data hw5;
input y1 y2 y3 group;
cards;
19.6 5.15 9.5 1
15.4 5.75 9.1 1
22.3 4.35 3.3 1
...
16.5 6.55 3.3 3
17.4 6.60 1.9 3
;
proc discrim manova canonical;
class group;
var y1 y2 y3;
run;
---------------------------------------------------------------------------------
Discriminant Analysis
33 Observations 32 DF Total
3 Variables 30 DF Within Classes
3 Classes 2 DF Between Classes
Class Level Information
Prior
GROUP Frequency Weight Proportion Probability
1 11 11.0000 0.333333 0.333333
2 11 11.0000 0.333333 0.333333
3 11 11.0000 0.333333 0.333333
Discriminant Analysis Pooled Covariance Matrix Information
Covariance Natural Log of the Determinant
Matrix Rank of the Covariance Matrix
3 5.90664529
Discriminant Analysis Pairwise Generalized Squared Distances Between Groups
2 _ _ -1 _ _
D (i|j) = (X - X )' COV (X - X )
i j i j
Generalized Squared Distance to GROUP
From GROUP 1 2 3
1 0 3.82929 3.47439
2 3.82929 0 0.03729
3 3.47439 0.03729 0
Discriminant Analysis
Multivariate Statistics and F Approximations
S=2 M=0 N=13
Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.52578837 3.5382 6 56 0.0049
Pillai's Trace 0.47667015 3.0248 6 58 0.0122
Hotelling-Lawley Trace 0.89723004 4.0375 6 54 0.0021
Roy's Greatest Root 0.89198797 8.6226 3 29 0.0003
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
NOTE: F Statistic for Wilks' Lambda is exact.
Canonical Discriminant Analysis
Adjusted Approx Squared
Canonical Canonical Standard Canonical
Correlation Correlation Error Correlation
1 0.686626 0.656071 0.093434 0.471455
2 0.072213 -.139365 0.175855 0.005215
Eigenvalues of INV(E)*H
= CanRsq/(1-CanRsq)
Eigenvalue Difference Proportion Cumulative
1 0.8920 0.8867 0.9942 0.9942
2 0.0052 . 0.0058 1.0000
Canonical Discriminant Analysis
Test of H0: The canonical correlations in the
current row and all that follow are zero
Likelihood
Ratio Approx F Num DF Den DF Pr > F
1 0.52578837 3.5382 6 56 0.0049
2 0.99478527 0.0760 2 29 0.9270
Canonical Discriminant Analysis
Total Canonical Structure
CAN1 CAN2
Y1 0.565965 0.545702
Y2 0.243728 0.693850
Y3 0.581791 -0.525512
Between Canonical Structure
CAN1 CAN2
Y1 0.994898 0.100888
Y2 0.957983 0.286823
Y3 0.995518 -0.094572
Canonical Discriminant Analysis
Pooled Within Canonical Structure
CAN1 CAN2
Y1 0.446970 0.591245
Y2 0.179960 0.702846
Y3 0.461776 -0.572231
Total-Sample Standardized Canonical Coefficients
CAN1 CAN2
Y1 1.155991773 0.260161821
Y2 -0.020572469 0.761278877
Y3 1.173241223 -0.572005336
Canonical Discriminant Analysis
Pooled Within-Class Standardized Canonical Coefficients
CAN1 CAN2
Y1 1.099060382 0.247349122
Y2 -0.020920448 0.774155771
Y3 1.109884712 -0.541116324
Raw Canonical Coefficients
CAN1 CAN2
Y1 0.3511025889 0.0790174213
Y2 -.0101965163 0.3773194483
Y3 0.2944046829 -.1435348897
Canonical Discriminant Analysis
Class Means on Canonical Variables
GROUP CAN1 CAN2
1 1.272352374 0.004139492
2 -0.682939550 0.082401706
3 -0.589412825 -0.086541197
Discriminant Analysis Linear Discriminant Function
_ -1 _ -1 _
Constant = -.5 X' COV X Coefficient Vector = COV X
j j j
GROUP
1 2 3
CONSTANT -42.50513 -27.30460 -27.52927
Y1 3.39524 2.71492 2.73441
Y2 0.69642 0.74589 0.68119
Y3 2.20958 1.62270 1.67448
Discriminant Analysis Classification Summary for Calibration Data: WORK.HW5
Resubstitution Summary using Linear Discriminant Function
Generalized Squared Distance Function:
2 _ -1 _
D (X) = (X-X )' COV (X-X )
j j j
Posterior Probability of Membership in each GROUP:
2 2
Pr(j|X) = exp(-.5 D (X)) / SUM exp(-.5 D (X))
j k k
Discriminant Analysis Classification Summary for Calibration Data: WORK.HW5
Resubstitution Summary using Linear Discriminant Function
Number of Observations and Percent Classified into GROUP:
From GROUP 1 2 3 Total
1 9 0 2 11
81.82 0.00 18.18 100.00
2 2 5 4 11
18.18 45.45 36.36 100.00
3 2 5 4 11
18.18 45.45 36.36 100.00
Discriminant Analysis Classification Summary for Calibration Data: WORK.HW5
Resubstitution Summary using Linear Discriminant Function
Number of Observations and Percent Classified into GROUP:
From GROUP 1 2 3 Total
Total 13 10 10 33
Percent 39.39 30.30 30.30 100.00
Priors 0.3333 0.3333 0.3333
Discriminant Analysis Classification Summary for Calibration Data: WORK.HW5
Resubstitution Summary using Linear Discriminant Function
Error Count Estimates for GROUP:
1 2 3 Total
Rate 0.1818 0.5455 0.6364 0.4545
Priors 0.3333 0.3333 0.3333