Example 81.1 Standardization of Variables in Cluster Analysis

To illustrate the effect of standardization in cluster analysis, this example uses the Fish data set described in the "Getting Started" section of Chapter 34, The FASTCLUS Procedure. The numbers are measurements taken on 159 fish caught from the same lake (Laengelmavesi) near Tampere in Finland; this data set is available from the Data Archive of the Journal of Statistics Education. The complete data set is displayed in Chapter 82, The STEPDISC Procedure.

The species (bream, parkki, pike, perch, roach, smelt, and whitefish), weight, three different length measurements (measured from the nose of the fish to the beginning of its tail, the notch of its tail, and the end of its tail), height, and width of each fish are recorded. The height and width are recorded as percentages of the third length variable.

Several new variables are created in the Fish data set: Weight3, Height, Width, and logLengthRatio. The weight of a fish indicates its size—a heavier pike tends to be larger than a lighter pike. To get a one-dimensional measure of the size of a fish, take the cubic root of the weight (Weight3). The variables Height, Width, Length1, Length2, and Length3 are rescaled in order to adjust for dimensionality. The logLengthRatio variable measures the tail length.

Because the new variables Weight3–logLengthRatio depend on the variable Weight, observations with missing values for Weight are not added to the data set. Consequently, there are 157 observations in the SAS data set Fish.

Before you perform a cluster analysis on coordinate data, it is necessary to consider scaling or transforming the variables since variables with large variances tend to have a larger effect on the resulting clusters than variables with small variances do.

This example uses three different approaches to standardize or transform the data prior to the cluster analysis. The first approach uses several standardization methods provided in the STDIZE procedure. However, since standardization is not always appropriate prior to the clustering (refer to Milligan and Cooper (1987) for a Monte Carlo study on various methods of variable standardization), the second approach performs the cluster analysis with no standardization. The third approach invokes the ACECLUS procedure to transform the data into a within-cluster covariance matrix.

The clustering is performed by the FASTCLUS procedure to find seven clusters. Note that the variables Length2 and Length3 are eliminated from this analysis since they both are significantly and highly correlated with the variable Length1. The correlation coefficients are 0.9958 and 0.9604, respectively. An output data set is created, and the FREQ procedure is invoked to compare the clusters with the species classification.

The DATA step is as follows, after the initial PROC FORMAT step:

   proc format;
      value specfmt
         1='Bream'
         2='Roach'
         3='Whitefish'
         4='Parkki'
         5='Perch'
         6='Pike'
         7='Smelt';

   data Fish (drop=HtPct WidthPct);
      title 'Fish Measurement Data';
      input Species Weight Length1 Length2 Length3 HtPct
            WidthPct @@;
   
      if Weight <= 0 or Weight=. then delete;
      Weight3=Weight**(1/3);
      Height=HtPct*Length3/(Weight3*100);
      Width=WidthPct*Length3/(Weight3*100);
      Length1=Length1/Weight3;
      Length2=Length2/Weight3;
      Length3=Length3/Weight3;
      logLengthRatio=log(Length3/Length1);
   
