Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [hotelling-t2]

Hotelling's T-squared test (one-sample or two-sample) is a generalization of a t-test for the multivariate case. It relies on Hotelling's T-squared distribution.

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Alternative distribution of $T^2$ statistic without Gaussian assumption

Background Let $p(x)$ be an arbitrary distribution defined on $\mathbb{R}^d$. Define $\mu = \mathbb{E}[x]$. Given an i.i.d. sample $x_1, \ldots, x_n \sim p(x)$, consider the following $T^2$ statistic for testing the hypotheses $H_0: \mu=0$ vs $H_1:…
wij
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Distribution of multivariate "$Z$-score"?

Suppose $\mathbf{X}_1, \dots, \mathbf{X}_n \sim N_p(\mathbf{\mu}, \Sigma)$ where $\mu \in \mathbb{R}^p$ and $\Sigma$ is a $p \times p$ covariance matrix. Suppose $\hat{\Sigma}$ is the sample covariance matrix, and $\bar{\mathbf{X}}$ is the sample…
Greenparker
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Confidence regions on bivariate normal distributions using $\hat{\Sigma}_{MLE}$ or $\mathbf{S}$

Given a $5 \times 2$ dataset $\mathbf{X} =\left( \begin{array}{rr}-0.9&0.2\\2.4&0.7\\-1.4&1.0\\2.9&-0.5\\2.0&-1.0 \end{array} \right)$. Assume that $X\sim N_2(\mu, \Sigma)$. Using $\hat{\mu}_{MLE}$ and $\hat{\Sigma}_{MLE}$, I need to calculate the…
Whizkid95
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Tutorial for performing Contrasts for MANOVA

Context I have a multivariate dataset with a test group and three control groups. I was thinking that the best way to determine if and how the test group differed from all of the control groups would be to perform a MANOVA, then perform contrast…
user3629
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Generating null distributions by a residual permutation procedure

I am trying to understand the method described in this paper which describes an hypothesis-testing framework for stable isotope ratios. The data are in a bivariate isotopic space and the metrics that are of use to me are the group centroid…
bobthechemist
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Proof for two-sample Hotelling $T^2$ statistic?

I've been reading "A primer of multivariate statistics" by Richard J. Harris, page 546, which shows how to derive the Hotelling $T^2$ statistic, after seeing this related but different question (I have the degrees of freedom $n_1$ and $n_2$ in…
Clair Crossupton
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Finding the probability density function of Hotelling's T-squared distribution

The following image is seen on wikipedia when searching for Hotelling's T-squared distribution This is apparently the pdf of the Hotelling T-squared distribution at different parameters. However, I am a bit suspicious of it, so I tried generate my…
Carl Näsvall Sindeby
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Hotelling's T-squared test

I am dealing with an $n$-dimensional random variable $\hat{P}$ for which I know that $$\sqrt{n}(\hat{P}-P) \to^d \mathcal{N}(\mathbf{0},\Sigma).$$ I could also estimate the covariance matrix, $\Sigma$. I am wondering if this information could help…
Nocturne
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Hotelling T^2 test derivation question

I am reading about the Hotelling $T^2$ test (A primer of multivariate statistic s by Richard J. Harris). It says here that the test can be seen as creating a linear combination of your variables and then obtaining the set of coefficients that…
Diego
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Proof that the Hotelling T$^2$ statistic is invariant under the choice of contrast matrices

Consider an one-way repeated measures design with $n$ subjects and $q$ measurements. It is assumed that $\mathbf{x}_{i}$ are $iid$ q x 1 random vectors that follow multivariate normal distribution, where $i$ is the index of subjects. Using the…
julyfire
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Hotelling's T$^2$ test when $p > n$

Suppose I have a data-matrix $\bf X$, which has more features than samples ($p > n$). I'd like to perform a Hotelling's T$^2$ test to determine whether or not to reject the null-hypothesis that the pop. mean is $\bf 0$. I essentially face two…
Thoth
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Relation between Two Sample Hotelling's T-test and Mahalanobis Distance?

Mahalanobis distance is a measure of distance between a point and distribution. So if we want to check if a point belongs to a particular distribution or not, we can use Hotelling's T-test, which is squared Mahalanobis distance. But if we have two…
tehseen fatima
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What is the distribution of this nearly-Hotelling statistic?

Let $X$ be an $n \times l$ matrix, and $F$ an $n \times p$ matrix, with the rows of $X$ and $F$ drawn i.i.d. from multivariate Gaussians. (The independence applies to rows: the $X$ and $F$ may be correlated.) Let $\Sigma_X, \Sigma_F, \Sigma_{XF}$ be…
shabbychef
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Using the Hotelling package in R

I have two samples of data in $\mathbb{R}^2$, assumed drawn from a gaussian distribution, and I would like to test whether the two samples have the same mean. I know that the right test to do this is the Hotelling T2 test and I would like to use the…
pitchounet
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Violation of the assumption of equal covariance matrices for two-sample Hotelling's test

I would like to compare nutrient intake of men and women. But the assumption of the same variance-covariance matrix is violated and therefore two-sample Hotelling's $T^2$ test cannot be applied. How else can I compare the intakes?
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