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.
55
questions
0
votes
0
answers
24
views
Confidence regions for multivariate normal distributions
From this post, Confidence regions on bivariate normal distributions using $\hat{\Sigma}_{MLE}$ or $\mathbf{S}$, the equation
$(\mathbf{x} - \boldsymbol{\mu})^{T} \mathbf{\Sigma}^{-1} (\mathbf{x} - \...
2
votes
1
answer
70
views
Testing that a multivariate mean approximately equals a vector of constants
Suppose we have a $p$-variate random vector that has a multivariate Normal distribution, $\boldsymbol{X}\sim MVN(\boldsymbol{\mu},\Sigma)$. My hypothesis is that the mean vector contains only zeros, $\...
- 62.1k
0
votes
0
answers
89
views
Should I favor the results of a Hotelling's $T^2$ or Bonferroni confidence interval?
I am trying to understand if my reasoning is correct about these two methods. Faced with a situation where it is impossible to get both methods to fail to reject the null hypothesis, which should I ...
0
votes
0
answers
44
views
Axes of confidence ellipse $\sqrt{\lambda_i}\sqrt{\frac{np-p}{n(n-p)}F^{-1}_{n,n-p}T_{\alpha}}e_i$
I am building my way up to creating a simple confidence region plot with HotellingT-test from scratch. The following information has been passed down to me:
I have attempted to reproduce the ...
- 23
0
votes
0
answers
28
views
Hotelling t at 0.95 interval
Suppose I want to test the hypothesis that the lengths of head size for first borns and seconds borns are $x = (182, 182)$ by using the Hotelling-T test.
So given our data:
...
- 23
0
votes
0
answers
47
views
Why "multivariate test takes advantage of the correlations between the marginal distributions?
It is claimed (in a comment here) that "The advantage of the multivariate test is that it takes advantage of the correlations between the marginal distributions". Why this is the case for ...
- 1,335
3
votes
0
answers
81
views
Why Hotelling's $T^2$ is taught in the book but it is not used anywhere so far?
I have seen various multivariate statistics book covering Hotelling $T^2$ statistics. However, when reading medical journals and papers, it does not show up anywhere.
Q1:Is this test of any practical ...
- 1,335
2
votes
1
answer
91
views
How to compare multiple Two-Sample Hotelling's T-squared tests performed on different sample sizes
It's my first question, so please be patient with me.
I'm working on the comparison of multiple datasets of different lengths (60 to ~10000). They share a set of 9 features which I would like to use ...
- 21
0
votes
0
answers
445
views
Trying to understand how to calculate a hotelling's T2 Confidence Ellipse
I have read probably too many things and maybe confused myself along the way :)
Anyway, let's say I had a data set that I have performed PCA analysis on.
Let's say a spectral dataset 100 observations ...
- 111
2
votes
0
answers
197
views
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 ...
- 14.2k
1
vote
0
answers
63
views
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 ...
- 165
4
votes
1
answer
2k
views
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 ...
- 485
2
votes
0
answers
69
views
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 ...
- 121
1
vote
0
answers
182
views
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, ...
- 281
5
votes
0
answers
314
views
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 ...
0
votes
1
answer
388
views
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 ...
- 31
1
vote
1
answer
54
views
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 ...
- 199
2
votes
0
answers
90
views
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 ...
- 14.2k
1
vote
0
answers
2k
views
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-...
- 21
6
votes
0
answers
2k
views
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 ...
- 14.9k
8
votes
1
answer
5k
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 $X\sim N_2(\mu, \Sigma)$.
...
- 292
1
vote
0
answers
109
views
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 ...
- 1,433
0
votes
1
answer
25
views
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 ...
- 101
1
vote
0
answers
176
views
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^...
6
votes
1
answer
776
views
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,
...
- 101
1
vote
0
answers
451
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 ...
- 191
1
vote
2
answers
2k
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 ...
- 207
5
votes
0
answers
180
views
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 ...
- 1,433
1
vote
0
answers
68
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:
$...
- 51
1
vote
1
answer
300
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^...
1
vote
0
answers
79
views
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:
...
- 5,360
2
votes
0
answers
216
views
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 ...
- 121
2
votes
0
answers
185
views
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.
...
- 680
4
votes
0
answers
559
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, $\Sigma$....
- 221
4
votes
1
answer
4k
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 ...
0
votes
0
answers
305
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 $\...
user72621
3
votes
1
answer
684
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 ...
- 571
7
votes
0
answers
137
views
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 ...
- 2,073
2
votes
0
answers
80
views
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 $\Sigma$ ...
- 1,545
2
votes
1
answer
423
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 ...
- 2,073
2
votes
0
answers
83
views
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 $i=1,2,\cdots, N$ ...
- 91
0
votes
0
answers
1k
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 $p$-...
- 1,516
1
vote
0
answers
3k
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 ...
- 1,025
4
votes
1
answer
1k
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 ...
- 325
3
votes
1
answer
592
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 ...
- 85
4
votes
1
answer
876
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 ...
- 273
3
votes
2
answers
664
views
What is the Hotelling $T^2$ used for?
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?...
- 403
3
votes
1
answer
1k
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 ...
- 983
4
votes
2
answers
2k
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 ...
- 699
2
votes
1
answer
938
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 ...
- 297