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I have data for the height of certain trees in 1996 (dependent variable) and their height in 1990 (explanatory variable). The question:

Is the value of $1$ included in the confidence interval for the slope? What does this tell you whether or not the trees are growing?

I found that the confidence interval for the slope is entirely above $1$, but I am not sure what that means about whether or not the trees are growing. My friend told me that this means that the height of the trees is not the same in 1996 as it was in 1990, but I don't see where this idea is coming from. Can someone please provide some hints on how this question can be solved?

Additional curiosity question: What would a slope of $0$ mean in this context?

Edit: Let $\beta_1$ denote the slope of the linear regression line predicting the height in 1996 from the height in 1990.

$H_0: \beta_1 = 0$ (There is no linear relationship between the height in 1996 and the height in 1990)

$H_a: \beta_1 \ne 0$ (There is some linear relationship between the height in 1996 and the height in 1990)

The regression equation is of the form: $\widehat{\text{Height in } 1996}= \hat{\beta_0} + \hat{\beta_1}\cdot \text{Height in } 1990$.

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  • $\begingroup$ Usually a confidence interval excluding zero means that the effect is “significant”. However, it depends on the regression equation, null hypothesis, and alternative hypothesis. Please post those. $\endgroup$
    – Dave
    Sep 12 '20 at 20:07
  • $\begingroup$ @Dave I've added the null and alternative hypotheses and the form of the regression equation. Does it matter what the exact regression equation is? It's just that the original question in the book didn't ask for it, but I can provide it if necessary. Please let me know. $\endgroup$ Sep 12 '20 at 20:13
  • $\begingroup$ What is $X$? The usual way to think about predicting the height in 1996 from the height in 1990 would involve testing something about the slope being $0$, not $1$, and while you've written that in your null hypothesis, asking about $1$ in the textbooks question is so off the wall to me that I think it's either a typo or you've omitted a piece of information. $\endgroup$
    – Dave
    Sep 12 '20 at 20:23
  • $\begingroup$ @Dave $X$ is the height in $1990$ and I've not omitted anything at all. I know, asking about $1$ was weird to me too. That's why I asked the question :) $\endgroup$ Sep 12 '20 at 20:30
  • $\begingroup$ Then I would say that having $1$ in the confidence interval tells you nothing. A confidence interval of $(0.1, 0.9)$ is equivalent to rejecting $H_0$ at the appropriate $\alpha$-level, same as a confidence interval of $(0.1, 1.1)$ or $(50, 100)$. $\endgroup$
    – Dave
    Sep 12 '20 at 20:32
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If $\beta_1=1$, the regression equation becomes $y=x+\beta_0+\epsilon$. That means, in average, all trees grow $\beta_0$ inches (or whatever unit you are using), regardless of their size in 1990.

If $\beta_1>1$, then the trees that started big in 1990 grew more than those that were small in 1990 (since the growth is $y-x=(\beta_1-1)x+\beta_0$). This could be due to the bigger trees getting more resources like sunlight and nutrients and therefore growing more.

On the other hand, if $\beta_1<1$, then the smaller trees grew more the big trees. This would be a odd behavior, but it could the case that the big trees from 1990 already reached their full height, while the little ones still have much growing to do. (disclaimer: I have no idea if this is biologically plausible, this is just a possible interpretation of such a result)

One could also check if $\beta_0=0$. That would mean $y=\beta_1x$, meaning that tree height at 1996 is directly proportional to tree height in 1990 (all trees would grow $(\beta-1)\times100\%$).

Usually, when doing regression, the hypothesis of interest is $H_0:\beta_1=0$..In this case, though, $\beta_1=0$ corresponds to a pretty odd scenario: $y=\beta_0$. It would mean that, whatever height the trees were in 1990, they all converge to an average height given by $\beta_0$. The hypothesis $\beta_1=1$ looks like a better null hypothesis, don't you think?

Now, about checking if the trees grew significantly, I would probably refer to a paired t-test instead of regression analysis. However, I see a few cases where regression could nicely answer your question: if you have $\beta_1\approx1$, you could simply check if $\beta_0>0$, and if you have $\beta_0\approx0$, you could check if $\beta_1>1$.

Hope I was helpful!

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