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Information on the Girth (), Height () and Volume () for 31 cherry trees yielded the following correlations : $r_{12}$ = 0.519, $r_{23}$= 0.598 and $r_{13}$= 0.967. If you were asked to setup a regression choosing a certain characteristic as the response and the other two as predictors, which characteristic will you choose as the response? Give your comments.

I think we choose Girth and Height as predictors for the response Volume because the correlation between Girth and Height is the least, so we're looking at how these individually affect the Volume (i.e. the factor with which their correlation is more than that between them).

Is my reasoning correct? This question was asked in the final examinations of the course on Multivariate Data & Linear Models and it had 10 marks. What more should I write?

Tapi
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    All three answers can be justified: as far as regression modeling is concerned, what aspect of a tree "affects" some other has no meaning. Indeed, even in the real world it isn't plausible to claim that any of these variables directly affects the others: they are all consequences of the size and shape of the tree. But, in practice, which of the three variables do you think might be the most difficult to measure in living trees? – whuber May 16 '22 at 16:52
  • @whuber the volume – Tapi May 16 '22 at 17:08
  • It is also interesting to note that a routine regression analysis would also consider the interaction term. When using girth and height as the regressors with an interaction, then (assuming the trees have similar shapes) you will get near-perfect fits for the volume, because that's essentially proportional to the interaction (girth times height). Even if your objective is to predict girth or height from the other two variables, this insight can lead you to a far better model than otherwise. (It will be either nonlinear or linear in the logarithms.) – whuber May 20 '22 at 15:43

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I might argue that it makes sense for volume to be the response, and then you can use direct measurements of how tall and wide a tree is in order to predict the volume (which is probably harder to measure).

However, we use regressions to predict or explain. If we want to predict or explain differences in girth, we would use girth as the response. If we want to predict or explain height, we would use height as the response. If we want to predict or explain volume, we would use volume as the response.

In all three cases, the other two variables are viable predictors.

I do not see the correlations between the marginal variables playing any role. If I wanted to measure volume and girth ($r=0.967$) in order to predict height, I would be totally comfortable doing so. One might argue that it might be worth it to drop one of those correlated predictors in order to have fewer parameters without losing much in terms of predictors (since the volume and girth have such a high correlation). While I understand this stance, it is not so simple.

Dave
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Predictors with high correlation are more linearly dependent. So they have a similar effect on the dependent variable. Thus doesn't make sense to use Height as the response. Choosing between Girth and Volume it seems more logical to use Volume. As you mentioned in the answer to @whuber, the volume might be the most difficult to measure in living trees.

vldud
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    Your logic fails at the first step, because correlated predictors do not necessarily have a "similar effect" on the dependent variable. For instance, consider three iid random variables $(X,Y,Z)$ and set $U=X,$ $V=X+\epsilon Y,$ and $W=Y+\delta Z$ for small numbers $\delta$ and $\epsilon.$ In this case $U$ is independent of the response $W$ even though the predictors $U$ and $V$ are arbitrarily highly correlated. – whuber May 20 '22 at 14:05
  • @whuber I'm afraid I don't quite understand. In your example *U* and *V* have a similar effect on *W*. – vldud May 20 '22 at 14:30
  • The point is that $V$ alone is associated with $W:$ $U$ has no association with $W$ at all. – whuber May 20 '22 at 15:31
  • But $U$ is associated with $W$, isn't it? $W$ = $V$/$\epsilon$ - $U$/$\epsilon$ + $\sigma$$Z$ – vldud May 20 '22 at 15:49
  • $U=X$ and $W=Y + \delta Z$ are independent in this example. That's the exact opposite of "associated with"! – whuber May 20 '22 at 15:51
  • I showed the dependence in my previous comment. – vldud May 20 '22 at 16:01
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    Unfortunately, that is erroneous: two independent variables cannot be dependent! – whuber May 20 '22 at 16:57
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The data alone does not suggest a model, just a correlation matrix. What is the purpose of the model, and in what conditions will it be applied? For example, if the trees are still standing in a grove or forest, it might be useful to estimate their height, based on girth (usually measured at 48" above ground). Of course, this is an opportunistic dataset collected from a plot of felled cherry trees, so the lumber mill model, volume ~ girth + height may be the one you want.

Mike Anderson
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