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So let's suppose that the normal error regression model $Y_i = \beta_0 + \beta_1X_i + \varepsilon_i$ is applicable except that the error variance is not constant; rather the variance is larger, the larger is X.

Can someone explain to me why $\beta_1$ = 0 still implies that there is no linear association between X and Y but does not imply that there is no association between X and Y? I'm a bit confused.

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  • $\begingroup$ I'm guessing it depends on exactly how we're defining "association". Were you given a definition? $\endgroup$ – Glen_b -Reinstate Monica Oct 1 '14 at 3:08
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Suppose you simulate $X_i\sim\mathcal{N}(0,1)$ and then you set $Y_i=1+\varepsilon_i$, with $\varepsilon_i\sim\mathcal{N}(0,X_i^2)$, for $i=1,\ldots,n$. A scatterplot of a random sample from the pair $(X,Y)$ will be something like:

enter image description here

It is clear that $X$ and $Y$ are not independent: although the conditional mean of $Y$ remains the same ($1$) regardless the value of $X$, the effect of $X$ on the conditional variance of $Y$ is evident. The linear regression model $Y=\beta_0+\beta_1X+\varepsilon$ holds (with $\beta_0=1$ and $\beta_1=0$) and the red line represents the fitted linear model to the sample.

So, if you understand "association" as "dependence", here you have a counterexample where $X$ and $Y$ are associated but linearly independent (or "no linearly associated"). Roughly speaking, the reason is that linear association looks only to relationships between $Y$ and $X$ that could be modeled by a straight line and it may produce this kind of effects when the dependence between them is more complex.

You can find more information and counterexamples on http://en.wikipedia.org/wiki/Correlation_and_dependence.

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