Example 92.1 Mammals’ Teeth

The following statements produce a data set containing the numbers of different kinds of teeth for a variety of mammals:

data teeth;
   title 'Mammals'' Teeth';
   input mammal $ 1-16 @21 (v1-v8) (1.);
   label V1='Right Top Incisors'
         V2='Right Bottom Incisors'
         V3='Right Top Canines'
         V4='Right Bottom Canines'
         V5='Right Top Premolars'
         V6='Right Bottom Premolars'
         V7='Right Top Molars'
         V8='Right Bottom Molars';
   datalines;
Brown Bat           23113333
Mole                32103333
Silver Hair Bat     23112333
Pigmy Bat           23112233
House Bat           23111233
Red Bat             13112233
Pika                21002233
Rabbit              21003233
Beaver              11002133
Groundhog           11002133
Gray Squirrel       11001133
House Mouse         11000033
Porcupine           11001133
Wolf                33114423
Bear                33114423
Raccoon             33114432
Marten              33114412
Weasel              33113312
Wolverine           33114412
Badger              33113312
River Otter         33114312
Sea Otter           32113312
Jaguar              33113211
Cougar              33113211
Fur Seal            32114411
Sea Lion            32114411
Grey Seal           32113322
Elephant Seal       21114411
Reindeer            04103333
Elk                 04103333
Deer                04003333
Moose               04003333
;
run;

The following statements use the CLUSTER procedure to cluster the mammals by average linkage and the TREE procedure to produce a horizontal tree diagram that uses the average-linkage distance as its height axis:

proc cluster method=average std pseudo noeigen outtree=tree;
   id mammal;
   var v1-v8;
run;

proc tree horizontal;
run;

Output 92.1.1 displays the information about how the clusters are joined. For example, the cluster history shows that the observations Wolf and Bear form cluster 29, which is merged with Raccoon to form cluster 11.

Output 92.1.2 shows the corresponding tree diagram.

Output 92.1.2 Tree Diagram of Mammal Teeth Clusters

As you view the diagram in Output 92.1.2 from left to right, objects and clusters are progressively joined until a single, all-encompassing cluster is formed at the right (or root) of the tree. Clusters exist at each level of the diagram, and every vertical line connects leaves and branches into progressively larger clusters. For example, the five bats form a cluster at the 0.6 level, while the next cluster consists only of the mole. The observations Reindeer, Elk, Deer, and Moose form the next cluster at the 0.6 level, the mammals Pika through House Mouse are in the fourth cluster, The observations Wolf, Bear, and Raccoon form the fifth cluster, while the last cluster contains the observations Marten through Elephant Seal.

The following statements create the same tree with line printer graphics in a vertical orientation:

proc tree lineprinter;
run;

The tree is displayed in Output 92.1.3.

                            Name of Observation or Cluster                      
                                                                                
              S                                                                 
              i                                                                 
              l                         G                                 E     
              v                         r                                 l     
              e                         a   H           R                 e     
              r                         y   o           i                 p     
            B   P   H                 G   P u         W v     S G         h     
            r H i   o   R             r S o s         o e     e r     F S a     
            o a g R u   e             o q r e     R   l r     a e     u e n     
            w i m e s   i         R B u u c       a M v   W B   y J C r a t     
            n r y d e   n     M   a e n i u M     c a e O e a O   a o           
                      M d   D o P b a d r p o W B c r r t a d t S g u S L S     
            B B B B B o e E e o i b v h r i u o e o t i t s g t e u g e i e     
            a a a a a l e l e s k i e o e n s l a o e n e e e e a a a a o a     
            t t t t t e r k r e a t r g l e e f r n n e r l r r l r r l n l     
    A  1.5 +                                                                    
    v      |                                                                    
    e      |                                                                    
    r      |XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX     
    a      |XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXX     
    g    1 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXX     
    e      |XXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXX     
           |XXXXXXXXXXX XXXXXXX XXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXX     
    D      |XXXXXXXXXXX XXXXXXX XXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXX     
    i      |XXXXXXXXX . XXXXXXX XXXXXXXXXXXXX XXXXX XXXXXXXXXXXXXXXXXXXXXXX     
    s  0.5 +XXXXXXXXX . XXXXXXX XXXXXXXXXXX . XXXXX XXXXXXXXXXXXXXXXX XXXXX     
    t      |XXXXXXXXX . XXX XXX XXXXXXXXXXX . XXXXX XXXXXXXXXXXXX XXX XXXXX     
    a      |XXX XXXXX . XXX XXX XXX XXXXXXX . XXX . XXXXXXXXXXX . XXX XXXXX     
    n      |XXX XXX . . XXX XXX XXX XXXXXXX . XXX . XXXXX XXXXX . XXX XXX .     
    c      |. . . . . . XXX XXX . . XXX XXX . XXX . XXX . XXX . . XXX XXX .     
    e    0 +. . . . . . XXX XXX . . XXX XXX . XXX . XXX . XXX . . XXX XXX .     
                                                                                
    B                                                                           

As you look up from the bottom of the diagram, objects and clusters are progressively joined until a single, all-encompassing cluster is formed at the top (or root) of the root. Clusters exist at each level of the diagram. For example, the unbroken line of Xs at the leftmost side of the 0.6 level indicates that the five bats have formed a cluster. The next cluster is represented by a period because it contains only one mammal, Mole. Reindeer, Elk, Deer, and Moose form the next cluster, indicated by Xs again. The mammals Pika through House Mouse are in the fourth cluster. The observations Wolf, Bear, and Raccoon form the fifth cluster, while the last cluster contains the observations Marten through Elephant Seal.

The next statements sort the clusters at each branch in order of formation and use the number of clusters as the height axis:

proc tree sort height=n horizontal;
run;

The resulting tree is displayed in Output 92.1.4.

Output 92.1.4 PROC TREE with SORT and HEIGHT= Options

Because the CLUSTER procedure always produces binary trees, the number of internal (root and branch) nodes in the tree is one less than the number of leaves. Therefore 31 clusters are formed from the 32 mammals in the input data set. These are represented by the 31 vertical line segments in the tree diagram, each at a different value along the horizontal axis.

As you examine the tree from left to right, the first vertical line segment is where Beaver and Groundhog are clustered and the number of clusters is 31. The next cluster is formed from Gray Squirrel and Porcupine. The third contains Wolf and Bear. Note how the tree graphically displays the clustering order information that was presented in tabular form by the CLUSTER procedure in Output 92.1.1.

The same clusters as in Output 92.1.2 and Output 92.1.3 can be seen at the six-cluster level of the tree diagram in Output 92.1.4, although the SORT and HEIGHT= options make them appear in a different order.

The following statements create these six clusters and save the result in the output data set part:

proc tree noprint out=part nclusters=6;
   id mammal;
   copy v1-v8;
proc sort;
   by cluster;

The PROC TREE statement produces no output but creates an output data set indicating the cluster to which each observation belongs at the six-cluster level in the tree. The following statements print part, with the results shown in Output 92.1.5:

proc print label uniform;
   id mammal;
   var v1-v8;
   format v1-v8 1.;
   by cluster;
run;