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Two-Sample Hotelling T2 1
OBS GROUP BM WP
1 1 190 90 2 1 170 80 3 1 180 80 4 1 200 120 5 1 150 60 6 1 180 70 7 2 160 120 8 2 190 150 9 2 150 90 10 2 160 130 11 2 140 110 12 2 145 130
Two-Sample Hotelling T2 2
General Linear Models Procedure Class Level Information Class Levels Values
GROUP 2 1 2 Number of observations in data set = 12
E = Error SS&CP Matrix
BM WP BM 3070.8333333 2808.3333333 WP 2808.3333333 4216.6666667
Two-Sample Hotelling T2 3
General Linear Models Procedure Multivariate Analysis of Variance Partial Correlation Coefficients from the Error SS&CP Matrix / Prob > |r|
DF = 10 BM WP BM 1.000000 0.780433
0.0001 0.0046
WP 0.780433 1.000000
0.0046 0.0001
Two-Sample Hotelling T2 4
General Linear Models Procedure Multivariate Analysis of Variance H = Type III SS&CP Matrix for GROUP
BM WP BM 1302.0833333 -2395.833333 WP -2395.833333 4408.3333333
Characteristic Roots and Vectors of: E Inverse * H, where H = Type III SS&CP Matrix for GROUP E = Error SS&CP Matrix Characteristic Percent Characteristic Vector V'EV=1 Root BM WP
6.41735719 100.00 -0.02640656 0.02380268 0.00000000 0.00 0.01164932 0.00633115
Manova Test Criteria and Exact F Statistics for the Hypothesis of no Overall GROUP Effect H = Type III SS&CP Matrix for GROUP E = Error SS&CP Matrix S=1 M=0 N=3.5 Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.13481891 28.8781 2 9 0.0001 Pillai's Trace 0.86518109 28.8781 2 9 0.0001 Hotelling-Lawley Trace 6.41735719 28.8781 2 9 0.0001 Roy's Greatest Root 6.41735719 28.8781 2 9 0.0001
Two-Sample Hotelling T2 5
Discriminant Analysis
12 Observations 11 DF Total 2 Variables 10 DF Within Classes 2 Classes 1 DF Between Classes
Class Level Information Output Prior GROUP SAS Name Frequency Weight Proportion Probability
1 _1 6 6.0000 0.500000 0.500000 2 _2 6 6.0000 0.500000 0.500000
Two-Sample Hotelling T2 6
Discriminant Analysis Within Covariance Matrix Information Covariance Natural Log of the Determinant GROUP Matrix Rank of the Covariance Matrix
1 2 10.39002 2 2 11.11642 Pooled 2 10.83209
Two-Sample Hotelling T2 7
Discriminant Analysis Test of Homogeneity of Within Covariance Matrices
Notation: K = Number of Groups P = Number of Variables N = Total Number of Observations - Number of Groups N(i) = Number of Observations in the i'th Group - 1 __ N(i)/2 || |Within SS Matrix(i)| V = ----------------------------------- N/2 |Pooled SS Matrix| _ _ 2 | 1 1 | 2P + 3P - 1 RHO = 1.0 - | SUM ----- - --- | ------------- |_ N(i) N _| 6(P+1)(K-1) DF = .5(K-1)P(P+1) _ _ | PN/2 | | N V | Under null hypothesis: -2 RHO ln | ------------------ | | __ PN(i)/2 | |_ || N(i) _| is distributed approximately as chi-square(DF) Test Chi-Square Value = 0.617821 with 3 DF Prob > Chi-Sq = 0.8923 Since the chi-square value is not significant at the 0.1 level, a pooled covariance matrix will be used in the discriminant function. Reference: Morrison, D.F. (1976) Multivariate Statistical Methods p252.
Two-Sample Hotelling T2 8
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
1 0 21.39119 2 21.39119 0
Two-Sample Hotelling T2 9
Discriminant Analysis Multivariate Statistics and Exact F Statistics
S=1 M=0 N=3.5
Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.13481891 28.8781 2 9 0.0001 Pillai's Trace 0.86518109 28.8781 2 9 0.0001 Hotelling-Lawley Trace 6.41735719 28.8781 2 9 0.0001 Roy's Greatest Root 6.41735719 28.8781 2 9 0.0001
Two-Sample Hotelling T2 10
Canonical Discriminant Analysis Adjusted Approx Squared Canonical Canonical Standard Canonical Correlation Correlation Error Correlation
1 0.930151 0.929263 0.040649 0.865181
Eigenvalues of INV(E)*H = CanRsq/(1-CanRsq) Eigenvalue Difference Proportion Cumulative
1 6.4174 . 1.0000 1.0000
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 0.13481891 28.8781 2 9 0.0001 NOTE: The F statistic is exact.
Total Canonical Structure CAN1
BM -0.586652 Basic Math WP 0.768607 Word Problems
Between Canonical Structure CAN1
BM -1.000000 Basic Math WP 1.000000 Word Problems
Pooled Within Canonical Structure CAN1
BM -0.257047 Basic Math WP 0.403622 Word Problems
Two-Sample Hotelling T2 11
Canonical Discriminant Analysis Total-Sample Standardized Canonical Coefficients CAN1
BM -1.664949888 Basic Math WP 2.107701136 Word Problems
Pooled Within-Class Standardized Canonical Coefficients CAN1
BM -1.463322437 Basic Math WP 1.545647499 Word Problems
Raw Canonical Coefficients CAN1
BM -.0835048900 Basic Math WP 0.0752706767 Word Problems
Class Means on Canonical Variables GROUP CAN1
1 -2.312530574 2 2.312530574
Two-Sample Hotelling T2 12
Discriminant Analysis Linear Discriminant Function
_ -1 _ -1 _ Constant = -.5 X' COV X Coefficient Vector = COV X j j j
GROUP 1 2 Label
CONSTANT -71.07648 -41.90798 BM 1.02321 0.63700 Basic Math WP -0.48384 -0.13571 Word Problems
Two-Sample Hotelling T2 13
OBS GROUP BM WP CAN1
1 1 190 90 -2.78495 2 1 170 80 -1.86756 3 1 180 80 -2.70261 4 1 200 120 -1.36188 5 1 150 60 -1.70287 6 1 180 70 -3.45531 7 2 160 120 1.97832 8 2 190 150 1.73129 9 2 150 90 0.55525 10 2 160 130 2.73102 11 2 140 110 2.89571 12 2 145 130 3.98360
Two-Sample Hotelling T2 14 Univariate t-test on Discrim scores
TTEST PROCEDURE Variable: CAN1 GROUP N Mean Std Dev Std Error Minimum Maximum ------------------------------------------------------------------------------
1 6 -2.31253057 0.79431921 0.32427946 -3.45531441 -1.36187838 2 6 2.31253057 1.17006709 0.47767789 0.55524582 3.98359734
Variances T DF Prob>|T| ---------------------------------------
Unequal -8.0108 8.8 0.0001 Equal -8.0108 10.0 0.0000 For H0: Variances are equal, F' = 2.17 DF = (5,5) Prob>F' = 0.4153