42
votes
Accepted
Can statistical units measured per thousand inhabitants be bigger than 1000?
This is not a rate per one thousand people, this is the absolute number of people, with one unit equating 1,000 people. So if you see something like 3,258.1, it simply means 3,258,100 people.
This is ...
- 3,238
15
votes
Pearson or Spearman?
Neither correlation coefficient presupposes normality. Marginal or bivariate normality is completely irrelevant to the choice between them.
They do differ in the questions they ask of the data. ...
- 118k
14
votes
Is it possible to have a multivariate random distribution with all its random variables (pair-wise) reverse correlated?
If $X \mathrel{:=} \left(X_1, \ldots, X_k\right)^\top \sim \mathop{\mathrm{Multinomial}}\left(n, \left(p_1, \ldots, p_k\right)^\top\right)$ with $n \in \mathbb N_{\geq 1}, k \in \mathbb N_{\geq 2},$ ...
- 5,350
13
votes
Accepted
correlation: the difference of two correlations is positive, but the correlation of the difference is negative?
It is possible. Simple algebra shows that
\begin{align}
\rho_{X_2 - X_1, Y} = \frac{\sigma_{X_2}\rho_{X_2, Y} - \sigma_{X_1}\rho_{X_1, Y}}{\sigma_{X_2 - X_1}}.
\end{align}
So although $\rho_{X_2, Y} &...
- 16.8k
13
votes
Accepted
Is it possible to have a multivariate random distribution with all its random variables (pair-wise) reverse correlated?
Mathematically, as I commented, you can use the equi-correlation matrix
\begin{align*}
\Sigma =
\begin{bmatrix}
1 & \rho & \cdots & \rho \\
\rho & 1 & \cdots & \rho \\
\vdots ...
- 16.8k
12
votes
Accepted
Calculate Spearman and Pearson correlation on variables of different units
Pearson correlation, $\rho_{XY}$, divides through by the product of the units and results in a unitless measure.
$$
\rho_{XY}=\dfrac{
\text{cov}\left(X,Y\right)
}{
\sigma_X\sigma_Y
}
$$
The covariance ...
- 58.4k
12
votes
Is Kendall's tau uniquely determined by Pearson rho?
Can two different data sets have the same Pearson's ρ, but different Kendall's τ?
Anscombe's quartet gives you four two-dimensional datasets with (almost) identical Pearson correlations, but their ...
- 118k
11
votes
Question about running Spearman's correlation instead of Pearson's
Pearson's correlation coefficient ($\boldsymbol{r}$) provides a measure of linear association between paired variables.
Spearman's correlation coefficient ($\boldsymbol{r_{\bf{S}}}$) provides a ...
- 29.3k
10
votes
How do I deal with many zero values in terms of correlation?
Because you are comparing simulated vs. true values, a correlation between the two is not the best way to evaluate the quality of your simulations.
This is easy to illustrate: imagine your model is ...
- 17.7k
9
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 ...
- 4,485
9
votes
How can I interpret it when correlation between two variables is significant, but not in a multiple regression?
Redundancy
As an example of Christian's point, lets say we create three variables $X$, $Y$, and $Z$ with this following simulated data in R:
...
- 8,834
8
votes
Accepted
Interpreting the results of different correlation methods
Correlation is not a binary yes/no property, but a continuous feature that can be weak or strong, which is indicated by the value of the correlation coefficient. Testing for "statistical ...
- 4,692
8
votes
Accepted
Can we consider the loadings as a proxy for correlation, in a Principal Component Analysis (PCA)?
You can answer the questions yourself if you look at how the PCA is defined. For this, let $\mathbb{X}$ denote the $n\times p$ data matrix, and let $S = [s_{ij}]$ be the sample covariance matrix, e.g. ...
- 11.2k
8
votes
How to handle multi-collinearity when all the variables are highly correlated?
The correlations and VIFs are telling you the story. All of your features are extremely related to each other, so any notion of increasing one while holding others constant is nonsense, and I see no ...
