What is the Hotelling $T^2$ used for?

Question

Hotelling's $T^2$ distribution arises in testing differences between means of different populations. But is it often used? Can it be implemented in a modeling procedure, let's say, logistic regression?

score 4 · Answer 1 · edited Aug 03 '13 at 08:29

4

It is used to test means of different populations in a multivariate sense. Hotelling's $T^{2}$ test is the multivariate analog of Student's $t$ test. So, the 'mean' is in the sense of a mean vector.

As far as logistic regression, I don't know. Perhaps you mean a general linear model with a logistic link function, since a logistic regression is a univariate procedure.

edited Aug 03 '13 at 08:29

Nick Cox

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answered Aug 03 '13 at 08:25

bill_e

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I wanted to know the applications of hotelling's t2 in modeling as well as in any other analysis since I haven't seen it being used often anywhere. What are the areas where I can use it and how? – Hassan Aug 03 '13 at 08:37
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Whenever it comes up, you are essentially comparing two means, like a t-test. It would come up in the analysis of a general linear model. My favorite book on multivariate analysis is online here: http://istics.net/stat/stat-text-booksfree-not-free/ – bill_e Aug 04 '13 at 05:25
@bill_e, the link is dead. Could you provide a full reference? – Richard Hardy Apr 25 '23 at 06:08

score 4 · Accepted Answer · answered Aug 03 '13 at 12:00

Peter O'Brien has shown that logistic regression has several advantages over Hotelling's $T^2$. You reverse the problem and predict group membership using as predictors the series of multivariate responses. This does not require the multivariate normality assumption required of $T^2$, and allows mixtures of continuous and categorical responses. If there is not just a difference in means but a difference in variance for a response across the groups, you include a square term in the logistic model for that response. I suppose that if skewness differs you could include a cube term.

@ARTICLE{obr88com,
  author = {{O'Brien}, Peter C.},
  year = 1988,
  title = {Comparing two samples: {Extensions} of the $t$, rank-sum, and
          log-rank test},
  journal = JASA,
  volume = 83,
  pages = {52-61},
  annote = {see Hauck WW, Hyslop T, Anderson S (2000) Stat in Med 19:887-899}
}

Thanks Frank, Hotelling's T 2 is used when the IV has only two groups so here in your example what would that mean? — Hassan, Aug 03 '13 at 13:00
That means that binary logistic regression is used with two levels for $Y$. For multiple groups you would use polytomous (multinomial) logistic regression. — Frank Harrell, Aug 03 '13 at 13:20

What is the Hotelling $T^2$ used for?

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