
Mode
Count
Count
Overlap
Count
Z Approx
P-value
1 156 0.00617328 17 0.00246931 39 15 0 3.130 0.1318
2 71 0.00571029 20 0.0043213 36 27 9 1.193 0.999
3 73 0.00478429 13 0.00293231 30 18 0 1.588 0.9778
4 30 0.00462996 0 . 29 0 . 3.611 0.0301

 Cluster 2 with P-value 0.9990 will be joined to cluster 3.

Cluster Statistics -Saddle Test: Version 92.7-
Cluster Frequency Maximum
Estimated
Density
Boundary
Frequency
Estimated
Density
Mode
Count
Count
Overlap
Count
Z Approx
P-value
1 156 0.00617328 17 0.00246931 39 15 0 3.130 0.1318
2 144 0.00571029 14 0.00293231 36 18 0 2.313 0.6447
3 30 0.00462996 0 . 29 0 . 3.611 0.0301

 Cluster 2 with P-value 0.6447 will be joined to cluster 1.

Cluster Statistics -Saddle Test: Version 92.7-
Cluster Frequency Maximum
Estimated
Density
Boundary
Frequency
Estimated
Density
Mode
Count
Count
Overlap
Count
Z Approx
P-value
1 300 0.00617328 0 . 39 0 . 4.246 0.0026
2 30 0.00462996 0 . 29 0 . 3.611 0.0301

 Cluster 2 with P-value 0.0301 will be dissolved.

Cluster Statistics -Saddle Test: Version 92.7-
Cluster Frequency Maximum
Estimated
Density
Boundary
Frequency
Estimated
Density
Mode
Count
Count
Overlap
Count
Z Approx
P-value
1 300 0.00617328 0 . 39 0 . 4.246 0.0026

 30 observations were unassigned.

 Cluster 1 with P-value 0.0026 will be dissolved.

 Modeclus Analysis with the JOIN= option A Normal Cluster Surrounded by a Ring Cluster

The MODECLUS Procedure

Cluster Summary
R Number of
Clusters
Joined
Maximum
P-value
Number of
Clusters
Frequency of
Unclassified
Objects
2.5 0 1.0000 6 0
2.5 1 0.9999 5 0
2.5 2 0.9990 4 0
2.5 3 0.6447 3 0
2.5 4 0.0301 2 0
2.5 5 0.0026 1 30

Output 78.3.3: Cluster Memberships When Number of Clusters Joined=0

Output 78.3.4: Cluster Memberships When Number of Clusters Joined=1

Output 78.3.5: Cluster Memberships When Number of Clusters Joined=2

Output 78.3.6: Cluster Memberships When Number of Clusters Joined=3

Output 78.3.7: Cluster Memberships When Number of Clusters Joined=4

Output 78.3.8: Cluster Memberships When Number of Clusters Joined=5