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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Using Hotelling's T-statistic to find an elliptic confidence set

The problem: We have samples of sizes ${n_1} = 25,{n_2} = 15,{n_3} = 30$ drawn independently from $N\left( {{\mu _i},{\sigma ^2}} \right),i = 1,2,3$ (normal distributions with same variance). We have $...
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Distribution of multivariate “$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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21 views

two-sample $T^2$ testing with $n=1$

Assuming I have two populations $X\sim\mathcal{N}(\mu_X,\Sigma)$ and $Y\sim\mathcal{N}(\mu_Y,\Sigma)$ of sizes $n_X,n_Y$, the hypotheses for equivalence of means (each of length $p$, and assuming ...
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240 views

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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21 views

Profile analysis for within group design?

So I have an experiment with a within-subject design that has two conditions with an independent variable (with 10 states) and one dependent variable. An analysis of variance gave me already the ...
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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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273 views

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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1answer
284 views

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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42 views

Alternate form of null hypothesis for equality of group means (Helmert matrix?)

I have two sets of hypotheses in a multivariate setting that are said to be equivalent. I am wondering if someone can explain why they are as opposed to me just taking it to be true. Version 1: $...
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66 views

Limit of Hotelling $T^2$ distribution

Suppose you have a sample $X_1, \ldots, X_n$, $n$ large, from a multivariate normal distribution $N_p(\mu, \Sigma)$. It is easy to show that $D_i := k(X_i - \bar{X}_n)'S^{1}_n(X_i-\bar{X}_n) \sim T^2_{...
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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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332 views

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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1k views

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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160 views

Multivariate confidence interval for means?

Let $\mathbf{X}=(X_1,X_2)$ a random sample, where $X_1=(X_{11},X_{21},\dots,X_{n1})$ and $X_2=(X_{12},\dots,X_{n2})$ are two random vectors. Find joint confidence intervals for $\mu_1$ and $\...
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1answer
357 views

What is “Discriminant hotelling”?

Recently I have encountered the term Discriminant hotelling. And I have no idea what this term means. All I know is that this is in context with multivariable data analysis. I tried googling but in ...
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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 ...
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Show that distribution of $\small(n-1)\overline{X}'(S^{-1}-\frac{S^{-1}\mu_0\mu_0'S^{-1}}{\mu_0'S^{-1}\mu_0})\overline{X}$ is $\small T^2(p-1,n-1)$

Show that the distribution of $(n-1)\overline{X}'\left(S^{-1}-\dfrac{S^{-1}\mu_0\mu_0'S^{-1}}{\mu_0'S^{-1}\mu_0}\right)\overline{X}$ is $T^2(p-1,n-1)$ where $X_i\sim N_p(\mu,\Sigma)$, where $\Sigma$ ...
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1answer
117 views

distribution of $T^2$ statistic without Gaussian assumption

I would like to know if the Gaussian assumption is needed on $x_i$ in deriving the asymptotic distribution of the $T^2$ statistic. Here is the presentation sequence I got from the Wiki page on ...
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Paired Hypothesis Test in TWO Linear Regressions

I am wondering how to perform a paired hypothesis test in a panel regression. I have been looking in many stats books but I have not found anything. The question goes: assume for $i=1,2,\cdots, N$ ...
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738 views

Hotelling's one-sample T2 statistic

I wanted to determine if a new sample is within reasonable bounds of the mean of an approximately normally distributed sample of points. I want to use Hotelling's $T^2$ statistic to obtain a $p$-...
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818 views

Which is right? Individual variable t test or multivariate Hotelling t test?

So, lets say a hypothetical example. You have data of women aged 25-50 on their nutrition activity and measured for 737 such women(n,sample size) on 5 variables(Calcium, Protein, Vitamin A, Vitamin ...
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682 views

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 ...
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1answer
444 views

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 ...
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1answer
597 views

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 ...
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1answer
884 views

Alternative tests for Hotelling's two-sample T-test

I have a bunch of vectors from two groups, $X$ and $Y$, and each vector in either $X$ or $Y$ groups has $m$ elements. Now I have $X_{1},\ldots,X_{8}$ and $Y_{1}, \ldots, Y_{8}$ in each group, and ...
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2answers
1k views

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 ...
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1answer
708 views

Correlation and Hotelling test

To find a Hotelling $T^2$ score it is necessary to calculate the covariance matrix and then invert it. Now, when the test is a two-sample $T^2$ test, the covariance matrix is a pooled matrix. When ...
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171 views

How do I find the coefficients of a two-sample T2 Hotelling Test?

Hi I am trying to figure out how to obtain the coefficients for the linear combination of variables determined by a Hotelling $T^2$ test. I know that for the one sample case it is $a = S^{-1} \Delta ...
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1answer
921 views

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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2answers
3k views

How exactly does Wilks' Lambda distribution generalize the Hotelling distribution?

According to Wikipedia, Wilks' Lambda distribution generalizes Hotelling's distribution. I am having some problems seeing how this works. I can see how Hotelling's distribution generalizes Student's t-...