Chapter Contents
Chapter Contents
Previous
Previous
Next
Next
The CLUSTER Procedure

Example 23.4: Evaluating the Effects of Ties

If, at some level of the cluster history, there is a tie for minimum distance between clusters, then one or more levels of the sample cluster tree are not uniquely determined. This example shows how the degree of indeterminacy can be assessed.

Mammals have four kinds of teeth: incisors, canines, premolars, and molars. The following data set gives the number of teeth of each kind on one side of the top and bottom jaws for 32 mammals.

Since all eight variables are measured in the same units, it is not strictly necessary to rescale the data. However, the canines have much less variance than the other kinds of teeth and, therefore, have little effect on the analysis if the variables are not standardized. An average linkage cluster analysis is run with and without standardization to allow comparison of the results. The results are shown in Output 23.4.1 and Output 23.4.2.

   title 'Hierarchical Cluster Analysis of Mammals'' Teeth Data';
   title2 'Evaluating the Effects of Ties';
   data teeth;
      input mammal $ 1-16
            @21 (v1-v8) (1.);
      label v1='Top incisors'
            v2='Bottom incisors'
            v3='Top canines'
            v4='Bottom canines'
            v5='Top premolars'
            v6='Bottom premolars'
            v7='Top molars'
            v8='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
   ;

   proc cluster data=teeth method=average nonorm outtree=_null_;
      var v1-v8;
      id mammal;
      title3 'Raw Data';
   run;

   proc cluster data=teeth std method=average nonorm outtree=_null_;
      var v1-v8;
      id mammal;
      title3 'Standardized Data';
   run;

Output 23.4.1: Average Linkage Analysis of Mammals' Teeth Data: Raw Data

Hierarchical Cluster Analysis of Mammals' Teeth Data
Evaluating the Effects of Ties
Raw Data

The CLUSTER Procedure
Average Linkage Cluster Analysis

Eigenvalues of the Covariance Matrix
  Eigenvalue Difference Proportion Cumulative
1 3.76799365 2.33557185 0.5840 0.5840
2 1.43242180 0.91781899 0.2220 0.8061
3 0.51460281 0.08414950 0.0798 0.8858
4 0.43045331 0.30021485 0.0667 0.9525
5 0.13023846 0.03814626 0.0202 0.9727
6 0.09209220 0.04216914 0.0143 0.9870
7 0.04992305 0.01603541 0.0077 0.9947
8 0.03388764   0.0053 1.0000

Root-Mean-Square Total-Sample Standard Deviation = 0.898027

Cluster History
NCL Clusters Joined FREQ RMS
Dist
T
i
e
31 BEAVER GROUNDHOG 2 0 T
30 GRAY SQUIRREL PORCUPINE 2 0 T
29 WOLF BEAR 2 0 T
28 MARTEN WOLVERINE 2 0 T
27 WEASEL BADGER 2 0 T
26 JAGUAR COUGAR 2 0 T
25 FUR SEAL SEA LION 2 0 T
24 REINDEER ELK 2 0 T
23 DEER MOOSE 2 0  
22 BROWN BAT SILVER HAIR BAT 2 1 T
21 PIGMY BAT HOUSE BAT 2 1 T
20 PIKA RABBIT 2 1 T
19 CL31 CL30 4 1 T
18 CL28 RIVER OTTER 3 1 T
17 CL27 SEA OTTER 3 1 T
16 CL24 CL23 4 1  
15 CL21 RED BAT 3 1.2247  
14 CL17 GREY SEAL 4 1.291  
13 CL29 RACCOON 3 1.4142 T
12 CL25 ELEPHANT SEAL 3 1.4142  
11 CL18 CL14 7 1.5546  
10 CL22 CL15 5 1.5811  
9 CL20 CL19 6 1.8708 T
8 CL11 CL26 9 1.9272  
7 CL8 CL12 12 2.2278  
6 MOLE CL13 4 2.2361  
5 CL9 HOUSE MOUSE 7 2.4833  
4 CL6 CL7 16 2.5658  
3 CL10 CL16 9 2.8107  
2 CL3 CL5 16 3.7054  
1 CL2 CL4 32 4.2939  

Output 23.4.2: Average Linkage Analysis of Mammals' Teeth Data: Standardized Data

Hierarchical Cluster Analysis of Mammals' Teeth Data
Evaluating the Effects of Ties
Standardized Data

