24
votes
Analogy of Pearson correlation for 3 variables
It is indeed something. To find out, we need to examine what we know about correlation itself.
The correlation matrix of a vector-valued random variable $\mathbf{X}=(X_1,X_2,\ldots,X_p)$ is the ...
- 316k
24
votes
Accepted
Completing a $3 \times 3$ correlation matrix — $2$ coefficients of the $3$ given
We already know $\gamma$ is bounded between $[-1,1]$
The correlation matrix should be positive semidefinite and hence its principal minors should be nonnegative
Thus,
\begin{align*}
1(1-\gamma^2)-0.6(...
- 3,185
20
votes
Is it meaningful to calculate Pearson or Spearman correlation between two Boolean vectors?
I would not advise to use Pearson's correlation coefficient for binary data, see the following counter-example:
...
- 3,493
18
votes
Accepted
Is it possible to have Pearson correlation coefficient values $< -1$ or values $> 1$?
The formulas you're using have long been known to be numerically unstable. If the squared means are large compared to the variances and/or products-of-means are large compared to the covariances, then ...
- 277k
17
votes
Accepted
What is the distribution of sample correlation coefficients between two uncorrelated normal variables?
As a general remark, your questions are usually very clear and well illustrated, but often tend to go too much into explaining your subject matter ("Q methodology" or whatever it is), ...
- 103k
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. ...
- 117k
13
votes
Pearson correlation coefficient is a measure of linear correlation - proof
It is indeed possible to show that the Pearson correlation is essentially the way to measure linearity of association when you elect to use standard deviations to measure the dispersion of random ...
- 316k
13
votes
Accepted
Difference between Pearson's r ~= 0 and p > 0.05
The p-values and Pearson's correlation coefficient $r$ measure different things.
$r$ measures the strength of the correlation. The p-value, on the other hand, measures how likely you would be to ...
- 3,340
12
votes
Is it meaningful to calculate Pearson or Spearman correlation between two Boolean vectors?
Arne's response above isn't quite right. Correlation is a measure of dependence between variables. The samples A and B are both independent draws, although they are from the same distribution, so we ...
- 121
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 ...
- 58.1k
11
votes
What is the difference between linear regression on y with x and x with y?
The insight that since Pearson's correlation is the same whether we do a regression of x against y, or y against x is a good one, we should get the same linear regression is a good one. It is only ...
- 1,157
11
votes
Spearman $\rho$ as a function of Pearson $r$
I think I found the answer. In Pearson's "On further methods of determining correlation" (1907) he derives the expression:
$$
r=2 \sin \Big(\frac{\pi}{6}\rho\Big),
$$
which implies,
$$
\rho= \frac{6}{\...
- 231
11
votes
Accepted
What is the 'right' slope formula of a regression? deltas or Pearson?
For only two points they are the same.
The slope of simple linear regression is
$$
\hat \beta = \frac{\sum_i (x_i - \bar x) (y_i - \bar y)}{\sum_i (x_i - \bar x)^2}
$$
that is the same form you ...
Tim♦
- 135k
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 ...
- 29.2k
10
votes
Why are $x$ and $x^2$ correlated?
The Pearson correlation measures the amount of linear relationship -- it doesn't ignore variables that have a relationship that's not perfectly linear. If things increase and decrease together, some ...
- 277k
10
votes
Completing a $3 \times 3$ correlation matrix — $2$ coefficients of the $3$ given
Here's a simpler (and perhaps more intuitive) solution:
Think of the covariance as an inner product over an abstract vector space. Then, the entries in the correlation matrix are $\cos\langle\mathbf{...
- 101
10
votes
Pearson correlation between a variable and its square
You are curious about whether your value of $r$ is "too high" — it seems you think that, as $X$ and $X^2$ do not have an exactly linear relationship, then the Pearson's $r$ should be rather low. ...
- 22.9k
10
votes
Accepted
Why the pearson correlation p-value doesn't fully correspond to CI in R?
Nearly all such questions are answered in the help on the relevant functions.
From the help for cor.test:
If method is ...
- 277k
10
votes
Accepted
What is a good journal for submitting my article on a conjecture in theoretical statistics, re: ancillary complement for correlation?
Although you describe your article as being "on a conjecture", from what you have described about the content, it looks to me like you do a lot of things first that are of definite substance ...
9
votes
Accepted
What is the explanation for having a Pearson's correlation coefficient significantly larger than the Spearman's rank correlation coefficient?
This is a simple dataset, where the points come alternating from two linear functions:
The pearson correlation detects, there is a general upwards motion in the combined data (red an black together) ...
- 8,345
9
votes
Accepted
A p-value greater than 0.05 means that my results are meaningless?
A p-value above 0.05 doesn't necessarily say 'your correlation is meaningless'.
However, there's more than a 5% chance that you could see a sample correlation at least as far from zero when the ...
- 277k
9
votes
Intraclass coefficient or Pearson coefficient
Pearson's $r$ is a linearity index that quantifies how well two variables $x$ and $y$ are related by the following equation: $y=mx+b$. In contrast, the consistency ICC is an additivity index that ...
- 4,667
9
votes
MSE as a proxy to Pearson's Correlation in Regression Problems
This is a good question and unfortunately unanswered for a long time, it seems that there was a partial answer given just a couple months after you asked this question here that basically just argues ...
- 405
9
votes
Accepted
Does data normalization and transformation change the Pearson's correlation?
Pearson's correlation measures the linear component of association. So you
are correct that linear transformations of data will not affect the correlation between them. However, nonlinear ...
- 55.1k
8
votes
Derivation of the standard error for Pearson's correlation coefficient
There are two equations here for computing the statistical significance of the correlation coefficient. The first is the variance of the true correlation coefficient $\rho$ of two bivariate normal ...
8
votes
Is it possible to have Pearson correlation coefficient values $< -1$ or values $> 1$?
The Pearson correlation coefficient is indeed between $-1$ and $+1$ (inclusive). This follows from the Cauchy-Schwarz inequality.
Getting a correlation coefficient of $1.0000000002$ is possibly (but ...
- 2,155
8
votes
If the Pearson r is .1, is there a weak relationship between the two variables?
Let me again post the same quote from the web:
I once asked a chemist who was calibrating a laboratory instrument to
a standard what value of the correlation coefficient she was looking
for. ...
Tim♦
- 135k
8
votes
Accepted
Is Spearman correlation never greater than Pearson correlation?
Simple example in which Spearman correlation is greater than Pearson correlation:
x = 1:10; y = x^2
cor(x,y, meth = "p")
[1] 0.9745586
cor(x,y, meth = "s")
[1] 1
- 55.1k
8
votes
Online update of Pearson coefficient
Recall the formula for the sample Pearson correlation between two vectors $x\in\mathbb{R}^n$ and $y\in\mathbb{R}^n$ (Eq. 3 in Wikipedia):
$$ r = \frac{\sum_{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}{...
- 117k
8
votes
How much can the Pearson and Spearman correlation coefficients differ in a dataset? (edited)
Sure. We can achieve this result by adding a single extreme data point to an otherwise uncorrelated, and nonmonotonically related, set of data:
...
- 37.6k
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