10 votes

Eigenvalues/Eigenvectors of Correlation and Covariance matrices

Expanding on my comment: Since $P = \text{diag}(\Sigma)^{-1/2} \Sigma \text{diag}(\Sigma)^{-1/2}$, where $\text{diag}(\Sigma)$ is the diagonal matrix obtained by considering only the diagonal entries ...
mhdadk's user avatar
  • 4,950
9 votes
Accepted

What advantages do we find when using a mixed model for nested data instead of multiple regression?

The primary reason why you want to use mixed models for repeated measurement data is that measurements taken on the same patient are correlated. Standard statistical regression models, such as linear ...
Dimitris Rizopoulos's user avatar
8 votes
Accepted

Eigenvalues/Eigenvectors of Correlation and Covariance matrices

If $\Sigma$ is diagonal (with arbitrary eigenvalues) then $P$ is just the unit matrix (all eigenvalues equal to one), so there cannot be any general relation between the eigenvalues of $\Sigma$ (alone)...
J. Delaney's user avatar
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6 votes
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Minimum number of observation in multivariate regression

I believe that it is better to think in terms of power to detect effects of interest, rather than rules of thumb. That said, if you want a quick-and-dirty baseline value, the standard rule of thumb ...
gung - Reinstate Monica's user avatar
6 votes

hundreds of linear mixed models

I agree with @rep_ho's "do whatever is commonly done with the data modality you are working with". However, if you are going to quote p-values (for example) you almost certainly need to do ...
Ben Bolker's user avatar
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6 votes

Univariate approach to a Bivariate logistic regression

This should be possible with bivariate probit regression. The observed bivariate binary response can be represented via thresholding a bivariate normal latent variable. A bivariate normal with ...
kjetil b halvorsen's user avatar
5 votes

hundreds of linear mixed models

TO answer your question: yes, from a scientific perspective it is ok to build hundreds of models. For example in fMRI data analysis we fit one model per voxel (3d pixel) per subject in the brain image,...
rep_ho's user avatar
  • 7,619
5 votes

Which variables I should include in the multivariable Cox regression model?

Categorization of continuous covariates is almost never a good idea. The reason is that you are basically throwing away valuable information, which may lead to a loss of statistical power or even ...
Denzo's user avatar
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5 votes
Accepted

How do we interpret the covariance matrices $\textbf{U}$ and $\textbf{V}$ in the Matrix Variate Normal Distribution?

In the context of the Matrix Normal Distribution, the entries $X_{ij}$ are samples from the Gaussian distribution with mean $M_{ij}$. To fully characterise their variance though, we need two matrices $...
usεr11852's user avatar
  • 44.2k
4 votes

Eigenvalues/Eigenvectors of Correlation and Covariance matrices

First question: given a vector $\lambda = (\lambda_1,\ldots,\lambda_n)$ of eigenvalues of a variance matrix $\Sigma,$ what are the possible eigenvalues $(\tau_1, \ldots, \tau_n)$ of a covariance ...
whuber's user avatar
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4 votes

What analysis to use for one quantitative independent variable and multiple quantitative dependent variables?

Your independent variable appears to be a composite of three items which is supposed to represent something intangible (e.g. anxiety, socioeconomic status) and four DVs which may also represent some ...
Shawn Hemelstrand's user avatar
4 votes

Sample size calculation for a multicenter RCT

Most statisticians ignore centers when doing sample size calculations. That doesn't cause too much of a problem. The best models for duration of treatment are the Cox proportional hazards model or ...
Frank Harrell's user avatar
3 votes
Accepted

How to prove Isserli's theorem $ E(x_1x_2x_3x_4) = E(x_1x_2)E(x_3x_4) + E(x_1x_3)E(x_2x_4) + E(x_1x_4)E(x_2x_3) $?

Isserli's Theorem equates various multivariate cumulants of a zero-mean multivariate Normal distribution. However, most of this theorem is a universal result, applicable to all multivariate ...
whuber's user avatar
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3 votes

Multiple predictors that measure the same concept in regression

If you can reasonably assume that the three questionnaires measure the same dimension/latent variable representing musical training (MT), you could aggregate the three scores and use a single ...
Christian Geiser's user avatar
3 votes

Material on Structural Equation Modelling (SEM)

There are two precursors to SEM: Factor Analysis, coming from psychology, and Path Analysis, coming from genetics. You might want to look at some of the materials on these two topics. Path analysis ...
3 votes
Accepted

Can you combine effects from both the composite and domain levels of questionnaires in a meta-analysis?

