The HPCANDISC Procedure

Example 5.1 Analyzing Iris Data with PROC HPCANDISC

The iris data that were 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 in 50 iris specimens from each of three species: Iris setosa, I. versicolor, and I. virginica. The iris data set is available from the Sashelp library.

This example is a canonical discriminant analysis that creates an output data set that contains scores on the canonical variables and plots the canonical variables. The ID statement is specified to add the variable Species from the input data set to the output data set.

The following statements produce Output 5.1.1 through Output 5.1.6:

title 'Fisher (1936) Iris Data';

proc hpcandisc data=sashelp.iris out=outcan distance anova;
   id Species;
   class Species;
   var SepalLength SepalWidth PetalLength PetalWidth;
run;

Output 5.1.1 displays performance information and summary information about the observations and the classes in the data set.

Output 5.1.1: Iris Data: Performance and Summary Information

Fisher (1936) Iris Data

The HPCANDISC Procedure

Performance Information
Execution Mode Single-Machine
Number of Threads 4

Total Sample Size 150 DF Total 149
Variables 4 DF Within Classes 147
Class Levels 3 DF Between Classes 2

Number of Observations Read 150
Number of Observations Used 150

Class Level Information
Species Frequency Weight Proportion
Setosa 50 50.00000 0.33333
Versicolor 50 50.00000 0.33333
Virginica 50 50.00000 0.33333


Output 5.1.2 shows results from the DISTANCE option in the PROC HPCANDISC statement, which display squared Mahalanobis distances between class means.

Output 5.1.2: Iris Data: Squared Mahalanobis Distances and Distance Statistics

Fisher (1936) Iris Data

The HPCANDISC 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, Num DF=4, Den DF=144 for Squared Distance to Species
From Species Setosa Versicolor Virginica
Setosa 0 550.18889 1098.27375
Versicolor 550.18889 0 105.31265
Virginica 1098.27375 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


Output 5.1.3 displays univariate and multivariate statistics. The ANOVA option uses univariate statistics to test the hypothesis that the class means are equal. The resulting R-square values 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 class levels (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 5.1.3: Iris Data: Univariate and Multivariate Statistics

Fisher (1936) Iris Data

The HPCANDISC 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 (mm) 8.28066 5.14789 7.95061 0.6187 1.6226 119.26 <.0001
SepalWidth Sepal Width (mm) 4.35866 3.39688 3.36822 0.4008 0.6688 49.16 <.0001
PetalLength Petal Length (mm) 17.65298 4.30334 20.90700 0.9414 16.0566 1180.16 <.0001
PetalWidth Petal Width (mm) 7.62238 2.04650 8.96735 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 0.023439 199.15 8 288 <.0001
Pillai's Trace 1.191899 53.47 8 290 <.0001
Hotelling-Lawley Trace 32.477320 582.20 8 203.4 <.0001
Roy's Greatest Root 32.191929 1166.96 4 145 <.0001
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
NOTE: F Statistic for Wilks' Lambda is exact.


Output 5.1.4 displays canonical correlations and eigenvalues. The R square between Can1 and the class variable, 0.969872, is much larger than the corresponding R square for Can2, 0.222027.

Output 5.1.4: Iris Data: Canonical Correlations and Eigenvalues

Fisher (1936) Iris Data

The HPCANDISC Procedure

  Canonical
Correlation
Adjusted
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 5.1.5 displays correlations between canonical and original variables.

Output 5.1.5: Iris Data: Correlations between Canonical and Original Variables

Fisher (1936) Iris Data

The HPCANDISC Procedure

Total Canonical Structure
Variable Label Can1 Can2
SepalLength Sepal Length (mm) 0.79189 0.21759
SepalWidth Sepal Width (mm) -0.53076 0.75799
PetalLength Petal Length (mm) 0.98495 0.04604
PetalWidth Petal Width (mm) 0.97281 0.22290

Between Canonical Structure
Variable Label Can1 Can2
SepalLength Sepal Length (mm) 0.99147 0.13035
SepalWidth Sepal Width (mm) -0.82566 0.56417
PetalLength Petal Length (mm) 0.99975 0.02236
PetalWidth Petal Width (mm) 0.99404 0.10898

Pooled Within Canonical Structure
Variable Label Can1 Can2
SepalLength Sepal Length (mm) 0.22260 0.31081
SepalWidth Sepal Width (mm) -0.11901 0.86368
PetalLength Petal Length (mm) 0.70607 0.16770
PetalWidth Petal Width (mm) 0.63318 0.73724


Output 5.1.6 displays canonical coefficients. The raw canonical coefficients for the first canonical variable, Can1, show that the class levels differ most widely on the linear combination of the centered variables: $-0.0829378 \times \Variable{SepalLength} - 0.153447 \times \Variable{SepalWidth} + 0.220121 \times \Variable{PetalLength} + 0.281046 \times \Variable{PetalWidth}$.

Output 5.1.6: Iris Data: Canonical Coefficients

Fisher (1936) Iris Data

The HPCANDISC Procedure

Total-Sample Standardized Canonical Coefficients
Variable Label Can1 Can2
SepalLength Sepal Length (mm) -0.68678 0.01996
SepalWidth Sepal Width (mm) -0.66883 0.94344
PetalLength Petal Length (mm) 3.88580 -1.64512
PetalWidth Petal Width (mm) 2.14224 2.16414

Pooled Within-Class Standardized Canonical Coefficients
Variable Label Can1 Can2
SepalLength Sepal Length (mm) -0.42695 0.01241
SepalWidth Sepal Width (mm) -0.52124 0.73526
PetalLength Petal Length (mm) 0.94726 -0.40104
PetalWidth Petal Width (mm) 0.57516 0.58104

Raw Canonical Coefficients
Variable Label Can1 Can2
SepalLength Sepal Length (mm) -0.08294 0.00241
SepalWidth Sepal Width (mm) -0.15345 0.21645
PetalLength Petal Length (mm) 0.22012 -0.09319
PetalWidth Petal Width (mm) 0.28105 0.28392


Output 5.1.7 displays class means on canonical variables.

Output 5.1.7: Iris Data: Canonical Means

Class Means on Canonical Variables
Species Can1 Can2
Setosa -7.60760 0.21513
Versicolor 1.82505 -0.72790
Virginica 5.78255 0.51277


The TEMPLATE and SGRENDER procedures are used to create a plot of the first two canonical variables. The following statements produce Output 5.1.8:

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 5.1.8: Iris Data: Plot of First Two Canonical Variables


The plot of canonical variables in Output 5.1.8 shows that of the two canonical variables, Can1 has more discriminatory power.