Hello again,
Thank you for your replies.
To give the most basic example of what I am talking about, let's say you have data on all countries from 1945-1995 and you want to run some regression where you could calculate/Stata will tell you p-values. Some articles will do this and then talk about their coefficients being statistically significant and imply that this matters for all the reasons statistical significance would matter (all I meant by "interpreting").
The problem here is that some people think p-values in this scenario would be either irrelevant or dubiously applied. If this data is best thought of as population data, p-values would be irrelevant.
When people decide to interpret it anyways, they will often make the appeal to the theoretically larger superpopulation argument which I mentioned in my original post. This goes basically as reasonablepie described it, that there are always future/past or even hypothetical cases that make it reasonable to talk about the statistical significance of what you have found.
The problem with this is that our "sample" of this population, all countries from 1945-1995, is in no way a random or even a probability sample of this theoretical superpopulation and measures like p-values are to my knowledge based on the properties of random sampling. For this reason, some would say interpreting p-values in this case would be dubious at best.
I have read in other forums that people use p-values in things like our example because the result is mathematically a reasonably good approximation of tests that we "should" be using because they are less problematic in their theoretical grounding. However, I have been unable to find a citation for this. I was hoping someone might have an answer and I was also legitimately curious what people were learning in other programs.
If I am still not communicating this in a way that makes any sense, here is a link to the debate I am talking about:
Critique: http://www.jstor.org/stable/30011818
Author's Reply: http://www.jstor.org/stable/30011819
And here is another helpful post:
http://stats.stackexchange.com/questions/2628/statistical-inference-when-the-sample-is-the-population
Thank you all so much!