Q:

Is it wrong to refer to results as "nearly" or "somewhat" significant?

The general consensus on a similar question, Is it wrong to refer to results as being "highly significant"? is that "highly significant" is a valid, though non-specific, way to describe the strength of an association that has a p-value far below your pre-set significance threshold. However, what about describing p-values that are slightly above your threshold? I have seen some papers use terms like "somewhat significant", "nearly significant", "approaching significance", and so on. I find these terms to be a little wishy-washy, in some cases a borderline disingenuous way to pull a meaningful result out of a study with negative results. Are these terms acceptable to describe results that "just miss" your p-value cutoff?

A:

If you want to allow "significance" to admit of degrees then fair enough ("somewhat significant", "fairly significant"), but avoid phrases that suggest you're still wedded to the idea of a threshold, such as "nearly significant", "approaching significance", or "at the cusp of significance" (my favourite from "Still Not Significant" on the blog Probable Error), if you don't want to appear desperate.

A:

From my perspective, the issue boils down to what it actually means to carry out a significance test. Significance testing was devised as a means of making the decision of either to reject the null hypothesis or to fail to reject it. Fisher himself introduced the infamous 0.05 rule for making that (arbitrary) decision. 
Basically, the logic of significance testing is that the user has to specify an alpha level for rejecting the null hypothesis (conventionally 0.05) before collecting the data. After completing the significance test, the user rejects the null if the p value is smaller than the alpha level (or fails to reject it otherwise). 
The reason why you cannot declare an effect to be highly significant (say, at the 0.001 level) is because you cannot find stronger evidence than you set out to find. So, if you set your alpha level at 0.05 before the test, you can only find evidence at the 0.05 level, regardless of how small your p values is. In the same way, speaking of effects that are "somewhat significant" or "approaching significance" also doesn't make much sense because you chose this arbitrary criterion of 0.05. If you interpret the logic of significance testing very literally, anything bigger than 0.05 is not significant. 
I agree that terms like "approaching significance" are often used to enhance the prospects of publication. However, I do not think that authors can be blamed for that because the current publication culture in some sciences still heavily relies on the "holy grail" of 0.05. 
Some of these issues are discussed in:
Gigerenzer, G. (2004). Mindless statistics. The Journal of Socio-Economics, 33(5), 587-606.
Royall, R. (1997). Statistical evidence: a likelihood paradigm (Vol. 71). CRC press.

A:

The difference between two p-values itself typically is not significant. So, it doesn't matter whether your p-value is 0.05, 0.049, 0.051...
With regards to p-values as  a measure of strength of association: A p-value is not directly a measure of strength of association. A p-value is the probability of finding as extreme or more extreme data as the data you have observed, given the parameter is hypothesized to be 0 (if one's interested in the null hypothesis -- see Nick Cox' comment). However, this is often not the quantity the researcher is interested in. Many researchers are rather interested in answering questions like "what's the probability of the parameter to be greater than some chosen cut-off value?" If this is what you're interested in, you need to incorporate additional prior information in your model.