      format Species specfmt.;
      symbol = put(Species, specfmt2.);
   datalines;
   1  242.0 23.2 25.4 30.0 38.4 13.4 1  290.0 24.0 26.3 31.2 40.0 13.8
   1  340.0 23.9 26.5 31.1 39.8 15.1 1  363.0 26.3 29.0 33.5 38.0 13.3
   1  430.0 26.5 29.0 34.0 36.6 15.1 1  450.0 26.8 29.7 34.7 39.2 14.2
   1  500.0 26.8 29.7 34.5 41.1 15.3 1  390.0 27.6 30.0 35.0 36.2 13.4
   1  450.0 27.6 30.0 35.1 39.9 13.8 1  500.0 28.5 30.7 36.2 39.3 13.7
   1  475.0 28.4 31.0 36.2 39.4 14.1 1  500.0 28.7 31.0 36.2 39.7 13.3
   1  500.0 29.1 31.5 36.4 37.8 12.0 1     .  29.5 32.0 37.3 37.3 13.6
   1  600.0 29.4 32.0 37.2 40.2 13.9 1  600.0 29.4 32.0 37.2 41.5 15.0
   1  700.0 30.4 33.0 38.3 38.8 13.8 1  700.0 30.4 33.0 38.5 38.8 13.5
   1  610.0 30.9 33.5 38.6 40.5 13.3 1  650.0 31.0 33.5 38.7 37.4 14.8
   1  575.0 31.3 34.0 39.5 38.3 14.1 1  685.0 31.4 34.0 39.2 40.8 13.7
   1  620.0 31.5 34.5 39.7 39.1 13.3 1  680.0 31.8 35.0 40.6 38.1 15.1
   1  700.0 31.9 35.0 40.5 40.1 13.8 1  725.0 31.8 35.0 40.9 40.0 14.8
   
   ... more lines ...   

   7   19.7 13.2 14.3 15.2 18.9 13.6 7   19.9 13.8 15.0 16.2 18.1 11.6
   ;

The following macro, Std, standardizes the Fish data. The macro reads a single argument, mtd, which selects the METHOD= specification to be used in PROC STDIZE.

   /*--- macro for standardization ---*/
   
   %macro Std(mtd);
   title2 "Data are standardized by PROC STDIZE with METHOD= &mtd";
      proc stdize data=fish out=sdzout method=&mtd;
         var Length1 logLengthRatio Height Width Weight3;
      run;
   %mend Std;

The following macro, FastFreq, includes a PROC FASTCLUS statement for performing cluster analysis and a PROC FREQ statement for crosstabulating species with the cluster membership information that is derived from the previous PROC FASTCLUS statement. The macro reads a single argument, ds, which selects the input data set to be used in PROC FASTCLUS.

   /*--- macro for clustering and crosstabulating ---*/
   /*--- cluster membership with species           ---*/
   %macro FastFreq(ds);
      proc fastclus data=&ds out=clust maxclusters=7 maxiter=100 noprint;
         var Length1 logLengthRatio Height Width Weight3;
      run;
   
      proc freq data=clust;
         tables species*cluster;
      run;
   %mend FastFreq;

The following analysis (labeled ‘Approach 1’) includes 18 different methods of standardization followed by clustering. Since there is a large amount of output from this approach, only results from METHOD=STD, METHOD=RANGE, METHOD=AGK(0.14), and METHOD=SPACING(0.14) are shown. The following statements produce Output 81.1.1 through Output 81.1.4.

   /**********************************************************/
   /*                                                        */
   /*     Approach 1: data are standardized by PROC STDIZE   */
   /*                                                        */
   /**********************************************************/
   
   %Std(MEAN);
   %FastFreq(sdzout);
   
   %Std(MEDIAN);
   %FastFreq(sdzout);
   
   %Std(SUM);
   %FastFreq(sdzout);
   
   %Std(EUCLEN);
   %FastFreq(sdzout);
   
   %Std(USTD);
   %FastFreq(sdzout);
   
   %Std(STD);
   %FastFreq(sdzout);
   
   %Std(RANGE);
   %FastFreq(sdzout);
   
   %Std(MIDRANGE);
   %FastFreq(sdzout);
   
   %Std(MAXABS);
   %FastFreq(sdzout);
   
   %Std(IQR);
   %FastFreq(sdzout);
   
   %Std(MAD);
   %FastFreq(sdzout);
   
   %Std(AGK(.14));
   %FastFreq(sdzout);
   
   %Std(SPACING(.14));
   %FastFreq(sdzout);
   
   %Std(ABW(5));
   %FastFreq(sdzout);
   
   %Std(AWAVE(5));
   %FastFreq(sdzout);
   
   %Std(L(1));
   %FastFreq(sdzout);
   
   %Std(L(1.5));
   %FastFreq(sdzout);
   
   %Std(L(2));
   %FastFreq(sdzout);