Example: Contrasting Univariate and Multivariate Analyses

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

The CANDISC Procedure

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

The CANDISC Procedure

Univariate Test Statistics
F Statistics, Num DF=1, Den DF=38
Variable	Total Standard Deviation	Pooled Standard Deviation	Between Standard Deviation	R-Square	R-Square / (1-RSq)	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 R-Square
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
Hotelling-Lawley Trace	0.55754252	10.31	2	37	0.0003
Roy's Greatest Root	0.55754252	10.31	2	37	0.0003

The CANDISC 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
	Canonical Correlation	Adjusted Canonical Correlation	Approximate Standard Error	Squared Canonical Correlation	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.

The CANDISC Procedure

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

The CANDISC Procedure

Total-Sample Standardized Canonical Coefficients
Variable	Can1
X	-2.625596855
Y	2.446680169

Pooled Within-Class 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 R²s 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 R² between Can1 and the class variable is 0.357963 as given by the Squared Canonical Correlation, which is much higher than either univariate R².

In this example, the variables are highly correlated within classes. If the within-class correlation were smaller, there would be greater agreement between the univariate and multivariate analyses.

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