The CLUSTER Procedure
Average Linkage Cluster Analysis

Eigenvalues of the Correlation Matrix
  Eigenvalue Difference Proportion Cumulative
1 4.74153902 3.27458808 0.5927 0.5927
2 1.46695094 0.70824118 0.1834 0.7761
3 0.75870977 0.25146252 0.0948 0.8709
4 0.50724724 0.30264737 0.0634 0.9343
5 0.20459987 0.05925818 0.0256 0.9599
6 0.14534169 0.03450100 0.0182 0.9780
7 0.11084070 0.04606994 0.0139 0.9919
8 0.06477076   0.0081 1.0000

The data have been standardized to mean 0 and variance 1

Root-Mean-Square Total-Sample Standard Deviation = 1

Cluster History
NCL Clusters Joined FREQ RMS
Dist
T
i
e
31 BEAVER GROUNDHOG 2 0 T
30 GRAY SQUIRREL PORCUPINE 2 0 T
29 WOLF BEAR 2 0 T
28 MARTEN WOLVERINE 2 0 T
27 WEASEL BADGER 2 0 T
26 JAGUAR COUGAR 2 0 T
25 FUR SEAL SEA LION 2 0 T
24 REINDEER ELK 2 0 T
23 DEER MOOSE 2 0  
22 PIGMY BAT RED BAT 2 0.9157  
21 CL28 RIVER OTTER 3 0.9169  
20 CL31 CL30 4 0.9428 T
19 BROWN BAT SILVER HAIR BAT 2 0.9428 T
18 PIKA RABBIT 2 0.9428  
17 CL27 SEA OTTER 3 0.9847  
16 CL22 HOUSE BAT 3 1.1437  
15 CL21 CL17 6 1.3314  
14 CL25 ELEPHANT SEAL 3 1.3447  
13 CL19 CL16 5 1.4688  
12 CL15 GREY SEAL 7 1.6314  
11 CL29 RACCOON 3 1.692  
10 CL18 CL20 6 1.7357  
9 CL12 CL26 9 2.0285  
8 CL24 CL23 4 2.1891  
7 CL9 CL14 12 2.2674  
6 CL10 HOUSE MOUSE 7 2.317  
5 CL11 CL7 15 2.6484  
4 CL13 MOLE 6 2.8624  
3 CL4 CL8 10 3.5194  
2 CL3 CL6 17 4.1265  
1 CL2 CL5 32 4.7753  


There are ties at 16 levels for the raw data but at only 10 levels for the standardized data. There are more ties for the raw data because the increments between successive values are the same for all of the raw variables but different for the standardized variables.

One way to assess the importance of the ties in the analysis is to repeat the analysis on several random permutations of the observations and then to see to what extent the results are consistent at the interesting levels of the cluster history. Three macros are presented to facilitate this process.

   /* --------------------------------------------------------- */
   /*                                                           */
   /* The macro CLUSPERM randomly permutes observations and     */
   /* does a cluster analysis for each permutation.             */
   /* The arguments are as follows:                             */
   /*                                                           */
   /*    data    data set name                                  */
   /*    var     list of variables to cluster                   */
   /*    id      id variable for proc cluster                   */
   /*    method  clustering method (and possibly other options) */
   /*    nperm   number of random permutations.                 */
   /*                                                           */
   /* --------------------------------------------------------- */
   %macro CLUSPERM(data,var,id,method,nperm);
   /* ------CREATE TEMPORARY DATA SET WITH RANDOM NUMBERS------ */
   data _temp_;
      set &data;
      array _random_ _ran_1-_ran_&nperm;
      do over _random_;
        _random_=ranuni(835297461);
      end;
   run;
   /* ------PERMUTE AND CLUSTER THE DATA----------------------- */
   %do n=1 %to &nperm;
       proc sort data=_temp_(keep=_ran_&n &var &id) out=_perm_;
          by _ran_&n;
       run;
       proc cluster method=&method noprint outtree=_tree_&n;
          var &var;
          id &id;
       run;
   %end;
   %mend;

