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Suppose one estimates a linear model $$ y=\beta_0+\beta_1 x+\varepsilon $$ and finds that $\hat\beta_1>0$ and the $p$-value associated with $\hat\beta_1$ is lower than the chosen significance level. Can one say that

$\hat\beta$ is statistically significantly positive?

I am more used to the expression

$\hat\beta$ is positive and statistically significant.

Is there a difference in the meanings of the two expressions? Can one use them interchangeably?

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    $\begingroup$ Piling on the adverbs is poor English, that's all. $\endgroup$ – whuber Oct 17 at 13:41
  • $\begingroup$ Everyone will know what you mean. Did someone ask you to write it the first way instead of the second? The best way might be to report $\hat{\beta}$ with some margin of error and a confidence level e.g. "We found $\hat{\beta} = 3 \pm 1.44$ with 95% confidence." Maybe throw on the p-value, too, in case someone is more skeptical and wants to evaluate your work at $\alpha = 0.01$ or something. $\endgroup$ – Dave Oct 17 at 14:35
  • $\begingroup$ @Dave, it is not me who is writing this. I am reading this and finding it a little unorthodox. $\endgroup$ – Richard Hardy Oct 17 at 15:36
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I don't think it is the same. If you say that $\hat \beta$ is statistically significant, that's a short way to say that it's significantly different from 0, and "different from 0" is not the same as "greater than 0", obviously.

If I read:

$\hat \beta$ is statistically significantly positive

I understand that it has been tested for being greater than 0 (one tailed test), if I read instead:

$\hat \beta$ is positive and statistically significant

I understand that it has been tested for being different from 0 (two tailed test), so in the end, even if the result is the same, the procedure seems to have been different.

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    $\begingroup$ A very interesting perspective. Thank you! $\endgroup$ – Richard Hardy Oct 17 at 15:37
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There is not a difference in meaning though the first one sounds strange to me. I would go with the second option or would word it as:

$\beta_1$ is statistically significantly greater than zero.

because that is the most common wording I've personally seen.

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    $\begingroup$ The definition of positive is literally greater than zero. Therefore, your alternative must have the exact same meaning as statistically significantly positive and it uses 5 words instead of 3. I wonder if it is actually better. Perhaps it is better in the sense of not piling on the adverbs (as per whuber's comment to the OP). $\endgroup$ – Richard Hardy Oct 17 at 14:13
  • $\begingroup$ Yes it's wordier though I find it easier to read. But that may be because it's just more common and so I'm used to seeing it. $\endgroup$ – Patrick Oct 17 at 14:21
  • $\begingroup$ Thank you, Patrick. $\endgroup$ – Richard Hardy Oct 17 at 14:23

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