# 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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### Manova, Hotelling's T^2 and follow-up?

For my final thesis, a few questions arose during the pre-registration process and I would be very happy to receive answers. I will conduct an experiment with three groups (1, 2, 3) and 3 dependent ...
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### Moments of the Hotelling Lawley Trace under the alternative

The Hotelling Lawley trace is used in multivariate ANOVA, or for tests of the Multivariate General Linear Hypothesis (regression coefficient testing in many-to-many linear regression). For our ...
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### In Hotelling's theory, why create a statistic and calculate the likelihood instead of using the maximum likelihood estimated normal distribution?

In Hotelling's theory, when data is assumed to be generated from a normal distribution, a statistic based on the Mahalanobis distance is calculated, and the degree of anomaly of the data is calculated ...
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### What is a generalisation of t-test to the case of multivariate distribution?

When we have a sample of numbers, one of the most basic tests is the t-test, in which we check the null hypothesis that the population mean is equal to zero. I am interested in a generalisation of ...
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### Intuition for Hotelling's T^2 Test

I have been learning about Hotelling's $T^2$ test from Multivariate Statistics: Old School. The test is given by $T^2 = \nu\cdot\text{trace}(\bf{W}^{-1}\bf{B})$. The author shows that in the case of ...
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### Hotelling T squared seemingly useless at detecting a mean shift

I am trying to develop a single-sample Hotelling $T^2$ test in order to implement a multivariate control chart, as described in Montgomery, D. C. (2009) Introduction To Statistical Quality Control, ...
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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 ...
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### Multiple comparisons problem one-sample t-test

I asked participants to rate the valence of 92 objects on a 7-point scale from negative to positive. I want to know for each object whether it's valence is significantly different from 4, i.e., the ...
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### Hotelling's $T^2$ test VS dendrogram

When i perform a Hotelling's $T^2$ test on a dataset, it stated that there is strong evidence that the mean vectors of the two groups differ. However, when I create a dendrogram, I got: where number ...
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### Hotelling's T for Matrix Normal Distribution

Suppose you observe matrix normal $\mathbf{X}_i \sim \mathcal{MN}_{m,p}\left(\mathbf{M},\mathbf{U},\mathbf{V}\right)$, for $i=1,2,\ldots,n$. That is $\mathbf{X}_i$ is an $m\times p$ matrix which is ...
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### 95% confidence ellipse using Hotelling T-squared in a score plot (R)

I am trying to draw a 95% confidence ellipse using Hotelling T-squared in a score plot of two principal components from a PCA. I have checked that: https://stackoverflow.com/questions/42637860/pca-...
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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 ...
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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 $X\sim N_2(\mu, \Sigma)$. ...
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### Minimum sample size heuristic for using Hotelling's T$^2$ test

My understanding is that for non-normal data, a widely accepted heuristic for using the t-test is a sample size of at least $n=30$. Since for smaller sample sizes the distribution of the sample mean ...
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### Testing the randomness of allocation of multivariate dataset

Everyday we allocate mortgages between two investor platforms. Within each risk segment and term (length of mortgage in years), mortgages are allocated based on investor demand randomly. ie 70/30 ...
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### Are there any good citations for the distribution of Hotelling's t-square statistic that don't assume a point null hypothesis?

Under standard assumptions, Hotelling's $t$-squared statistic, \begin{align*} t^2=(\overline{x}-\mu)'\hat{\Sigma}_\overline{x}^{-1} (\overline{x}-\mu), \end{align*} is distributed as \begin{align*} t^...
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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, ...
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### How to interpret Hotelling T2 results

I've been presented with report that compares two items, each having over 400 data points. Because it's multivariate, the Hotelling T2 test was used and the result was the Hostelling ellipsoid with ...
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### is the computed Hoteling's T square correct?

I am testing the Hoteling Tsquare test on a toy example using scikit learn. I am following the description in this this link. Is the size of the obtained Tsquare values correct? Shouldn't it be a ...
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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 ...
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### Is $T^2$ equivalent to 1 Way MANOVA when we have just two populations?

When we're in the presence of two populations, for the same assumptions (random samples, multivariate distribution, same covariance matrix), and we want to test the mean vectors, we have two options: ...
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### Should we use Hotelling $T^2$ test or something else?

As part of our project, we have used a combination of three machine learning classifiers combined with a voting algorithm over it to obtain reasonably good results. The input data for the classifiers ...
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### Given $\{\vec{x}_i\}$ i.i.d $N(\vec{\mu}, \Sigma)$ find confidence ellipse for $\mu$ (unknown $\Sigma$)

Searching online, I was not able to find construction of a confidence ellipse for $\mu$ in this case. Any help would be appreciated. Below is my attempt to construct $1- \alpha$ confidence ellipse. ...
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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, $\Sigma$....
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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 ...
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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 ...
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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 ...
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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 ...
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