   /* --------------------------------------------------------- */
   /*                                                           */
   /* The macro PLOTPERM plots various cluster statistics       */
   /* against the number of clusters for each permutation.      */
   /* The arguments are as follows:                             */
   /*                                                           */
   /*    stats   names of variables from tree data set          */
   /*    nclus   maximum number of clusters to be plotted       */
   /*    nperm   number of random permutations.                 */
   /*                                                           */
   /* --------------------------------------------------------- */
   %macro PLOTPERM(stat,nclus,nperm);
   /* ---CONCATENATE TREE DATA SETS FOR 20 OR FEWER CLUSTERS--- */
   data _plot_;
      set %do n=1 %to &nperm; _tree_&n(in=_in_&n) %end; ;
      if _ncl_<=&nclus;
      %do n=1 %to &nperm;
         if _in_&n then _perm_=&n;
      %end;
      label _perm_='permutation number';
      keep _ncl_ &stat _perm_;
   run;
   /* ---PLOT THE REQUESTED STATISTICS BY NUMBER OF CLUSTERS--- */

   proc plot;
      plot (&stat)*_ncl_=_perm_ /vpos=26;
   title2 'Symbol is value of _PERM_';
   run;
   %mend;

   /* --------------------------------------------------------- */
   /*                                                           */
   /* The macro TREEPERM generates cluster-membership variables */
   /* for a specified number of clusters for each permutation.  */
   /* PROC PRINT lists the objects in each cluster-combination, */
   /* and PROC TABULATE gives the frequencies and means.  The   */
   /* arguments are as follows:                                 */
   /*                                                           */
   /*    var     list of variables to cluster                   */
   /*            (no "-" or ":" allowed)                        */
   /*    id      id variable for proc cluster                   */
   /*    meanfmt format for printing means in PROC TABULATE     */
   /*    nclus   number of clusters desired                     */
   /*    nperm   number of random permutations.                 */
   /*                                                           */
   /* --------------------------------------------------------- */
   %macro TREEPERM(var,id,meanfmt,nclus,nperm);
   /* ------CREATE DATA SETS GIVING CLUSTER MEMBERSHIP--------- */
   %do n=1 %to &nperm;
      proc tree data=_tree_&n noprint n=&nclus
                out=_out_&n(drop=clusname 
                              rename=(cluster=_clus_&n));
         copy &var;
         id &id;
      run;
      proc sort;
         by &id &var;
      run;
   %end;
   /* ------MERGE THE CLUSTER VARIABLES------------------------ */
   data _merge_;
      merge
         %do n=1 %to &nperm;
            _out_&n
         %end; ;
      by &id &var;
      length all_clus $ %eval(3*&nperm);
      %do n=1 %to &nperm;
         substr( all_clus, %eval(1+(&n-1)*3), 3) =
            put( _clus_&n, 3.);
      %end;
   run;
   

   /* ------PRINT AND TABULATE CLUSTER COMBINATIONS------------ */
   proc sort;
      by _clus_:;
   run;
   proc print;
      var &var;
      id &id;
      by all_clus notsorted;
   run;
   proc tabulate order=data formchar='           ';
      class all_clus;
      var &var;
      table all_clus, n='FREQ'*f=5. mean*f=&meanfmt*(&var) /
         rts=%eval(&nperm*3+1);
   run;
   %mend;

To use these, it is first convenient to define a macro, VLIST, listing the teeth variables, since the forms V1-V8 or V: cannot be used with the TABULATE procedure in the TREEPERM macro:

   /* -TABULATE does not accept hyphens or colons in VAR lists- */
   %let vlist=v1 v2 v3 v4 v5 v6 v7 v8;

The CLUSPERM macro is then called to analyze ten random permutations. The PLOTPERM macro plots the pseudo F and t2 statistics and the cubic clustering criterion. Since the data are discrete, the pseudo F statistic and the cubic clustering criterion can be expected to increase as the number of clusters increases, so local maxima or large jumps in these statistics are more relevant than the global maximum in determining the number of clusters. For the raw data, only the pseudo t2 statistic indicates the possible presence of clusters, with the 4-cluster level being suggested. Hence, the TREEPERM macro is used to analyze the results at the 4-cluster level:

   title3 'Raw Data';

   /* ------CLUSTER RAW DATA WITH AVERAGE LINKAGE-------------- */
   %clusperm( teeth, &vlist, mammal, average, 10);

   /* -----PLOT STATISTICS FOR THE LAST 20 LEVELS-------------- */
   %plotperm( _psf_ _pst2_ _ccc_, 20, 10);

   /* ------ANALYZE THE 4-CLUSTER LEVEL------------------------ */
   %treeperm( &vlist, mammal, 9.1, 4, 10);

The results are shown in Output 23.4.3.

