# 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"?

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## 2 Answers

Under the definitions you've listed, "association" and "relationship" would not be interchangeable. However, I would argue that a better use of the term "relationship" would make them fairly synonymous in this application. I think that your teacher was making an important, and correct, point about correlation and regression, but that the way it was done (at least according to your memory) used the term "relationship" in a non-standard way. I think you are on solid footing to make the claim as you do in your last paragraph. For more info on the asymmetrical vs. symmetrical nature of regression and correlation, see here.

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We were answering this question at the same time and you finished first. We reached the same conclusion. –  Michael Chernick May 8 '12 at 17:17
@MichaelChernick, that happens sometimes. I just finished reading your answer, +1. –  gung May 8 '12 at 17:20
I however do not mind the term relationship. I think in general the regression problem for Y a dependent and X an independent variable is a functional relationship with Y=f(x) + e where f(x) = E(Y|X=x]. in the special case of linear regresion f(x) is linear in the model parameters a and b and often but not always is linear in x. So for example Y=bX^2 +a +e is a linear regression even though it is quadratic in X. –  Michael Chernick May 8 '12 at 17:24
@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 May 8 '12 at 17:51
Thanks for clarifying your thinking. You make a good point. I don't think it is nitpicky. My wife tells me I am nitpicky and I think many in the general public consider statisticians to be nitpicky. If that is the case I think it is good to be nitpicky. –  Michael Chernick May 8 '12 at 18:23
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Yes. You are basically correct. Regression is used when you want to show how a dependent variable $Y$ is related to one or more dependent variables. Whem 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 momnet correlation measures the strength of the linear relationship between $X$ amd $Y$. 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.

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