# 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 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 at 14:35
• @Dave, it is not me who is writing this. I am reading this and finding it a little unorthodox. – Richard Hardy Oct 17 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! – Richard Hardy Oct 17 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). – Richard Hardy Oct 17 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. – Patrick Oct 17 at 14:21
• Thank you, Patrick. – Richard Hardy Oct 17 at 14:23