The CANDISC Procedure

## Example 27.1 Analysis of Iris Data With PROC CANDISC

The iris data published by Fisher (1936) have been widely used for examples in discriminant analysis and cluster analysis. The sepal length, sepal width, petal length, and petal width are measured in millimeters on 50 iris specimens from each of three species: Iris setosa, I. versicolor, and I. virginica.

This example is a canonical discriminant analysis that creates an output data set containing scores on the canonical variables and plots the canonical variables. The following statements produce Output 27.1.1 through Output 27.1.6:

```   title 'Fisher (1936) Iris Data';

proc format;
value specname
1='Setosa    '
2='Versicolor'
3='Virginica ';
run;

data iris;
input SepalLength SepalWidth PetalLength PetalWidth
Species @@;
format Species specname.;
label SepalLength='Sepal Length in mm.'
SepalWidth ='Sepal Width in mm.'
PetalLength='Petal Length in mm.'
PetalWidth ='Petal Width in mm.';
symbol = put(Species, specname10.);
datalines;
50 33 14 02 1 64 28 56 22 3 65 28 46 15 2 67 31 56 24 3
63 28 51 15 3 46 34 14 03 1 69 31 51 23 3 62 22 45 15 2

... more lines ...

63 33 60 25 3 53 37 15 02 1
;
```
```   proc candisc data=iris out=outcan distance anova;
class Species;
var SepalLength SepalWidth PetalLength PetalWidth;
run;
```

PROC CANDISC first displays information about the observations and the classes in the data set in Output 27.1.1.

Output 27.1.1 Iris Data: Summary Information
 Fisher (1936) Iris Data

The CANDISC Procedure

 Total Sample Size DF Total 150 149 4 147 3 2

 Number of Observations Read 150 150

Class Level Information
Species Variable
Name
Frequency Weight Proportion
Setosa Setosa 50 50.0000 0.333333
Versicolor Versicolor 50 50.0000 0.333333
Virginica Virginica 50 50.0000 0.333333

The DISTANCE option in the PROC CANDISC statement displays squared Mahalanobis distances between class means. Results from the DISTANCE option are shown in Output 27.1.2.

Output 27.1.2 Iris Data: Squared Mahalanobis Distances and Distance Statistics
 Fisher (1936) Iris Data

The CANDISC Procedure

Squared Distance to Species
From Species Setosa Versicolor Virginica
Setosa 0 89.86419 179.38471
Versicolor 89.86419 0 17.20107
Virginica 179.38471 17.20107 0

F Statistics, NDF=4, DDF=144 for Squared Distance
to Species
From Species Setosa Versicolor Virginica
Setosa 0 550.18889 1098
Versicolor 550.18889 0 105.31265
Virginica 1098 105.31265 0

Prob > Mahalanobis Distance for Squared Distance
to Species
From Species Setosa Versicolor Virginica
Setosa 1.0000 <.0001 <.0001
Versicolor <.0001 1.0000 <.0001
Virginica <.0001 <.0001 1.0000

The ANOVA option uses univariate statistics to test the hypothesis that the class means are equal. The resulting R-square values (see Output 27.1.3) range from 0.4008 for SepalWidth to 0.9414 for PetalLength, and each variable is significant at the 0.0001 level. The multivariate test for differences between the classes (which is displayed by default) is also significant at the 0.0001 level; you would expect this from the highly significant univariate test results.

Output 27.1.3 Iris Data: Univariate and Multivariate Statistics
 Fisher (1936) Iris Data

The CANDISC Procedure

Univariate Test Statistics
F Statistics, Num DF=2, Den DF=147
Variable Label Total
Standard
Deviation
Pooled
Standard
Deviation
Between
Standard
Deviation
R-Square R-Square
/ (1-RSq)
F Value Pr > F
SepalLength Sepal Length in mm. 8.2807 5.1479 7.9506 0.6187 1.6226 119.26 <.0001
SepalWidth Sepal Width in mm. 4.3587 3.3969 3.3682 0.4008 0.6688 49.16 <.0001
PetalLength Petal Length in mm. 17.6530 4.3033 20.9070 0.9414 16.0566 1180.16 <.0001
PetalWidth Petal Width in mm. 7.6224 2.0465 8.9673 0.9289 13.0613 960.01 <.0001

Average R-Square
Unweighted 0.7224358
Weighted by Variance 0.8689444

Multivariate Statistics and F Approximations
S=2 M=0.5 N=71
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda 0.02343863 199.15 8 288 <.0001
Pillai's Trace 1.19189883 53.47 8 290 <.0001
Hotelling-Lawley Trace 32.47732024 582.20 8 203.4 <.0001
Roy's Greatest Root 32.19192920 1166.96 4 145 <.0001

The R square between Can1 and the class variable, 0.969872, is much larger than the corresponding R square for Can2, 0.222027. This is displayed in Output 27.1.4.