- 58.4k
8
votes
Accepted
Sign of Correlation between $X$ and $f(X)$ for strictly monotonic $f$
Let $f$ be strictly increasing. Then
$$\operatorname{Cov}(X, f(X)) =\mathbb E[Xf(X) ]-\mathbb E[X]\mathbb E[f(X) ]=\mathbb E[(X-\mathbb E[X])(f(X) -f(\mathbb E[X]))].\tag 1\label 1$$
Now $$X\gtreqless\...
- 7,022
8
votes
Accepted
How do I deal with many zero values in terms of correlation?
Zero is a value like any other value to each kind of correlation. Each correlation takes zeros into account in its way:
as implying a deviation from the mean of either variable in the case of Pearson ...
- 54.1k
8
votes
Can statistical units measured per thousand inhabitants be bigger than 1000?
(THIS IS NO LONGER THE CORRECT ANSWER BUT IS KEPT UP AS AN INTERESTING IDEA. THIS IS THE CORRECT ANSWER.)
I find it plausible that the calculation, while reported as number of self-employed people per ...
- 58.4k
8
votes
Accepted
Is it wrong to trust Pearson Correlation output of r=.830 (sig at 0.01 2-tailed) with n=15?
At the first level, I'd say there's nothing wrong with interpreting a significant correlation even if your sample size is small. There are some caveats/reasons to take the results with a slightly ...
- 41.6k
8
votes
The meaning of the p-value for a correlation coefficient
The p value does not tell you how strong the relationship is. Not for correlations or anything else. That's what parameter estimates do. The p value answers this question (general case, with your ...
- 110k
8
votes
Hypotheses Testing - Correlation vs. Regression
Welcome to CV.
For question A: No, you don't need to run correlation before regression, but you do need to check the assumptions of the model. Most of these checks are done after running the model.
...
- 110k
8
votes
What does reported "r" mean in the context of a t-test?
Without any other information, I would say $r$ is correlation. And that is an effect size.
- 110k
8
votes
Accepted
What does reported "r" mean in the context of a t-test?
The reported $r$ is usually referring to an effect size. It is calculated from the t-statistic and the degrees of freedom
$$
r = \sqrt{\frac{t^2}{t^2 + df}}
$$
where $t$ is the t-statistic and $df$ is ...
- 57.1k
7
votes
Accepted
How to determine the correlation between two normal random variables conditioned on their sum being negative?
The conditional expectations of $X$ and $Y$ are obviously equal. Moreover, because $(X+Y)/\sqrt 2$ has a standard Normal distribution, its conditional expectation is the negative of $-|Z|$ where $Z$ ...
- 316k
7
votes
Accepted
Variance of sample autocorrelation (Ljung-Box)
Edit (05/07/2023)
When answering this question, I realized the job actually can be done by only summoning Lemma 1 below (hence avoid touching the much more difficult Lemma 2), which will substantially ...
- 16.8k
7
votes
Why is the p-value of Pearson's correlation test large even when the sample size is sufficient?
The p-value is a function of both the effect size and the sample size. If you have a gigantic sample size but a tiny (or zero) effect size, then you still do not have to wind up with small p-values.
...
- 58.4k
7
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)...
- 5,370
7
votes
Can statistical units measured per thousand inhabitants be bigger than 1000?
The specific question from the body text is answered by J-J-J but the title question can have more explanations
Can statistical units measured per thousand inhabitants be bigger than 1000?
The ...
- 71.9k
7
votes
Measuring level of uncorrelation from correlation matrix?
Perhaps the tolerance statistic is something that could be helpful for you. It is defined as 1 - R^2 for a given variable, where R^2 is calculated based on a linear regression of that variable on all ...
- 1,821
7
votes
Accepted
Measuring level of uncorrelation from correlation matrix?
It turns out that if you take the inverse of the correlation matrix, then take the reciprocal of the diagonal elements of the inverse, the result is one minus the $R^2$ values from the regression ...
- 50.8k
7
votes
Accepted
How can I interpret it when correlation between two variables is significant, but not in a multiple regression?
This can happen when N and G are redundant (overlapping, correlated) predictors of X. Once G is in the regression model, N may no longer be needed because N may not account for a unique variance ...
- 1,821
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