difference between correlation and coefficient in a regression

What's the difference between correlation and coefficient in a regression? If the author writes "there is a positive relationship between x and y" or "there is a positive correlation between x and y" is he taking about correlation or the coefficient in a regression?

Il your regression is $$y=\alpha+\beta x+\epsilon$$, then $$\hat\beta = \frac{\sigma_{xy}}{\sigma^2_x},\quad\text{while}\quad\rho_{xy}=\frac{\sigma_{xy}}{\sigma_x\sigma_y},\quad\text{so}\quad\hat\beta=\rho_{xy}\frac{\sigma_y}{\sigma_x}$$ Since $$\sigma_y/\sigma_x>0$$, $$\hat\beta$$ and $$\rho_{xy}$$ have the same sign: a positive coefficient implies a positive correlation, a negative coefficient implies a negative correlation.

An example in R:

> set.seed(1234)
> x <- 1:10
> error <- rnorm(10)
> y <- 3 - 2*x + error
> fit <- lm(y ~ x)
> beta_hat <- fit$coefficients[2]; beta_hat x -2.035185 > cov(x,y) / var(x) # == beta_hat [1] -2.035185 > rho <- cor(x,y); rho # same sign as beta_hat [1] -0.9873359 > rho * sd(y) / sd(x) # == beta_hat [1] -2.035185 • Please define your notation!$\sigma_{xy}\$ is what (etc.)? Commented Aug 21, 2020 at 11:21
• PS I did it in my response, maybe that's enough. Commented Aug 21, 2020 at 11:46
• @Lewian I suppose it is a standard notation, see <en.wikipedia.org/wiki/Covariance#Definition> Commented Aug 21, 2020 at 12:15
• Yes it is, and I have no problem understanding it, but you can't assume that the person who asked the question knows it as well. Commented Aug 21, 2020 at 12:48

If the author writes "there is a positive relationship between x and y" or "there is a positive correlation between x and y" then they would be talking about the correlation, not necessarily about the sign of the coefficient. A positive correlation means that Y increases with increasing X. I can't comment on what test the author used, but many coefficients also reflect the sign of the correlation. For example a Pearson correlation coefficient of -1 means perfect linear anti-correlation. Still, there might be some coefficients that don't behave like this, so a "positive relationship" always refers to the sign of the correlation and not necessarily to the sign of the coefficient of the specific test.

edit: For example a Chi-Square test would not tell you the sign of the correlation, just a presence of a correlation. So if the author referred to a negative correlation it would not mean that the Chi-Square coefficient is negative.

• If you want to compute the sign of a correlation, you only need to compute the correlation, not a test. Commented Aug 21, 2020 at 11:56

If you have two variables $$x$$ and $$y$$, correlation measures the relation between $$x$$ and $$y$$ in a way that is symmetric between $$x$$ and $$y$$, i.e., with $$\rho_{xy}$$ denoting the correlation between $$x$$ and $$y$$, $$\rho_{xy}=\rho_{yx}$$. I'm using the notation of the answer of Sergio here. A standard regression model looks like this: $$y=\alpha+\beta x+\epsilon,$$ with $$\beta$$ the regression coefficient of $$x$$, $$\alpha$$ a constant intercept term, and $$\epsilon$$ a random error term independent of $$x$$. $$\beta$$ is related to how strongly $$y$$ depends on $$x$$. The answer of Sergio states that $$\beta=\rho_{xy}\frac{\sigma_{y}}{\sigma{x}}$$, $$\sigma_x$$ and $$\sigma_y$$ being the standard deviations of $$x$$ and $$y$$, which says how they are formally related. Note that this means that $$\beta$$ can be large if $$\rho_{xy}$$ is not that large in case $$\sigma_{y}$$ is far bigger than $$\sigma_x$$. The correlation $$\rho_{xy}$$ measures the connection between $$x$$ and $$y$$ free of the influence of $$\sigma_x$$ and $$\sigma_y$$, whereas the regression coefficient $$\beta$$ depends on these.

This also implies that the regression coefficent $$\beta$$ is almost always different from the coefficient $$\gamma$$ of a regression of $$x$$ on $$y$$ in a model $$x=\delta+\gamma y+\eta,$$ i.e., regression coefficients are not symmetric in $$x$$ and $$y$$.

In fact, the idea of regression of $$y$$ on $$x$$ (first regression) is that $$y$$ is a function of $$x$$ plus some error (random deviation), and the error appears in $$y$$, not in $$x$$ (estimating the regression will minimise the squared error of the regression line comparing the predicted $$y$$ by the regression with the observed $$y$$), whereas the second regression of $$x$$ on $$y$$ implies that there is random error in $$x$$, not in $$y$$, leading to a different minimisation problem for estimation.

I don't have time to produce and embed a graph, but regression of $$y$$ on $$x$$ puts a line in a scatterplot between $$x$$ and $$y$$ that minimises the sum of squared distances of all $$y$$-observations to the line in y-direction, regression on $$x$$ on $$y$$ produces a line that minimises the sum of squared distances to the line in x-direction, and correlation essentially gives the slope of a line that minimises the sum of squared distances to the line of all $$(x,y)$$ orthogonally to the line, after having standardised the $$x$$ and $$y$$ to remove the influence of $$\sigma_x$$ and $$\sigma_y$$.

If an author writes about correlation, chances are they mean correlation ;-). If an author writes "there is a positive relationship between x and y" it doesn't matter because correlation and regression coefficient are positive in the same cases, although they won't have the same values, see Sergio's response.

PS: Note that this becomes more difficult and somewhat different in multiple regression. If there is $$y$$ and more than one $$x$$-variable, $$x_1,x_2, x_3,\ldots$$, say, it actually can happen that the correlation between $$y$$ and $$x_1$$ is positive whereas the regression coefficient of $$x_1$$ in the multiple regression for $$y$$ on all $$x$$-variables simultaneously is negative, due to potential dependence between different $$x$$-variables. If an author in such a situation says "there is a positive relationship between $$x_1$$ and $$y$$", chances are they mean the regression coefficient, however this is imprecise terminology and should be avoided.