Introduction to Discriminant
Procedures 
Example: Contrasting Univariate and Multivariate Analyses
Consider the two classes indicated by
`H' and `O' in Figure 7.1.
The results are shown in Figure 7.2.
data random;
drop n;
Group = 'H';
do n = 1 to 20;
X = 4.5 + 2 * normal(57391);
Y = X + .5 + normal(57391);
output;
end;
Group = 'O';
do n = 1 to 20;
X = 6.25 + 2 * normal(57391);
Y = X  1 + normal(57391);
output;
end;
run;
symbol1 v='H' c=blue;
symbol2 v='O' c=yellow;
proc gplot;
plot Y*X=Group / cframe=ligr nolegend;
run;
proc candisc anova;
class Group;
var X Y;
run;
Figure 7.1: Groups for Contrasting Univariate and
Multivariate Analyses
Observations 
40 
DF Total 
39 
Variables 
2 
DF Within Classes 
38 
Classes 
2 
DF Between Classes 
1 
Class Level Information 
Group 
Variable Name 
Frequency 
Weight 
Proportion 
H 
H 
20 
20.0000 
0.500000 
O 
O 
20 
20.0000 
0.500000 

Figure 7.2: Contrasting Univariate and Multivariate Analyses
Univariate Test Statistics 
F Statistics, Num DF=1, Den DF=38 
Variable 
Total Standard Deviation 
Pooled Standard Deviation 
Between Standard Deviation 
RSquare 
RSquare / (1RSq) 
F Value 
Pr > F 
X 
2.1776 
2.1498 
0.6820 
0.0503 
0.0530 
2.01 
0.1641 
Y 
2.4215 
2.4486 
0.2047 
0.0037 
0.0037 
0.14 
0.7105 
Average RSquare 
Unweighted 
0.0269868 
Weighted by Variance 
0.0245201 
Multivariate Statistics and Exact F Statistics 
S=1 M=0 N=17.5 
Statistic 
Value 
F Value 
Num DF 
Den DF 
Pr > F 
Wilks' Lambda 
0.64203704 
10.31 
2 
37 
0.0003 
Pillai's Trace 
0.35796296 
10.31 
2 
37 
0.0003 
HotellingLawley Trace 
0.55754252 
10.31 
2 
37 
0.0003 
Roy's Greatest Root 
0.55754252 
10.31 
2 
37 
0.0003 


Canonical Correlation 
Adjusted Canonical Correlation 
Approximate Standard Error 
Squared Canonical Correlation 
Eigenvalues of Inv(E)*H = CanRsq/(1CanRsq) 
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 
0.598300 
0.589467 
0.102808 
0.357963 
0.5575 

1.0000 
1.0000 
0.64203704 
10.31 
2 
37 
0.0003 
NOTE: 
The F statistic is exact. 


Total Canonical Structure 
Variable 
Can1 
X 
0.374883 
Y 
0.101206 
Between Canonical Structure 
Variable 
Can1 
X 
1.000000 
Y 
1.000000 
Pooled Within Canonical Structure 
Variable 
Can1 
X 
0.308237 
Y 
0.081243 

TotalSample Standardized Canonical Coefficients 
Variable 
Can1 
X 
2.625596855 
Y 
2.446680169 
Pooled WithinClass Standardized Canonical Coefficients 
Variable 
Can1 
X 
2.592150014 
Y 
2.474116072 
Raw Canonical Coefficients 
Variable 
Can1 
X 
1.205756217 
Y 
1.010412967 
Class Means on Canonical Variables 
Group 
Can1 
H 
0.7277811475 
O 
.7277811475 

The univariate R^{2}s are very small, 0.0503 for X and
0.0037 for Y, and neither variable shows a significant
difference between the classes at the 0.10 level.
The multivariate test for differences between
the classes is significant at the 0.0003 level.
Thus, the multivariate analysis has found a highly
significant difference, whereas the univariate
analyses failed to achieve even the 0.10 level.
The Raw Canonical Coefficients for the first canonical variable, Can1,
show that the classes differ most widely on the linear combination
1.205756217 X + 1.010412967 Y or approximately Y 
1.2 X.
The R^{2} between Can1 and the class variable is
0.357963 as given by the Squared Canonical Correlation,
which is much higher than either univariate R^{2}.
In this example, the variables are
highly correlated within classes.
If the withinclass correlation were smaller,
there would be greater agreement between
the univariate and multivariate analyses.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.