# st: Hotelling's T Squared

 From "Abrams, Judith" To Subject st: Hotelling's T Squared Date Mon, 24 Nov 2003 17:46:27 -0500

```Using the following dataset I compared the 2 groups with respect to x1 and x2 using Hotelling's T squared statistic.

group  animal   x1     x2     x3
|--------------------------------------
1. | 0        1    52.2   35.4   12.4
2. | 0        2    32.6   42.3   25.1
3. | 0        3    48.5   37.0   14.5
4. | 0        4    32.1   46.5   21.4
5. | 0        5    38.4   46.0   15.6
|--------------------------------------
6. | 1        6    19.2   50.1   30.7
7. | 1        7    11.7   55.0   33.3
8. | 1        8     7.0   64.5   28.5
9. | 1        9     6.4   50.2   43.4
10. | 1       10     6.2   50.0   43.8
+--------------------------------------

I got the following results:

2-group Hotelling's T-squared = 40.80087
F test statistic: ((10-2-1)/(10-2)(2)) x 40.80087 = 17.850381

H0: Vectors of means are equal for the two groups
F(2,7) =   17.8504
Prob > F(2,7) =    0.0018

I didn't include all 3 x variables because x1+x2+x3=100. Since I had given the investigator a long sermon about why I couldn't include x1 and x2 and x3, I decided to run Hotelling's T square with all 3 variables just to demonstrate that it would fail because of a singular matrix. Instead of failing, I got these results:

2-group Hotelling's T-squared = 40.80087
F test statistic: ((10-3-1)/(10-2)(3)) x 40.80087 = 10.200218

H0: Vectors of means are equal for the two groups
F(3,6) =   10.2002
Prob > F(3,6) =    0.0090

Shouldn't I have received an error message?

The regression command operated as expected. When I executed regress x1 x2 x3, x1 was dropped. The results of Manova were consistent with Hotelling's T-square.

Judith Abrams, PhD
Director, Integrated Biostatistics Unit
Phone 313 966 8657
Fax 313 966 8334

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```