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 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 VariableName 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 TotalStandardDeviation PooledStandardDeviation BetweenStandardDeviation 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

 CanonicalCorrelation AdjustedCanonicalCorrelation ApproximateStandardError SquaredCanonicalCorrelation Eigenvalues of Inv(E)*H = CanRsq/(1-CanRsq) Test of H0: The canonical correlations inthecurrent row and all that follow are zero Eigenvalue Difference Proportion Cumulative LikelihoodRatio ApproximateF 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 CanonicalStructure Variable Can1 X -1.000000 Y 1.000000

 Pooled Within CanonicalStructure Variable Can1 X -0.308237 Y 0.081243

 The CANDISC Procedure

 Total-Sample StandardizedCanonical Coefficients Variable Can1 X -2.625596855 Y 2.446680169

 Pooled Within-ClassStandardized CanonicalCoefficients Variable Can1 X -2.592150014 Y 2.474116072

 Raw Canonical Coefficients Variable Can1 X -1.205756217 Y 1.010412967

 Class Means on CanonicalVariables 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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