Output from hotel.sas
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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