I would say, yes, one can do this. However, I would not include subdomain level results and the composite result from the same study, since most 'composites' are just sum scores or means based on the ...
Wolfgang's user avatar
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3 votes
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Copula to ensure one team wins and the other loses (Bernoulli margins)

A copula is just a way to construct a joining, or coupling, of two distributions. Let $X$ and $Y$ be two random variables. There is a notion of maximal coupling of (the distributions of) $X$ and $Y$. ...
Stéphane Laurent's user avatar
3 votes

Copula to ensure one team wins and the other loses (Bernoulli margins)

You've set up the problem to preserve the teams' historical win probabilities in the final result, but with $p_1 \not = 1 - p_2$ this is impossible. If two teams with such high win probabilities play ...
Nobody's user avatar
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3 votes

What analysis to use for one quantitative independent variable and multiple quantitative dependent variables?

Useful comments/answer were already given to make you think if you really should do what you planned to do. Still, if you would nevertheless like to continue, I'll briefly mention below how that can ...
BenP's user avatar
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3 votes
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Univariate approach to a Bivariate logistic regression

Let's see if this leads anywhere. From an econometrics point of view, I see two latent regressions, \begin{align} \begin{cases} y^*_1 = \beta x_1 + u_1\\ y^*_2 = \beta x_2 + u_2 \end{cases} \end{align}...
Alecos Papadopoulos's user avatar
3 votes

Multivariate Normal Distribution. How do we apply this to dataset?

A Brief Explanation of the MVN Distribution A multivariate normal (MVN) distribution is simply a mathematical definition of two or more normally distributed variables. For simplicity, we can just ...
Shawn Hemelstrand's user avatar
3 votes
Accepted

How to code for a mulitgroup analysis concerning two catagorical variables

One option could be to combine the two grouping variables (sex, country) into a single grouping variable, for example, as follows: Category 1: US males Category 2: US females Category 3: Canadian ...
Christian Geiser's user avatar
3 votes

How do we interpret the covariance matrices $\textbf{U}$ and $\textbf{V}$ in the Matrix Variate Normal Distribution?

The $i$th column of $X$ has a covariance matrix $V_{ii}U$, and the $j$th row of $X$ has a covariance matrix $U_{jj}V$. Hence, you can think of $U$ as the covariance of the columns, and of $V$ as the ...
Car Loz's user avatar
  • 850
3 votes

Sample size calculation for a multicenter RCT

Given your non-trivial design, I would suggest you turn this task a bit onto its head and use a simulation-based sample size planning approach. A nice intro to this would be: External validation of ...
usεr11852's user avatar
  • 44.2k
2 votes

Material on Structural Equation Modelling (SEM)

This video by Johny Lin at UCLA is great. Its basically a full structural equation model (SEM) course in a single video.
2 votes

Covariance of a linear and quadratic form of a multivariate normal

The covariance between a linear and a quadratic form of a multivariate normal vector is given in Mathai and Provost, page 74, Theorem 3.2d.2. Let $\mathbf{y} \in \mathbb{R}^p \sim \mathcal{N}(\mathbf{\...
dherrera's user avatar
  • 1,268
2 votes

How to prove Isserli's theorem $ E(x_1x_2x_3x_4) = E(x_1x_2)E(x_3x_4) + E(x_1x_3)E(x_2x_4) + E(x_1x_4)E(x_2x_3) $?

$\newcommand{\bt}{\boldsymbol t}$ Without attempting to the reach the generalities as in whuber's answer, below is a proof for $n = 4$ by directly differentiating the MGF $M(\bt)$ of $(X_1, X_2, X_3, ...
Zhanxiong's user avatar
  • 18.9k
2 votes
Accepted

Gumbel distribution conditional on exceeding a threshold

The quantile function for a $\text{Gumbel}(0,1)$ is $$-\ln(-\ln(p))$$ So to sample $Y|Y>u$, first sample $X\sim\text{U}(e^{-e^{-u}},1)$ then use the transformation $$Y=-\ln(-\ln(X))$$
jblood94's user avatar
  • 1,479
2 votes
Accepted

Multiple group comparisons in a linear mixed-effect model

You did not get gender- or age-specific estimates because you didn't ask for them. The call emmeans(mod, pairwise ~ group asks for marginal means for each group, ...
Russ Lenth's user avatar
  • 20.3k
2 votes

Univariate vs. Multivariate Standardization

Consider why someone would standardize at all. Two reasons come to mind. Computational When people standardize to ease the computations, it is typically because numerical optimization gets confused ...
Dave's user avatar
  • 62.5k

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