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