Output 23.4.3: Analysis of Ten Random Permutations of Raw Mammals' Teeth Data: Indeterminacy at the 4-Cluster Level

Hierarchical Cluster Analysis of Mammals' Teeth Data
Symbol is value of _PERM_

                                    Plot of _PSF_*_NCL_.  Symbol is value of _perm_.                                    
                                                                                                                        
            |                                                                                                           
        100 +                                                                                                           
            |                                                                                                           
            |                                                                                                           
      P     |                                                                                                           
      s     |                                                                                                           
      e     |                                                                                                 5         
      u  80 +                                                                                                           
      d     |                                                                                                           
      o     |                                                                                                           
            |                                                                                                           
      F     |                                                                                                           
            |                                                                                                 2         
      S  60 +                                                                                                           
      t     |                                                                                            5    4         
      a     |                                                                                            2              
      t     |                                                                                       9    9    1         
      i     |                                                                                  3    3    1    6         
      s     |                                                                        2    2    1    1    4              
      t  40 +                                                                   2    4    1                             
      i     |                                                         1    1    1    1                                  
      c     |                                                    2    3                                                 
            |            1                   2         2    1    1                                                      
            |       1         1    1    1         1    1                                                                
            |            2                   1                                                                          
         20 +                                                                                                           
            ---+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+--       
               1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16   17   18   19   20         
                                                                                                                        
                                                      Number of Clusters                                                
                                                                                                                        
NOTE: 10 obs had missing values.  151 obs hidden.                                                                       


Hierarchical Cluster Analysis of Mammals' Teeth Data
Symbol is value of _PERM_

                                   Plot of _PST2_*_NCL_.  Symbol is value of _perm_.                                    
                                                                                                                        
       P    |                                                                                                           
       s 30 +                                                                                                           
       e    |                                                                                                           
       u    |  1                                                                                                        
       d    |                                                                                                           
       o 25 +                                                                                                           
            |                                                                                                           
       T    |                                                                                                           
       -    |            1                                                                                              
       S 20 +                                                                                                           
       q    |       1                                                                                                   
       u    |                                                                                                           
       a    |                                                                                                           
       r 15 +                                                                                                           
       e    |                                                                                                           
       d    |            2                                                                                              
            |       2                                  1                                                                
       S 10 +                                                                                                           
       t    |                                                                                                           
       a    |                 2                        2                        3                                       
       t    |                 1    2    2    1    1              1                                                      
       i  5 +                           1                                       2    5                                  
       s    |                      1                        2    3              4    1                                  
       t    |                                               1                                                           
       i    |                                                                   1    2    1                             
       c  0 +                                                                                                           
            ---+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+--       
               1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16   17   18   19   20         
                                                                                                                        
                                                      Number of Clusters                                                
                                                                                                                        
NOTE: 69 obs had missing values.  104 obs hidden.                                                                       


Hierarchical Cluster Analysis of Mammals' Teeth Data
Symbol is value of _PERM_

                                    Plot of _CCC_*_NCL_.  Symbol is value of _perm_.                                    
                                                                                                                        
C   |                                                                                                                   
u 4 +                                                                                                                   
b   |                                                                                                                   
i   |                                                                                                                   
c   |                                                                                                                   
    |                              2                                                                                    
C   |                                                                                                                   
l 3 +                                                                                                                   
u   |                        1     1                                                                                    
s   |                                                                                                                   
t   |                                                                                                                   
e   |                                                                                                                   
r   |                        2                                                                                          
i 2 +                                                                                                                   
n   |                  1                                                                                                
g   |            1     2                                                                                                
    |                                                                                                                   
C   |                                                                                                                   
r   |                                                                                                                   
i 1 +                                                                                                                   
t   |                                                                                                                   
e   |                                                                                                                   
r   |            2                                                                                                      
i   |                                                                                                                   
o   |      1                                                                                                            
n 0 +1                                                                                                                  
    -+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
     1     2     3     4     5     6     7     8     9    10    11    12    13    14    15    16    17    18    19    20
                                                                                                                        
