Consider an artificial data set with two classes of observations indicated by ‘H’ and ‘O’. The following statements generate and plot the data:
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; proc sgplot noautolegend; scatter y=y x=x / markerchar=group group=group; run;
The plot is shown in Figure 10.1.
Figure 10.1: Groups for Contrasting Univariate and Multivariate Analyses
The following statements perform a canonical discriminant analysis and display the results in Figure 10.2:
proc candisc anova; class Group; var x y; run;
Figure 10.2: Contrasting Univariate and Multivariate Analyses
Total Sample Size  40  DF Total  39 

Variables  2  DF Within Classes  38 
Classes  2  DF Between Classes  1 
Number of Observations Read  40 

Number of Observations Used  40 
Class Level Information  

Group  Variable Name 
Frequency  Weight  Proportion 
H  H  20  20.0000  0.500000 
O  O  20  20.0000  0.500000 
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 squares 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 square between Can1
and the class variable is 0.357963 as given by the squared canonical correlation, which is much higher than either univariate
R square.
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.