# Association or relationship

I remember being taught that correlation is a test of association and regression is test of relationship.

In addition, I remember learning that association means that no assumption is made on which variable is independent and which variable is dependent, whereas relationship implies such a distinction.

My question: Are the terms "association" and "relationship" interchangeable?

More specifically, if I interpret my findings from a bivariate linear correlation analysis, would it be appropriate to use words such as "there was a strong positive relationship between A and B"?

• I really don't see how "relationship" implies such a distinction as they made. I'd be pretty content with the conclusion in your final sentence. – Glen_b -Reinstate Monica Jul 5 '14 at 7:39

• @MichaelChernick, you're right, this is all a little slippery, & maybe I'm making too big a deal out of it. What I'm thinking of is this: if there's a relation b/t $x$ & $y$, then there's a relation b/t $y$ & $x$; on the other hand, $y$ can be a function of $x$, w/o $x$ being a function of $y$ (eg, $y=x^2$). Amarald's teacher was trying to get at the asymmetry of regression; I would rather convey that idea via some other term than "relationship", but I could also edit if I'm being too nitpicky. – gung - Reinstate Monica May 8 '12 at 17:51
Yes. You are basically correct. Regression is used when you want to show how a dependent variable $Y$ is related to one or more independent variables. When we refer to correlation we are taking about an association. Regression is often used to predict future responses for $y$ based on given values for $x$. In least squares regression the predictor variables are assumed to be observed without error and $Y$ has an independent random error term. There is also error in variables regression where both $X$ and $Y$ are assumed to be observed with error. For that problem least squares is not the appropriate was to estimate the regression function. The function $f(x) =E(Y\vert X=x)$ for the model is called the regression function. Nevertheless the two ideas are intertwined. The Pearson product moment correlation measures the strength of the linear relationship between $X$ and $Y$. If you are using a simple linear regression model $Y=bX+a+\epsilon$ where $\epsilon$ is the independent error term in $Y$ and $a$ and $b$ are the intercept and slope parameters respectively then there is a direct relationship between the parameter $b$ and the Pearson correlation coefficient.