                                                     Number of Clusters                                                 
                                                                                                                        
NOTE: 140 obs had missing values.  50 obs hidden.                                                                       


all_clus=' 1 3 1 1 1 3 3 3 2 3'

mammal v1 v2 v3 v4 v5 v6 v7 v8
DEER 0 4 0 0 3 3 3 3
ELK 0 4 1 0 3 3 3 3
MOOSE 0 4 0 0 3 3 3 3
REINDEER 0 4 1 0 3 3 3 3

all_clus=' 2 2 2 2 2 2 1 2 1 1'

mammal v1 v2 v3 v4 v5 v6 v7 v8
BADGER 3 3 1 1 3 3 1 2
BEAR 3 3 1 1 4 4 2 3
COUGAR 3 3 1 1 3 2 1 1
ELEPHANT SEAL 2 1 1 1 4 4 1 1
FUR SEAL 3 2 1 1 4 4 1 1
GREY SEAL 3 2 1 1 3 3 2 2
JAGUAR 3 3 1 1 3 2 1 1
MARTEN 3 3 1 1 4 4 1 2
RACCOON 3 3 1 1 4 4 3 2
RIVER OTTER 3 3 1 1 4 3 1 2
SEA LION 3 2 1 1 4 4 1 1
SEA OTTER 3 2 1 1 3 3 1 2
WEASEL 3 3 1 1 3 3 1 2
WOLF 3 3 1 1 4 4 2 3
WOLVERINE 3 3 1 1 4 4 1 2

all_clus=' 2 4 2 2 4 2 1 2 1 1'

mammal v1 v2 v3 v4 v5 v6 v7 v8
MOLE 3 2 1 0 3 3 3 3

all_clus=' 3 1 3 3 3 1 2 1 3 2'

mammal v1 v2 v3 v4 v5 v6 v7 v8
BEAVER 1 1 0 0 2 1 3 3
GRAY SQUIRREL 1 1 0 0 1 1 3 3
GROUNDHOG 1 1 0 0 2 1 3 3
HOUSE MOUSE 1 1 0 0 0 0 3 3
PORCUPINE 1 1 0 0 1 1 3 3

all_clus=' 3 4 3 3 4 1 2 1 3 2'

mammal v1 v2 v3 v4 v5 v6 v7 v8
PIKA 2 1 0 0 2 2 3 3
RABBIT 2 1 0 0 3 2 3 3

all_clus=' 4 4 4 4 4 4 4 4 4 4'

mammal v1 v2 v3 v4 v5 v6 v7 v8
BROWN BAT 2 3 1 1 3 3 3 3
HOUSE BAT 2 3 1 1 1 2 3 3
PIGMY BAT 2 3 1 1 2 2 3 3
RED BAT 1 3 1 1 2 2 3 3
SILVER HAIR BAT 2 3 1 1 2 3 3 3

 


  FREQ Mean
Top incisors Bottom incisors Top canines Bottom canines Top premolars Bottom premolars Top molars Bottom molars
all_clus 4 0.0 4.0 0.5 0.0 3.0 3.0 3.0 3.0
1 3 1 1 1 3 3 3 2 3
2 2 2 2 2 2 1 2 1 1 15 2.9 2.6 1.0 1.0 3.6 3.4 1.3 1.8
2 4 2 2 4 2 1 2 1 1 1 3.0 2.0 1.0 0.0 3.0 3.0 3.0 3.0
3 1 3 3 3 1 2 1 3 2 5 1.0 1.0 0.0 0.0 1.2 0.8 3.0 3.0
3 4 3 3 4 1 2 1 3 2 2 2.0 1.0 0.0 0.0 2.5 2.0 3.0 3.0
4 4 4 4 4 4 4 4 4 4 5 1.8 3.0 1.0 1.0 2.0 2.4 3.0 3.0


 


  FREQ Mean
Top incisors Bottom incisors Top canines Bottom canines Top premolars Bottom premolars Top molars Bottom molars
all_clus 4 0.0 4.0 0.5 0.0 3.0 3.0 3.0 3.0
1 3 1 1 1 3 3 3 2 3
2 2 2 2 2 2 1 2 1 1 15 2.9 2.6 1.0 1.0 3.6 3.4 1.3 1.8
2 4 2 2 4 2 1 2 1 1 1 3.0 2.0 1.0 0.0 3.0 3.0 3.0 3.0
3 1 3 3 3 1 2 1 3 2 5 1.0 1.0 0.0 0.0 1.2 0.8 3.0 3.0
3 4 3 3 4 1 2 1 3 2 2 2.0 1.0 0.0 0.0 2.5 2.0 3.0 3.0
4 4 4 4 4 4 4 4 4 4 5 1.8 3.0 1.0 1.0 2.0 2.4 3.0 3.0


From the TABULATE and PRINT output, you can see that two types of clustering are obtained. In one case, the mole is grouped with the carnivores, while the pika and rabbit are grouped with the rodents. In the other case, both the mole and the lagomorphs are grouped with the bats.

