The DISTANCE Procedure
 Creating a Distance Matrix as Input for a Subsequent Cluster Analysis

The following example demonstrates how you can use the DISTANCE procedure to obtain a distance matrix that will be used as input to a subsequent clustering procedure.

The following data, originated by A. Weber and cited in Hand et al. (1994, p. 297), measure the amount of protein consumed for nine food groups in 25 European countries. The nine food groups are red meat (RedMeat), white meat (WhiteMeat), eggs (Eggs), milk (Milk), fish (Fish), cereal (Cereal), starch (Starch), nuts (Nuts), and fruits and vegetables (FruitVeg). Suppose you want to determine whether national figures in protein consumption can be used to determine certain types or categories of countries; specifically, you want to perform a cluster analysis to determine whether these 25 countries can be formed into groups suggested by the data.

The following DATA step creates the SAS data set Protein:

```data Protein;
input Country \$14. RedMeat WhiteMeat Eggs Milk
Fish Cereal Starch Nuts FruitVeg;
datalines;
Albania        10.1  1.4  0.5   8.9  0.2  42.3  0.6  5.5  1.7
Austria         8.9 14.0  4.3  19.9  2.1  28.0  3.6  1.3  4.3
Belgium        13.5  9.3  4.1  17.5  4.5  26.6  5.7  2.1  4.0
Bulgaria        7.8  6.0  1.6   8.3  1.2  56.7  1.1  3.7  4.2
Czechoslovakia  9.7 11.4  2.8  12.5  2.0  34.3  5.0  1.1  4.0
Denmark        10.6 10.8  3.7  25.0  9.9  21.9  4.8  0.7  2.4
E Germany       8.4 11.6  3.7  11.1  5.4  24.6  6.5  0.8  3.6
Finland         9.5  4.9  2.7  33.7  5.8  26.3  5.1  1.0  1.4
France         18.0  9.9  3.3  19.5  5.7  28.1  4.8  2.4  6.5
Greece         10.2  3.0  2.8  17.6  5.9  41.7  2.2  7.8  6.5
Hungary         5.3 12.4  2.9   9.7  0.3  40.1  4.0  5.4  4.2
Ireland        13.9 10.0  4.7  25.8  2.2  24.0  6.2  1.6  2.9
Italy           9.0  5.1  2.9  13.7  3.4  36.8  2.1  4.3  6.7
Netherlands     9.5 13.6  3.6  23.4  2.5  22.4  4.2  1.8  3.7
Norway          9.4  4.7  2.7  23.3  9.7  23.0  4.6  1.6  2.7
Poland          6.9 10.2  2.7  19.3  3.0  36.1  5.9  2.0  6.6
Portugal        6.2  3.7  1.1   4.9 14.2  27.0  5.9  4.7  7.9
Romania         6.2  6.3  1.5  11.1  1.0  49.6  3.1  5.3  2.8
Spain           7.1  3.4  3.1   8.6  7.0  29.2  5.7  5.9  7.2
Sweden          9.9  7.8  3.5   4.7  7.5  19.5  3.7  1.4  2.0
Switzerland    13.1 10.1  3.1  23.8  2.3  25.6  2.8  2.4  4.9
UK             17.4  5.7  4.7  20.6  4.3  24.3  4.7  3.4  3.3
USSR            9.3  4.6  2.1  16.6  3.0  43.6  6.4  3.4  2.9
W Germany      11.4 12.5  4.1  18.8  3.4  18.6  5.2  1.5  3.8
Yugoslavia      4.4  5.0  1.2   9.5  0.6  55.9  3.0  5.7  3.2
;
```

The data set Protein contains the character variable Country and the nine numeric variables representing the food groups. The \$14. in the INPUT statement specifies that the variable Country has a length of 14.

The following statements create the distance matrix and display part of it:

```title 'Protein Consumption in Europe';
proc distance data=Protein out=Dist method=Euclid;
var interval(RedMeat--FruitVeg / std=Std);
id Country;
run;
```
```proc print data=Dist(obs=10);
title2 'First 10 Observations in Output Data Set from PROC DISTANCE';
run;
```

An output SAS data set called Dist that contains the distance matrix is created through the OUT= option. METHOD=EUCLID requests that Euclidean (which also is the default) distances should be computed.

