# Use of expression "statistically significantly positive"

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?

• Piling on the adverbs is poor English, that's all.
– whuber
Oct 17, 2019 at 13:41
• 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.
– Dave
Oct 17, 2019 at 14:35
• @Dave, it is not me who is writing this. I am reading this and finding it a little unorthodox. Oct 17, 2019 at 15:36

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.

$$\hat \beta$$ 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.

• A very interesting perspective. Thank you! Oct 17, 2019 at 15:37

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.

• 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). Oct 17, 2019 at 14:13
• 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. Oct 17, 2019 at 14:21
• Thank you, Patrick. Oct 17, 2019 at 14:23