Next, the analysis is repeated with the standardized data. The pseudo F and t2 statistics indicate 3 or 4 clusters, while the cubic clustering criterion shows a sharp rise up to 4 clusters and then levels off up to 6 clusters. So the TREEPERM macro is used again at the 4-cluster level. In this case, there is no indeterminacy, as the same four clusters are obtained with every permutation, although in different orders. It must be emphasized, however, that lack of indeterminacy in no way indicates validity. The results are shown in Output 23.4.4.

   title3 'Standardized Data';

   /*------CLUSTER STANDARDIZED DATA WITH AVERAGE LINKAGE------*/
   %clusperm( teeth, &vlist, mammal, average std, 10);

   /*------PLOT STATISTICS FOR THE LAST 20 LEVELS--------------*/
   %plotperm( _psf_ _pst2_ _ccc_, 20, 10);

   /*------ANALYZE THE 4-CLUSTER LEVEL-------------------------*/
   %treeperm( &vlist, mammal, 9.1, 4, 10);

Output 23.4.4: Analysis of Ten Random Permutations of Standardized Mammals' Teeth Data: No Indeterminacy at the 4-Cluster Level

Hierarchical Cluster Analysis of Mammals' Teeth Data
Symbol is value of _PERM_

                                    Plot of _PSF_*_NCL_.  Symbol is value of _perm_.                                    
                                                                                                                        
            |                                                                                                           
        100 +                                                                                                           
            |                                                                                                           
            |                                                                                                           
      P     |                                                                                                           
      s     |                                                                                                           
      e     |                                                                                                 1         
      u  80 +                                                                                                           
      d     |                                                                                            1              
      o     |                                                                                       1                   
            |                                                                                                           
      F     |                                                                                  1                        
            |                                                                             1                             
      S  60 +                                                                                                           
      t     |                                                                                                           
      a     |                                                                                                           
      t     |                                                                                                           
      i     |                                                                   1    1                                  
      s     |                                                                                                           
      t  40 +                                                         1    1                                            
      i     |                                                    1                                                      
      c     |                 1                             1                                                           
            |            1                        1    1                                                                
            |       1              1    1    1                                                                          
            |                                                                                                           
         20 +                                                                                                           
            ---+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+--       
               1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16   17   18   19   20         
                                                                                                                        
                                                      Number of Clusters                                                
                                                                                                                        
NOTE: 10 obs had missing values.  171 obs hidden.                                                                       


Hierarchical Cluster Analysis of Mammals' Teeth Data
Symbol is value of _PERM_

                                   Plot of _PST2_*_NCL_.  Symbol is value of _perm_.                                    
                                                                                                                        
       P    |                                                                                                           
       s    |                                                                                                           
       e    |                                                                                                           
       u    |                                                                                                           
       d    |                                                                                                           
       o    |                                                                                                           
         30 +                                                                                                           
       T    |  1                                                                                                        
       -    |                                                                                                           
       S    |                                                                                                           
       q    |                                                                                                           
       u    |                                                                                                           
       a 20 +                                                                                                           
       r    |                                                                                                           
       e    |       1                                                                                                   
       d    |                                                                                                           
            |            1                                                                                              
       S    |                                                                                                           
       t 10 +                                               1                                                           
       a    |                                                                                                           
       t    |                 1    1         1         1                             1                                  
       i    |                                                                                                           
       s    |                           1                             1    1                                            
       t    |                                                                             1                             
       i  0 +                                                                                                           
       c    |                                                                                                           
            ---+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+--       
               1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16   17   18   19   20         
                                                                                                                        
                                                      Number of Clusters                                                
                                                                                                                        
NOTE: 70 obs had missing values.  117 obs hidden.                                                                       


Hierarchical Cluster Analysis of Mammals' Teeth Data
Symbol is value of _PERM_

                                    Plot of _CCC_*_NCL_.  Symbol is value of _perm_.                                    
                                                                                                                        