Output 27.1.4 Iris Data: Canonical Correlations and Eigenvalues
 Fisher (1936) Iris Data

The CANDISC Procedure

Canonical
Correlation
Canonical
Correlation
Approximate
Standard
Error
Squared
Canonical
Correlation
Eigenvalues of Inv(E)*H
= CanRsq/(1-CanRsq)
Test of H0: The canonical correlations in the current row and all that follow are zero
Eigenvalue Difference Proportion Cumulative Likelihood
Ratio
Approximate
F Value
Num DF Den DF Pr > F
1 0.984821 0.984508 0.002468 0.969872 32.1919 31.9065 0.9912 0.9912 0.02343863 199.15 8 288 <.0001
2 0.471197 0.461445 0.063734 0.222027 0.2854   0.0088 1.0000 0.77797337 13.79 3 145 <.0001

Output 27.1.5 Iris Data: Correlations between Canonical and Original Variables
 Fisher (1936) Iris Data

The CANDISC Procedure

Total Canonical Structure
Variable Label Can1 Can2
SepalLength Sepal Length in mm. 0.791888 0.217593
SepalWidth Sepal Width in mm. -0.530759 0.757989
PetalLength Petal Length in mm. 0.984951 0.046037
PetalWidth Petal Width in mm. 0.972812 0.222902

Between Canonical Structure
Variable Label Can1 Can2
SepalLength Sepal Length in mm. 0.991468 0.130348
SepalWidth Sepal Width in mm. -0.825658 0.564171
PetalLength Petal Length in mm. 0.999750 0.022358
PetalWidth Petal Width in mm. 0.994044 0.108977

Pooled Within Canonical Structure
Variable Label Can1 Can2
SepalLength Sepal Length in mm. 0.222596 0.310812
SepalWidth Sepal Width in mm. -0.119012 0.863681
PetalLength Petal Length in mm. 0.706065 0.167701
PetalWidth Petal Width in mm. 0.633178 0.737242

The raw canonical coefficients (shown in Output 27.1.6) for the first canonical variable, Can1, show that the classes differ most widely on the linear combination of the centered variables: .

Output 27.1.6 Iris Data: Canonical Coefficients
 Fisher (1936) Iris Data

The CANDISC Procedure

Total-Sample Standardized Canonical Coefficients
Variable Label Can1 Can2
SepalLength Sepal Length in mm. -0.686779533 0.019958173
SepalWidth Sepal Width in mm. -0.668825075 0.943441829
PetalLength Petal Length in mm. 3.885795047 -1.645118866
PetalWidth Petal Width in mm. 2.142238715 2.164135931

Pooled Within-Class Standardized Canonical Coefficients
Variable Label Can1 Can2
SepalLength Sepal Length in mm. -.4269548486 0.0124075316
SepalWidth Sepal Width in mm. -.5212416758 0.7352613085
PetalLength Petal Length in mm. 0.9472572487 -.4010378190
PetalWidth Petal Width in mm. 0.5751607719 0.5810398645

Raw Canonical Coefficients
Variable Label Can1 Can2
SepalLength Sepal Length in mm. -.0829377642 0.0024102149
SepalWidth Sepal Width in mm. -.1534473068 0.2164521235
PetalLength Petal Length in mm. 0.2201211656 -.0931921210
PetalWidth Petal Width in mm. 0.2810460309 0.2839187853

Output 27.1.7 Iris Data: Canonical Means
Class Means on Canonical Variables
Species Can1 Can2
Setosa -7.607599927 0.215133017
Versicolor 1.825049490 -0.727899622
Virginica 5.782550437 0.512766605

The TEMPLATE and SGRENDER procedures within ODS Graphics are invoked to create a plot of the first two canonical variables. For general information about ODS Graphics, see Chapter 21, Statistical Graphics Using ODS. The following statements produce Output 27.1.8:

```   proc template;
define statgraph scatter;
begingraph;
entrytitle 'Fisher (1936) Iris Data';
layout overlayequated / equatetype=fit
xaxisopts=(label='Canonical Variable 1')
yaxisopts=(label='Canonical Variable 2');
scatterplot x=Can1 y=Can2 / group=species name='iris';
layout gridded / autoalign=(topleft);
discretelegend 'iris' / border=false opaque=false;
endlayout;
endlayout;
endgraph;
end;
run;

proc sgrender data=outcan template=scatter;
run;
```

Output 27.1.8 Iris Data: Plot of First Two Canonical Variables

The plot of canonical variables in Output 27.1.8 shows that of the two canonical variables, Can1 has more discriminatory power.

 Previous Page | Next Page | Top of Page