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

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```