C   |                                                                                                                   
u 4 +                                                                                                                   
b   |                  1           1                                                                                    
i   |                                                                                                                   
c   |                                                                                                                   
    |                                                                                                                   
C   |                                                                                                                   
l 3 +                        1                                                                                          
u   |                                                                                                                   
s   |                                                                                                                   
t   |                                                                                                                   
e   |                                                                                                                   
r   |            1                                                                                                      
i 2 +                                                                                                                   
n   |                                                                                                                   
g   |                                                                                                                   
    |                                                                                                                   
C   |                                                                                                                   
r   |                                                                                                                   
i 1 +                                                                                                                   
t   |                                                                                                                   
e   |                                                                                                                   
r   |                                                                                                                   
i   |                                                                                                                   
o   |      1                                                                                                            
n 0 +1                                                                                                                  
    -+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
     1     2     3     4     5     6     7     8     9    10    11    12    13    14    15    16    17    18    19    20
                                                                                                                        
                                                     Number of Clusters                                                 
                                                                                                                        
NOTE: 140 obs had missing values.  54 obs hidden.                                                                       


all_clus=' 1 3 1 1 1 3 3 3 2 3'

mammal v1 v2 v3 v4 v5 v6 v7 v8
DEER 0 4 0 0 3 3 3 3
ELK 0 4 1 0 3 3 3 3
MOOSE 0 4 0 0 3 3 3 3
REINDEER 0 4 1 0 3 3 3 3

all_clus=' 2 2 2 2 2 2 1 2 1 1'

mammal v1 v2 v3 v4 v5 v6 v7 v8
BADGER 3 3 1 1 3 3 1 2
BEAR 3 3 1 1 4 4 2 3
COUGAR 3 3 1 1 3 2 1 1
ELEPHANT SEAL 2 1 1 1 4 4 1 1
FUR SEAL 3 2 1 1 4 4 1 1
GREY SEAL 3 2 1 1 3 3 2 2
JAGUAR 3 3 1 1 3 2 1 1
MARTEN 3 3 1 1 4 4 1 2
RACCOON 3 3 1 1 4 4 3 2
RIVER OTTER 3 3 1 1 4 3 1 2
SEA LION 3 2 1 1 4 4 1 1
SEA OTTER 3 2 1 1 3 3 1 2
WEASEL 3 3 1 1 3 3 1 2
WOLF 3 3 1 1 4 4 2 3
WOLVERINE 3 3 1 1 4 4 1 2

all_clus=' 3 1 3 3 3 1 2 1 3 2'

mammal v1 v2 v3 v4 v5 v6 v7 v8
BEAVER 1 1 0 0 2 1 3 3
GRAY SQUIRREL 1 1 0 0 1 1 3 3
GROUNDHOG 1 1 0 0 2 1 3 3
HOUSE MOUSE 1 1 0 0 0 0 3 3
PIKA 2 1 0 0 2 2 3 3
PORCUPINE 1 1 0 0 1 1 3 3
RABBIT 2 1 0 0 3 2 3 3

all_clus=' 4 4 4 4 4 4 4 4 4 4'

mammal v1 v2 v3 v4 v5 v6 v7 v8
BROWN BAT 2 3 1 1 3 3 3 3
HOUSE BAT 2 3 1 1 1 2 3 3
MOLE 3 2 1 0 3 3 3 3
PIGMY BAT 2 3 1 1 2 2 3 3
RED BAT 1 3 1 1 2 2 3 3
SILVER HAIR BAT 2 3 1 1 2 3 3 3

 


  FREQ Mean
Top incisors Bottom incisors Top canines Bottom canines Top premolars Bottom premolars Top molars Bottom molars
all_clus 4 0.0 4.0 0.5 0.0 3.0 3.0 3.0 3.0
1 3 1 1 1 3 3 3 2 3
2 2 2 2 2 2 1 2 1 1 15 2.9 2.6 1.0 1.0 3.6 3.4 1.3 1.8
3 1 3 3 3 1 2 1 3 2 7 1.3 1.0 0.0 0.0 1.6 1.1 3.0 3.0
4 4 4 4 4 4 4 4 4 4 6 2.0 2.8 1.0 0.8 2.2 2.5 3.0 3.0

Chapter Contents
Chapter Contents
Previous
Previous
Next
Next
Top
Top

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.