The VAR statement lists the variables (RedMeatFruitVeg) along with their measurement level to be used in the analysis. An interval level of measurement is assigned to those variables. Since variables with large variances tend to have more effect on the proximity measure than those with small variances, each variable is standardized by the STD method to have a mean of 0 and a standard deviation of 1. This is done by adding "/ STD=STD" at the end of the variables list.

The ID statement specifies that the variable Country should be copied to the OUT= data set and used to generate names for the distance variables. The distance variables in the output data set are named by the values in the ID variable, and the maximum length for the names of these variables is 14.

There are 25 observations in the input data set; therefore, the output data set Dist contains a 25-by-25 lower triangular matrix.

The PROC PRINT statement displays the first 10 observations in the output data set Dist as shown in Figure 32.1.

Figure 32.1 First 10 Observations in the Output Data Set from PROC DISTANCE
 Protein Consumption in Europe First 10 Observations in Output Data Set from PROC DISTANCE

Obs Country Albania Austria Belgium Bulgaria Czechoslovakia Denmark E_Germany Finland France Greece Hungary Ireland Italy Netherlands Norway Poland Portugal Romania Spain Sweden Switzerland UK USSR W_Germany Yugoslavia
1 Albania 0.00000 . . . . . . . . . . . . . . . . . . . . . . . .
2 Austria 6.12388 0.00000 . . . . . . . . . . . . . . . . . . . . . . .
3 Belgium 5.94109 2.44987 0.00000 . . . . . . . . . . . . . . . . . . . . . .
4 Bulgaria 2.76446 4.88331 5.22711 0.00000 . . . . . . . . . . . . . . . . . . . . .
5 Czechoslovakia 5.13959 2.11498 2.21330 3.94761 0.00000 . . . . . . . . . . . . . . . . . . . .
6 Denmark 6.61002 3.01392 2.52541 6.00803 3.34049 0.00000 . . . . . . . . . . . . . . . . . . .
7 E Germany 6.39178 2.56341 2.10211 5.40824 1.87962 2.72112 0.00000 . . . . . . . . . . . . . . . . . .
8 Finland 5.81458 4.04271 3.45779 5.74882 3.91378 2.61570 3.99426 0.00000 . . . . . . . . . . . . . . . . .
9 France 6.29601 3.58891 2.19329 5.54675 3.36011 3.65772 3.78184 4.56796 0.00000 . . . . . . . . . . . . . . . .
10 Greece 4.24495 5.16330 4.69515 3.74849 4.86684 5.59084 5.61496 5.47453 4.54456 0 . . . . . . . . . . . . . . .

The following statements produce the tree diagram in Figure 32.2:

```proc cluster data=Dist method=Ward outtree=Tree noprint;
id Country;
run;

axis1 order=(0 to 1 by 0.1);
proc tree data=Tree haxis=axis1 horizontal;
height _rsq_;
id Country;
run;
```

The CLUSTER procedure performs a Ward’s minimum-variance cluster analysis based on the distance matrix created by the PROC DISTANCE. The printed output has been omitted, but the output data set Tree is created (through OUTTREE=TREE) and used as input to the TREE procedure, which produces the tree diagram shown in Figure 32.2. The HEIGHT statement specifies the variable _RSQ_ (the squared multiple correlation) as the height variable.

Figure 32.2 Tree Diagram of Clusters versus -Squared Values

After inspecting the tree diagram in Figure 32.2, you will see that when the countries are grouped into six clusters, the proportion of variance accounted for by these clusters is slightly less than 70% (69.3%). The 25 countries are clustered as follows:

• Balkan countries: Albania, Bulgaria, Romania, and Yugoslavia

• Mediterranean countries: Greece and Italy

• Iberian countries: Portugal and Spain

• Western European countries: Austria, Netherlands, Switzerland, Belgium, former West Germany, Ireland, France, and U.K.

• Scandinavian countries: Denmark, Norway, Finland, and Sweden

• Eastern European countries: former Czechoslovakia, former East Germany, Poland, former U.S.S.R., and Hungary

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